∫ K∫ K∫ Chaplygin Equations and an Infinite Set of Uniformly Divergent Gas-Dynamics Equations

Home , Bondarenko (crater), Chaplygin (crater)

PHYSICS

Chaplygin Equations and an Infinite Set of Uniformly Divergent Gas-Dynamics Equations A. I. Rylov Presented by Academician G.G. Chernyœ September 19, 2001

Received September 24, 2001

1. We consider two-dimensional potential ideal-gas (ϕ and ψ) and, as a result, a conservation law in the ﬂows. At the hodograph plane, they are described by physical plane: the Chaplygin equations kβϕ +0,αψ ==βψ Ð0;αϕ (3) ϕ ψ ϕ ψ z +0,k θ ==θ Ð0z (1) ()αρu + βv + ()αρv Ð βu or the Chaplygin equations of the second order for the x y ()ρ ()ρ stream function [1Ð4]: = f ug+ v x +0.f v Ð gu x = (4)

ρ 1MÐ 2 Remark. The solutions ϕ = f = const and ψ = g = kψθθ +0,ψ ==z ---dq, k =------. (2) zz ∫ q ρ2 const also allow us to obtain certain conservation laws on the (x, y)-plane. For example, substituting either Hereafter, ϕ is the potential; ψ is the stream function; q, f = 1 and g = 0 or f = 0 and g = 1 into (4), we obtain the θ, u, and v are the modulus, angle of inclination, and conservation law for either mass or circulation, respec- horizontal and vertical components of the velocity vec- tively. tor, respectively; M is the Mach number; p is pressure; 2. We consider several tentative examples of exact and ρ is the density. solutions. Eqs. (1) and (2) have an infinite set of exact solu- A flow with a source (radial motion). In this case, tions [1Ð4]. In this paper, we show that each solution is associated with a proper set of gas-dynamics equations, g = θ and f = −∫k dz = – K(z). Then, one of them being divergent (see Theorem 1). Our detailed analysis of the solutions to system (1), which kθϕ Ð0,Kψ ==θψ +0;Kϕ are found by the method of separation of variables led to one more algorithm for constructing an infinite set of ()Kρu Ð θv +0.()Kρv + θu = conservation laws (see Theorem 2). The set constructed x y includes, as a particular case, the set of the conservation A flow similar to a potential vortex. In this case, laws found in [5, 6]. They can be applied to two-dimen- θ sional steady flows (mass, momentum, and angular- f = and g = z. Then, momentum conservation laws). It is worth noting that a kzϕ +0,θψ ==zψ Ð0,θϕ finite set of the conservation laws was found in [5, 6] (5) ()θρ ()θρ for the cases of both steady and unsteady three-dimen- uz+ v z +0.v Ð zu y = sional flows. Theorem 1. For each solution ϕ = f(z, θ) ≠ const It is worth noting that system (5) is well known in gas and ψ = g(z, θ) ≠ const to Chaplygin equations (1), the dynamics [4], but irrespectively of the potential-vortex introduction of variables α and β dependent on the flow. θ θ α θ functions f(z, ) and g(z, ), respectively, = f(z, ) and 2 β θ θ = g(z, ) yields a system of uniformly divergent gas- The solution ψ = g = θz and ϕ = f = ----- – F(z), with dynamics equations in the plane of the potentials 2 F(z) = ∫k zdz, leads to the equations

Sobolev Institute of Mathematics, Siberian Division, θ2 θ2 ()θ  ()θ  Russian Academy of Sciences, pr. Koptyuga 4, k z ϕ +0,----- Ð F ==z ψ Ð0;----- Ð F Novosibirsk, 630090 Russia 2 ψ 2 ϕ

1028-3358/02/4703-0173 $22.00 © 2002 MAIK “Nauka/Interperiodica” 174 RYLOV α β and the variables ( = fi, = gi) dependent on corresponding functions and an infinite set of conservation laws θ2 θ2 ρ θ ρ θ acquiring the following form: ----- Ð F u + zv +0.----- Ð F v Ð zu = 2 x 2 y () () f i ϕ Ð0,gi ψ = θ2 ()ρ ()ρ The solution ϕ = f = –θK(z) and ψ = g = ----- – G(z), f i ug+ iv x +0.f i v Ð giu y = 2 In the general case, the functions h(1) and h(2) are with G(z) = ∫K (z)dz, leads to the equations expressed in terms of the hypergeometric functions [1Ð4]. In the case of λ = 1, they take the known explicit form: θ2 θ2 ()θ ()θ k----- Ð G Ð0,K ψ ==----- Ð G +0;K ϕ ()1 1 ()1 1 2 ϕ 2 ψ h ()1; z = ---, h ()1; z = Ð,------q z ρq and 2 ()2 p ()2 p + ρq 2 2 () () θ θ h 1; z = ---, hz 1; z = Ð.------ρ θKρu Ð ----- Ð G v +0.θKρv + ----- Ð G u = q q 2 2 x y The functions h(1) correspond to the Ringleb flow [1, It should be noted that the angle θ of inclination of 3, 8]. The streamline pattern for the flow corresponding the velocity vector enters into the conservation laws to the function h(2) is very similar to that for the Ringleb under consideration as both an explicit argument and flow [1]. We now present the relations that describe the argument of the trigonometric functions. Moreover, such flows, the systems of gas-dynamics equations in the two last solutions depend on the variable θ2. This the form of Eqs. (3), and the corresponding conserva- fact can be used to carry out certain estimates. For tion laws in the physical plane. example, using the known asymptotic form of the sym- Ringleb flow. (a): metric flow around a body [7] and the conservation law cosθ sinθ presented above, we can prove that the following rela- ϕ ==f ------, ψ ==q ------; tionships are satisfied at an arbitrary streamline over its 1 ρq 1 q entire length: θ θ θ θ +∞ +∞ sin cos sin cos 2 k------+0,------= ------Ð0;------= θ q ϕ ρq ψ q ψ ρq ϕ ()Gz()Ð Gz()∞ qld = -----qld > 0. ∫ ∫ 2 ∞ ∞ θ θ θ θ Ð Ð cos ρ sin cos ρ sin ------ρ - u + ------v + ------v Ð ------u 3. In order to construct new systems of equations q q x pq q y and conservation laws, we now consider known solutions to Eqs. (1), which are found by the method of sep- = 1x +0.0y = aration of variables. The four independent solutions are Ringleb flow. (b): of the form sinθ cosθ ϕ ==f Ð------, ψ ==g ------; 1 ()1 λθ ()1 λθ 2 ρq 2 q f 1 ==Ðλ---hz cos , g1 h sin ; cosθ sinθ cosθ sinθ 1 ()1 λθ ()1 λθ k------Ð0,------ρ - ==------+0;------ρ- f 2 ==λ---hz sin , g2 h cos ; q ϕ q ψ q ψ p ϕ (6) () () sinθ cosθ sinθ cosθ 1 2 λθ 2 λθ Ð------ρu + ------v Ð ------ρv + ------u f 3 ==Ð---hz cos , g3 h sin ; ρ ρ λ q q x q q y

() () = 0 Ð0.1 = 1 2 λθ 2 λθ x y f 4 ==λ---hz sin , g4 h cos . Modified Ringleb flow. (a): Here, λ is an arbitrary constant, and the functions 2 p + ρq p h(1) λ z and h(2) λ z are two independent solutions to ϕ θ ψ θ ( ; ) ( ; ) ==f 3 ------ρ cos , ==g3 --- sin ; the ordinary differential equation of the second order: q q λ2 hzz(z) = kh(z). The subscript z stands for differentia- ρ 2 p θ p + q θ tion with respect to z. k--- sin +0,------cos = q ρq ψ Since the solutions depend on the arbitrary constant ϕ λ , it is possible to construct an infinite set of the func- ρ 2 tions (f , g ). Therefore, according to Theorem 1, we can p θ p + q θ i i --- sin Ð0;------ρ cos = find an infinite set of gas-dynamics equations (3) with q ψ q ϕ

DOKLADY PHYSICS Vol. 47 No. 3 2002 CHAPLYGIN EQUATIONS AND AN INFINITE SET 175

()ρ 2 ()ρ (), p + u x +0.uv y = R 41 ==Ypx∫ d , Modiﬁed Ringleb ﬂow. (b): with the integrals taken along streamlines. In the case of λ = 1, in the potential plane and the p + ρq2 p ϕ ==f Ð------sinθ, ψ ==g --- cosθ; physical plane, conservation law (7) takes the following 4 ρq 4 q forms, respectively:

2 2 2 ρ ρ ρ p + q θ p + q θ p θ p + q θ ------cos y Ð ------sin x k--- cos Ð0,------sin = ρq ρq ϕ q ϕ ρq ψ p p Ð --- sinθy + --- cosθx = 0, ρ 2  p θ p + q θ q q ψ --- cos +0;------sin = q ψ ρq ϕ 2 2 ()()p + ρu y Ð ρuvx x +0.()ρuvyppÐ ()+ v x y = ()()ρ 2 puv x +0.p + v y = This is simply the angular-momentum conservation law. λ The Ringleb flow was comprehensively discussed in For = 1, the functions f and g entering into (7) cor- the literature [1, 3, 8]. It represents one of a few exact respond to the modified Ringleb flow, while the func- solutions to the problem on a flow with a local super- tion R corresponds to the Ringleb flow. At the same sonic area. However, the Ringleb flow leads to no time, these flows interchange in conservation law (8). instructive conservation law in the (x, y)-plane. At the As a result, we have same time, the modified Ringleb flow was only briefly θ θ θ θ mentioned in the literature [1], not being discussed in cos sin sin cos ------ρ -X Ð ------ρ -Y Ð0,------X + ------Y = detail. However, as was shown above, this is the flow q q ϕ q q ψ that is associated with the momentum conservation law. Nevertheless, we prove below that a combination of the Xx Ð0.Y y = functions describing the Ringleb flows of both types λ enters into the known angular-momentum conservation As is seen, for = 1 in the physical plane, conservation law. law (8) has an extremely simple form. It is evident that Theorem 1 does not exhaust all possible conservation laws, in particular, the angular- ACKNOWLEDGMENTS momentum conservation law. The following theorem, which to a large extent generalizes the conservation law This work was supported in part by the Russian Foun- mentioned above, is of interest. dation for Basic Research, project no. 01-01-00851. Theorem 2. For arbitrary values of λ, the following conservation laws hold: REFERENCES ()(), λ (), λ 1. R. von Mises, Mathematical Theory of Compressible f 3 R 1 + f 4 R 2 ϕ Fluid Flow (Acad. Press, New York, 1958; Inostrannaya ()(), λ (), λ Literatura, Moscow, 1961). Ð g3 R 1 + g4 R 2 ψ = 0, (7) 2. L. I. Sedov, Plane Problems of Hydrodynamics and ()(), λ (), λ Aerodynamics (Nauka, Moscow, 1966, 2 ed.; Academic f 1 R 3 + f 2 R 4 ϕ Press, New York, 1959). ()(), λ (), λ Ð g1 R 3 + g2 R 4 ψ = 0. (8) 3. G. G. Chernyœ, Gas Dynamics (Nauka, Moscow, 1988). 4. L. V. Ovsyannikov, Lectures on Basic Principles of Gas Here, fi and gi are the functions given by Eqs. (6), and Dynamics (Nauka, Moscow, 1981). the function R(i, λ) is defined by its derivatives: λ λ 5. N. Kh. Ibragimov, Dokl. Akad. Nauk SSSR 210, 1307 Rϕ(i, ) = gi and Rψ(i, ) = fi . (1973). Remark. In contrast to Theorem 1, the expressions in 6. E. D. Terent’ev and Yu. D. Shmyglevskiœ, Zh. Vych. Mat. the parentheses in (7) and (8) do not satisfy system (1). Mat. Fiz. 15, 1535 (1975). Corollary. For λ = 1, the function R can be written 7. A. I. Rylov, Prikl. Mat. Mekh. 63, 405 (1999). out in the explicit form 8. F. Ringleb, Z. Angew. Math. Mech. 20, 85 (1940).

, (), , R()11 ====y, R 21 x, R()31 Xpy∫ d , Translated by V. Chechin

DOKLADY PHYSICS Vol. 47 No. 3 2002

PHYSICS

Behavior of the Parameters of the Thermal Surface Decomposition of Polymers near the Temperature Boundary of Thermodynamic Stability O. F. Shlensky and A. S. Lyapin Presented by Academician V.V. Osiko September 20, 2001

Received October 11, 2001

The upper temperature phase-state boundary (PSB), being examined. These are (1) polymethylmethacrylate below which substances in their condensed state are (PMMA), (2) high-density polyethylene (LPP), and thermodynamically stable, can be reached by both (3) polyethylene glycol (PEG-40000) with a molecular high-intensity heating and a sharp decrease in pressure mass of 40000 amu. The plots clearly demonstrate that (see, e.g., [1Ð4]). Thermal processes accompanying the the thermal decomposition of the first two polymers is decomposition of materials under one-sided heating of a multistage process that can be treated with allowance their surface layers are of the most interest for practice, for the evaporation of the resulting products of complex in particular, from the standpoint of both the stability of chemical composition [11, 12]. To develop a mathemat- heat-resistant coatings and burning [5]. A new method ical model for processes occurring at the thermal- of contact thermal analysis, described in [6, 7], makes decomposition front under intense heating, it is suffi- it possible to record the kinetics of thermal decomposi- cient to determine by recording photodetector signals tion with a duration from several seconds to several tens the thermal-decomposition duration at a certain tem- of microseconds and to determine kinetic parameters in perature. In other words, we need to measure the time the immediate vicinity of the PSB of materials. interval td between the first dip (the moment when a In this study, we propose a mathematical model for sample is set onto a substrate) and the moment when describing the results of kinetic investigations and sim- the td line becomes horizontal. Figure 2 shows the time ulating processes at the front of polymer decomposition td in the semilogarithmic scale as a function of the under high-intensity thermal attacks on surface layers reciprocal absolute temperature attained when testing for temperatures up to the PSB. samples. The lower branches of the plots have a shape The position of the PSB is thermodynamically close to inclined straight lines and can be described by determined from the condition that the second variation the Arrhenius equation. The left upper branches are of one of the thermodynamic potentials is zero; e.g., bent upward and asymptotically approach vertical 2 1 δ G = 0, where G is the Gibbs free energy [1]. At the straight lines with abscissas ---- . This behavior is incon- ∂ p ∂T T l stability boundary, the derivatives ------and ------vanish, ∂V ∂V sistent with Arrhenius kinetics. and, therefore, the PSB can be calculated from the To describe the experimental results, various modi- equation of state. The wide-range equations of state fied Arrhenius equations were proposed in [10, 11], proposed recently in [8, 9] allow accurate calculations whose additional multiplier takes into account the of the parameters of both polymers and other materials. peculiar behavior of systems near the PSB. We use the Experimental methods enable us to determine the following equation to obtain this multiplier in the ana- kinetic analogs of the temperatures at the PSB, the lytical form: attainable superheating temperatures Tl [2Ð4, 6, 7], which are only slightly (2–5°ë) lower than corre- n sponding temperatures on the PSB at a given pres- () T E kT = exp---- BexpÐ------, (1) sure. Figure 1 shows the simplified block diagram of T l RT a setup for contact thermal analysis and signal recording from a photodetector for three linear polymers where Tl, E, and B are the problem parameters and the exponent n lies within the range from 3 to 20, depending on the substance properties. Equation (1) for n = 0 Mendeleev University of Chemical Technology, transforms into the ordinary Arrhenius equation. The Miusskaya pl. 9, Moscow, 125190 Russia preexponential factor is responsible for the accelerated

BEHAVIOR OF THE PARAMETERS OF THE THERMAL SURFACE DECOMPOSITION 177

Video camera (a)

Filter set Illuminating HeÐNe laser

Sample

Converter Substrate ~ 220 V Monitor Heater

I, arb. units 25 (b)

20 1

2 10

5 3

0 100 200 300 400 t, ms Fig. 1. Determination of the parameters for the upper temperature boundary of thermodynamic stability of polymeric materials: (a) block diagram of the experimental setup; (b) recording signals from a photodetector for (1) polymethylmethacrylate, (2) low- density polyethylene, and (3) polyethylene glycol. decomposition of polymers due to weakening the inter- heat-absorption function usually taken in the form ρ molecular interaction and the intensiﬁcation of homo- F(T) = 0Qk(T) [5] under the assumption that thermol- geneous nucleation near the PSB [2Ð4, 6, 10]. ysis is a zero-order reaction, Q is the thermal effect of We consider the thermal decomposition of a mate- the reaction with the rate constant k(T), and ρ is the rial half-space, which is described by the heat-conduc- 1 tion equation density. For these reactions, we have k(T) = --- , which td d dT dT allows the use of Eq. (1). ------λ------Ð FT()+0,ρC u Ð ------= (2) dx dx p dx The last term on the left-hand side of Eq. (2) is usually ignored because of its smallness [5]. Integrating where u is the decomposition-front velocity, F(T) is the Eq. (2) with allowance for the boundary condition

DOKLADY PHYSICS Vol. 47 No. 3 2002

178 SHLENSKY, LYAPIN

dT 1/2 log(1/td) ------= ÐDJT()() . (3) 500°C dx 2.0 5

530°C 470°C 400°C 380°C T ρ 1/2 () 2 0Q ρ λ 1.5 6 Here, J(T) = ∫ kTdT , D = ------λ - , and 0 and 0 1 0 T0 1.0 are the original values of ρ and λ, respectively. 3 7 Furthermore, we equate the heat ﬂow qw arriving at 9 0.5 10 the surface to the general heat spent for heating and 515°C 4 decomposing the material:

0 540°C 1.25 1.30 1.35 1.40 1.45 1.50 103/T, K–1 dT 5 q ==Ðλ ------uρ[]C ()ξT Ð T + Q , (4) w wdx p w 0 –0.5 11 3 x = 0 1 8 2 ρ()T –1.0 λ λ χλ χ Ӎ w ξ where w = (Tw) = 0, ------ρ , and is the frac-

= 380°C 0 i t –1.5 tion of the reacted substance. Substituting the temperature gradient from Eq. (3) into Eq. (4), taking into account Eq. (1), and approximately calculating the integral J(Tw) for n < 55 with an accuracy of 5% (see Fig. 2. Results of studying the kinetics of polymer thermal decomposition near the PSB: (1) high-impact polystyrene, Appendix), we arrive at the equation (2) block polystyrene, (3) high-density polyethylene, n/2 1/2 (4) lavsan (dacron), (5) low-density polyethylene, (6) poly- Dλ exp()ÐE/2RT exp()T /T B u = ------w w w l . (5) vinyl chloride, (7) polymethylmethacrylate, (8) polycapro- ρ []()ξ[]( n Ð 1 n ) 2 lactam, (9) polyethylene glycol, (10) borax, (11) Alanin- 0 C p T w Ð T 0 + Q nTw /T l + E/RTw skoe-deposit oil. For n = 0 and T < Tl, Eq. (5) goes over into the well- v, mm/s known MerzhanovÐDubovitskiœ formula [12] that describes the linear dependence of logu on Tw: 100 III 2 () aRTw BEexp Ð /RTw u1 = ------[]() -, (6) ETw Ð T 0 + Q/2C p –1 λ λ ξ 10 II where a is the thermal diffusivity, w = 0, and = 1. We compare the result obtained with available 1 experimental data. Figure 3 shows the rate of thermal 2 decomposition and gasiﬁcation of linear PMMA and 10–2 3 I spatially cross-linked polymers produced by the copo- 512 °C lymerization of methyl methacrylate and diethyl ether or triethylene glycol (TEG) in the case of burning and 1.1 1.2 1.3 1.4 1.5 linear pyrolysis. The samples were heated by the linear- (1/T) × 103, K–1 pyrolysis method through contacting their end surface with a hot metallic plate containing numerous holes for Fig. 3. (1, 2) Decomposition-front velocity and (3) burning the removal of pyrolysis products. rate as a function of the reciprocal temperature for (1) polymethylmethacrylate and (2, 3) polymethylmethacrylate + The lower sections of the plots (Fig. 3, regime I) cor- triethylene glycol; IÐIII are regimes of thermal decomposi- respond to low temperatures at which sample cooling by tion. Experimental points are obtained by linear pyrolysis, a surrounding medium affects the result. In region II, and dashed lines are calculated by Eqs. (5) and (7) with experimental data are approximated by an inclined n = 48 for heating regimes II and III. For n = 100, the dashed lines merge with the vertical dashed-dotted straight line. straight line [Eq. (6) and Eq. (5) for T < Tl]. In this case, the kinetic parameters determined by solving the inverse problem of nonisothermal kinetics agree well with those ∞ obtained by the thermal analysis (see Table 1). T = T0 as x and taking into account variations of λ ρ() Regime III observed for higher temperatures λ 0 T the thermal conductivity (T) = ------ρ caused by the sharply differs from the two other regimes: the thermal- 0 decomposition rate increases abruptly with a minor rise appearance of secondary porosity [6], we obtain in temperature. Experimental points deviate upward

DOKLADY PHYSICS Vol. 47 No. 3 2002 BEHAVIOR OF THE PARAMETERS OF THE THERMAL SURFACE DECOMPOSITION 179 from the straight sloping Arrhenius line that corre- Table 1. Kinetic parameters of the polymer thermal decom- sponds to Eq. (6). At the same time, this deviation composition for polymethylmethacrylate (PMMA) and methyl pletely corresponds to Eq. (5), which includes the PSB methacrylateÐtriethylene glycol (MMA + TEG), which are parameters. For this case, the maximum temperature of determined by methods of thermal analysis (TA) and linear pyrolysis (LP) the surface for PMMA samples (Tw ~ 512°ë) corresponds to data shown in Fig. 2, where the temperature E, kcal/mol logB [sÐ1] Tl of various PMMA trade marks (shaded region) lies Polymer within the range 500 to 515°ë [12]. TA LP TA LP The relation of Eqs. (5) and (6) is descriptively rep- ± ± ± ± resented as the product PMMA 42.6 2 43 3 12.47 0.7 13.32 1 ± ± ± ± u = K(T )u , (7) MMA + 43.5 2 43 3 12.14 0.7 12.51 1 w 1 2% TEG-3 where MMA + 43.8 ± 2 43 ± 3 12.3 ± 0.7 12.51 ± 1 ⁄ exp()T /T n 2 10% TEG-3 KT()= ------w l -. w ()n nRT Tw/T l /E + 1 Introducing the coefficient K(T ) provides the Table 2. Values of the K(TW) coefficient for polymethyl- w methacrylate (PMMA) description of the transition from regime II to regime III and a further increase in the thermal-decomposition T K (TW) rate. According to data of [2Ð4], the frequency of nucle------ation rises by twoÐnine orders of magnitude with an Tl n = 24 n = 48 increase in temperature near the PSB by only 1°C. This rise can be described by Eq. (1) with n = 20Ð100. 0.8 1.066 1.004 Table 2 presents the values of K(Tw) for PMMA at E = 0.85 1.133 1.013 43.0 kcal/mol and Tl = 773 K for n = 24 and 48 for var- 0.9 1.289 1.071 T ious ratios ---- . 0.95 1.552 1.159 T l 1.0 1.995 1.575 This table demonstrates that the coefficient K(Tw) is 1.1 5.042 112.5 close to unity for T < T , and Eq. (5) approximates the l 1.2 249.6 3.06 × 1030 linear section of the plots. The dashed line in Fig. 2 shows an increase in the decomposition-front velocity u according to Eqs. (5) and (7) for n = 24. A similar cal- under one-sided heating allow us to make important culation was performed for other polymers of the spatially cross-linked structure, PMMA + TEG (2 and conclusions. These studies have demonstrated that het- 10%, respectively). erogeneous nucleation changes to homogeneous nucleation near the PSB with an increase in the heating inten- According to Eq. (3), the highest temperature gradi- sity. The Arrhenius equation traditionally applied for ents arise near the heated surface. In particular, the ther- mathematically simulating polymer thermal decompo- TTÐ mal-decomposition depth l ~ ------0 is equal to 1.3 × sition does not involve information on the PSB param- dx ------eters and, therefore, cannot allow for a change in the dT nucleation mechanism near the PSB. The above-pro- Ð3 ≈ dT × 8 posed modification of the Arrhenius equation, in con- 10 mm for PMMA at Tw Tl and ------= 3 10 K/m. dx trast to other attempts [10, 11], provides not only For this value of l, Eqs. (5) and (7) are applicable for numerical calculation of the macroscopic kinetics of small polymeric particles, polymer granules, and poly- thermal-decomposition processes for polymers near the meric drops of various shapes with the characteristic PSB but also analytical description of these processes. size L ӷ l. The limitation of an increase in surface tem- In particular, this modification makes it possible to perature (stabilization) with qw is predicted by Eqs. (4) derive the expression for the linear rate and the mass and (7) and is corroborated in experiments with various rate of the thermal decomposition with the variation in methods of thermal supply, e.g., heating by an electric the parameters near the PSB taken into account. The arc, radiant and convective heating, diffusion burning in expression obtained is corroborated by the data of pre- media enriched in an oxidant, laser-beam action, etc. vious experimental studies. The above-proposed rela- [5, 11Ð15]. tions considerably improve the accuracy of calculations Thus, we may state that theoretical and experimen- for temperature fields and other characteristics of the tal studies [10, 12, 13] involving microscopic and mac- thermal decomposition under the highly intense heating roscopic kinetics of polymer thermal decomposition of polymeric materials.

DOKLADY PHYSICS Vol. 47 No. 3 2002 180 SHLENSKY, LYAPIN

Appendix. Calculation of the J(T) integral. We 2. V. P. Skripov, E. N. Sinitsyn, P. A. Pavlov, et al., Thermal () and Physical Properties of Liquids in Metastable State approximately calculate the integral ∫ expFxdx by (Atomizdat, Moscow, 1980). the following method. For a linear function F(x), the 3. V. P. Skripov, Metastable Liquid (Nauka, Moscow, integral is exactly calculated as 1972). 4. P. A. Pavlov, Boiling Dynamics for Highly Superheated ∫ expFx()dx = expFx()/F'()x + const. Liquids (Izd. Akad. Nauk SSSR, Sverdlovsk, 1988). 5. Yu. V. Polezhaev and F. B. Yurevich, Thermal Resistance This expression is used as the ﬁrst approximation (Énergiya, Moscow, 1976). for integrating the nonlinear function F(x) = k(T) with a 6. O. F. Shlensky, Dokl. Akad. Nauk 378, 179 (2001) certain error ∆, whose upper estimate is [Dokl. Phys. 46, 317 (2001)]. 7. O. F. Shlensky, Zh. Fiz. Khim. 75, 636 (2001). Bnn()()Ð 1 ()T/T n Ð 2E/RT ∆ = ------l 8. I. V. Lomonosov, V. E. Fortov, and K. V. Khishchenko, 1 ()n ()2 Khim. Fiz. 14, 47 (1995). nT/Tl + E/RT 9. I. V. Lomonosov, K. V. Khishchenko, V. E. Fortov, and n × T E O. F. Shlensky, Dokl. Akad. Nauk 349, 322 (1996) exp---- expÐ------. [Phys. Dokl. 41, 304 (1996)]. T l RT 10. O. F. Shlensky, N. V. Afanas’ev, and A. G. Shashkov, The corresponding expression for the desired integral Thermal Decomposition of Materials (Énergoatomizdat, can be written out in the form Moscow, 1996). 11. N. A. Zyrichev and O. F. Shlensky, Khim. Vys. Énerg. n () BT T E 34, 46 (2000). JT = ------n exp---- expÐ------, nT()/T + E/RT T l RT 12. R. M. Aseeva and G. E. Zaœkov, Combustion of Poly- l meric Materials (Nauka, Moscow, 1981). dT 13. A. S. Shteœnberg, Preprint, IKhF RAN (Inst. of Chemical which is used to calculate the gradient ------and the dx Physics, Russian Academy of Sciences, 1977). decomposition-front velocity u. 14. A. G. Gal’chenko, N. A. Khalturinskiœ, and A. A. Berlin, Vysokomol. Soedin. 22, 125 (1980). 15. A. V. Butenin and B. Ya. Kagan, Zh. Tekh. Fiz. 49, 870 REFERENCES (1979). 1. A. Münster, Chemische Thermodynamik (Akademie, Berlin, 1970; Mir, Moscow, 1971). Translated by R. Tyapaev

DOKLADY PHYSICS Vol. 47 No. 3 2002

Doklady Physics, Vol. 47, No. 3, 2002, pp. 181Ð183. Translated from Doklady Akademii Nauk, Vol. 383, No. 3, 2002, pp. 322Ð325. Original Russian Text Copyright © 2002 by Koronovskiœ, Trubetskov, Khramov, Khramova.

PHYSICS

Universal Scaling Laws of Transients

A. A. Koronovskiœ, Corresponding Member of the RAS D. I. Trubetskov, A. E. Khramov, and A. E. Khramova Received September 25, 2001

The intricate behavior of nonlinear dynamical sys- trolling parameter λ in the range from 1.0 to 3.57. In tems currently attracts the attention of researchers. To a other words, the subharmonic-cascade region is consid- large extent, it is associated with the fundamental prob- ered [1]. lem of studying the laws of dynamical chaos. Systems Figure 1 illustrates the dependence of the transient with discrete time (of mapping) are the simplest nonlin- duration for λ values corresponding to cases where a ear dynamical systems exhibiting many typical phe- fixed stable point exists in the system. It is seen that the nomena. dependence of the transient duration K on the initial However, there are only a few results on analyzing conditions x0 is of a complex “jagged” form. The transients, even though dynamical chaos and scenarios dependence K(x0) is regularly complicated as the con- of the orderÐchaos transition are actively studied. It is trolling parameter λ increases. generally thought that transients are unessential features of the system dynamics and do not carry any par- Global minima in the transient-duration dependence ticular information. Stable regimes (periodic, quasi- correspond to the initial conditions x0 coinciding with periodic, chaotic) are a major focus of interest, whereas the points of an attractor that exists in the system phase space for a given value of the controlling parameter λ. no consideration is given to transients. When transients λ are prolonged processes, they may be perceived as In other words, for = 1Ð3, the global minimum of the obstacles to efficient investigations of the system under transient-duration dependence is attained at the fixed λ Ð 1 consideration. At the same time, the role of transients in stable point x0 = ------, which is an attractor (Fig. 1). As nonlinear-system dynamics is of great importance. For λ a number of dynamical systems (systems under an the parameter λ increases and the fixed point x0 loses its external impulse action, systems with very prolonged stability, the global minima of the transient duration ini- transients, systems with multistability, and distributed tially correspond to the elements of the stable 2-cycle oscillatory systems), a transient substantially deter- λ mines their behavior. Thus, an important class of phe- [ = 3–(1 + 6 )], and then to the elements of the stable nomena was actually disregarded. As will be shown cycle with a period of 4, etc. below, a number of scaling laws exist for transients, and Since logistic mapping (1) is irreversible, two processes occurring in a system affect the transient- {}()0 ∞ {}()1 ∞ duration dependence on initial conditions under the sequences of points xi i = 1 and xi i = 1 exist for variation of controlling parameters. λ = 1Ð3.1 These points are mapped into the fixed stable In this paper, transients in a nonlinear discrete-time point x0 in a finite number of iterations: dynamical system are analyzed in detail for the first () () () 0 ()0 ()()0 ()()1 time, and mechanisms complicating the transient-dura- x ==fx1 ffx2 =ffx2 (2) tion dependence on initial conditions are disclosed. ()n ()()0 ()n ()()1 Fundamentally new scaling laws typical for transients = … ==f xn f xn . are found. () ∞ The sequence {}x 0 converges to the boundary of We study transients by the example of the classic i i = 1 nonlinear-dynamic model, logistic mapping [1]: the attraction pool of the mapping attractor x = 0 as λ ()0 xn + 1 = f(xn) = xn(1 – xn). (1) ()0 xi x = ------, i ∞ . (3) The dependence of the transient duration on initial i + 1 λ conditions is investigated for various values of the con- {}()1 ∞ The sequence xi i = 1 converges to the other bound-

Saratov State University, 1 With the exception of the case λ = 2 for which x0 = 0.5, the condi- ul. Universitetskaya 42, Saratov, 410601 Russia tion f(x) = x0 is satisﬁed only for x = x0.

f(x) K 80 0.8 60

20 0.4

0 0.005 0.010 0.015 0.020

80 0 (0) (0) (0) (1) (1) x 2 x 1 x x 2 x 3 60 K(x0) 40

40 20

0 0.002 0.004 0.006

20 80 60

0 0.2 0.4 0.6 0.8 x0 20 Fig. 1. The Lameray diagram and the transient-duration dependence for the logistic mapping at the controlling parameter λ = 2.75 corresponding to the case where a ﬁxed 0 0.001 0.002 x0 stable point is realized in the system. Fig. 2. Scaling for the transient-duration dependence K(x0) with the scale factor λ with respect to the attraction-pool boundary x = 0 of the attractor. The region to the left of the ary of the attraction pool of the mapping attractor x = 1 as dashed line is exaggerated by a factor of λ. On rescaling, the transient-duration dependence is repeated by shifting to greater values by +1. The results are obtained for the con- ()1 () x trolling-parameter value λ = 2.75 when a ﬁxed stable point 1 i ∞ xi + 1 = 1 Ð ------λ-, i . (4) is realized in the system.

() ∞ 2 Since the points of the sequences {}x k , k = 0, 1 with the elements of these unstable cycles. Simulta- i i = 1 neously, the transient duration is also infinite at the set are mapped into the fixed stable point x in a finite num- 0 {}k ∞ ber of iterations, the minima of the transient durations of points yi i = 1 , which, by virtue of the irreversibil- ()k are also observed at these points (Fig. 1), and K()xi + 1 =ity of mapping (1), are mapped into elements of unsta- ()k ble cycles in a finite number of iterations. Second, the K(x ) + 1. The maxima of the transient duration () ∞ ∞ i number of sequences {}x k and {}yk whose behave similarly. Therefore, the dependence of the tran- i i = 1 i i = 1 sient duration K on the initial conditions x0 in the sys- elements are mapped in a finite number of iterations tem under consideration exhibits scaling with the scale into elements of either stable or unstable cycles factor λ relative to the boundaries x = 0 and x = 1 of the increases.3 attraction pool of the attractor (Fig. 2). When stable n-cycles, n , are realized in λ 2 = 1, 2, … As the controlling parameter increases, a cascade of the system, the transient-duration dependence also period-doubling bifurcations occurs in the system [1], and the dependence of the transient duration K on the 2 Maxima of the transient durations are of different height and do initial conditions x0 is complicated for few reasons. not turn to infinity in numerical simulation, because the initial First, unstable cycles appear, and the transient duration conditions x0 are given with a finite accuracy. 3 λ is infinite for the initial-condition points x0 coinciding Generally, the number of these sequences is infinite for > 3.

DOKLADY PHYSICS Vol. 47 No. 3 2002 UNIVERSAL SCALING LAWS OF TRANSIENTS 183 exhibits scaling in the vicinity of the attraction-pool K boundary points of the attractor. However, after the fixed point x0 has lost its stability (and a stable cycle with a 60 period of 2 appears), a scaling in the vicinity of this fixed unstable point is observed with the scale factor µ = f '(x0), which is a multiplier of this fixed point. To illustrate this 40 behavior, Figure 3 shows the transient-duration dependences near an unstable fixed point for λ = 3.25. 20 The existence of the unstable fixed point implies that () ∞ there are two sequences {}z k , k = 0, 1, converging 0 i i = 1 0.8936 0.8944 from the left and right to the unstable point x0 as ()k 0 60 ()k 0 zi Ð x → ∞ zi + 1 = x + ------µ -, i . (5) 40 A local minimum of the dependence K(x0) corre- () ∞ sponds to every element of the sequences {}z k i i = 1 20 ()k (Fig. 3). Similar to Eq. (2), every element zi is mapped into elements of the stable 2-cycle in a finite 0 number of iterations. Moreover, every element of the 0.8938 0.8940 0.8942 () ∞ sequences {}z k generates one or several i i = 1 60 {}()k ∞ sequences xi i = 1 converging to the attraction-pool boundary points of the attractors x = 0 and x = 1. In 40 other words, scaling with respect to the unstable point x0 is transferred to scaling in the vicinity of the attraction-pool boundaries of the attractor. The exception is 20 the case where the maximum-stability cycle with a period of 2 is realized in the system. In this case, only 0 these two points are mapped into the elements of the 0.8940 0.8941 0.8942 x stable 2-cycle in a finite number of iterations. 0 0 λ Fig. 3. Scaling in the vicinity of the unstable fixed point x It is apparent that, as the controlling parameter for λ = 3.25. The region between the dashed lines is exag- increases, similar phenomena will be also observed on gerated with the scale factor µ = 2.562. On rescaling, the the basis of the elements of 2n-cycles, n = 0, 1, …, los- transient-duration dependence K(x0) is repeated by shifting ing their stability. Thus, the transient-duration depen- to greater values by +1. λ dence K(x0) is regularly complicated as the parameter increases. The elements of unstable 2n-cycles, which when the controlling parameter varies and found uni- appear due to the cascade of the period-doubling bifur- versal scaling laws for such a complication. cations, are responsible for this complication. The scaling of the transient dependence K(x ) 0 ACKNOWLEDGMENTS described above is not unique. There is one more type of scaling caused by the behavior of the logistic map- This work was supported by the Russian Foundation ping at the critical point [1]. The renormalization-group for Basic Research (project no. 01-02-17392) and by analysis [2] demonstrates that the dependences of the the US Civilian Research and Development Foundation duration of transients for maximum-stability cycles for the Independent States of the Former Soviet Union exhibit scaling laws with the scaling constants a = (grant no. REC-006). −2.503 and b = 2. When a selected segment is scaled by a factor of a with respect to the point x = 0.5 and the transient duration is increased by a factor of b for the REFERENCES 2n-cycle (n = 1, 2, ...), the dependence K(x0) corre- 1. A. P. Kuznetsov and S. P. Kuznetsov, Izv. Vyssh. sponds to the similar dependence for the (2n Ð 1)-cycle. Uchebn. Zaved. Prikl. Nelin. Dinamika 1, 15 (1993). Thus, in this paper, using a logistic equation as an 2. M. J. Feigenbaum, J. Stat. Phys. 21, 669 (1979). example, we revealed the reasons complicating the transient-duration dependence on initial conditions Translated by V. Tsarev

DOKLADY PHYSICS Vol. 47 No. 3 2002

Doklady Physics, Vol. 47, No. 3, 2002, pp. 184Ð187. Translated from Doklady Akademii Nauk, Vol. 383, No. 3, 2002, pp. 326Ð329. Original Russian Text Copyright © 2002 by Krasnoslobodtsev, Lyakhov, Suyazov.

PHYSICS

Generalized Sine-Gordon Equation in Wave Dynamics and the Geometry of Surfaces A. V. Krasnoslobodtsev, G. A. Lyakhov, and N. V. Suyazov Presented by Academician F.V. Bunkin May 17, 2001

Received May 21, 2001

1. The sine-Gordon (sG) equation F(η) are arbitrary functions. The substitution

∂ξ∂ηΦΦ= sin , (1) ϕ ()ξ i + A3 + iA1 = G exp------, where ξ = x – ct and η = x + ct are the light-front vari- 2 ables, describes numerous physical phenomena [1Ð5]. ϕ ()η i Ð It was first applied in the differential geometry of sur- A4 + iA2 = F exp------faces with constant negative curvature (see, e.g., [6]). 2 Among other applications, this equation provides a with the real-valued functions ϕ± reduces Eqs. (3) to the rather general scheme of three-frequency interaction involving a counterrunning wave [7]. desired set of equations 2. Equation (1) can be generalized by analyzing ∂±ϕ± = sinϕ+− , (4) four-frequency interaction. When frequencies ω and j ∂ wave numbers kj satisfy resonance conditions where ∂± = ------are the operators of differentiation with ∂x± ω ω ω ω 1 + 2 ==3 ++4, k1 k2 k3 + k4, respect to the new independent variables the wave equation for a field reduces [8] to the set of η 2()η ξ 2()ξ x+ ==d F , xÐ Ð d G . equations for slow amplitudes aj . This set consists of ∫ ∫ four similar equations, the first of which is Differentiation of set (4) yields a pair of second- 4 order equations ()∂ + c ∂ a = ia ∑ g a 2 + ih a*a a , (2) t 1 x 1 1 1 j j 1 2 3 4 ∂ ∂ ϕ ()∂ ϕ 2 ϕ j = 1 + Ð ± = 1 Ð ± ± sin ±. (5) where cj is the group velocity and hj and gjl are the non- Each of these equations is a peculiar generalization linearity coefficients proportional to the components of of the sG equation. Indeed, these equations for 2 the cubic susceptibility. Furthermore, we assume that a (∂±ϕ±) Ӷ 1 (particular linearization) reduce to Eq. (1). pair of waves propagates contrary to another pair and In the three-frequency problem [7], sG equation (1) that the absolute values of the group velocities are equal arises from the set ∂ Φ = sinΦ , ∂−Φ = Φ . Therefore, to each other, c = c = – c = –c = c. In this case, under + + – – + 1 3 2 4 set (4) can be referred to as the symmetric sine-Gordon the physically realizable conditions g g and g 1j = 3j 2j = (SsG) equation. Set (4) was first derived in [9, 10] for g4j, and under the assumption of the equiphase condi- the case of the interaction of a pair of counterrunning tion, Eqs. (2) (in the natural normalization to the non- waves with a standing low-frequency wave. linearity coefficients and for the real-valued amplitudes A ) are reduced to the set 3. The set of equations (4) has a class of self-similar j solutions depending only on the linear combination S = ∂ ± ∂ − λ nλ–1 λ η A13, ==A2 A4 A31, , ξ A24, +A1 A3 A42, . (3) x+ + (–1) x– of the variables x±, where and n are real-valued and integer-valued parameters, respec- The set of equations (3) has two independent integrals, tively. Set (4) reduced for this class of solutions has the 2 2 2 ξ 2 2 2 η ξ A1 + A3 = G ( ) and A2 + A4 = F ( ), where G( ) and integral (Hamiltonian) ()nλϕλÐ1 ϕ Ð1 cos + Ð cos Ð = H. | | | | λ λ–1 Institute of General Physics, Russian Academy of Sciences, For H > Hs , where Hs = – , both components ul. Vavilova 38, Moscow, 117942 Russia ϕ± of a self-similar solution periodically oscillate. For

| | | | − + H < Hs , one component oscillates, whereas the other ϕ ϕ unboundedly increases. 2 2π m For H = (–1) Hs, where m is an integer, the separa- t = –40 trices of above regimes are the solitons

m S S ϕ± = ()±1 ×Λ2[]arctan()Ce ± arctan()Ce ()11± n (6) v + π ------+,m 2

1 + λ 0 –2π where Λ = ------and C is a constant. In the limiting 1 Ð λ 2 2π case of λ = ±1, solution (6) describes a smooth drop between two constant levels, i.e., a simple π kink. For t = 0 λ ≠ ±1, solution (6) is the superposition of two kinks with the constant relative shift ∆S = ln|Λ|. The sum and difference of two kinks is a two-step kink with a total amplitude of 2π and a pulse, respectively. In the coordinate system (x, t) (x± = x ± t), the amplitude and width of this pulse are determined by the parameter λ and 2 0 –2π []λ2 Ð ()Ð1 n depend on the pulse velocity v = ------. For 2 2π 1 Ð λ4 λ2 ϕ t = 40 < 1, the component – has the pulse form. In contrast to Eqs. (4), the classical sine-Gordon equation has only simple-kink self-similar solitons. 4. The sum and difference of the solutions to Eqs. (4), Φ ϕ ± ϕ ± = + Ð , (7) 0 –2π ∂ ∂ Φ Φ satisfy Eq. (1) in the form + – ± = sin ±. Therefore, –30 0 30 x set (4) describing actual phenomena [e.g., those Fig. 1. Interaction between two composed symmetric sine- described by Eq. (2)] is a dynamical realization of the 1 5 Gordon solitons (6) for λ = Ð--- , λ = Ð--- , C = –1, and m = formal Bäcklund transformation (BT) relating two 1 5 2 7 solutions to the sG equation (see, e.g., [1, 4]) with the ϕ n = 0: the elastic collision of two pulses ( –, solid line) and ϕ transformation parameter equal to 1. The pairs of solu- of two double kinks ( +, dashed line). tions to the sG equation do not all generate the solution to set (4) according to Eq. (7). However, the aforementioned relation can be used to construct those exact without introducing a simple kink, it is necessary to solutions to set (4) that describe interactions between remove this kink from the interaction, shifting the kink solitons (6). to ±∞. This result is attained by taking 0 or ∞ for the To construct these exact solutions, we apply the free constant of the first transformation (with λ = 1) in method that was developed for constructing many-soli- the chain (i.e., taking the degenerate first transforma- ton solutions of the classical sine-Gordon equation and tion). This modified method provides many-soliton is based on the commutativity of Bäcklund transforma- solutions of set (4), including solutions describing tions. The method consists in constructing closed breather-type solitary waves and their interaction. chains or networks of successive BTs [1, 4]. To obtain solutions to the SsG set, this method should be adapted Figure 1 shows an example of the exact solution to to Eqs. (4) using chains that end by the Bäcklund trans- set (4). This solution describes a collision of two com- 2 formation with the parameter λ = 1. In this case, the posite solitons (λ ≠ 1) [see Eqs. (6)]. For the motion in components of a solution to set (4) are constructed as the same direction in the (x, t) coordinate system, each ϕ the half-sum and half-difference [see Eq. (7)] of two of the components of the solution ± describes interac- solutions to the sG equation related by the above trans- tion between equal-type solitons. Here, one of the com- ϕ formation. However, this transformation corresponds to ponents ( – when v > 0) presents a most interesting introducing a simple kink to the interaction between case of the interaction between pulse solitons. After solitons [see Eq. (6) as λ 1]. In order to obtain gen- interaction, the shape of the solitons is completely eral many-soliton solutions to set (4), in particular, recovered.

DOKLADY PHYSICS Vol. 47 No. 3 2002 186 KRASNOSLOBODTSEV et al.

(a) (a)

r3 r3

1 1

0 0

–2 –2 –1 –1 0 0 r1 0 0 r r 1 2 2 r z, KG 2 1 z, KG (b)2 (b) 2 1 z z 1 0

KG 0 –1 0 2 KG ρ –1 0 2 ρ Fig. 2. (a) Surface of revolution (10) (contours u1, 2 = const are shown) and (b) generatrix z(ρ) (solid line) and curvature ρ KG (dashed line) as a function of radius . Fig. 3. The same as in Fig. 2 for surface of revolution (11).

5. We now discuss the application of SsG equations We now discuss surfaces for which the angle φ is ϕ φ in the geometry of surfaces. determined by Eq. (4) for x± = u1, 2 and + = . It follows ϕ from Eqs. (8) and (4) that the curvature KG = –cos – of Let a surface in the three-dimensional space with these surfaces can vary. We seek a corresponding exam- Cartesian coordinates r = {r1, r2, r3} be speciﬁed by ple among surfaces of revolution, equations rk = rk(u1, u2), k = 1, 2, 3. The Gaussian coor- r ==ρβsin , r ρβcos , r =z()ρ . dinates (u1, u2) of the surface are chosen in such a man- 1 2 3 ner that its metric tensor gij has the form g11 = g22 = 1 and In addition, we suppose that the angle φ and radius φ ρ σ g12 = g21 = cos . In this case, the Gaussian curvature of depend only on the sum = u1 + u2. Substituting these the surface is [11] relations into the deﬁnition for gij yields φ ∂2φ β ()ρ Ð1 1 ===bu1 Ð u2 , b sin--- , b const. KG = Ð------φ- ∂------∂ -. (8) 2 sin u1 u2 The generatrix of the surface of revolution, z(ρ), is This equation indicates that the angle φ between the determined by the equation contours u1, 2 = const for a surface of a constant negative 2 Ð2 dz 2 2 dρ curvature KG = Ð1 is described by the sG equation. An ------= ()1 Ð b ρ ------Ð 1. example of such a surface is the Beltrami surface [6]. dρ dσ

DOKLADY PHYSICS Vol. 47 No. 3 2002 GENERALIZED SINE-GORDON EQUATION IN WAVE DYNAMICS 187

To construct a surface corresponding to set (4), we tionary boundary conditions, the squared amplitudes of use simple-kink solution (6) for λ = ±1, φ = the interacting waves are proportional to the functions ±() u1 + u2 ± ϕ 2arctan[]e . In this case, the dependence of the 1 cos ±, +− . Therefore, the squared amplitudes of waves curvature on surface coordinates also has the form of a for each solution to set (4) are proportional to the Gauss- surface-curvature kink ian curvature of a surface generated by this solution. ± () KG = tanh u1 + u2 (9) ACKNOWLEDGMENTS describing a variation in the curvature K from 1 to Ð1. G This work was supported by the Russian Foundation In order to determine the surface, we should also for Basic Research, project nos. 00-15-99636 and 1 specify the parameter b. For b = --- , the surface is 99-02-16415. 2 expressed in elementary functions as REFERENCES +− σ Ð1/2 ρ = 2[] 1 + e 2 , 1. A. Seeger, H. Donth, and A. Kochenforfe, Z. Phys. 134, (10) 173 (1953). ()+−σ []±2σ Ð1/2 r3 = arcsinhe Ð 2 1 + e 2. J. K. Perring and T. H. R. Skyrme, Nucl. Phys. 31, 550 (1962). and is shown in Fig. 2. 3. S. L. McCall and E. L. Hahn, Phys. Rev. 183, 457 For b = 1, the generatrix of the surface is speciﬁed (1969). by the quadrature 4. G. L. Lamb, Rev. Mod. Phys. 43, 99 (1971). ρ []+−2σ Ð1/2 5. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, = 1 + e , Phys. Rev. Lett. 31, 125 (1973). σ ()±2s ()±2s ±4s (11) 6. B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko, Mod- +− 1 + e 1 ++e e ern Geometry: Methods and Applications (Nauka, Mos- r3 = ∫------± 2 ds ()1 + e 2s cow, 1986). 0 7. A. V. Krasnoslobodtsev, G. A. Lyakhov, and N. V. Suya- and is shown in Fig. 3. In the vicinity of ρ = 1, both of zov, Bull. Rus. Acad. Sci. Physics, Suppl. Phys. of the curvature radii of surface (11) have almost the same Vibrations 61, 197 (1997). magnitudes and this surface is close to the unit-radius 8. S. A. Akhmanov and R. V. Khokhlov, Problems of Non- hemisphere. As ρ 0, this surface degenerates to a linear Optics (VINITI, Moscow, 1965). Beltrami pseudosphere. Thus, kink (9) here describes a 9. K. I. Volyak, G. A. Lyakhov, and I. V. Shugan, Phys. ≈ variation in the curvature from the region with KG 1, Lett. A 96A, 53 (1983). in which the surface is locally spherical, to the region 10. K. I. Volyak, G. A. Lyakhov, and I. V. Shugan, Tr. Inst. ≈ with KG –1, in which the surface behaves as a Bel- Obshch. Fiz., Ross. Akad. Nauk 1, 114 (1986). trami pseudosphere. 11. B. Blyashke, Einfürung in die Differentialgeometrie The above applications of SsG equations are evi- (Springer, Berlin, 1950; Gostekhizdat, Moscow, 1957). dently related to each other. For the four-wave problem described by these equations [see Eqs. (3), (4)] with sta- Translated by R. Tyapaev

DOKLADY PHYSICS Vol. 47 No. 3 2002

Doklady Physics, Vol. 47, No. 3, 2002, pp. 188Ð191. Translated from Doklady Akademii Nauk, Vol. 383, No. 3, 2002, pp. 330Ð333. Original Russian Text Copyright © 2002 by Rudenko, A. Sobisevich, L. Sobisevich, Hedberg.

PHYSICS

Enhancement of Energy and Q-factor of a Nonlinear Resonator with an Increase in Its Losses

Corresponding Member of the RAS O. V. Rudenko*, A. L. Sobisevich**, L. E. Sobisevich**, and C. M. Hedberg*** Received November 6, 2001

A phenomenon that is paradoxical at first sight is Here, ρ, ε, and b are the density, nonlinearity, and effec- studied: an appropriately-organized energy outflow from tive viscosity of the medium, respectively [1]. a resonator cavity results not in the attenuation of nonlinear vibrations but in their noticeable enhancement. Recall that the quantity Q is the ratio of the characteristic amplitude of field oscillations in the resonator to An increase in the resonator Q-factor and in the the amplitude of the oscillations of the boundary. energy accumulated in it is well pronounced when the frequencies of higher harmonics generated in a nonlin- The necessity of increasing the energy stored in a ear medium are close to the natural frequencies of the resonator arises in many applications (see, e.g., [4, 5]). resonator. The ways of increasing Qnl were discussed in [6]. One important method based on the introduction of selective An important example of nonlinear systems with the ω necessary properties is an acoustic resonator with losses at the second-harmonic frequency 2 0 is ana- selective losses. First, we consider a conventional reso- lyzed below. nator whose right boundary x = L is fixed and whose left The general ideas of controlling the nonlinear inter- boundary x = 0 oscillates with the velocity actions by introducing selective losses were reported in (), ()ω [7, 8]. In this case, the absorption at the frequency 2ω ux= 0 t = Asin 0t . (1) 0 suppresses the generation of the second harmonic and, ω π Here, 0 = c/L, where c is the speed of sound. Since the therefore, interrupts the cascade of the nonlinear trans- ω spectrum of the natural frequencies is equidistant, n = fer of energy upwards over the spectrum. In practice, ω ω n 0 (n = 2, 3, …) and a cascade of nonlinear processes losses at the frequency 2 0 can be realized either by takes place in the system, resulting in an effective trans- introducing resonant scatterers to the medium (e.g., gas fer of energy to the high-frequency region (higher har- bubbles to a fluid) or by using selective boundaries ω monics). In this region, the energy of oscillations is (e.g., transmitting the frequency 2 0 and reflecting all heavily absorbed due to the high-frequency dissipation other frequencies [9]). effects, which are usually related to the viscosity and thermal conductivity of the medium [1]. A standing- We analyze a nonlinear resonator in an approximate wave field formed in the resonator involves shock approach. As was shown in [2], an oscillating field fronts traveling between the walls [2]. Nonlinear between the walls x = 0 and x = L can be represented as absorption occurring in the narrow region of the front the superposition of two mutually opposing waves. Each of them can be strongly distorted due to nonlinear determines the Qnl-factor of the resonator in the strong- oscillation mode. This Q-factor is much less than the self-action, but the contributions of cross interactions conventional linear quality Q [3]: are nonresonant and can be neglected [3]. The velocity lin u of oscillating medium particles in one of the opposing 8c 1/2 c2ρ waves obeys the equation [3, 7, 10]: Qnl ==πε------, Qlin ------π ω-. (2) A b 0 1∂u ε ∂u b ∂2u ------Ð ---- u------Ð ------c ∂t 2 ∂τ 3ρ∂τ2 c 2c (3) A α * Moscow State University, Vorob’evy gory, = ------sin()ω τ Ð --- b ()t sin()2ω τ . Moscow, 119899 Russia 2L 0 c 2 0 ** Schmidt Joint Institute of Physics of the Earth, Russian Academy of Sciences, Here, for definiteness, the wave traveling in the positive Bol’shaya Gruzinskaya ul. 10, Moscow, 123995 Russia x-axis direction is considered; t is the slow time corre- *** Institutionen for Maskinteknik, sponding to the transition to a steady state in the reso- Karlskrona, 37179 Sweden nator; the fast time τ describes the oscillations; α is the

ENHANCEMENT OF ENERGY AND Q-FACTOR OF A NONLINEAR RESONATOR 189 selective-absorption coefficient; and the amplitude of Vst the second harmonic, D = 20 4 π 10 () 2 (), τ ()ω τ ()ω τ 3 4 b2 t = ---∫ut sin 2 0 d 0 , (4) π 2 1 0 1 D = 0 is not known in advance. Thus, Eq. (3) with Eq. (4) is a nonlinear integro-differential equation [7]. When the –π –2 –1 1 2 πθ right-hand side of Eq. (3) is defined, Eq. (3) goes over –1 into a Burgers-type inhomogeneous equation [11]. For convenience, we use the dimensionless vari- –3 ables –4 u t V ==-----, θωτ, T =---, (5) 0 Fig. 1. One-period profiles of oscillations for various values u0 ts of selective absorption D. where ts is the characteristic nonlinear time of forming a discontinuity in the wave and u0 is the characteristic when the wall motion is described by Eq. (1), has the amplitude: form V ()θ c Ac ------st -1= ±()+ cosθ 1/2 ts ==------εω , u0 ------πε . (6) 2 0u0 2 π 1/2 (9) 2 Equation (3) with Eq. (4) in view of Eqs. (5) and (6) × 1 + D()1 Ð cosθ --- V ()θ' sin2θ'dθ' . takes the form π∫ st 0 ∂V ∂V ∂2V The signs + and Ð in solution (9) are taken for the half------Ð V------Ð Γ------periods 0 < θ ≤ π and –π ≤ θ < 0, respectively. In the ∂T ∂θ 2 ∂θ vicinity of θ = 0, a shock front is formed. Ignoring its π (7) structure, we set Γ = 0 in solution (9). Allowance for 2 Γ = sinθ Ð Dsin2θ--- VT(), θ' sin2θ'dθ'. nonzero values can be made by the method of π∫ matched asymptotic expansions (see, e.g., [12]) and 0 will give only small corrections (in the strongly pronounced-nonlinearity mode) for the energy characteris- Here, the dimensionless numbers tics of the field. ω The one-period profiles of oscillations are shown in b 0 ts αc Γ α Fig. 1 for the D selective absorption values [Eq. (8)] ==------ερ - --- and D ==------εω ts (8) 2 cu0 ta 0u0 equal to 0, 1, 4, 10, and 20. In the presence of only nonlinear absorption, the profile has the form are the ratios of the nonlinear time t to the time of the s θ ()θ θ conventional absorption ta and to the characteristic time V st = 2cos--- sgn , (10) α–1 of selective losses, respectively. 2 which is the known solution of the nonuniform Burgers To analyze the excitation of forced oscillations in equation [11] and follows from (9) for D = 0. With an the resonator, Eq. (7) should be solved with the zero ini- θ ∞ increase in the selective absorption D, the dimension- tial conditions V(T = 0, ) = 0. For T , a balance less amplitude of the discontinuity does not increase, between the energy inflow from the source (oscillating but a noticeable increase in the perturbation V is wall) and the losses of three types—viscous, nonlinear, st observed in smooth sections of the profile. For D ӷ 1, and selective—is achieved. Steady-state oscillations ≈ θ the oscillation is almost harmonic, Vst V0sin , and the are most easily analyzed. At the same time, steady Ӷ states, where nonlinearity is most pronounced, are of jump of the small relative magnitude 2 V0 only θ the most interest. The steady-state solution satisfying remains at the point = 0. ω the periodical conditions, which reduces to the homo- Thus, the amplitude B2 of the second harmonic 2 0 geneous boundary conditions decreases noticeably with increasing D. The onset of this process is shown in Fig. 2. The suppression of the ()θπ ()θπ ω V st = ==V st = Ð 0 2 0 wave retards the energy transfer to higher harmon-

DOKLADY PHYSICS Vol. 47 No. 3 2002 190 RUDENKO et al.

DB2 also increases with an increase in DB2(D), i.e., with the 6 2B enhancement of selective absorption. 1 ӷ ≈ In the limit D 1, we obtain the formula DB2 21/3D2/3 and the expressions K 4 ≈ 2/3 1/3 θ 1/3 Ð1/3 θ V st 2 D sin+ 2 D sin2 , (13) 10B 2 ≈≈2/3 1/3 1/3 2/3 V max 2 D , and I 2 D (14) 2 for the steady-state profile, maximum (11), and mean intensity (12), respectively. The profiles of standing waves between the walls of resonator 0 < x ≤ L are plotted by combining two func- 0510 15 20 D tions (9) shifted relative to another [2, 3]: Fig. 2. The solid lines are the amplitudes B1 and B2 of the first and second harmonics, respectively, and the corre- , ω ω ππx ω π π x Vx(0t ) = V0t + Ð --- Ð V0t Ð + --- . (15) sponding product DB2 for various values of selective L L absorption D and (dashed line) the amplification coefficient K of the oscillation energy accumulated in the cavity of res- The shapes of oscillations measured in the cavity onator. sections x = L/8, L/4, L/2, 3L/4, and 7L/8 are shown in Fig. 3 for D = 20. Similar profiles in the absence of V selective absorption (D = 0) were plotted earlier (see 10 Fig. 4 in [3]) The comparison of results shows that for D = 20 the 8 7L/8 3L/4 “traveling discontinuities” in the profiles of standing 6 waves are less pronounced than in the absence of selec- 4 tive losses. Consequently, nonlinear attenuation is partially suppressed. As a result, the maximum V values in 2 Fig. 3 are approximately as high as the corresponding 0 peaks for D = 0. The Q-factor of the resonator in the nonlinear oscil- –2 lation mode with a complicated spectrum can be –4 defined as the ratio of the maximum velocity perturba- –6 L/8 tion 2umax in the standing wave to the velocity ampli- /2 L/4 tude A of the boundary oscillations: –8 L umax 2c –10 Q ==2------Φ()DB ; –4–20246810 A πεA 2 θ 12+ DB Φ 2 > (16) = ------for 2 DB 2 1, Fig. 3. Shapes of oscillations in the cavity sections x = L/8, 2DB L/4, L/2, 3L/4, and 7L/8 for D = 20. 2 Φ ≤ and = 2 for 2 DB 2 1. ω ω ics 3 0, 4 0, … . Therefore, energy is accumulated at It is also possible to define Q in terms of the ratio of the ω average intensities of these oscillations: the fundamental frequency 0, which virtually does not attenuate. The first-harmonic amplitude B1(D) is also 2 shown in Fig. 2. This figure shows product DB D 2 2u 2c () 2( ) Q ==------2 + DB2 . (17) appearing in the basic formulas below. A2/2 πεA In particular, perturbation (9) attains the maximum θ ≤ Formulas (16) and (17) both describe an increase value Vst = 2 at = 0 for DB2 0.5, and for DB2 > 0.5 in the Q-factor with enhancement the selective θ the peak (see Fig. 1) is shifted to the point max at which absorption D.

1 12+ DB2 “Ampliﬁcation coefﬁcient” K(D) of the oscillation V ==V θ = arccos------. (11) max stmax 2DB energy accumulated in the resonator cavity is shown in 2 2DB2 Fig. 2 by the dashed line.

The mean intensity over period The estimation of the nonlinear factor Qnl by Eq. (2) for a conventional air-ﬁlled resonator whose boundary 2 () Ϸ IV==st 2 + DB2 D (12) oscillates with the velocity A = 10 cm/s yields Qnl 85.

DOKLADY PHYSICS Vol. 47 No. 3 2002 ENHANCEMENT OF ENERGY AND Q-FACTOR OF A NONLINEAR RESONATOR 191

If the right wall x = L of the resonator transmits 98% of 4. A. B. Coppens and A. A. Atchley, in Encyclopedia of the incident radiation power at the second harmonic Acoustics (Wiley, New York, 1997), pp. 237Ð246. frequency, it follows from Eq. (16) that Q ≈ 300; i.e., 5. R. Lanbury, Phys. Today 51 (2), 23 (1998). selective losses should increase the resonator Q-factor 6. O. V. Rudenko, Akust. Zh. 45, 397 (1999) [Acoust. Phys. by a factor of about 3.5 and the energy of oscillations 45, 351 (1999)]. by more than an order of magnitude. 7. O. V. Rudenko, Akust. Zh. 29, 398 (1983) [Sov. Phys. We note that the problems of nonlinear systems with Acoust. 29, 234 (1983)]. selective losses are recently of interest in connection 8. O. V. Rudenko, Priroda (Moscow), No. 7, 16 (1986). with a number of applications [13, 14]. 9. V. G. Andreev, V. É. Gusev, A. A. Karabutov, et al., Akust. Zh. 31, 275 (1985) [Sov. Phys. Acoust. 31, 162 ACKNOWLEDGMENTS (1985)]. 10. O. V. Rudenko and A. V. Shanin, Akust. Zh. 46, 392 This work was supported by INTAS, the US Civilian (2000) [Acoust. Phys. 46, 334 (2000)]. Research and Development Foundation for the Inde- 11. A. A. Karabutov, E. A. Lapshin, and O. V. Rudenko, pendent States of the Former Soviet Union, and Rus- Zh. Éksp. Teor. Fiz. 71, 111 (1976) [Sov. Phys. JETP 44, sian Foundation for Basic Research (including the grant 58 (1976)]. for support of Leading Scientiﬁc Schools). 12. O. A. Vasil’eva, A. A. Karabutov, E. A. Lapshin, and O. V. Rudenko, The Interaction of One-Dimensional REFERENCES Waves in Dispersion-Free Media (Mosk. Gos. Univ., Moscow, 1983). 1. O. V. Rudenko and S. I. Soluyan, Theoretical Founda- 13. V. E. Gusev, H. Balliet, P. Lotton, et al., J. Acoust. Soc. tions of Nonlinear Acoustics (Nauka, Moscow, 1975; Am. 103, 3717 (1998). Consultants Bureau, New York, 1977). 14. V. A. Khokhlova, S. S. Kashcheeva, M. A. Avierkiou, 2. V. V. Kaner, O. V. Rudenko, and R. V. Khokhlov, Akust. and L. A. Crum, AIP Conf. Proc. 524, 151 (2000). Zh. 23, 756 (1977) [Sov. Phys. Acoust. 23, 432 (1977)]. 3. O. V. Rudenko, C. M. Hedberg, and B. O. Enﬂo, Akust. Zh. 47 (2001) [Acoust. Phys. 47, 452 (2001)]. Translated by T. Galkina

DOKLADY PHYSICS Vol. 47 No. 3 2002

TECHNICAL PHYSICS

The Selection of an Efficient Blast-Reduction Method when Detonating Explosives B. E. Gelfand* and M. V. Sil’nikov** Presented by Academician Yu.S. Vasil’ev September 3, 2001

Received September 17, 2001

The choice of simple and efficient means for protec- In order to obtain a sufficient concentration of a tion from blast loads when detonating explosives is a blast-consuming material to provide reliable damping topical problem in fundamental and applied studies. In of explosive waves, gas-filled covers consisting of searching for these means, sources of blast loads in the water foam [6] or polyurethane foam [7] were pro- form of explosive charges were placed inside various posed. For these materials, σ = 10Ð20 kg/m3 and 50Ð containment shells or immersed in diverse media con- 100 kg/m3, respectively, while ρ = 900 kg/m3. suming blast energy. In practice, the application of gas-filled shells to There exist two directions aimed at creating means confine the demolition effects of explosive charges for blast protection. The first is associated with the (with a weight not more than 3 kg) has shown that the obvious attempt to surround an explosive charge by a energy of explosion products is spent mainly for the solid impermeable shell. Here, we deal with a trivial kinematic acceleration of containment shells. In addi- case when an explosive charge is placed into a closed or tion, the role of the transformation of energy flows at half-closed container [1]. In practice, it was shown that the interfaces of explosion products, the blast-consum- this method of charge localization is poorly efficient. ing medium, and the atmosphere is also significant. Afterwards, in order to improve the exploitation prop- Attempts associated with regulating blast loads at the erties of a protective container, its inner cavity came to expense of energy loss for evaporating or spraying pro- be filled with a compressible substance. In [2], [3], and tective-medium materials have demonstrated little [4], a liquid foam, a liquid containing gas bubbles, and effect. a granular or fibrous material, respectively, were used as By virtue of the dominating role of inertial proper- fillers of containers. Each of these blast-consuming ties of containment shells when suppressing explosion media can be characterized by an effective apparent den- σ shock waves, the change in the kinematic characteris- sity , which differs from the skeleton-material density tics of a medium that damped the expansion of explo- ρ σ 3 ρ . For example, for water foam, = 10Ð30 kg/m at = sion products turned out to be an efficient means of 1000 kg/m2; for water saturated with gas bubbles, σ = blast reduction. Note that, for the medium surrounding 900Ð990 kg/m3; and for fibrous and granular fillers, σ ≥ an explosive charge, the compressibility is the most 1000 kg/m3 at ρ ≥ 2000 kg/m3. important characteristic. A reliable reduction of explosion effects is attained when the protective medium is a The second line of studies aimed at creating means liquid that assures (at least, at initial expansion stages) of protection is specified by the complete rejection of against the penetration of explosion products into the an impermeable solid shell around an explosive charge. atmosphere being protected. In this case, a variant of blast suppression in the volume of a sprayed liquid (most often, water) can be consid- As is known, the measure of the compressibility of ered as an ultimate case [5]. However, due to the impos- a liquid is the speed of sound in it. We can control the sibility of obtaining highly concentrated liquid sprays speed of sound in a liquid by incorporating into its vol- with σ ≥ 1–2 kg/m3, this means of blast suppression has ume porous elements, gas bubbles, or hollow polymeric proved unfeasible. inclusions. The embedding of compressible inclusions is especially significant in affecting the propagation of moderate-intensity shock waves with a frontal pressure

P1 * Semenov Institute of Chemical Physics, drop ----- ≤ 1000 . For higher-intensity shock waves, the Russian Academy of Sciences, P0 ul. Kosygina 4, Moscow, 117977 Russia presence of compressible elements is of less impor- ** Scientiﬁc and Industrial Enterprise of Special Materials, tance. The immersion of explosive charges in a com- B. Sampsonievskiœ pr. 28A, St. Petersburg, pressible two-phase medium critically changes the 194044 Russia characteristic time scales of all wave processes and

THE SELECTION OF AN EFFICIENT BLAST-REDUCTION METHOD 193 enhances the possibilities of efﬁcient energy exchange 4 between the explosion products and the containment (a) shell [8]. 3

These arguments make it possible to propose a new 2 blast-reduction method. Figure 1a shows such a scheme for an explosive charge (1), which involves a thin 1 layer of a liquid dispersion medium confined in the volume (2) between elastic shells (3). The liquid layer is separated from the explosive charge by a gas-filled 3 spacing [4]. The arrangement of the basic elements dif- (b) fers from that in the case of a traditional scheme [9] 2 (Fig. 1b). The crucial role of the air spacing consists in reducing the amplitude of the shock wave (due to the 1 difference in the explosive-charge volume Rch and the shell inner cavity R ) before its meeting with the liquid in 3 layer. (c) On the basis of published [9] and new experimental 4 data, we can compare the efficiency of the blast reduc- 1 tion for the schemes shown in Figs. 1a and 1b. Informa- tion on the blast reduction with the help of a water shell contacting the explosive charge (i.e., for the scheme shown in Fig. 1b) was taken from [9]. Data on the blast Fig. 1. Possible schemes of blast reduction, which use elastic containers filled with liquids: (1) explosive charge, reduction according to the scheme shown in Fig. 1a (2) dispersion medium, (3) elastic shells, and (4) air were obtained in a series of experiments similar to [10]. spacing. The ratio of the water mass (å) to the explosive-charge mass (G) was 50 (G = 40 and 100 g in [9] and [10], respectively). î Φ Figure 2 shows the amplitude variation for the 1 coefficient of the blast reduction as a function of the 0.9 R normalized distance R* = ------1/3- . The distance R was G ~ measured from the blast epicenter to the measurement ~ ∆ () Φ PR* point. The blast-reduction coefficient = ------∆ ()- P0 R* 0.4 was evaluated using the amplitude of the shock wave ∆P(R*) for the explosion in the liquid shell and the 2 ∆ amplitude P0(R*) for the explosion in the gas-filled shell of the same size according to the scheme shown in 0.3 ∆ 3 Fig. 1c. This is justified by the fact that the value of P0 (crosshatched region 1 in Fig. 2) is by 10Ð20% smaller ∆ than that of P1(R*) for a completely open charge. 0.2 The regions 2 and 3 for values of Φ in Fig. 2 were obtained, respectively, for explosions corresponding to the schemes shown in Figs. 1b and 1a. 0.1 1.2 1.7 2.2 2.7 R*, m/kg1/3 Figure 3 shows the variation in the pressure pulse Fig. 2. Dependence of the coefficient of the shock-wave for the shock wave in the case of compression phase I+ amplitude reduction on the normalized distance in the case when detonating an explosive in the gas-filled air shell of various blast schemes. (curve 1 according to [9]). The bands of values denoted as 2 and 3 for I+ were obtained for explosions using the schemes shown in Figs. 1b and 1a, respectively. Our investigations are based on the rational use of propagation features for pressure waves and rarefaction The plots in Figs. 2 and 3 confirm that the scheme waves when detonating explosives in a medium charac- presented in Fig. 1a for reducing the demolition effect terized by elevated (compared not only to a liquid, but of an explosion has the highest efficiency. also to a gas) compressibility.

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194 GELFAND, SIL’NIKOV

I, kPa ms/kg1/3 The described method of blast reduction was used in designing a series of devices for the rapid and reliable protection of equipment and personnel in the process of 100 the conservation and/or elimination of explosive substances and systems.

REFERENCES 80 1. J. M. Powers and H. Krier, J. Haz. Mater 13, 121 (1986). 2. R. Raspet, P. B. Butler, and F. Yahani, Appl. Acoust. 22, 1 35 (1987). 3. A. A. Borisov, B. E. Gelfand, and E. I. Timofeev, Int. J. 60 Multiphase Flow 9, 531 (1983). 2 4. R. Raspet, P. B. Butler, and F. Yahani, Appl. Acoust. 22, 243 (1987). 40 5. A. A. Buzukov, Fiz. Goreniya Vzryva 36, 120 (2000). 3 6. V. M. Kudinov, B. I. Palamarchuk, B. E. Gelfand, and S. A. Gubin, Dokl. Akad. Nauk SSSR 228, 555 (1976) [Sov. Phys. Dokl. 21, 256 (1976)]. 20 7. H. Kleine, K. Disconescu, and J. H. S. Lee, in Proc. 20 1.6 2.0 2.4 2.8 R*, m/kg1/3 Intern. Symp. on Shock Waves, Singapore, 1996 (World Sci., Singapore, 1996), Vol. 2, p. 1351. Fig. 3. Dependence of the pressure pulse for the compression phase on the normalized distance in the case of various 8. I. M. Voskoboœnikov, B. E. Gelfand, S. A. Gubin, et al., blast schemes. Inzh.-Fiz. Zh. 31, 674 (1976). 9. M. A. Rinaudo, P. D. Smith, T. A. Rose, in Proc. of the 15 Int. Symp. on Military Aspects of Blast and Shock, The essential distinction in the above method of Sufﬁeld, 1997 (Sufﬁeld, 1997), p. 225. blast reduction is the preliminary correction of initial 10. B. E. Gelfand, M. V. Sil’nikov, A. I. Mikhaœlin, and conditions for the expansion of explosion products due A. V. Orlov, Fiz. Goreniya Vzryva 37, 128 (2001). to the attached volume of a thin gas layer between the explosive-charge surface and a compressible medium. Translated by Yu. Vishnyakov

DOKLADY PHYSICS Vol. 47 No. 3 2002

TECHNICAL PHYSICS

A New Method of Multidimensional-Signal Processing with the Use of R Functions and Atomic Functions V. F. Kravchenko* and M. A. Basarab** Presented by Academician Yu.V. Gulyaev May 24, 2001

Received May 25, 2001

INTRODUCTION ∪ (association), and ¬ (complementation). We can write out this assumption in the form Employment of atomic functions (AF) in the digital processing of one-dimensional signals was thoroughly Ω = F(){Ω ,,,Ω …Ω }, {} ∩∪¬,, . analyzed in [1, 2]. Many operations of one-dimensional 1 2 m (1) signal processing are considered to be applicable to the multidimensional case [3]. For example, employing In this case, we also assume that the initial regions have operations of the direct product or rotation of one- simpler shapes than Ω and that, for each of them, the ω dimensional windows, it is possible to synthesize mul- equation for its boundary i(x) = 0, i = 1, 2, …, m is tidimensional atomic-function windows having rectan- known. The theory (method) of R functions was devel- gular, hexagonal, or circular apertures. While using oped by Rvachev [4]. In the framework of the set the- windows with an arbitrary reference region, certain ory, this method makes it possible (on the basis of complications of both a qualitative and quantitative describing the Ω region) to obtain the equation ω(x) = 0 nature arise, which are associated with a necessity of in the analytical form for the boundary of this region. analytically describing the geometry of this region. According to [4], one of the basic complete systems of This problem can be solved with the help of application R functions is the ᑬα system of the form of the R-function method [4]. In the present paper, we consider for the first time problems of constructing ∧ ≡ 1 ()2 2 α multidimensional windows based on both R functions x α y ------α- xy+ Ð x+ y Ð 2 xy , and atomic functions, as well as on the concepts devel- 1 + oped in [1]. 1 2 2 (2) x ∨α y ≡ ------()xy++ x+ y Ð 2αxy , 1 + α ¬ ≡ R FUNCTION AND THE INVERSE PROBLEM xxÐ . OF ANALYTICAL GEOMETRY Here, α = α(x, y) is an arbitrary function satisfying the α ≤ α ≡ Let a region Ω with the boundary ∂Ω be given in the condition –1 < 1. Usually, it is assumed that 0 ᑬ α ≡ ᑬ space Rn. Let it be necessary to construct a function ( 0 system) or 1 ( 1 system). The correspondence ω (x), x = (x1, x2, …, xn). This function must be strictly established by Rvachev between logical-algebra func- positive inside Ω, negative outside Ω, and equal to zero tions and R functions makes it possible to find a con- on ∂Ω. The equation ω(x) = 0 determines in the implicit structive solution to the inverse problem of analytical form the boundary of the region. We assume that Ω can geometry. To do this, it is sufficient to formally Ω ω Ω ω be represented as a combination of certain initial exchange in (1) by (x), i by i(x), i = 1, 2, …, m, Ω Ω Ω regions 1, 2 , …, m on the basis of the following and to use symbols of R operations {∧, ∨, –) in system (2) logical operations over sets, namely, ∩ (intersection), instead of {∩, ∪, ¬}, respectively. As a result, we obtain the analytical expression that determines the equation for the boundaries of the region Ω in terms of elementary functions * Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Mokhovaya ul. 18, Moscow, 103907 Russia ω()x = 0. (3) ** Bauman Moscow State Technical University, Vtoraya Baumanskaya ul. 5, Moscow, In this case, ω(x) > 0 for inner points and ω(x) < 0 for 107005 Russia outer points of the region Ω.

1028-3358/02/4703-0195 $22.00 © 2002 MAIK “Nauka/Interperiodica” 196 KRAVCHENKO, BASARAB ω ω ATOMIC FUNCTIONS H*( 1, 2) or that the pulse characteristic is symmetric The class of atomic functions [1, 2] includes finite with respect to the origin of a coordinate system; i.e., solutions to functional differential equations (FDE) of h[n1, n2] = h*[–n1, –n2]. There exist several methods for the form calculating two-dimensional FPC filters [3]; among them, the window-function method is the most widely N M propagated in practice. In correspondence with this ()n () () ∑ dn y x = ∑ cm yaxÐ bm . method, the desired two-dimensional frequency char- n = 1 m = 1 acteristic of the filter is represented in the form of the Fourier series Here, a, dn, cm, and bm are numerical parameters, and ∞ ∞ |a| > 1. One of basic types of AF is determined by the ()ω ω ()ω , ω [], Ð j 1n1 + 2n2 following FDE: H0 1 2 = ∑ ∑ h0 n1 n2 e , (6) ∞ ∞ 2y'()x n1 = Ð n2 = Ð ------= yax()+ 1 Ð yax()Ð 1 , (4) a2 where π π ()ω ω where a > 1. Finite solutions (4) with the carrier 1 j 1n1 + 2n2 h []n , n = ------H ()ω , ω e dω dω . 1 1 0 1 2 π2 ∫ ∫ 0 1 2 1 2 −------, ------are usually denoted as h (x) [5]. Their 4 π π a Ð 1 a Ð 1 a Ð Ð Fourier transform has the form Here, h0[n1, n2] is the infinite pulse characteristic of a two-dimensional filter that corresponds to the given fre- ∞ quency characteristic H (ω , ω ). In order to realize the () ()Ðk 0 1 2 Fa p = ∏sinc pa , (5) two-dimensional FPC filter, the summation in series (6) k = 1 should be limited. This results in deteriorating the convergence of truncated series (6) to the given frequency sinx where sinc(x) ≡ ------. Using expression (4), we can characteristic in its discontinuity points (the so-called x Gibbs effect). To improve the convergence, the coeffi- express the derivatives of the nth order in terms of the cients h0[n1, n2] should be multiplied by the two- values of functions themselves. For a = 2, we obtain the dimensional window function w[n , n ]. In other ≡ 1 2 most well-known and investigated function h2(x) words, the functions up(x). The principal properties of both the parent’s h[n1, n2] = w[n1, n2] á h0[n1, n2] function up(x) and the function ha(x) are described in [1, 2, 4, 5]. As a rule, atomic functions and Rn functions are used as filter coefficients. are constructed on the basis of the direct product of For filters with the zero phase shift, the window one-dimensional AF, e.g., function must satisfy the condition w[n1, n2] = w*[–n1, n –n2]. Two-dimensional FPC filters are synthesized on up ()x = ∏up()x . the basis of two-dimensional window functions defined n j on rectangular, hexagonal, or circular reference regions j = 1 (apertures) [3], which we call regular. In the simplest case of a square aperture, the two- SYNTHESIZING TWO-DIMENSIONAL FILTERS dimensional weight window is formed on the basis of a WITH A FINITE PULSE CHARACTERISTIC direct product of one-dimensional windows AND A REGULAR APERTURE ON THE BASIS w[n , n ] = w[n ] á w[n ]. (7) OF ATOMIC FUNCTIONS 1 2 1 2 One promising direction in the application of R The window with a hexagonal reference region is functions and atomic functions is the digital processing formed in a similar manner: of multidimensional signals. As is well known [3], in [], []⋅⋅n1 + n2 3 n1 Ð n2 3 problems of the digital filtration of two-dimensional wn1 n2 = wn1 w ------w ------. (8) signals, filters with a finite pulse characteristic (FPC fil- 2 2 ters) gained the most acceptance. Their principal Finally, in the case of a circular aperture, the two- advantage compared to filters with an infinite pulse dimensional window is obtained by rotating a one- characteristic is the possibility to synthesize filters with dimensional window about the symmetry axis and is a zero phase shift. In addition, a number of difficulties described by the expression in constructing two-dimensional filters with an infinite pulse characteristic are associated with the necessity to wn[], n = wn[]2 + n2 . (9) provide their stability. A two-dimensional FPC filter 1 2 1 2 ensures the zero phase shift, provided that its frequency A large number of one-dimensional window weight ω ω characteristic is a real-valued function H( 1, 2) = functions are known [1, 15]. In addition, in solving

DOKLADY PHYSICS Vol. 47 No. 3 2002 A NEW METHOD OF MULTIDIMENSIONAL-SIGNAL PROCESSING 197

Table 1. Basic physical parameters of two-dimensional KravchenkoÐRvachev windows constructed on the basis of the atomic function h0(x) Equivalent Correlation of Parasitic Maximum Maximum Window Coherent noise overlapping segments amplitude transformation level of side width at the Parameter ampliﬁcation a band, bin (50% overlap), % modulation, dB loss, dB lobes, dB 6-dB level, bin

b1 b2 b3 b4 b5 b7 b8 1.1 20.7 2 × 10Ð7 0.2 13.4 Ð154.3 6 0.03 1.2 10.24 8 × 10Ð3 0.3 10.4 Ð86.1 4.2 0.05 1.3 7.02 0.18 0.5 9 Ð55.2 3.5 0.07 1.5 4.49 2 0.7 7.2 Ð36.3 2.8 0.12 2 2.62 12 1.2 5.4 Ð23.3 2.1 0.25 3 1.74 29 1.8 4.2 Ð17 1.7 0.45 5 1.35 40 2.5 3.8 Ð14.5 1.5 0.64 problems of one-dimensional signal processing, new 4. The maximum transformation loss (expressed in widow classes, e.g., RvachevÐKravchenko windows decibels), and Kravchenko windows (both based on atomic func- ( tions), as well as hybrid windows, e.g., KravchenkoÐ b4 = 10log b1 ) + b3. Hamming windows, KravchenkoÐKaiser windows, and 5. The maximum level of side lobes (expressed in KravchenkoÐBlackmanÐHarris windows, find applica- decibels), tion [1, 2]. Using expressions (7)Ð(9), we can synthe- (), 2 size their two-dimensional analogs with regular refer- Wuk 0 b5 = 10 logmax ------, ence regions. Below, we thoroughly analyze windows k W()00, with square apertures, which are based on the atomic where {u } are points of local maxima (excluding u ). function ha(x). To compare the characteristics of two- k 0 ω dimensional windows (in the plane 2 = 0) defined on 6. The asymptotic-decrease velocity for side lobes the square aperture (Ð1 ≤ x ≤ 1, −1 ≤ y ≤ 1), we employ (expressed in decibels per octave), the following set of physical parameters. W()2u, 0 2 b6 = 10 log lim ------. 1. The equivalent noise band, u → ∞ Wu(), 0 1 1 7. The window width at the 6-db level, ∫ ∫ w2()xy, dyxd b7 = 2u, ------Ð1 Ð1 b1 = 4 1 1 2. where u is the maximum frequency such that (), W()00, 2 ∫ ∫ wxydyxd 10 log ------= 6. Wu(), 0 Ð1 Ð1 8. The coherent amplification, 2. The correlation of overlapping segments, 1 1 1 1 1 (), b8 = --- ∫ ∫ wxydyxd . ∫∫wxy(), wx()Ð 1, y dyxd 4 Ð1 Ð1 Ð1 0 b2 = ------1 1 -100%. Here, the normalization conditions w(x, y) = 0 for | | | | w2()xy, dyxd x > 1 or y > 1; w(0, 0) = 1; and w(–x, y) = w(x, y) = ∫ ∫ w(x, −y) = w(–x, –y) are fulfilled. By virtue of the con- Ð1 Ð1 ω dition of symmetry in the plane 1 = 0, the window has 3. The parasitic modulation amplitude (expressed in the same characteristics. decibels), The principal physical characteristics of the normal- h ()x/()a Ð 1 W()π/2, 0 2 ized windows ------a - as functions of the param- b3 = Ð10 log ------, h ()0 W()00, a eter a are presented in Table 1. As in the one-dimen- where W(p, q) is the two-dimensional Fourier trans- sional case, owing to the infinite differentiability of form for the window function. atomic weight functions, the value of b5 for all taken

DOKLADY PHYSICS Vol. 47 No. 3 2002 198 KRAVCHENKO, BASARAB

(a)

1.0 0.8 0.6 0.4 0.2

0 (b)

–20 –40

Fig. 1. Two-dimensional KravchenkoÐRvachev window with a square reference region, which is constructed on the basis (a) of the ᑬ 0 system, and (b) of the logarithm of its frequency characteristic. windows is equal to infinity. In the case of ignoring this The numerical experiment has shown that as far as the ≥ parameter, windows with a 2 are similar to well- b5 parameter (equal to Ð31.1 and Ð52.7 dB) is con- known Tukey windows [1]. cerned the synthesized windows are comparable to the In the one-dimensional case, modified (M + 1)-term classical Hamming windows (cosine squared) and windows of the form [1, 2] Blackman windows; however, they considerably M exceed these windows with respect to the parameter b6. () () () 2k () The parameters of synthesized windows are: b1 = 2.25 w˜ x = wx + ∑ ckw x (10) and 3.65 bin; b2 = 17 and 4.7%; b3 = 1.4 and 0.9 dB; k = 1 b4 = 4.9 and 6.5 dB; b7 = 2 and 2.5 bin; and b8 = 0.25 are used with the intent of improving the b5 parameter. and 0.14. The choice of a larger number of terms in We now employ a similar approach for the two-dimen- expansion (10) allows two-dimensional weight win- sional case restricting this approach by the two-term dows with higher characteristics to be synthesized. windows (M = 1). We consider two-dimensional weight functions with a square aperture, which are based on one-dimensional windows [1]: SYNTHESIZING TWO-DIMENSIONAL FPC w ()x = up()x + 0.01up"()x , FILTERS WITH AN ARBITRARY AMPLITUDE 1 ON THE BASIS OF R FUNCTIONS 1 w ()x = 1.5249 h ()2x + ------h" ()2x . The above approaches (i.e., direct product and rota- 2 1.5 128 1.5 tion) do not allow us to synthesize two-dimensional fil-

DOKLADY PHYSICS Vol. 47 No. 3 2002 A NEW METHOD OF MULTIDIMENSIONAL-SIGNAL PROCESSING 199

(a)

1.0 0.8 0.6 0.4 0.2 0

(b) 0

–20

–40

Fig. 2. Two-dimensional KravchenkoÐRvachev window with a square reference region, which is constructed on the basis (a) of the ᑬ 1 system, and (b) of the logarithm of its frequency characteristic. ters with reference regions of an arbitrary configurations (windows) proposed having an arbitrary aperture, tion. This statement especially relates to nonconvex we call two-dimensional KravchenkoÐRvachev win- regions, e.g., cross-shaped and star-shaped ones. The dows, or KR2 windows. As examples, contour images only exception belongs to filters based on the simplest and perspective projections of square KR21 windows rectangular Dirichlet window. Employing the R-func- and KR22 windows constructed with the help of R oper- tion method, we can efficiently realize the synthesis of ations of system (2) in the case of α ≡ 0 and α ≡ 1 are two-dimensional windows for apertures of an arbitrary shown in Figs. 1 and 2. The logarithms of their fre- shape. quency characteristics (expressed in decibels) are also Let Ω be the required reference region for a window. given in the same figures. Characteristics of these win- Using one of the complete systems of R functions, we dows are presented in Table 2. Using the diversity of construct an equation for its boundary ∂Ω, ω(x, y) = 0. R-function systems [4], we can obtain the equation for It is evident that the function ω˜ (x, y) that provides the most suitable frequency char- 1 acteristic of the desired FPC filter. ω˜ ()xy, ≡ ------()ω()ωxy, + ()xy, (11) 2 max ω()xy, We now describe a new method for constructing a ()Ωxy, ∈ two-dimensional analog w[n1, n2] of a one-dimensional coincides with ω(x, y) inside Ω, becomes zero outside prototype window w[n] on an arbitrary reference region it, and max ω˜ ()xy, = 1. Here, ω˜ (x, y) is the window whose boundary is described by algebraic curves with ()Ωxy, ∈ an order not exceeding two. Initially, on the basis of a function finite in R2. The new class of the weight func- complete system (2) of R functions for α ≡ 1 with the

DOKLADY PHYSICS Vol. 47 No. 3 2002 200 KRAVCHENKO, BASARAB

Table 2. Basic physical parameters of two-dimensional KravchenkoÐRvachev windows Equivalent Correlation of Parasitic Maximum Maximum Window Coherent noise overlapping segments amplitude transformation level of side width at the ampliﬁcation Window band, bin (50% overlap), % modulation, dB loss, dB lobes, dB 6-dB level, bin

b1 b2 b3 b4 b5 b7 b8

KR21 1.27 40 2.4 7.5 Ð17 1.4 0.55 KR22 1.5 31 2.1 7.2 Ð20 1.4 0.33 help of (11), we compose the equation for the reference i.e., when the direct product in the forms of (7) or (8) is region ω(x, y) ≥ 0 such that maxω(x, y) = ω(0, 0) = 1. unacceptable. Furthermore, the two-dimensional window is represented in the form CONCLUSIONS [], ()ω[], wn1 n2 = w 1 Ð n1 n2 . (12) Thus, new designs of two-dimensional windows proposed and substantiated in this paper can ﬁnd wide In the case of a circular reference region for the unit application in solving problems of processing multidi- radius and of a center in the origin, described by the for- mensional digital signals of Doppler radars and radar ω 2 stations with digital synthesis of the antenna’s aperture, mula (x, y) = 1 – x2 Ð y = 0, relationship (12) coincides with expression (9). Thus, formula (12) represents in the case of signal isolation and compression, and in the generalization of the expression (9) for the case of the solving problems of medical telemetry, mathematical reference region of an arbitrary conﬁguration. modeling of a heart bioelectric generator, computer thermography, and tomography as well. We consider the algorithm proposed for the case of a square aperture represented by the equation ACKNOWLEDGMENTS ω(), xy The authors are grateful to Academician Yu.V. Gulyaev, () ∧() ∧() ∧() = 1Ð x 1 1+ x 1 1Ð y 2 1+ y = 0 Academician of the National Academy of Sciences of the Ukraine V.L. Rvachev, Corresponding Member of or, with allowance for (2) in the complete form, the RAS V.I. Pustovoœt, and Professor A.V. Sokolov for discussing the results of this work. 1 ω()xy, ==1 Ð0.---()xy++ yxÐ 2 REFERENCES Then, formula (12) acquires the form 1. V. F. Kravchenko, Dokl. Akad. Nauk 382, 190 (2002) [Dokl. Phys. 47 (2002)].

n1 ++n2 n1 Ð n2 2. E. G. Zelkin and V. F. Kravchenko, Radiotekh. Élektron. wn[], n = w ------. (13) 46, 903 (2001). 1 2 2 3. D. E. Dudgeon and R. M. Mersereau, Multidimensional We now compare formulas (7) and (13). It is evident Signal Digital Processing (Prentice-Hall, Englewood Cliffs, London, 1984; Mir, Moscow, 1988). that when n1 or n2 are equal to zero they yield the same result transforming into the one-dimensional prototype 4. V. L. Rvachev, R-Function Theory and Applications weight function. However, in contrast to expression (7), (Naukova Dumka, Kiev, 1982). formula (13) also yields the same function at the diag- | | | | 5. V. L. Rvachev and V. A. Rvachev, Nonclassical Methods onals of the square ( n1 = n2 ). This fact can turn out to in Theory of Approximations for Boundary Value Prob- be useful, e.g., for image processing, when four rather lems (Naukova Dumka, Kiev, 1979). than two directions in the image plane are equivalent. In addition, the algorithm indicated is applicable for the case of a reference region with an arbitrary geometry, Translated by G. Merzon

DOKLADY PHYSICS Vol. 47 No. 3 2002

TECHNICAL PHYSICS

Application of Singular Integral Equations for Solving Problems of Wave Diffraction on an Array Composed of Irregular Planar Strips V. A. Doroshenko* and V. F. Kravchenko** Presented by Academician V.A. Sadovnichiœ September 28, 2001

Received September 28, 2001

INTRODUCTION angles formed by the planes passing through the OZ-axis and the edge of neighboring strips (see figure). While studying the diffraction of electromagnetic We introduce a spherical coordinate system (r, ϑ, ϕ) waves on ideally conducting arrays, the solution to  electrodynamic boundary value problems is tradition- with the origin at the vertex of the strips the array ally reduced to that of the first or second boundary  value problem of mathematical physics. However, vari- π ϑ  ation in the geometry of the structure that we deal with plane is defined by the equation = --- . A source of and allowance for its physical parameters (e.g., the 2 ϑ impedance) that were not previously taken into account spherical waves is located at the point B(r0), r0 = (r0, 0, ϕ result in the complication of the mathematical model 0), and the wave field varies according to a harmonic under consideration. Solving electrodynamic boundary law. We need to find a potential u(r), r = (r, ϑ, ϕ) cor- value problems for superconductors and superconduct- responding to the total field and satisfying the following coatings suggests introducing impedance boundary ing equations and conditions. conditions [1]. This corresponds to solving the third 1. Helmholtz equation (everywhere outside the and fourth boundary value problems for such structures strips and the source), i.e., (with allowance for connection of the normal and tangential derivatives). In this paper, an approach is pro- ∆uqÐ0,2u = q > 0; posed based on employing the KontorovichÐLebedev 2. The boundary condition on the array strips Σ integral transformation and singular integral equations. ( ( ∂ ( ( This approach is used for solving problems of wave dif- ξ ζ u ξ , ϕ ζ , ϕ ≠ u + ∂------= 0, ()r ()0;r (1) fraction on a three-dimensional array consisting of n Σ irregular planar impedance strips on which the third and fourth boundary conditions are given. 3. The condition of the energy boundedness ∫()u 2 + ∇u 2 dV < ∞; FORMULATION OF THE PROBLEM: D THE THIRD BOUNDARY CONDITION 4. The condition at infinity. ON STRIPS z We consider a scalar problem for wave diffraction ␽ ϕ on a periodic array composed of N infinitely thin B(r0, 0, 0) unbounded imperfectly conducting irregular (angular) planar strips having a common vertex. The array is located in the plane z = 0 of a Cartesian coordinate sys- 2π tem. The array period is l = ------, the width of the strips n N α α y is , and the slit width is d = l – and defines dihedral 0 * Kharkov Technical University of Radio Electronics, n pr. Lenina, Kharkov, 61276 Ukraine x ** Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Mokhovaya ul. 18, Moscow, 103907 Russia Geometry of the problem.

202 DOROSHENKO, KRAVCHENKO

The validity of the conditions 2Ð4 provides the ()0 U τ()ϑϑ,,,m τ uniqueness of the solution to the problem formulated. m 0 With due regard for n = eϑ, boundary condition (1) can m ()ϑ m ()ϑ ϑϑ<  PÐ1/2 + iτ cos PÐ1/2 + iτ Ðcos 0 , 0 be written as =  m m P τ()Ðcosϑ P τ()cosϑ , ϑϑ< , ( ( ∂ Ð1/2 + i Ð1/2 + i 0 0 ξ ζ1 u u + ---∂ϑ------= 0, r Σ 1 Γ --- Ð m + iτ ϕ  1 Ðim 0 m K τ()qr 1 2 π ------()------i 0------Σ (),,ϑϕ ∈ 3 ∈ , ∞), ϑ , ϕ ∈ amτ = e Ð1 πτ , = r R : r [0 + = --- L , 4r0 r cosh 1 2 (2) 0 Γ --- ++miτ 2 N d d LL==∪ , L ()s Ð 1 l + ---, sl Ð --- , Γ m ϑ s s 2 2 where (z) is the gamma function and PÐ1/2 + iτ (cos ) is s = 1 the associated Legendre function of the ﬁrst kind, we [], π CL = 02 \L. seek the potential u1 in the form of the KontorovichÐ We now assume that one of the cases Lebedev integral (7)Ð(9): ∞

( ( ∞ 1 + + ξ ==---ξ, ξ const, ζζ ==, ζ const 2 ()1 Kiτ()qr (3) u = ----- τπτsinh ∑ b τU τ ------dτ, (10) r 1 π2 ∫ m m m = Ð∞ r or 0

( ( m π ζζ ζ ξξξ b τ = Ða τP τ()cosϑ , ϑ < ---, ==r, const, ==, const (4) m m Ð1/2 + i 0 0 2 takes place. Then, the condition (2) acquires the form ()1 Umτ ∂u ξu + ζ------= 0, (5)  ∞ ∂ϑ + mnN+ ()ϑ Σ  τ PÐ1/2 + iτ cos inN()ϕ+ m  ∑ xmn, + m()------e , 0 d mnN+ where ξ and ζ are constant quantities. Furthermore, we  ∞ ()ϑ n = Ð ϑPÐ1/2 + iτ cos consider boundary condition (5) instead of (1), assum-  d ϑπ= /2 ing the validity of formulas (3) or (4). The desired  π potential u can be represented in the form 0 <<ϑ ---  2 =  uu= 0 + u1. (6)  +∞ mnN+ ()ϑ PÐ1/2 + iτ Ðcos inN()ϕ+ m  ∑ y , ()τ ------e , () mn+ m0 exp Ðqr  d mnN+ ()ϑ Here, u0 = ------corresponds to the source field n = Ð∞ PÐ1/2 + iτ Ðcos 4πr rrÐ  dϑ ϑπ 0 0  = /2 (primary field), while the potential u1 is caused by the π existence of the array and corresponds to the secondary --- <<ϑπ. field. 2 To solve the problem formulated, we make use of a pair of KontorovichÐLebedev integral transformations with respect to the radial coordinate [2]: FUNCTIONAL RELATIONSHIPS. THE SINGULAR INTEGRAL EQUATION +∞ ( K τ()qr G()τ = ∫ Gr()------i -dr, (7) In order to determine the unknown coefficients r xmn, + m and ymn, + m , we use the boundary condition (5) 0 0 0

+∞ ∂ ∂ u1 u1 ( K τ()qr ξ ζ ξ ζ 2 τπττ i τ u1 + ------∂ϑ- = u1 + ------∂ϑ- Gr()= -----2 ∫ sinh G()------d . (8) ϑπ= /2+ 0 ϑπ= /2Ð 0 π r (11) 0 u ξ ζ------0- ϕ ϕ ∈ Here, K τ(z) is the Macdonald function. With allowance = Ð u0 + = g(), L i ∂ϑ ϑπ for the representation = /2 and the conjugation condition in slits +∞ +∞ 2 ()0 imϕ Kiτ()qr ∂u ∂u u = ----- τπτsinh a τU τ e ------dτ, 1 1 ϕ ∈ 0 2 ∫ ∑ m m (9) ------π = ------π , CL. (12) π r ∂ϑ ϑ = ---0Ð ∂ϑ ϑ = ---0+ 0 m = Ð∞ 2 2

DOKLADY PHYSICS Vol. 47 No. 3 2002 APPLICATION OF SINGULAR INTEGRAL EQUATIONS FOR SOLVING PROBLEMS 203

Extracting from formulas (11) and (12) conditions for +∞ ()n ψπ their transforms (7), and allowing for representation (10), ∑ Ð1 zn ==0, . (17) we arrive at the system of equations with respect to the n = Ð∞ coefﬁcients zn connected with the desired coefﬁcients (by virtue of the array periodicity, these equations are We now introduce the function considered for a period) +∞ ψ ()ν in()ψ+ ν ψππ∈ [], +∞ F()= iNn∑ + zne , Ð , (18) 2 1 n ()1 inψ ξ ∑ ------()1 Ð ε z e n = Ð∞ Nn()+ ν n n n n = Ð∞ where +∞ π

() ψ ( ζ2 ()ν n ()ε 2 in ξ ψ (13) νψ ψ Ð ∑ Nn+ ----- 1 Ð n zne = g(), 1 ψ Ði Ðin ψ n zn = ------π ()ν ∫ F()e e d , n = Ð∞ 2 Ni n + (19) π πα Ð ψ < ------, n + ν ≠ 0. l ν ν For n + = 0, which is possible at n = 0 and = 0, z0 is +∞ ψ πα determined from relationship (17). In accordance with ∑ z ein = 0, ------< ψπ≤ . (14) relationship (16), n l n = Ð∞ απ F()ψ = 0, ------< ψπ≤ , m ν m 1 ≤ l Here, ---- = m0 + , m0 is an integer closest to ---- , –--- N N 2 so that we ﬁnd from (19) 1 ν < --- , 2 1 iνψ Ðinψ z = ------F()ψ e e dψ, n 2πNi() n + ν ∫ 1 n ()1 1 Nn()+ ν + 1 S (20) ------()1 Ð ε = ---()Ð1 Nn()+ ν n n π απ S: ψδ< , δ = ------. 1 l Γ --- ++iτ Nn()+ ν 2 With allowance for ×πτcosh ------(15) ∞ 1 + n Γ --- + iτ Ð Nn()+ ν π Ðiνβ ()Ð1 inβ 2 ------e = ∑ ------e , sinπν n + ν 1 n = Ð∞ × ------, 2 we have from relationships (17) and (20) d Nn()+ ν ()ϑ PÐ1/2 + iτ cos dϑ ϑπ = /2 π 1 β 1 Ðiνβ β z0 = Ð------π -∫F()------πν Ð --ν-e d . (21) ()2 1 n 2 iN sin 1 Ð ε ==------, z ()Ð1 ()y , Ð x , , S n ε()1 n mn mn 1 Ð n Thus, we can derive a singular integral equation for ϕ determination of the unknown function F(ψ) (18) from ψ ϕ π = N Ð -----ϕ- , Eq. (15) by substituting into it the representation (20), (21) for zn: ( ε()j and g is transform (7) of the function g. For n , the Ðiνβ 1 2 F()β e estimate ---ζ ------dβ π ∫ βψÐ S ()j 1 ε ==O ------, Nn()+ ν ӷ 1, j 12, n 2 2 (22)

()ν ( N n + 1 ()βψ β Ðiνβ β ξ ( ψ + π---∫Kmτ Ð F()e d = g(), takes place. After differentiating both parts of Eq. (14) S with respect to ψ and complementing the additional ψ ∈ S, condition at ψ = π, we obtain the relations where +∞ inψ πα ()ν < ψπ≤ ( θ ∑ n + zne = 0, ------, (16) θ ζ21 1 l Kmτ()= --- cot--- Ð --- n = Ð∞ 2 2 θ

DOKLADY PHYSICS Vol. 47 No. 3 2002 204 DOROSHENKO, KRAVCHENKO ζ χ µ , ϕ χ 1 2 ν 2 1 1 π iνθ (iii) 1 ==1 ϕ(),r 1 const, + ------ξ Aτ Ð ζ ------Ð ------e 2iNν ν sinπν ζ χ Aτ 2 ==2 const. In each of these cases, this condition is transformed 1 2 1 n Ðinθ + -----ξ ∑ ------e to the form 2i 2()ν 2 n n ≠ 0 N n + ∂ν ∂ν π χ χ ϑ ∈ (), ∞ 1------+0,2------==---, r 0+ . 1 n ()1 Ðinθ 1 2 n ()2 Ðinθ ∂ϑ ∂ϕ Ð ∑ ------ε e + -----ζ ∑ ----- ε e , 2 (27) 2()ν 2 n n 2i n n ϕ ∈ n ≠ 0 N n + n ≠ 0 L.

ν 1 n ()1 Aτ = ------()1 Ð ε . Here, χ and χ are constants. Under the assumptions Nn()+ ν n n 1 2 n = 0 made above, initial boundary condition (23) is reduced The singular integral equation obtained (22) with the to the form (27), which is analyzed below. We seek the ( θ potential w1 corresponding to the secondary ﬁeld in Cauchy kernel and the smooth function Kmτ() can be solved numerically by its discretization and the form (10): employment of Gaussian quadratures. +∞ +∞ 2 ()1 Kiτ()qr w = ----- τπτsinh ∑ b τV τ ------dτ, (28) 1 π2 ∫ m m THE FOURTH BOUNDARY CONDITION m = Ð∞ r ON STRIPS 0 m π b τ = Ða τP τ()cosϑ , ϑ < ---, We now analyze the problem of wave diffraction on m m Ð1/2 + i 0 0 2 a periodic array composed of strips for which the fourth boundary condition is given. Everywhere outside the ()1 strips and the source, the desired potential w(r) satisﬁes V mτ the Helmholtz equation with the boundary conditions +∞ mnN+

( ()ϑ ∂ ∂  PÐ1/2 + iτ cos inN()ϕ+ m w w ∑ x , ()τ ------e , ζ ------+ ζ ------= 0, ζζ ≠ 0, (23)  mn+ m0 mnN+ 1 ∂ 2 ∂ς 2 P τ()0 n Σ n = Ð∞ Ð1/2 + i  π constraints of the ﬁnite energy, and conditions at inﬁn- 0 <<ϑ --- ∂w ∂w  2 ity. In condition (23), ------and ------are the normal and =  ∂n ∂ς  +∞ mnN+ ()ϑ ( PÐ1/2 + iτ Ðcos inN()ϕ+ m  ∑ y , ()τ ------e , tangential derivatives of w(r), respectively, mn+ m0 mnN+  P τ()0 ζ ==ζ (),r, ϕ ζ ζ (),r, ϕ n = Ð∞ Ð1/2 + i 1 1 2 2 π (24)  <<ϑπ ne==ϑ, ς v e + µϕeϕ. --- . r r 2 Taking into account relationships (24), we can write out conditions (23) in the form As a result of the employment of the boundary con- ∂ν ∂ν ∂ν dition ζ v ζ ζ µ 2r r-----∂ - ++1∂ϑ------2 ϕ∂ϕ------= 0. (25) r Σ ∂w ∂w χ ------1 + χ ------1 Assuming vr = 0, we obtain from boundary condi- 1 ∂ϑ 2 ∂ϕ tions (23) ϑπ= /2+ 0 ∂ ∂ ∂ν ∂ν w1 w1 ζ ζ µ = χ ------+ χ ------1∂ϑ------+ 2 ϕ∂ϕ------= 0. (26) 1 ∂ϑ 2 ∂ϕ Σ ϑπ= /2Ð 0 ∂ Next, we consider the following particular cases of ∂w w1 χ ------χ ------ϕ condition (26): = Ð 1 ∂ϑ + 2 ∂ϕ = f (), ϑπ= /2 ζ χ ζ µ χ (i) 1 = 1 = const, 2 ϕ = 2 = const, ϕ ∈ L

µϕ()0;r, ϕ ≠ ϕ ∈ and the condition w1 ϑπ= /2+ 0 = w1 ϑπ= /2+ 0 , CL (ii) ζ ==χ κ(),r, ϕ χ const, 1 1 1 for the conjugation in slits, we arrive at the system of ζ µ χ κ , ϕ χ κ , ϕ ≠ 2 ϕ ==2 (),r 2 const, ()0;r equations

DOKLADY PHYSICS Vol. 47 No. 3 2002 APPLICATION OF SINGULAR INTEGRAL EQUATIONS FOR SOLVING PROBLEMS 205

+∞ 2

( χ 2 n ()2 inψ 1 n ()2 1 Ðinθ χ ∑ Nn()+ ν ----- ()1 Ð ε z e + ----- ∑ ----- ε , Ð ------e 1 n n n 2i n mn 2()ν 2 n = Ð∞ n ≠ 0 N n + +∞

( (29) χ2 ν n ()ε()2 ( inψ χ ψ χ2 Ð 2 ∑ Nn()+ ----- 1 Ð n zne = 1 f (), 2 n ε()1 1 Ðinθ n + ----- ∑ ----- mn, Ð ------e . n = Ð∞ 2i n 2()ν 2 n ≠ 0 N n + ψδ< , Singular integral equation (31) has the Cauchy kernel, +∞ θ ( Qντ( ) being the smooth function. ()ν inψ δψπ< ≤ ∑ n + zne = 0, , n = Ð∞ +∞ CONCLUSIONS

( (30) ()n ψπ ∑ Ð1 zn = 0, =, Thus, as a result of employing KontorovichÐLebe- n = Ð∞ dev integral transformation, the third and fourth bound- n ( ( ary value problems of the Helmholtz equation for the z = ()Ð1 ()y , Ð x , . n mn mn three-dimensional array consisting of planar angular ( Here, f is the transform of the function f. After intro- strips are reduced to singular integral equations of the ducing the function ﬁrst kind with the Cauchy kernel. The unambiguous solvability of singular integral equations (22) and (31) +∞

( follows from their equivalence to the pair summator Φψ ()ν in()ψ+ ν ()= iNn∑ + zne , equations (13), (14) and (29), (30), respectively, the lat- n = Ð∞ ter being adequate to the initial boundary value problem. The algorithms developed imply the necessity of ψππ∈ []Ð , numerically solving singular integral equations. The and using formulas (19)Ð(21), we arrive at approach can be realized by discretizing singular integral equations and using Gaussian quadratures. 2 2 νβ χ + χ Φβ()eÐi ------1 -2 ------dβ π ∫ βψÐ S ACKNOWLEDGMENTS

( The authors are grateful to Academician V.A. Sado- 1 βψΦβ Ðiνβ β χ ψ + π---∫Qmτ()Ð ()e d = 2 f (), vnichiœ, Corresponding Member of the RAS V.I. Pusto- S voœt, Professor Yu.V. Gandel, and Professor Ya.S. Shifrin for discussing the results of this work. ψ ∈ S,

2 2 1 θ 1 (31) Q τ()θ = ()χ + χ --- cot--- Ð --- m 1 2 2 2 θ REFERENCES 1. V. T. Erofeenko and V. F. Kravchenko, Radiotekh. Élek- 1 1 2 2 2 ν 2 π iνθ 1 tron. 45, 1 (2000). + ------χ + N ν A τχ ------e Ð --- 2iν 1 m 2 sinπν ν NAmτ 2. V. A. Doroshenko and V. F. Kravchenko, Dokl. Akad. Nauk 375, 611 (2000) [Dokl. Phys. 45, 659 (2000)]. 1 2 2 1 n Ðinθ + -----()χ + χ ∑ ------e 2i 1 2 2()ν 2 n n ≠ 0 N n + Translated by G. Merzon

DOKLADY PHYSICS Vol. 47 No. 3 2002

TECHNICAL PHYSICS

Propagation of an Acoustic Wave in a Thin Elastic Strip Embedded in a Low-Modulus Matrix S. L. Bazhenov Presented by Academician N.F. Bakeev October 31, 2001

Received October 31, 2001

In connection with the practical problem of measur- Here, u is the displacement of the strip element from its ing the elastic modulus for rubber deposited on a metal equilibrium position, h is the strip thickness, τ is the or glass substrate, we theoretically solve the problem of shear stress at the rubberÐstrip interface, and y is the the propagation of longitudinal acoustic waves in a thin strip axis along which the acoustic wave propagates. elastic strip-shaped rod embedded in an unbounded The factor 2 corresponds to the two stripÐmatrix inter- bulk low-modulus matrix. The speed of the propagation faces. of acoustic waves in an elastic rod is determined by the τ density ρ and elastic modulus E of the rod [1]: To determine the quantity , we consider the shear 0 vibrations of the matrix in the elastic half-plane x > 0. E These vibrations are described by the equation [1] c = ρ-----. (1) 0 ∂2u ∂2u ρ------= G------. (3) The velocity of longitudinal ultrasonic waves in a ∂ 2 ∂ 2 thin strip-shaped rod was found to decrease upon dip- t x ping the rod into a fluid [2, 3]. This decrease is caused by a boundary layer involved in joint vibrations with Here, ρ and G are the density and shear modulus of the the rod. The thickness of the boundary layer depends on matrix, respectively, and x is the axis perpendicular to the vibration frequency and the density and viscosity of the strip plane. the fluid [4–6]. This effect was used for determining The boundary conditions are determined by the fluid viscosity, and the results of these acoustic mea- requirement that displacements of the matrix at the x = 0 surements are consistent with those obtained by con- interface are equal to those of the strip. We seek a solu- ventional methods [3]. tion of Eq. (3) in the form of a traveling wave: Let the elastic modulus of the strip be much greater than the modulus of the matrix. In this case, according uu= exp[]Ði()ωtkÐ ⊥ x , (4) to Eq. (1), the longitudinal-wave velocity in the rod is 0 much higher than that in the matrix. Being in contact with the strip, the boundary layer of the matrix is where k⊥ is the wave number of a transverse wave in the involved in the motion. It is evident that the matrix matrix and ω is the angular frequency of the strip vibra- motion is of a shear origin and that a longitudinal wave tions. Substituting Eq. (4) into Eq. (3), we have in the strip excites a shear wave in the matrix. We assume that the strip width significantly exceeds ρ k⊥ = ω ----. (5) the length of the transverse wave excited in the matrix. G Under this assumption and taking into account that a strip element is subjected to the elastic tensileÐcom- pressive forces and forces of shearing interaction with The length of the transverse wave in the matrix is the latter caused by the matrix, one can describe the given by the formula longitudinal vibrations of the strip by the equation 1 G ∂2 ∂2 λ⊥ = ------, ρ u u τ f ρ 0h------= Eh------+ 2 . (2) ∂t2 ∂y2 where f = ω/2π is the vibration frequency. For G = 3 3 Institute of Synthetic Polymeric Materials, 1 MPa and ρ = 10 kg/m , which are typical values for Russian Academy of Sciences, rubber, f = 250 kHz, and the transverse wave length is Profsoyuznaya ul. 70, Moscow, 117393 Russia estimated as 125 µm.

PROPAGATION OF AN ACOUSTIC WAVE IN A THIN ELASTIC STRIP 207

Stress in the matrix at the stripÐmatrix interface is Ò, km/s ∂u 5.2 deﬁned as G------at x = 0 and is equal to ∂x

τ = iGk⊥u. (6) 4.8

We seek a solution of Eq. (2) in the form

[]()ω 4.4 uu= 0 exp Ði tkÐ || y , (7) where k|| is the wave number. 0 5 10 15 20 Assuming that the longitudinal wavelength in the G, MPa rod is much greater than the transverse wavelength in rubber, and substituting Eqs. (6) and (7) into Eq. (2), we Fig. 1. Speed of sound vs. the shear modulus G. The calculations were carried out for the strip thickness h = 0.1 mm, obtain the following expression for the wave number of c = 5.1 km/s, vibration frequency f = 250 kHz, ρ = the longitudinal wave in the rod: 0 0 2.6 g/cm3, and ρ = 0.93 g/cm3.

2 2 ω ρ λ ϕ ⊥ i –1 k|| = ---- 1 + ------e , (8) β, m c 4 π2ρ2 2 0 0h 120 ϕ where c0 is the speed of sound in the rod and = ρλ 1 ⊥ 80 --- arctan------πρ - . 2 0h The phase velocity of the longitudinal wave is equal to ω/k||. Using the equality λ⊥ = 2π/k⊥ and taking Eq. (5) 40 into account, we obtain the following expression for the phase velocity of the longitudinal wave:

c 0510 15 20 c = ------0 -. (9) ρG G, MPa 1 + ------4 π2 2ρ2 2 Fig. 2. The same as in Fig. 1, but for the damping coefﬁ- f 0h cient.

When λ⊥ Ӷ h, the second term in the radicand is much less than unity. Retaining the first two terms in the amplitude of the longitudinal wave in the strip is the expansion of Eq. (9) in a power series of this term, given by the relationship we have the relation ()β uu= 0 exp Ð y , (12) ρG Gρ cc= 0 1 Ð ------, (10) β ------π2 2ρ2 2 where = ρ is the damping coefficient for the lon- 4 f 0h 0c0h gitudinal wave. It is worth noting that the coefficient β which can be written in the form is equal to the ratio m1/m0 of the boundary-layer mass to the strip mass. If the wave damping factor is small, the coefficient β can be considered as a small parame- m2 cc= 1 Ð ------1 , (11) ter. In this case, the decrease in velocity is a second- 0π2 2 order correction in the damping coefficient. 4 m0 The results calculated by Eq. (9) for the velocity of the propagation of acoustic waves are shown in Fig. 1 where m = ρλ⊥ and m = ρ h are the masses of the 1 0 0 as a function of the matrix shear modulus G. The veloc- boundary layer and strip, respectively. ity of sound decreases monotonically with the increas- The complex-valued component of the wave num- ing shear modulus of the matrix. A similar dependence ber determines wave damping. It is easy to prove that for the damping coefficient is presented in Fig. 2. In the inequality λ⊥ Ӷ h is satisfied for ϕ Ӷ 1. In this case, the case of the matrices with small coefficients of stiff-

DOKLADY PHYSICS Vol. 47 No. 3 2002 208 BAZHENOV ness, the damping coefﬁcient increases very rapidly as 2. A. K. Rogozinskiœ, S. L. Bazhenov, and A. A. Berlin, β ∝ Dokl. Akad. Nauk 362, 618 (1998) [Dokl. Phys. 43, 623 G . (1998)]. Thus, the results of embedding a thin strip in a low- modulus matrix are similar to the results of the immer- 3. A. K. Rogozinskiœ, S. L. Bazhenov, and A. A. Berlin, Dokl. Akad. Nauk 362, 481 (1998) [Dokl. Phys. 43, 616 sion of the strip into a ﬂuid [4–6]. Namely, longitudinal (1998)]. vibrations of the strip excite a transverse wave in the matrix. There is also an inverse effect, which becomes 4. L. D. Landau and E. M. Lifshitz, Fluid Mechanics substantial when the strip mass is commensurable with (Nauka, Moscow, 1988; Pergamon, Oxford, 1987). the mass of the matrix boundary layer. Because of the 5. V. A. Krasil’nikov and V. V. Krylov, Introduction to presence of the matrix, the longitudinal wave is damped Physical Acoustics (Nauka, Moscow, 1984). and its velocity decreases. 6. L. F. Lependin, Acoustics (Vysshaya Shkola, Moscow, 1978). REFERENCES 1. L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Nauka, Moscow, 1986; Pergamon, Oxford, 1986). Translated by V. Chechin

DOKLADY PHYSICS Vol. 47 No. 3 2002

TECHNICAL PHYSICS

A New Class of Self-Similar Antenna Arrays V. F. Kravchenko* and M. A. Basarab** Presented by Academician V.A. Sadovnichiœ October 25, 2001

Received November 6, 2001

INTRODUCTION Solutions (1) for the function ha(x) are determined on –1 –1 Currently, new approaches in the analysis and synthe- the carrier [–(a – 1) , (a – 1) ], and their Fourier trans- sis of antenna systems, which are based on concepts of forms have the form: fractal geometry, have received wide recognition [1Ð8]. ∞ As is well known, the simplest mathematical abstrac- F ()p = ∏sinc()paÐk , tion for a self-similar set is a Cantor set (dust is an ana- a log) [9Ð12]. On this basis, one-dimensional and two- k = 1 dimensional antenna radiators were developed and sinx where sinc(x) ≡ ------. The derivatives of the nth order their properties were thoroughly investigated [6, 7]. x In the present study, a new class of self-similar sets can be expressed in terms of the function ha(x) in itself as based on a simple recurrence procedure is considered. nn()+ 3 ------In this case, terms of the sequence being formed alter- ()n () Ðn 2 nate in accordance with the distribution law for signs of ha x = 2 a derivatives of certain atomic functions (AF) [13]. These 2n n (2) p ()k Ð 1 functions are finite solutions to functional differential ×δ∑ h an xa+ ∑ j Ð 1()Ð1 j , equations relating shifts and compressions of such k a functions to their derivatives. Note that atomic func- k = 1 j = 1 tions turn out similar to linear combinations of their where derivatives. Therefore, their multiple differentiation δ δ δ δ δ ,,… makes it possible to design a new type of self-similar 1 ==1, 2k Ð k, 2k Ð 1 =k, k = 1 2 . (3) antenna arrays. Here, pj(k) denotes the jth bit in the binary representation × j of the number k; i.e., pj(k) = [k 2 ] mod 2. For a = 2, we THE ha(x) FAMILY OF ATOMIC FUNCTIONS obtain the simplest atomic function up(x) with the carrier (–1, 1), which satisfies the equation Atomic functions are finite solutions to linear functional differential equations with constant coefficients y'()x ------= y()2x + 1 Ð y()2x Ð 1 . N M 2 ()n () () ∑ dn y x = ∑ cm yaxÐ bm . Expression (2) for the derivatives of up(x) has the form

n = 1 m = 1 2n ()n nn()+ 1 /2 n n Here, a, dn, cm, and bm are numerical parameters, and up ()x = 2 ∑ δ up() 2 x ++2 12Ð k . | | k a > 1. One of the basic classes of atomic functions is k = 1 determined by the following expression: In Fig. 1, plots are presented for the function up(x), 2y'()x its ﬁrst two derivatives, and the Fourier transform as ------= yax()+ 1 Ð yax()Ð 1 . (1) 2 well. As is seen from Figs. 1b and 1c, the derivatives a consist of equal parts, similar to the function itself in the shiftÐcompression sense. In addition, in each of these segments, the derivatives have a sign determined * Institute of Radio Engineering and Electronics, by the recurrence sequence (3). Russian Academy of Sciences, Mokhovaya ul. 18, Moscow, 103907 Russia DESIGN OF A SELF-SIMILAR ANTENNA ARRAY ** Bauman Moscow State Technical University, Vtoraya Baumanskaya ul. 5, Moscow, Certain questions associated with the application of 107005 Russia atomic functions in problems of antenna-array synthe-

(a) (b) 1.0 2 1 0.5 –1.0 –0.5 0.5 1.0 –1 –2 –1.0 –0.5 0 0.5 1.0 (c) (d) 8 4 0.8 0.6 0.4 –1.0–0.5 0.5 1.0 –4 0.2 0 12.57 –8 –0.2 6.28 Fig. 1. (a) Atomic function up(x), (b) its ﬁrst derivative, and (c) second derivatives; as well as (d) the Fourier transform. sis are considered in [1Ð4]. In the present study, we use ment of generating sequence (3). It is well known that an original approach for creating a non-Cantor antenna the multiplier of a linear equidistant array is repre- array of a new type. This array is based on the employ- sented in the form [14]

 N ()ψ I0 + 2 ∑ In cos n for 2N + 1 element  AF()ψ =  n = 1 (4) N  1 2 ∑ I cos n Ð --- ψ for 2N elements.  n 2 n = 1

2π δ + 1 Here, ψ = kd(cosθ – cosθ ), k = ------, and d is the dis- ments to be even, we use the relationship I = ------n - as 0 λ n 2 δ tance between array elements. Let certain of them be a law for the electric-current distribution, where n are switched off or removed so that recursively determined from relationships (3). Let N = 4k. Then, the expression for the multiplier of array (3) can 1if the element is switched on be written out in the form In =  0if the element is switched off. N 1 AF ()ψ = ∑ () δ+ 1 cos n Ð --- ψ . (5) As a result, a nonequidistant equal-amplitude line of k n 2 discrete radiators is formed. n = 1 δ δ δ δ The simplest scheme (based on the Cantor-set con- It is evident that N – i + 1 = i and N/2 – i + 1 = – i . Thus, cept) for constructing a fractal array is analyzed in [5]. it can be shown that the preceding relationship has a The 101 equidistant generating sequence consisting of simpler form three elements (with the central one switched off) is N/4 Nψ δ considered as an initial sequence. The triad Cantor AF ()ψ = 4cos ------∑ cos 2n Ð 1 + -----n ψ . (6) array is obtained recursively by the sequential k 2 2 exchange of unity by 101 and zero by 000 at each n = 1 design stage [6]. Possible generalizations of Cantor The number of arithmetic operations needed for radiators (two-scale arrays) and methods for improving ﬁnding AFk in accordance with formula (6) is less than their characteristics are considered in [7]. in the case of employing formula (5) by a factor of four. We now investigate one possible design of a self- We now denote bn ° bn – 1 the concatenation of two ﬁnite similar antenna array. Assuming the number of its ele- sequences bn and bn – 1, where n is the recursion index.

DOKLADY PHYSICS Vol. 47 No. 3 2002 A NEW CLASS OF SELF-SIMILAR ANTENNA ARRAYS 211

(a) 1.0 | ψ | AF1( ) | ψ | AF2( ) 0.8 | ψ | AF3( ) 0.6

0.4

0.2

(b) 1.0

0.8 | ψ | AF2( ) | ψ || ψ | AF1( ) AF2( ) 0.6

0.4

0.2

–1.5 –1.0 –0.5 0 0.5 1.0 1.5 ψ Fig. 2. (a) Moduli of the antenna-array multiplier for the ﬁrst three stages of construction, k = 1, 2, 3, and (b) multipliers for an array of the second level (solid line) and combined array (dashed line) formed by the convolution of arrays of the ﬁrst and second levels.

Then, the following recurrence formula for the ordered It is of interest to compare expression (7) with the set of included (1) and excluded (0) array elements takes Fibonacci sequence place: F0 ===1, F1 0, Fn Fn Ð 1 ° Fn Ð 2, ,,… n = 23,,…, F0 ==1, Fn Fn Ð 1 ° Fn Ð 1, n =12 . (7) which is formed from the initial unity by the recurrent exchange 1 by 10 and 0 by 1. Here, the bar from above denotes the logical negation. The desired current distribution has the form I = F We consider a number of properties for the new 2n ° array. Furthermore, we use the normalized expression F n . Thus, we have obtained the following 2n, = 0, 1, … N/4 algorithm for constructing a discrete radiator line. 4 Nψ δ AF ()ψ = ---- cos ------∑ cos 2n Ð 1 + -----n ψ (8) Beginning from the initial sequence 11, we should sub- k N 2 2 stitute 1001 instead of 1 and 0110 instead of 0. In par- n = 1 ticular, at the second step, we have 10011001. instead of (6).

DOKLADY PHYSICS Vol. 47 No. 3 2002 212 KRAVCHENKO, BASARAB

DEC (a) 20 (b) k = 1 k = 2 0

–20

–40

–60

–80 (c) (d) 20 k = 3 comb. 0

–20

–40

–60

–80 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 θ

Fig. 3. Directional-effect coefﬁcients (DEC) as functions of the angle θ for (a) k = 1, (b) 2, and (c) 3, and (d) DEC for the combined array formed by the convolution of arrays of the ﬁrst and second levels.

In Fig. 2, we show the results for the multiplier (8), We now consider the procedure of constructing a which are obtained at the three first stages of designing combined two-scale array [7], which is defined as a the array. Let the distance between its three elements be convolution of two arrays with a different number of λ elements N = 4k and N = 4l. In this case, the array mul- a quarter of the wavelength; i.e., d = --- . In this case, for 1 2 4 tiplier is a product of the multipliers corresponding to θ initial radiators AF (ψ) = AF (ψ)AF (ψ). This combi- 0 = 90°, we arrive at the following expression for the k, l k l array directional-effect coefficient (DEC): nation makes it possible to optimize the array design from the standpoint of decreasing the level of side 2()ψ lobes. Figure 2b exhibits moduli of the multipliers for () AFk Dk u = 2------1 -, the initial array of the second level (N1 = 16) and an 2()ψ array represented as a convolution of two sequences for ∫ AFk du two linear discrete radiators of the second and first Ð1 (N2 = 4) levels. The directional-effect coefficient of the where ψ = πu/2 and u = cosθ. The maximum value of resulting array is shown in Fig. 3d. this coefficient corresponds to u = 0. It turns out that in ≈ the asymptotic limit, Dk(0) 4Dk – 1(0). The values k → ∞ A TWO-DIMENSIONAL ARRAY of the DEC are presented for k = 1, …, 5 in the following table: The Sierpinski carpet is a generalization of the Can- tor set to the two-dimensional case. In [6], problems are analyzed associated with constructing a fractal antenna k 123 4 5array based on the Sierpinski carpet. In our case, the Dk(0) 3.771 13.258 52.282 208.422 832.983 process of constructing a plane system of discrete radi-

Plots of the DEC for k = 1, 2, 3 are shown in Figs. 3a, ators is quite similar. We take the matrix 11 as an ini- 3b, and 3c. 11

DOKLADY PHYSICS Vol. 47 No. 3 2002 A NEW CLASS OF SELF-SIMILAR ANTENNA ARRAYS 213

z AF1

z AF2

z AF3 Fig. 4. Two-dimensional antenna arrays and their corresponding multipliers for the three ﬁrst construction stages. tial sequence and then sequentially change each 1 or 0 by respectively. The expression for the array multiplier takes the form N N 1111 0000 ()ψ , ψ AFk 1 2 = ∑ ∑ Imn, 1001 0110 or , m = ÐN + 1 n = ÐN + 1 1001 0110 1 1 × exp imÐ --- ψ +,n Ð --- ψ 1111 0000 2 1 2 2

DOKLADY PHYSICS Vol. 47 No. 3 2002 214 KRAVCHENKO, BASARAB where N = 4k. In Fig. 4 (on the right), we demonstrate 3. V. F. Kravchenko and A. A. Potapov, in Proc. of the IV the general view of two-dimensional arrays (discrete Intern. Symp. on Physics and Engineering of Millimeter radiators are situated in the center of black squares). On and Sub-Millimeter Waves, Kharkov, Ukraine, 2001, the left, their multipliers are displayed as functions of Vol. 1, p. 102. ψ ψ 4. M. A. Basarab, V. F. Kravchenko, and V. M. Masyuk, angles 1 and 2, which vary within the limits from 0 to π. Zarub. Radioélektron. Usp. Sovrem. Radioélektron., No. 8, 5 (2001). Thus, in the present study, a new approach associ- 5. H. P. Peitgen, H. Jurgens, and D. Saupe, Chaos and ated with the concept of atomic functions is proposed Fractals: New Frontiers of Science (Springer, New York, for constructing one-dimensional and two-dimensional 1992). antenna arrays. The construction is based on the recur- 6. D. H. Werner, R. L. Haupt, and P. L. Werner, IEEE rence procedures; however, in contrast to the traditional Antennas and Propag. Mag. 41, 37 (1999). Cantor sets it cannot be called fractal. It is easy to verify that the new type of array is based on the HausdorffÐ 7. S. E. El-Khamy and M. I. Elkashlan, in Proc. of the URSI Besikovich sets of unit dimensionality [8]. In spite of Int. Symp. on Electromag. Theory, Victoria, 2001, p. 16. this fact, these arrays possess self-similar properties, 8. B. B. Mandelbrot, The Fractal Geometry of Nature which allows us to considerably simplify the analysis (Freeman, San Francisco, 1983). of their properties and characteristics. Possible general- 9. A. V. Arkhangel’skiœ, Theory of Cantor Sets (Mosk. Gos. izations of the arrays proposed, such as combined two- Univ., Moscow, 1988). scale ones and two-dimensional discrete radiators, are 10. M. Schroeder, Fractals, Chaos, Power Laws: Minutes also considered. from an Inﬁnite Paradise (Freeman, New York, 1990). 11. J. L. McCauley, Chaos, Dynamics, and Fractals: An Algorithmic Approach to Deterministic Chaos (Cam- ACKNOWLEDGMENTS bridge Univ. Press, Cambridge, 1995). The authors are grateful to Academician V.A. Sado- 12. A. N. Kolmogorov and S. V. Fomin, Elements of the The- vnichiœ, Academician of the National Academy of Sci- ory of Functions and Functional Analysis (Nauka, Mos- ences of the Ukraine V.L. Rvachev, and Corresponding cow, 1981; Vol. 1, Graylock Press, Rochester, New York, Member of the RAS V.I. Pustovoœt for discussing the 1957; and Vol. 2, Graylock Press, Albany, New York, results of this work. 1961). 13. E. G. Zelkin and V. F. Kravchenko, Radiotekh. Électron. 46, 903 (2001). REFERENCES 14. W. L. Stutzman and G. A. Thiele, Antenna Theory and 1. V. F. Kravchenko, A. A. Potapov, and V. M. Masyuk, Design (Wiley, New York, 1981). Zarub. Radioélektron. Usp. Sovrem. Radioélektron., 15. M. A. Basarab and V. F. Kravchenko, Élektromag. Volny No. 6, 4 (2001). i Élektron. Sistemy 6, 31 (2001). 2. V. F. Kravchenko and A. A. Potapov, in Proc. of the URSI Intern. Symp. on Electromagn. Theory, Victoria, 2001, p. 660. Translated by G. Merzon

DOKLADY PHYSICS Vol. 47 No. 3 2002

ENERGETICS

Acoustic Instability while Combusting Aluminum Particles in MHD Facilities V. R. Pesochin Presented by Academician V.E. Fortov September 26, 2001

Received October 5, 2001

In this paper, the theoretical study of acoustic-vibra- der-like fuel in the combustion camber of an MHD gen- tion excitation while combusting aluminum particles in erator is an urgent problem. a gas mixture is carried out. At long distances, alumi- Usually, in studies of acoustic-vibration excitation, num particles are considered to be monodisperse and it is suggested that, in the case of combusting a disperse immobile with respect to the gas. In determining the fuel, the burning zone is substantially smaller than the excitation conditions for acoustic vibrations, the gas length of the combustion chamber [3, 4]. In actual con- mixture used was considered as a perfect gas. Expres- ditions, the burning zone has a certain extension along sions for the frequency and excitation increment of the longitudinal axis of the combustion chamber, which acoustic vibrations are obtained, characteristics of a is comparable with its length [6]. Therefore, in the case fuel and of an oxidizer explicitly entering into these of acoustic vibrations of pressure, disturbances of the expressions. velocity and temperature of the gas mixture along the In [1], powders prepared on the basis of carbon and length of the combustion chamber are different, which metals were suggested as a fuel for pulsed MHD gener- affects the stability of the combustion process [3, 4]. In ators. Air or oxygen are oxidizers in this case. Charac- this paper, in studies of the excitation of acoustic vibra- teristics of MHD generators while operating with these tions, a distributed combustion model is used. In other fuels, in fact, do not differ from those obtained using words, we assume that the zone of burning aluminum solid rocket propellants. In this case, the powder-like particles is approximately equal to the length of the fuel presents the possibility for profound control of the combustion chamber. device power when changing the mass flow-rate of components and provides for long-term operation of Here, in calculations of the burning process in a for- powerful MHD generators. It was shown in [2] that alu- ward-flow combustion chamber, the aluminum parti- minum fuel is optimal for pulsed MHD generators, pro- cles are considered to be monodisperse and immobile vided that the air oxidizer is employed. with respect to the gas mixture. It is assumed that the particle combustion occurs in the vapor-phase diffusion It is known [3] that in the process of burning a dis- regime. The specific heat c and the adiabatic index γ of perse fuel the acoustic vibrations can be excited, which p results in violation of the normal operation of combus- the gas mixture were assumed to be constant and inde- tion chambers and even in their destruction. In the prac- pendent of its temperature and component concentra- tice of rocket-engine development [4], engines having tions. The flow in the combustion chamber was pressure-vibration amplitudes of not more than 5% assumed to be one-dimensional under complete mixing from the nominal value are related to those exhibiting in the transverse direction and in the absence of mixing stable combustion. It was shown in [5] that the require- in the longitudinal direction. Such assumptions are ments for stability of the combustion process in the common in calculations of the design of combustion MHD-generator combustion chamber should be more chambers [6]. While analyzing acoustic vibrations, the rigorous than in rocket engines. This is caused by the gas mixture was considered as a perfect gas, and the fact that for such pressure vibrations electric-power presence of a disperse phase in it was ignored. pulsation phenomena hinder the normal operation of an Under these assumptions, the continuity equation, MHD generator. Therefore, consideration of acoustic- the equation of motion, and the energy equation can be vibration excitations in the case of combusting a pow- written in the form ∂ρ ∂ 3M ------+ ----- ()ρu = Ð------1W, (1) ∂t ∂x 4 M Institute of High-Energy Densities, Joint Institute for High Temperatures, ∂u ∂u 1∂p Russian Academy of Sciences, ----- + u----- = Ð------, (2) ul. Fadeeva 6, Moscow, 125047 Russia ∂t ∂x ρ ∂x

216 PESOCHIN

∂T ∂T Q ∂p with respect to t, and excluding u', we ﬁnd ρc ------+ u------= ------W + ------, (3) p∂t ∂x 4M ∂t ∂2 p' 1 ∂2 p' γQ ∂W' ---------- = ---------- Ð ------, (10) where W is the combustion rate for aluminum particles ∂x2 p a2 ∂t2 p 4Mρc Ta2 ∂t in the volume unit, Q is the thermal effect of the reac- s p s tion of the aluminum oxidation, and M and M1 are the where as is the adiabatic sound velocity. atomic and molecular masses of aluminum and oxygen, respectively. To solve the last equation, it is necessary to obtain the expression for W'. As a rule, for description of the Below, we consider that a local thermodynamic combustion of metal particles, the combustion theory equilibrium is conserved in the gas phase. This assump- for drops of a liquid fuel complemented with the allow- tion is common in studies of the excitation of acoustic ance for the formation of condensed products (oxides) vibrations in combustion chambers [3]. Then, we can is used [7]. It is considered that in the diffusion vapor- write the equation of state for the gas mixture phase model of combusting metal particles, as in the ρρ= ()pT, . (4) combustion theory of liquid-fuel drops, chemical transformations including the formation of condensed com- It is known [3, 4] that high-frequency acoustic bustion products (oxides) occur in a narrow zone above vibrations are the most dangerous and difﬁcult for elim- the particle surface. This theory leads to a satisfactory ination. As follows from [3], they are characterized by agreement for the calculated time of combusting a the fact that for them, the parameter Sh ӷ 1. Here, Sh = metal particle with the experimental data in the low ωl/u, ω is the angular frequency, and l is the chamber pressure zone (p < 30 atm). In the zone of high pres- length. Thus, linearizing equations (1)Ð(3) with allow- sures characteristic for the conditions in the combustion ance for Sh ӷ 1, we arrive at chamber of a pulsed MHD generator, a noticeable discrepancy with the vapor-phase diffusion model is ∂ρ' ∂u' ∂ρ 3M1 observed. The combustion time for an aluminum parti------++ρ------u'------= Ð------W', (5) ∂t ∂x ∂x 4 M 3/2 2 cle becomes proportional to r0 but not to r0 (the ∂u' 1∂p' Sreznevskiœ law). Here, r0 is the initial radius of a parti------= Ð------, (6) cle. This is associated with the fact that, in the case of ∂t ρ ∂x high pressures, the liquid aluminum oxide is accumulated at the surface of a particle, thus hampering its ∂ ∂T Q ∂p' ρ T' evaporation. At the present time, there is no theoretical c p------∂ - + u'------∂ = ------W' + ------∂ . (7) t x 4M t model of the vapor-phase combustion of aluminum particles with allowance for the accumulation of the oxide Here and below, the perturbations are indicated by at their surfaces. Therefore, in this paper, an empirical primes. time dependence for the size of a burning particle is Linearizing (4), we obtain a thermodynamic con- employed [7]: nection between perturbations of p, ρ, and T. r3/2 = r3/2 Ð αct. (11) ρ' p' T' 0 ---- = ------Ð -----. (8) ρ ρ 2 T α aT Here, is an empirical constant and c is the dimensionless mass concentration of oxygen far away from the Here, aT is the isothermal sound velocity. burning particle. Substituting expression (8) into Eq. (5) with account For combustion of a monodisperse system of alumi- of (7), we obtain num particles in the combustion chamber of a pulsed MHD generator, there exists a correlation between the 1 ∂ p' u'∂T ∂u' u'∂ρ QW' particle sizes and oxygen concentration: --γ-∂---- ---- +++------∂ ------∂ - ---ρ------∂ - = ------ρ . t p T x x x 4M c pT ()r/r 3 = c/c , Then, using the relation 0 0 where c is the initial dimensionless mass concentration ∂ρ ρ∂T 0 ------= Ð------, of oxygen. ∂x T ∂x Differentiating (11) with respect to t and substitut- we write out the last equation in the form ing the last relation into the equation obtained, we have 1 ∂ p' ∂u' QW' ------+ ------= ------. 3 3 Ð5/2dr α γ ∂  ∂ ρ (9) ---r0r ----- = Ð c0. (12) t p x 4M c pT 2 dt

Differentiating Eq. (6) with respect to x and Eq. (9) The initial condition is r = r0 for t = 0.

DOKLADY PHYSICS Vol. 47 No. 3 2002 ACOUSTIC INSTABILITY WHILE COMBUSTING ALUMINUM PARTICLES 217

The solution to this equation has the form When deriving the expression for 〈T〉, it was taken into account that the following inequalities take place: r 3/2 1 ---- = ------. (13) α  3/2 Qc0 c0l r0 1 + αc t/r ------ӷ 1; ------ӷ 1. 0 0 3M c T 3/2 1 p 0 u0r0 The number of particles N per volume unit of the combustion chamber is determined from the balance Furthermore, averaging expression (15) for the equation steady-state combustion rate, we obtain with the help of (16) T M T ρ c NN==-----0 ------0 ------0 0-, (14) ρ c u 0 T M T π 3ρ 〈〉0 M 0 0 0 1 r0 p W = ------. M1 l ρ where 0 is the density of an aluminum particle. In [7], the results of experiments on the combustion When deriving this expression, the change of the gas rate of finely dispersed aluminum particles as a func- mixture flow-rate as a result of combustion of alumi- tion of temperature and pressure for the invariable com- num particles was ignored. We should note that this position of the gas mixture are given. According to assumption is common in calculations of combustion these data, the combustion rate practically does not chambers [3]. With the help of Eqs. (12)Ð(14), we can change within the temperature range from 1600 to obtain the expression for the steady-state combustion 3300 K, and the pressure dependence ceases to mani- rate of aluminum particles in the volume unit fest itself for p > 25 atm. In this case, we can obtain an approximate expression for W' ρ 2α 0 2 dr 8 M T 0 0c0 ρ W ==Ð4πr ρ N------. (15) p' M 0c0u0 p' p 3/2 3 W' = 〈〉W ---- ≈ ------. dt 3M1 T ()1 + αc t/r 0 γ 0 0 p M1 l p As is seen from this expression, in contrast to the high- It is taken into account in the last expression that for temperature combustion of the monodisperse system of acoustic vibrations in gas mixtures the following rela- carbon particles [8], in the case of the combustion of tionships takes place the (quasi-adiabatic approach) [3] aluminum particles, the boundary of the burning zone ρ' 1 p' N' ρ' is absent and the process is continued beyond the com- ---- ==------, ------. bustion chamber (in the nozzle and in the channel of a ρ γ p N ρ MHD generator). Substituting the expression for W' into (10), we The energy equation for the steady-state combustion obtain an equation of the telegraph-type, which approx- of aluminum particles in the combustion chamber has imately describes the excitation of acoustic vibrations the form while combusting aluminum particles in the combustion chamber of a pulsed MHD generator: ρ dT Q 0 c pu------= ------W . dx 4M ∂2 p' 1 ∂2 p' δ ∂ p' ------= ------Ð 2------, (17) ∂ 2p 〈〉α2 ∂ 2p 〈〉α2 ∂tp The boundary condition is T = T0 for x = 0. x s t s The approximated solution to the last equation has Qc u the form where δ = ------0 ------0 . 8M1c pT 0 l Qc0 1 TT= + ------1 Ð;------To determine frequencies of acoustic vibrations as a 0 3/2 2 3M1c p ()α 〈〉 1 + c0 x/ u r0 result of their excitation, it is necessary to set boundary conditions for Eq.(17). The question on the boundary here, 〈u〉 is the average velocity of the gas mixture in the conditions at the nozzle inlet is rather complicated and 〈 〉 〈 〉 〈 〉 combustion chamber: u = u0 T /T0, T is the gas-mix- can be the object of a special study [3, 4]. In the present ture temperature averaged over the combustion cham- paper, similar to [4], the simplest assumption is made ber length. that the flow-velocity perturbations in the critical cross section of the nozzle are absent. Then, the boundary Averaging the last expression over the chamber conditions can be written in the form length, we arrive at the expression: ∂p' 〈〉α2 α ------= 0 for x = 0; (17.1) 〈〉T s c0 l ∂ ------x ------==------2 - 3/2, (16) T 0 α u0 r s0 0 ∂p' ------= 0 for x = l. (17.2) where l is the length of the combustion chamber. ∂x

DOKLADY PHYSICS Vol. 47 No. 3 2002 218 PESOCHIN Solving Eq. (17) by the method of separation of In conclusion, we note that the condition of the variables with allowance for the boundary conditions, vibration excitation, which is known in thermal acous- we obtain tics as the Rayleigh criterion [3], allows us to obtain ∞ sufficient conditions for acoustic stability and reveal p' δt πkx sources of the vibration energy in the case when they ---- = e ∑ B cos------p k l are not evident, e.g., for phase transitions in vaporÐgas k = 1 (18) mixtures [9]. However, this criterion does not allow us π 2 to determine the increment and the vibration frequency. × k 〈〉δ2 2 ϕ cos ------as Ð t +.k The model of the acoustic-vibration excitation, which l is suggested in this paper as distinct from the Rayleigh ϕ criterion, makes it possible to obtain (with invoking a Here, k = 1, 2, 3, … . The quantities Bk, k, and the eigenfrequency are determined from the initial condi- known empirical dependence for the rate of steady- tions and boundary conditions, respectively. This fre- state combustion of an individual aluminum particle) quency is the expression for the increment and frequency, which explicitly depends on the fuel and oxidant parameters. π 〈〉 ω kas k = ------. l REFERENCES δ It is seen from expression (18) that for > 0 the exci- 1. Yu. M. Volkov, Yu. G. Degtev, V. V. Krikun, et al., in tation of acoustic vibrations caused by the combustion Proc. of 8th Intern. Conf. on MHD Power Transforma- of aluminum particles can occur with the increment tion, Moscow, Russia, 1983 (Moscow, 1983), p. 104. 2πδ 2. Yu. G. Degtev and V. P. Panchenko, Teplofiz. Vys. Temp. µ = ------. (19) 31, 229 (1993). 2 πk 2 2 ------〈〉δα Ð 3. K. I. Artamonov, Thermohydroacustical Stability (Mashi- l s nostroenie, Moscow, 1982). 4. Liquid Propellant Rocket Combustion Instability, Ed. by As follows from formula (18), excitation of the D. T. Harrje and F. G. Reardon (NASA, Washington, acoustic vibrations results in a decrease of the vibration 1972; Mir, Moscow, 1975). frequency of the combustion chamber compared to the 5. E. L. Borblik, R. V. Ganefel’d, V. V. Naletov, et al., Pre- eigenfrequency. However, this decrease is insignificant. print No. 67, IED AN UkrSSR (Inst. of Electrodynamics, For example, for combustion of aluminum particles µ δ Akad. Nauk UkrSSR, 1973). with r0 = 30 m, T0 = 400 K in air, the value of is equal Ð1 Ð1 6. F. A. Williams, Combustion Theory (AddisonÐWesley, to ~70 s for the principal harmonic ∼3000 s (l ~ 1 m). Reading Mass., 1968; Nauka, Moscow, 1971). As is seen from expressions (18) and (19), the excita- 7. N. F. Pokhil, A. F. Belyaev, Yu. V. Frolov, et al., The tion conditions for acoustic vibrations are improved Combustion of Powder-Like Metals in Active Media with an increase in the initial concentration of the oxi- (Nauka, Moscow, 1972). dant and a decrease in the initial sizes of aluminum par- 8. A. M. Golovin and V. R. Pesochin, Fiz. Goreniya Vzryva ticles (or a reduction in the length of the combustion 21, 59 (1985). chamber). These conclusions are confirmed by the known fact of the excitation of acoustic vibrations by 9. V. R. Pesochin, Teplofiz. Vys. Temp. 32, 791 (1994). enhancement of the afterburn in the combustion chamber [3]. Translated by T. Galkina

DOKLADY PHYSICS Vol. 47 No. 3 2002

Doklady Physics, Vol. 47, No. 3, 2002, pp. 219Ð223. Translated from Doklady Akademii Nauk, Vol. 383, No. 1, 2002, pp. 46Ð50. Original Russian Text Copyright © 2002 by Akulenko, Georgievskiœ, Kumakshev.

MECHANICS

New Asymmetric and Multimode Solutions of the Problem on Viscous-Fluid Flow inside a Plane Confusor L. D. Akulenko*, D. V. Georgievskiœ**, and S. A. Kumakshev* Presented by Academician D.M. Klimov October 2, 2001

Received October 3, 2001

The problem on steady viscous-fluid flows in a According to Eqs. (1) providing the incompressibility plane confusor (the JeffreyÐHamel problem [1Ð3]) condition, we write the nonzero components of the again attracts attention due to increased numerical and strain-rate and stress tensors analytical possibilities of currently available comput- ers, as well as to the development of corresponding Q ()θ Q ()θ v rr ==Ðv θθ ----V , v rθ = Ð------V' , (2) software [4, 5]. This attention is also induced by practi- r2 2r2 cal needs for solving a wider class of problems on flows ρ 2 of a viscoplastic medium with a low yield stress in a σ ± 2 Q ()θ rr, θθ = Ð p ------V , plane confusor [6, 7]. In this paper, we report the results r2Re of constructing and investigating multimode and asym- (3) metric solutions in a wide region of physical and geo- ρQ2 σ ==Ðp, σ θ Ð------V'()θ . metric parameters of the system. New qualitative zz r 2 r Re mechanical effects were established and discussed. The substitution of Eqs. (3) into two NavierÐStokes 1. We consider a steady flow of an incompressible equations [1] leads to the following ordinary differen- ρ fluid with density in a plane confusor, which has tial equation for the function V(θ) appearing in angle 2β (0 < β < π) and outflow power Q (Q > 0) at the Eqs. (1)Ð(3) and yields the pressure p: point O. It is convenient to describe the motion in the polar coordinates (r, θ), where the region occupied by the V''+ 4V Ð ReV 2 ==C, C const; (4) fluid has the form |θ| < β, r > 0. In addition to the density, ν ρQ2 the fluid is characterized by the kinematic viscosity . p = ------()C Ð 4V . ρ ν 2 (5) For the plane confusor, three quantities { , Q, } are 2r Re dimensionally dependent ([Q] = [ν]). Since there are no other dimensional quantities in the problem, the subse- The mechanical meaning of the constant C is clear quent equations cannot be made totally dimensionless. from the condition of adhering the fluid to the confusor |θ| β Q walls = The Reynolds number Re = ---- is a dimensionless com- ν V()±β ==0, CV"()±β . (6) bination. Furthermore, the integral condition of constant For a certain distribution of the pressure p in the flow rate closes the formulation of the JeffreyÐHamel angle θ (for r ∞, this distribution can be found on problem: the basis of kinematic relationships), the velocity field β v v ( r, θ) is radial [1, 3]: ∫ V()θθ d = 1. (7) Q Ðβ v = Ð----V()θ , v θ ≡ 0. (1) r r The analytical investigations of the problem, Eqs. (4), (6), and (7), were extensively considered and reported in special papers [4, 5] and manuals [1Ð3]. The order of Eq. (4) can be reduced by multiplying both * Institute for Problems of Mechanics, sides by V' and isolating the total derivatives. The equa- Russian Academy of Sciences, tion has the first integral pr. Vernadskogo 101, Moscow, 117526 Russia 1 2 2 1 3 1 2 ** Moscow State University, ---V' + 2V Ð ---ReV Ð CV = ---V' ()+−β . (8) Vorob’evy gory, Moscow, 119899 Russia 2 3 2

It follows from Eq. (8) that the derivatives V ' are 2. On the basis of the high-accuracy numericalÐana- θ θ equal in magnitude at the zeros i of the function V( ); lytical method with improved convergence [4] in a wide 2 θ 2 −β range of parameters (a, b), we constructed numerical i.e., we have V' ( i) = V' ()+ . Equations (8) and (4) can be integrated in elliptic functions [1Ð3]. However, solutions to problem (9) in the class of symmetric sin- it is difficult to analyze the elliptic integrals obtained gle-mode profiles y(x) [5]. We also analytically investigated the following asymptotic cases of the parameters and to solve the set of two transcendental equations β with respect to the constants of integration [1]. One fails (a, b) or ( , Re): to construct an explicit solution of the boundary value π problem for arbitrary values of parameters β and Re. (i) Re Ӷ 1, 0 < β < --- ; 2 It should be noted that Re can formally be negative; in this case, a diffuser flow takes place (Q < 0). Such a (ii) Re ~ 1, 0 < β Ӷ 1; flow was considered as more complicated and diverse (iii) β Ӷ 1, βRe ~ 1; than the confusor one. Asymmetric (about the axis θ = 0) regimes and regimes admitting the alternation of flow- (iv) Re ӷ 1, β ~ 1. out and flow-into zones (multimode regimes) were In case (iii), numerical methods should be applied known for this flow [1Ð3]. The presence of these [5]; in case (iv), there exists a formal passage to the dis- regimes is the manifestation of the fact that the nonlin- continuous-profile limit for an inviscid fluid. The pres- ear boundary value problem with the condition of con- ence of the large parameter b requires the application of stant flow rate has several solutions. With increasing methods of singular perturbations [8]. By means Re, a steady flow in a diffuser loses its stability and of these methods, the asymptotic behavior of the solu- becomes turbulent. tion [1] in case (iv) can be represented in the very con- However, the asymmetric and multimode character venient form [5, 7] was not explicitly established for a confusor flow. 6 Therefore, the following questions remained unan- y∞()x ≈ 1 Ð ------, swered: (a) Are the asymmetric or multimode velocity b 15+ cosh arccosh + ---1()Ð 2x Ð 1 profiles possible in the confusor along with the sym- 2 metric and single-mode ones, and if so, under what conditions? (b) If these profiles are possible, what are their y∞()0 ==y∞()1 0, (11) properties? The boundary value problem, Eqs. (4) and (6), with 4b y∞' ()0 = γ∞ ≈ ------, y∞''()0 = λ∞ ≈ Ðb, b ∞ . condition (7) is two-parametric; it admits variational 3 treatment [4, 5]. For further numericalÐanalytical investigation, it is convenient to change the parameters Expressions (11) provide satisfactory accuracy for (β, Re) to the pair (a, b), where a = 4β and b = 2βRe. b ~ 103–104. θ Changing V( ) to the new unknown function y(x), where Further, we present certain integral estimates for y λ γ V = ------and the argument θ = (2x – 1)β (0 < x < 1), y(x), , and following from Eq. (9). It should be noted 2β that, for an arbitrary function f(x) continuously differ- the initial problem is represented in the form entiable on the interval 0 < x < 1, the following equality is valid: y" + a2 ybyÐ 2 ===λ, y()0 y()1 0, ()2 2 λ[]() () 1 ∫ y" + a ybyÐ f 'dx = fx2 Ð fx1 , (12) y'0()== γ, yx()dx 1, (9) ∫ where two arbitrary zeros x1 and x2 of the function y(x) ≤ ≤ 0 satisfy the conditions 0 x1 < x2 1. In Eq. (12) and λ ===8β3Cy"0() y"1(). below, we omit the limits of integration with respect to x from x1 to x2. Along with the variables y and y', we introduce the A. Let f(x) = y(x); in this case, integration of Eq. (12) variable z characterizing the fluid flow rate according to by parts yields two important equalities Eq. (8): () ± () z' ===y Ð0,1, z()0 z()1 0. (10) y' x2 = y' x1 . (13)

It is necessary to solve the boundary value problem, If x1 = 0 and x2 = 1, the lower sign in Eq. (13) corre- Eqs. (9) and (10), for the variables y and z. The param- sponds to the symmetrical single-mode regime includ- eter λ is unknown and must be calculated. This param- ing the classical one; the upper sign corresponds to the eter can be excluded by differentiating Eq. (9); how- asymmetric regime. It follows from the condition ever, a third-order nonlinear equation is obtained as a y'(1) = y'(0) that the multimode velocity profile is pos- result. sible.

DOKLADY PHYSICS Vol. 47 No. 3 2002 NEW ASYMMETRIC AND MULTIMODE SOLUTIONS OF THE PROBLEM 221

B. Let f ' = y and f(0) = 0. In this case, taking into certain fixed value of the parameter b = b0 (convention- ≈ account the Friedrichs inequality ally, b0 10) and the number of the chosen mode (n = 2, 3, 4, ), we seek unknown values γ and λ and calcu- 2 … 2 π 2 late them with a high accuracy, providing the above ∫y' dx ≥ ------∫y dx , ()x Ð x 2 residual with respect to the boundary conditions. Then, 2 1 by means of the procedure of continuation with respect γ we obtain the upper estimate of the parameter λ from to the parameter b, we plot universal curves (b) and λ ∞ Eq. (12): (b) for 0 < b < b0 and b0 < b < . The difficulties in these calculations are aggravated by the fact that the λ ()2 2 2 3 γ ӷ γ λ ӷ λ ∫yxd = ∫ Ðy' + a y Ð by dx values n 1 and n 1 (by several orders of magnitude) and attain high values for b = 1, for example, γ ~ (14) 5 π2 104 and λ ~ 105. These circumstances can explain why ≤ a2 Ð ------y2dx Ð by3dx, 5 ()2 ∫ ∫ conclusive results on determining and analyzing multi- x2 Ð x1 mode flows in the JeffreyÐHamel problem have not which is very accurate for the case of the single-mode been obtained yet. flow (on the entire interval 0 ≤ x ≤ 1). Below, we outline and discuss the graphic represen- C. Let f(x) = y'(x); in this case, integrating (12) by tation of the results of numericalÐanalytical investiga- parts, we write the equality tion of multimode flows for n = 2, 3, 4, 5. They are compared with the corresponding curves for the classical nλγ = ∫()y"2 Ð a2 y'2 + 2byy'2 dx, (15) solution (n = 1), which was studied in detail in [5], in particular, for various values of the parameter a. It was where n takes the values Ð2, 0, and 2 depending on the established that the odd n modes for n ≥ 2 correspond choice of the sign in Eq. (13). For n = 0, i.e., for the to symmetric (about x = 1/2, i.e., θ = 0) solutions, functions with the boundary conditions whereas the even n modes correspond to asymmetric () () solutions. yx1 ==yx2 0, () () () () The analysis shows that the flows have certain struc- y' x1 ==y' x2 , y" x1 y" x2 , tural properties. Namely, the positive maxima (n ≥ 3) ≥ one more Friedrichs inequality is valid: and the negative minima (n 4) of the functions yn(x, b) have the same magnitudes. Furthermore, in all the zeros 2 2 4π 2 x , i = 1, …, n + 1, of the function y (x, b) (for fixed n, y" dx ≥ ------y' dx. i n ∫ ()2∫ x2 Ð x1 b), the derivatives yn' are equal in magnitude; i.e., γ ± This inequality and Eq. (15) lead to the estimate n(b) = yn' (xi, b). Thus, in a certain sense, a multimode flow is a combination (aggregate) of single-mode and 2 4π 2 2 two-mode flows. This property likely follows from the ------Ð2a + by y' dx ≤ 0. (16) ∫ 2 fact that the flows are radial. ()x Ð x 2 1 γ λ Figures 1 and 2 show the functions n(b) and n(b) 3. For definiteness, we consider a fixed value of the in different scales for 1 ≤ b ≤ 10 and 10 ≤ b ≤ 200. The aperture angle β = 10°, i.e., a ≈ 0.7, that is widely met functions determine the solutions to the boundary value in the applied problems [5, 7]. Numerous calculations problem, Eqs. (9) and (10), by integrating the Cauchy were also carried out for other values of a. The value b problems for a fixed value of the parameter a (i.e., the is considered to vary within wide limits: 0 < b ≤ 200, angle β). The curves are grouped in pairs n = 2 and 3, i.e., 0 < Re Շ 600. A high-accuracy solution of Eqs. (8) and n = 4 and 5. Interesting properties of these curves and (10) with a relative error of 10Ð7Ð10Ð8 {absolute manifest themselves near b = 0, and the graphs have –5 γ ∞ λ ∞ error O(10 )} can be constructed using the modified vertical asymptotes: n + and n – for γ λ method of improved convergence of the Newton algo- b +0. We recall that finite values 1 and 1 corre- rithm type and the procedure of continuation with spond to the classical single-mode solution [5]. Large γ λ respect to a parameter [4]. It was successfully applied magnitudes of n and n for b ~ 1 are also remarkable. γ γ λ λ to the classical problem of investigating single-mode The differences n – n – 1 and n – 1 – n increase indefi- symmetric flows [5] for which an exact limiting solu- nitely for b → +0 and with the number n. Every curve tion is known in the analytical form for b 0 and γ γ arbitrary a. n attains a minimum for a certain large value b = bn ~ 2 3 Substantial difficulties in constructing multimode 10 –10 and tends very slowly to asymptote γ∞ = velocity profiles are caused by the degeneration of the λ 4b/3 (11) from above. Similarly, the curves n attain problem for b 0. This fact leads to unlimited values λ 2 of the desired parameters γ(b) and λ(b) necessary for maxima for certain b = bn ~ 10 and also tend very integrating the corresponding Cauchy problem. For a slowly to asymptote λ∞ = –b (11) from below. On the

DOKLADY PHYSICS Vol. 47 No. 3 2002 222 AKULENKO et al.

γ γ n n

5 4 4000 500

3 4 2 3 0 2 2 0 2 3 3 4

5 –60 000 –7000 4

–120 000 λ 05b 10 –14 000 n λ 10 100b 200 n γ λ ≤ ≤ ≤ ≤ Fig. 1. n and n vs. the parameter b, 1 b 10. Fig. 2. The same as in Fig. 1, but for 10 b 200. basis of these facts, we arrive at the important mechan- numerous characteristics of steady flows in a confusor ical conclusion that all the modes of steady flows, by integrating the Cauchy problem for Eqs. (9) and including the fundamental one (n = 1, see [5]), tend to (10). These characteristics, cited in Section 1, are a perfect-fluid flow in the limit b ∞ (Re ∞) in velocity profile (1), pressure (5), components of the a certain metric for 0 < x < 1. strain-rate tensor (2) and strain tensor (3), etc. The γ λ γ λ The curves n(b) and n(b) are essentially the prin- shape of the curves n and n is rather simple; however, cipal result of these investigations and provide the their construction requires very cumbersome high-

y2, 3 y4, 5 300 3 350 5 2 4 0 0

–300 –350

–600 –700 0 0.5 1 0 0.5 1 x

Fig. 3. Velocity proﬁles yn(x) for b = 1.

DOKLADY PHYSICS Vol. 47 No. 3 2002 NEW ASYMMETRIC AND MULTIMODE SOLUTIONS OF THE PROBLEM 223

y2, 3 y4, 5 3.5 3.5 3 5 2 4

0 0

–3.5 –3.5

–7.0 –7.0 0 0.5 1 0 0.5 1 x

Fig. 4. The same as in Fig. 3, but for b = 200. accuracy calculations, which are considerably compli- REFERENCES cated for b +0 and b ∞. Pronounced boundary layer effects take place in the problem. Calculating algo- 1. N. E. Kochin, I. A. Kibel’, and N. V. Roze, Theoretical rithms on the basis of the known functional-analysis Hydromechanics (Fizmatlit, Moscow, 1963). methods (Bubnov–Galerkin), finite-element and finite- 2. L. D. Landau and E. M. Lifshitz, Course of Theoretical difference methods does not provide satisfactory results. Physics: Fluid Mechanics (Nauka, Moscow, 1986; Per- To illustrate multimode flows, Figs. 3 and 4 show gamon, New York, 1987). the velocity profiles yn(x), n = 2, 3, 4, 5, for a relatively small value of b = 1 and a relatively large value of b = 3. L. G. Loœtsyanskiœ, Mechanics of Fluids and Gases 200, respectively. For small values of the parameter (Nauka, Moscow, 1987). b (Re), large oscillations of positive and negative veloc- 4. L. D. Akulenko and S. V. Nesterov, Dokl. Akad. Nauk ity values, i.e., the functions yn (x), are observed. The 374, 624 (2000) [Dokl. Phys. 45, 543 (2000)]. inflow (y > 0) and outflow (y < 0) regions correspond n n 5. L. D. Akulenko, D. V. Georgievskiœ, S. A. Kumakshev, to these values. An increase in the parameter b(Re) and S. V. Nesterov, Dokl. Akad. Nauk 374, 44 (2000) reduces the amplitude of oscillations and reverse flows. [Dokl. Phys. 45, 467 (2000)]. For large values b ~ 102–103, the pronounced forms corresponding to confusor flow (11) are observed for a 6. D. M. Klimov, S. V. Nesterov, L. D. Akulenko, D. V. Geor- weakly viscous fluid. A deviation from the rectangular gievskiœ, and S. A. Kumakshev, Dokl. Akad. Nauk 375, profile of a flow of a nonviscous (perfect) fluid in a cer- 37 (2000) [Dokl. Phys. 45, 601 (2000)]. tain metric tends to zero for 0 < x < 1; near x = 0 and 1 − 7. D. V. Georgievskiœ, Stability of Deformation Processes (θ = +β ), typical boundary layer phenomena arise. of Viscoplastic Bodies (Izd. URSS, Moscow, 1998). 8. S. A. Lomov, Introduction to the General Theory of Sin- ACKNOWLEDGMENTS gular Perturbations (Nauka, Moscow, 1981). This work was supported by the Russian Foundation for Basic Research, project nos. 99-01-00222, 99-01-00276, 01-01-06306, and 99-01-00125. Translated by V. Bukhanov

DOKLADY PHYSICS Vol. 47 No. 3 2002

MECHANICS

On an Extreme Feature of Detached Flows upon the Interaction of a Shock with a Boundary Layer M. A. Zubin and N. A. Ostapenko Presented by Academician G.G. Chernyœ September 20, 2001

Received September 27, 2001

Calculations carried out with empirical relations tems [4, 7, 8]. The formula for calculating the angle ϕ under certain assumptions have revealed nontrivial reg- has the form [4] ularities inherent in λ-configurations of shocks produced at the developed separation of a turbulent bound- ϕ ps ϕ = hlog----- + k, ary layer in cone flows and are characterized by the pk minimum (or near-minimum) production of entropy. h = 5.1exp() Ð0.89M + 0.71, (1) Experimental studies [1Ð4] of the separation of a M ϕ = arcsin ------nk . turbulent boundary layer in cone flows under the action k M of shocks revealed that a disturbed flow has a number of fundamental features that have not been theoretically Here, the quantities with the index k correspond to the explained yet. Among them are the following ones. A formation of separating the turbulent boundary layer, flow with a boundary layer separation has a cone char- and pk = 1.6 is the minimum value of the intensity ps of acter, which is disturbed only in the regions of laminar- the shock C generated by a plate A, which makes an to-turbulent state transition [2Ð4]. The quantitative angle α with the flow direction at infinity (Fig. 1). The characteristics of a detached flow for the artificially shock is incident at a right angle on the plate B posi- induced turbulence of a boundary layer [1] coincide tioned along the flow direction. The Mach number with those under natural conditions of varying the local Reynolds number along the line of turbulent boundary M × –6 ∈ nb layer separation in the range Re 10 (1.5; 20) [4]. 2.4 Some other features will be mentioned later. 4 p2

Semiempirical relations for angles ϕ and γ between 8 3 the direction of an undisturbed flow with the Mach 2.2 6 number M and the lines of separation and attachment, respectively (see Fig. 1, the flow diagram), are obtained 7  π  2 in [4] for regimes of “free” interaction ϕ < --- – ε 2.0 5  2  6 [5, 6]. These relations are applicable for calculating the A angular dimensions and position of the region of sepa- 1.8 4 rating the turbulent boundary layer under the model- 1 problem conditions (a flow in a right dihedral angle, C Fig. 1). They may also be used for similar calculations 5 D 1.6 ϕ 3 in a shock layer at the interaction of a boundary layer θ α with curvilinear shocks and compression-shock sys- B M ε 1.4 90°C 2 23456 7 ps Institute of Mechanics, Moscow State University, Michurinskiœ pr. 1, Moscow, 117192 Russia Fig. 1.

ON AN EXTREME FEATURE OF DETACHED FLOWS UPON THE INTERACTION 225

Mnk = 1.23 of the undisturbed-flow velocity component Nevertheless, the effect of various processes on normal to the wave corresponds to p . entropy production can be estimated. Let us consider k ∆ the entropy increment Sw = Sw – S∞ per unit mass of a The values of pk and Mnk agree with data on the formation of a plane separation (see, e.g., [9]). This is one gas (S∞ is the specific entropy in an undisturbed flow) of the general features of separating the turbulent when one or several shock waves of local supersonic boundary layer in flows with various dimensions. Among regions located in the separation-zone outline travel λ these features is the fact that the pressure downstream through the shock -configuration and move further from the oblique shock above the region of the developed downstream [2Ð6]. We suppose that this entropy increment is about the increment in a single incident wave separation is equal to the pressure plateau p1. The latter corresponds to the free-interaction regimes [2, 3]. characterized by the Mach number Mn of the velocity normal to the wave front: According to [10], p1 (divided by the pressure p∞ in the undisturbed flow) can be calculated by the formula  p = 0.287+ 0.713M, (2)  2γ 2 ( 1 ∆ ∼  () Sw cv ln 1 + ------γ - Mn Ð 1 -- which yields adequate results for M ∈ (1.75, 5). For  + 1 ( cone flows with boundary-layer separation, in Eq. (2)  (5) M should be replaced by the Mach number Mns = ϕ γ  Msin (ps) [2Ð4] of the free-flow velocity normal to the 2 2γ ()M Ð 1  separation line. × 1 Ð ------n . γ + 1 2 For “restricted” interaction, the indicated property Mn  holds as before, but with other pressure levels [5, 6]. For  ε ∈ θ M = 3.04 and (36°, 46°), the slope s of the oblique shock above the separation region is calculated as Here, cv is the constant-volume specific heat for ideal gas, and γ is the specific heat ratio. θ ()ρε []θ()ε 0[]()ε s = K s Ð ps + s ps , Let the entropy per unit mass of the gas increase due to viscosity (in the presence of velocity gradients) by p ≥≥p p ()ε , (3) sm s s ∆Sµ in a characteristic time of T = L/U∞, where L is the K()ε = Ð1.03ε2 +,1.774ε Ð 0.577 characteristic dimension of the reference volume in the flow direction or the length of the experimental model π [2Ð4] and U∞ is the velocity magnitude for the undis- ϕ[]p ()ε = --- Ð ε . (4) s 2 turbed flow. The similar increment determined by heat conductivity (in the presence of the temperature gradi- ε Here, the intensity ps( ) of the incident shock C (Fig. 1) ent) is denoted by ∆Sλ. These increments are about is such that, according to Eq. (4), the separation line coincides with the leading edge of the plate B having γγ()Ð 1 M2 γ ∆Sµ ∼ c ------, ∆Sλ ∼ c ------. (6) ε θ 0[]()ε v v the glancing angle ; the slope s ps of the shock Re PrRe above the separation region corresponds to this case Here, Re is the Reynolds number calculated on the and is calculated using both Eq. (2) (with M replaced by length L, and Pr is the Prandtl number. We take into Mcosε) and the above pressure-plateau property; and account the foregoing intervals of varying the quanti- the intensity psm of the incident shock C corresponds to ties appearing in Eqs. (6), as well as the fact that Mn > π M for a developed detached flow. Then, we conclude the maximum possible value of the angle θ = --- – ε nk 2 that the entropy increment in compression shocks of the λ ∆ (Fig. 1) at restricted interaction for fixed sweep angle ε. shock -configuration, i.e., Sw from (5), makes the Under certain assumptions, the above data enable us greatest contribution to entropy production. to calculate the entropy variation induced by passing To find the entropy production in compression the shock λ-configuration of a developed detached shocks situated under the contact-discontinuity surface flow. It is impossible to calculate the entropy-flow emanating from the shock branch line, one should cal- increment by introducing a reference volume including culate the entropy flow transported in a unit of time by the characteristic region of the disturbed flow produced the gas particles passing through the oblique shock by the interaction of the incident shock with the bound- above the separation region and other downstream ary layer (Fig. 1). The first reason is the lack of neces- waves. sary qualitative data on the flow parameters in a conic We assume that the isentropic-deceleration parame- separation zone and in adjacent flow regions. More- ters (and, consequently, entropy) on all current surfaces over, it is difficult to simulate three-dimensional mixing between the indicated contact-discontinuity surface processes in such a way as to provide an efficient calcu- and conic separation region (Fig. 1), after the passage lation of the entropy increment caused by viscosity and of all discontinuities, are close to (or coincide with) the heat conductivity. corresponding parameters for gas particles passing

DOKLADY PHYSICS Vol. 47 No. 3 2002 226 ZUBIN, OSTAPENKO

M2 the separation region interacts with the incident shock Mn2 on the section of the subsonic cone flow downstream of 3.0 the shock. The branch-line position is considered to be little different from the position of the line of the oblique- shock intersection with the undisturbed shock C incident on the plate B at a right angle (Fig. 1). This assumption 2.5 is strongly supported by experimental results obtained 5 for a disturbed flow by a special optic method [2Ð6]. 6 Thus, we accept that Mnb is equal to the Mach number 2.0 of the velocity normal to the line of intersection of the incident shock and the oblique shock specified by ~ empirical relations (1), (2) or (3), (4). 7 8 The dependencies Mnb calculated in such a way are 1.00 shown in Fig. 1 for the undisturbed-flow Mach number 1 2 4 3 M = 3.04 corresponding to approximations (3). Curve 1 and α ∈ [12°, 30°] correspond to the free-interaction case, whereas curves 2Ð4 for ε = 46°, 41°, and 36°, 0.50 234567respectively, correspond to the restricted interaction. ps The upper circle, triangle, and diamond denote the ori- gins of respective curves 2–4, which show the solutions Fig. 2. of Eq. (4). The terminal points of these curves correspond to the values of the intensity of the incident shock C, when its position coincides with the leading through the oblique shock and breakdown shock wave π in the branch-line vicinity. There is only indirect sup- edge of the plate B: θ = --- Ð ε (Fig. 1, the flow dia- port for this speculation. Such an assumption was made 2 in [2] for particles on the current surface coming to the gram). Curves 2Ð4 differ little from curve 1 for the same attachment line (Fig. 1), and this allowed description of ps value, because the angle between the line of intersec- those observed qualitative changes in the internal flow of tion of the oblique shock with the incident shock and the the separation region that are caused, among other fac- plate B is small compared with the angle θ. tors, by the transonic transition of the backward flow. The branch-point structure is calculated for a three-

Since the entropy flow downstream of the shock shock configuration, which is supplemented with a cen- λ-configuration is generally described by an improper tered rarefaction wave when the corresponding solution integral, one should consider the ratio of this flow to the is absent. In this case, the parameters of the breakdown entropy flow transported in unit time by the correspond- shock wave at the branch point correspond to the acous- ing particles of the undisturbed flow. Under the above tic point in the interior polar constructed for the Mach assumption, this ratio reduces to the ratio of the values number of the uniform-flow velocity component down- Sb and S∞ of the specific entropy in the shock branch stream of the oblique shock above the flow separation line under the contact-discontinuity surface and in the region normal to the branch line. undisturbed flow, respectively. The reason is that, under Figure 1 also shows the data calculated for the pres- the accepted assumption, the integrals are the mass sure p2 (divided by the pressure p∞) downstream from flows upstream and downstream of the shock λ-config- the shocks at the branch point for (line 5) free and uration and are equal and cancel each other according (lines 6Ð8 corresponding to lines 2Ð4 for Mnb) to the mass conservation law. In turn, the calculation of restricted interactions. Figure 2 shows the Mach num- the ratio S /S∞ is equivalent to the calculation of the b bers on the sphere Mn2 and the total velocity M2 for ρ γ parameters under the contact discontinuity at the pb b entropy function s = ------ ρ------, which will be dis- branch point (curves 1 , 5 and 2 Ð 4 , 6Ð 8 correspond to p∞ ∞ free and restricted interactions, respectively). The cir- cussed later. cles, triangles, and diamonds have the same meaning as To calculate gas parameters under the contact-dis- in Fig. 1. continuity surface emerging from the shock branch The results obtained for p2 (Fig. 1) and M2 (Fig. 2) line, one needs to determine the Mach number Mnb of with empirical relations (1)Ð(3) point to an unexpected the undisturbed-flow velocity component normal to the fact: the total pressure recovery factor η in the flow branch line. Relations (1), (2) (free interaction) or (3), downstream from the shock λ-configuration takes close (4) (restricted interaction) are inadequate to calculate values for two fundamentally different interaction α Mnb for the given M and . The quantity Mnb should be regimes if the incident shocks have the same intensity. determined from a solution of the corresponding The same conclusion also holds for the entropy func- boundary value problem, since the oblique shock above tion related to the recovery factor as s = η1 – γ.

DOKLADY PHYSICS Vol. 47 No. 3 2002 ON AN EXTREME FEATURE OF DETACHED FLOWS UPON THE INTERACTION 227 γ Let us consider the dependence s( , M, pss), where s pss is the arbitrary intensity of the oblique shock rami- 1.2 fying in the three-shock configuration in a flow with the (a) Mach number M. For γ = 1.4 (air), Figs. 3a and 3b show γ A the curve nets s( , M, pss) corresponding to the Mach 6 numbers M = (1) 1.5, (2) 1.7, (3) 1.9, (4) 2.1, (5) 2.3, and (6) 2.4 and M = (1) 1.5, (2) 2, (3) 2.5, (4) 3, (5) 3.5, 5 and (6) 4, respectively. From the right, the curves are 1.1 11 7 s 8 p1 + ps 4 bounded by the values pss = ------, where p1 is 2 3 10 s determined by Eq. (2) and ps is the pressure at the 2 9 sound point of the interior polar constructed for the 1 1.0 number M. For each M value, the point A indicates the ~ 234 s value corresponding to the normal shock wave. These ~ (b) values do not coincide with the ordinates of the curves γ s( , M, pss) for pss 1, because shock disturbance for a subsonic flow downstream from it depends irregularly 6 on the intensity of the interacting oblique shock [11]. 1.4 As is seen in Fig. 3, for each number M, there is an 5 11 oblique shock with an intensity of p such that, as it 4 7 ss 10 ramifies, the entropy increment for gas particles which 3 have passed the shock λ-configuration under the contact discontinuity is minimal. Curves 7 in Figs. 3a and 2 8 9 1 3b correspond to the values of the entropy function for 1.0 pss = p1 given by Eq. (2) [10]. Thus, curves 7 give 1357 insight into entropy production in the shock λ-configu- pss ration, which is realized in a two-dimensional problem when a direct shock wave interacts with a turbulent Fig. 3. boundary layer. The other curves in Fig. 3 correspond to the results Indeed, curves 7 in Fig. 3 indicate that entropy pro- of calculations for the case of separation in a cone flow duction in the shock λ-configuration is not minimal. In (Fig. 1). Here, the net of the curves M = const should be order for it to reach a minimum, a considerable increase considered as Mnb = const, and the value of the abscissa of oblique-shock intensity above the separation region pss should be considered as the intensity of the oblique is required. This increase is incompatible with available shock interacting with the normally incident shock C. experimental data and the physical meaning of the the- γ In Fig. 3a, curve 8 corresponds to s( , Mnb, p1) for free oretical results [13]. In other words, the existing spread interaction. Here, p1 is determined by Eqs. (1) and (2), of experimental data [9] for the pressure plateau in the ϕ separation region of a turbulent boundary layer where M should be replaced by Mns = Msin (ps). The Mach number for the undisturbed flow is M = 3.04, excludes the values pss = p1 that exceed certain critical α ∈ ∈ ∈ values p M and are close to the minimum points of ° [12.5°, 30°], Mns [1.74; 2.64], Mnb [1.51, cr( ) 2.39]. In Fig. 3a, curves 9Ð11 correspond to the case of curves 1Ð6. This is the essence of the foregoing restric- restricted interaction, when the slope of the oblique tion. At the same time, in the case of a turbulent bound- λ shock is determined by Eq. (3) with ε = 46°, 41°, and ary layer, pcr(M) is large enough to allow the shock - 36°, respectively. The use of Eqs. (1) and (2) allows cal- configuration to considerably reduce entropy production γ culations of s( , Mnb, p1) for various M values. In in contrast to the case of the separation of a laminar Fig. 3b, the entropy function is shown for M = (8) 3.04, boundary layer. (9) 4, (10) 5, and (11) 6. For the free interaction of shocks with a turbulent The results presented in Fig. 3 support the hypothesis boundary layer in cone flows, the two-dimensional sep- that the principle of minimum entropy production [12] aration has the foregoing fundamental properties in the with restriction is realized in the flow formed by the plane normal to the separation line. However, an addi- ϕ interaction of shocks with a boundary layer. In this tional degree of freedom, the angle (ps) calculated case, the states of particles at the entry and exit of the from semiempirical formula (1), arises in the disturbed reference volume outside the layers where viscosity flow. It is precisely this degree of freedom combined and heat conductivity are substantial should be with dependence (2) that makes it possible to choose accepted as the stationary states. the value Mnb in such a way that the corresponding p1

DOKLADY PHYSICS Vol. 47 No. 3 2002 228 ZUBIN, OSTAPENKO

s 4b show results for M = 4 and 5, respectively. Curves 7 1.2 correspond to the separation in the two-dimensional (a) problem. Curves 9 and 10 correspond to the respective lines in Fig. 3b. The dashed lines bound the regions 7 involving all the pressure plateau values found in various 9 experiments [9]. It is seen that the results of the calcula- 1.1 γ tions with the possible spread of s( , Mnb, p1) values do not change the above conclusions. In conclusion, we note that the functional corresponding to entropy production can have no minimum because 1.0 of conditions excluding process-parameter values falling ~ (b) outside the restrictions imposed, as in the above case of the two-dimensional separation, by the properties of the 7 local or global structure of a disturbed ﬂow. As a conse- 10 quence of these restrictive conditions, which are some- 1.2 times difﬁcult to reveal, only a certain boundary extremum for the indicated functional is realized.

ACKNOWLEDGMENTS 1.0 12345This work was supported by the Russian Foundation pss for Basic Research (project no. 00-01-00234) and the Fig. 4. program “Universities of Russia—Fundamental Inves- tigations” (project no. 99-1653). value provides the minimum value of s(γ, M , p ) nb 1 REFERENCES (Fig. 3b, curves 8Ð11). For restricted interaction, the angle of the oblique shock above the separation region 1. V. S. Dem’yanenko and V. A. Igumnov, Izv. Sib. Otd. rises linearly with the intensity of incident shock (3), Akad. Nauk SSSR, Ser. Tekh. Nauk, No. 8, 56 (1975). 2. M. A. Zubin and N. A. Ostapenko, Izv. Akad. Nauk and Mnb is virtually unaffected (Fig. 1, curves 1Ð4). In this case, the disturbed flow loses one degree of freedom SSSR, Ser. Mekh. Zhidk. Gaza, No. 3, 51 (1979). and its equilibrium state becomes further from the state 3. M. A. Zubin and N. A. Ostapenko, Jet and Detached with minimum entropy production (Fig. 3a, curves 9Ð Flows (Mosk. Gos. Univ., Moscow, 1979). 11). However, the corresponding increase in the entropy 4. M. A. Zubin and N. A. Ostapenko, Izv. Akad. Nauk function is small, as was indicated above, when the SSSR, Ser. Mekh. Zhidk. Gaza, No. 6, 43 (1983). results for p (Fig. 1) and M (Fig. 2) were discussed. 5. M. A. Zubin and N. A. Ostapenko, Dokl. Akad. Nauk 2 2 368, 50 (1999) [Dokl. Phys. 44, 626 (1999)]. Under the assumption that the principle of minimum 6. M. A. Zubin and N. A. Ostapenko, Izv. Akad. Nauk, entropy production with restriction is realized for the Ser. Mekh. Zhidk. Gaza, No. 3, 57 (2000). flows under consideration, it is easy to show that the cone character of a flow for free interaction is the key 7. M. A. Zubin and N. A. Ostapenko, Izv. Akad. Nauk SSSR, Ser. Mekh. Zhidk. Gaza, No. 3, 68 (1989). manifestation of this principle. Indeed, if the disturbed flow were not a cone flow but became a quasi-two- 8. M. A. Zubin and N. A. Ostapenko, Izv. Akad. Nauk, dimensional flow beginning with a certain local Re Ser. Mekh. Zhidk. Gaza, No. 2, 137 (1992). number, the values of the entropy function in the corre- 9. G. N. Abramovich, Applied Gas Dynamics (Nauka, sponding region of the shock-wave configuration Moscow, 1976). would lie not in curves 8Ð11 but in curve 7 (Fig. 3b); 10. G. I. Petrov, V. Ya. Likhushin, I. P. Nekrasov, and L. I. Sorkin, Effect of Viscosity on Supersonic Stream i.e., entropy production would increase. The cone char- with Compression Shocks, Preprint No. 224, TsIAM acter of the flow can only be violated in regions of the (Tsentral’nyœ Inst. Aviatsionnykh Motorov, Moscow, laminarÐturbulent transition, where pcr(M) changes. 1952). This behavior was observed experimentally. 11. N. A. Ostapenko, Dokl. Akad. Nauk 372, 181 (2000) In summary, we conclude that entropy production [Dokl. Phys. 45, 225 (2000)]. realized in the shock λ-configuration with the restric- 12. I. R. Prigogine, Introduction to Thermodynamics of Irre- tion on p1 is analogous to the attainment of a boundary versible Processes (Interscience, New York, 1968). extremum in optimization problems. 13. L. V. Gogish, V. Ya. Neœland, and G. Yu. Stepanov, in Since Eq. (2) [10] is generally one of the available Itogi Nauki Tekh., Ser.: Gidromekh. (VINITI, Moscow, approximations for the pressure plateau [9], it is 1975), Vol. 8. instructive to consider the effect of the p1 value on the results calculated for the entropy-function. Figures 4a and Translated by V. Tsarev

DOKLADY PHYSICS Vol. 47 No. 3 2002

MECHANICS

Stability of Steady-Flows of Two Ideal Fluids with Different Constant Densities V. I. Sedenko Presented by Academician I.I. Vorovich July 4, 2001

Received August 22, 2001

In this paper, the stability of steady flows in the sys- the kinematic boundary conditions tem of two ideal fluids with different constant densities (), ∈ is analyzed with allowance for surface tension at their V in xt = 0, xSi, (2) interface. The analysis is based on the abstract scheme (), χ(), ∈ () proposed by Arnol’d [1, 2]. The fluids in this system V in xt = in xt, xS3 t ; can have free boundaries. It is shown that the well- known RayleighÐTaylor instability is compensated for and the dynamic boundary conditions certain vortex flows by vorticity and surface tension. (), σ (), ∈ () pi xt = 2 Hxt, xS2 t , The method reported in [1, 2] was used in [3] to analyze the stability of steady flows of an ideal fluid with a p ()xt, = p ()xt, + 2σHxt(), , (3) free boundary. i j ∈ () , , It was shown that, among all fields with the same xS3 t , ij= 12. vorticity, only the velocity fields for steady flows correspond to the critical points of kinetic energy (as well as Here, Hk(x, t) is the average curvature of the surfaces to the critical points of the momentum and angular Sk(t) at the point x [the average curvature is expressed momentum functionals if the flow is invariant under in terms of the principal radii of curvature by the rela- translations and rotations, respectively). If the critical –1 Ð1 Ð1 χ point is the nondegenerate maximum or minimum for tionship H = 2 (R1 + R2 )], in denotes the normal the connective of integrals at the layer of flows with the component of the velocity of a point at the boundary of same vorticity, the corresponding flow is stable. domain Dk(t), and the subscript takes the values k = 2. 3.

1. THE EULER EQUATIONS FOR FLOWS 2. CONFIGURATION SPACE OF AN IDEAL INHOMOGENEOUS FLUID WITH A PIECEWISE CONSTANT DENSITY A flow is referred to as steady flow when domains 0 Let a time-dependent domain D(t) in the three- Di filled with corresponding homogeneous fluids and dimensional space R3 have the boundary ∂D(t) = S ∪ 0 1 velocity fields V , i = 1, 2, are independent of time t. S (t) and be filled with two ideal fluids with the inter- i 2 Now, we define the configuration space M0 correspond- face S3(t), where S1, S2(t), and S3(t) at any t consist of 0 0 a finite number of connected closed disjoint surfaces. ing to the steady flow {(Di , V i ): i = 1, 2}, the stability Here, S1 is a set of rigid walls and S2(t) is the free of which will be studied. Let M be a manifold formed boundary. The fluid is subjected to the bulk forces with by sets of pairs {(Di, Vi): i = 1, 2}, where Di is a potential U(x) and to the capillary forces with the sur- domain, Vi is a solenoidal vector field in the domain Di, face tension σ ≥ 0. Then, velocity V (x, t) and pressure ∈ ∩ i i = 1, 2, and Vim(x) = Vjn(x), x Di Dj . We have pi (x, t) in the domain Di (t) for each t value satisfy the ∈ 0 {(Di, Vi)} M if there exists a set of maps gi, i = 1, 2, Euler equation such that ρ ()(), ∇ i V it + V i V i ==Ð0;grad(),Up+ i div V i (1) 0 (1) gi diffeomorphism of the domain Di into Di conserves a volume element;

Rostov State University of Economics, (2) giS1 = S1; ul. Bol’shaya Sadovaya 69, ⊂ ∂ 0 ∩ ∂ 0 Rostov-on-Don, 34400 Russia (3) giS = gjS, where S Di D j , i, j = 1, 2.

3. FUNCTIONALS OF ENERGY, MOMENTUM, have equal vorticities, and the map gi transforms xi(0) AND ANGULAR MOMENTUM into xi(t). The subscript takes the values i = 1, 2. Set (1) with boundary conditions (2) and (3) has the total-energy integral 5 VARIATIONS OF ELEMENTS 2 3 FROM M0 ALONG THE LAYERS 1 2 E = ∑ ---ρ V dx ++Uxd ∑ σ ds, (4) OF THE INVARIANT FOLIATION ∫ 2 i i ∫ °∫ i = 1 D ()t Dt() k = 2 S ()t τ i k Let solenoidal vector fields fi(x, ) defined in the τ which is the sum of the kinetic energy, potential energy neighborhoods of Di(0) be such that giτ(xi(0)) = xi( ) is of the field of bulk forces, and surface energy of capil- a solution of the set of differential equations lary forces. For each element {(D , V ): i = 1, 2}, Eq. (4) i i dx determines the energy functional. If the rigid walls are ------i ==f ()x, τ , i 12., invariant under translations along the OX axis or to dτ i rotations about the OX3 axis, it is possible to define the τ The following conditions are satisfied: fin(x, ) = 0, momentum functional with respect to the OX1 axis ∈ τ τ ∈ ∂ τ ∩ ∂ τ x S1; fin(x, ) = fjn(x, ); x Di( ) Dj( ); i, j = 1, 2. 2 ρ Similar to [3], it can be shown that velocity varia- L1 = ∑ ∫ iV ie1dx tions have the form i = 1 Di ∂ 1 δV ==V ()x, τ f × r ++gradα' gradβ ,(5) or the angular momentum functional with respect to the i ∂-----τ i i i i i τ = 0 OX3 axis

2 2 2 1 ∂ ρ × ⋅ (),, δ V = ------V ()x, τ K3 ∑ ∫ iV i Re3dx, R = x1 x2 x3 , i 2∂τ i τ = 0 i = 1 () (6) D1 t 1 2 2 respectively. = ---[]f ×ϕ{}f , r + × r ++gradα gradβ . 2 i i i i i i i

0 α1 α2 4. THE INVARIANT FOLIATION ON M Here, i and i are determined by the conditions Generalizing the definition from [3], we refer to ele- δ δ2 div V i ==div V i 0, (7) ments {(Di, Vi)} {(Di' , V i' )} as elements with equal vorticity if there exists a set of maps gi, i = 1, 2, such d V ()g τ x, τ ng()τ x, τ = 0, (8) that -----τ i i i d τ = 0 (1) gi diffeomorphism of the domain Di into Di' con- 2 serves a volume element; d (), τ (), τ ------V i giτ x ngiτ x = 0, (9) (2) g S = S ; 2 i i 1 dτ τ ⊂ ∂ ∩ ∂ = 0 (3) giS = gjS, where S Di Dj, i, j = 1, 2; (4) the equality where ri(x) = curlVi(x, 0), {fi(x, 0), ri(x)} is the Poisson ϕ bracket for the vector fields fi(x, 0) and ri(x); i(x) = ' , ∂ °∫V idx ==°∫ V idx, i 12 f (x, τ); and i = 1, 2. In Eqs. (8) and (9), ∂-----τ i r g r τ = 0 i i i ∈ ∂ τ γ x Di(0), and n(giτx, ) is the unit vector of the outer is satisfied for any closed path i in the domain Di. ∂ τ normal to Di( ) at the point giτx, i = 1, 2. In Eqs. (5) Thus, we define the invariant HelmholtzÐThomson 1 2 0 β β foliation on M , i.e., the partition into the classes of and (6), i and i are harmonic functions in the equivalence: two elements belong to one layer if and domain Di, i = 1, 2. At the interfaces between fluid par- only if their vorticities are equal. The invariance of this ticles of different densities, the following conditions foliation for the aforementioned dynamic system is the should be satisfied: essence of the HelmholtzÐThomson theorem on the β1()⋅ () β1()⋅ () vortex conservation. grad i x nx = grad j x nx, (10) Theorem 1. Let Vi(x, t) satisfy Eqs. (1)Ð(3) in the 2 2 ∈ δ δ gradβ ()x ⋅ nx()= gradβ ()x ⋅ nx(), domain Di(t) for all t (– and ) and xi(t) be the path i j of a fluid particle. Then, ∈ ∂ ∩ ∂ where x Di Dj and n(x) is the unit vector of the {}()(), (), {}() (), (), ∂ Di 0 V i x 0 and D i t V i xt outer normal to D i at the point x .

DOKLADY PHYSICS Vol. 47 No. 3 2002 STABILITY OF STEADY-FLOWS OF TWO IDEAL FLUIDS 231

6. VARIATIONAL PRINCIPLES 8. THE STABILITY CRITERION FOR THE ENERGY FUNCTIONAL Theorem 3. If some linear combination of second 0 0 variations, Theorem 2. Let {(Di , V i ): i =1, 2} be a steady flow µ δ2 µ δ2 µ δ2 0 0 i E ++2 L1 3 K3, (11) of an ideal fluid. Then, {(Di , V i ): i =1, 2} is a critical point of the energy functional E at a layer of the invari- 0 0 taken along the Helmholtz layer at the point {(,Di V i ): ant foliation; conversely, any critical point of the i =1, 2} corresponding to a steady flow of an ideal fluid energy functional corresponds to a steady flow. If the with piecewise constant density is a fixed-sign qua- steady flow is invariant under translations along the β1 OX1 axis, this flow is a critical point of the momentum dratic form of fi and gradi , i = 1, 2, then the steady functional L1 at this layer. If the steady flow is invariant flow is stable under small finite perturbations of the under rotations about the OX axis, this flow is a criti- 0 0 ∂ 0 3 velocity V i , vortex ri , and domain boundary Di , cal point of the angular momentum functional K3 at this i = 1, 2. layer. 9. STABILITY OF FLOWS 7. THE SECOND VARIATIONS OF ENERGY, IN A TWO-LAYER FLUID MOMENTUM, AND ANGULAR MOMENTUM Let us consider a plane-parallel flow in a layer where the straight lines y = 0 and y = a correspond to Taking into account Theorem 2 and using Eqs. (5)Ð rigid walls and the straight line y = y1 (0 < y1 < a) is the (10), we derive the following formulas for the second ρ interface between fluid particles with the densities 1 variation of the energy functional: ρ and 2 and velocity profiles u1(y) and u2(y) in the upper 2 and lower layers, respectively. We analyze the stability 2 2δ2E = ∑ ρ [()f × r + grad()α1 + β1 of this flow under perturbations periodic in x with the ∫ i i i i i period X. i = 1 Di ×]⋅ {}, + V i f i f i ri dx 9.1 Flows in a Rigid-Wall Channel with High Surface Tension at the Interface 2 In this case, the second energy variation has the  ρ ()× ()α1 β1 +2∑ ∫  iV i f i ri + grad i + i --- form i = 1 ∂ 2 Di δ2 ρ δ()0 2 ui ()δ 0 2 2 E = ∑ ∫ i V + ----- ri dyxd 2 ui" V  i = 1 D + grad ρ ------i + U ⋅ f f i i 2 i in ρ ()δ0 ρ ()δ0 +2∫ 1u1 y1 V 11 f 12 Ð 2 2u2 y1 V 21 f 22 σ∇()(), ()2 ()2 --+ f in f in + 2K Ð 4H f in ds yy= 1 (12) 2 2 ∂ u 2 ∂ u 2 and momentum functional + ----- ρ -----1 ()f Ð ----- ρ -----2 ()f ∂y1 2 12 ∂y2 2 22 2  2 2 2 ∂ f ∂ f 2δ L = ∑  ρ e × f ⋅ {}f , r dx + σ ------12 + ------22 dx. 1  ∫ i i i i i ∂x ∂x i = 1 D i For Eq. (12) to be positive definite, the condition [ ρ ()× ()α1 β1 ⋅ () () +2∫ i f i ri + grad i + i e1 u1 y > ∈ [], u2 y > ∈ [], ------0, y 0 y1 , ------0, yy1 a ∂ () () Di u1" y u2" y

 and the following conditions at the y = y1 interface + grad()ρ V e f ] f ds , i i 1 i in σ > ()(),,,()ρ ,  CX; ui y1 ui" y1 i i = 12 where e1 is the unit vector of the OX1 axis. should be satisfied. If surface tension increases at the interface, we will eventually obtain all flows with pro- δ2 The formula for K3 is similarly derived and has a files minimizing the energy functional in the situation similar form. when the interface is replaced by a rigid wall.

DOKLADY PHYSICS Vol. 47 No. 3 2002 232 SEDENKO 9.2. Flows in a Rigid-Wall Channel at the interface. Thus, bulk forces that can stabilize a with High Surface Tension at the Interface flow with a free boundary [3] cannot stabilize a flow µ µ µ with an interface. Let us set 1 = 2, 2 = –2c, and 3 = 0 in Eq. (11) assuming that u1(y1) = u2(y1) = c. Then, we have the square form ACKNOWLEDGMENTS 2 The work was supported in part by the U. S. Civilian δ2 δ2 ρ ()δ 0 2 2 E Ð 2c L1 = ∑ ∫ i V i --- Research and Development Foundation for the Inde- i = 1 D pendent States of the Former Soviet Union, grant no. j (13) RM1-2084. I am grateful to I.I. Vorovich, V.I. Yudovich, ∂ 2 ∂ 2 u ()y Ð c 0 2 f f and A.B. Morgulis for helpful discussions. + ------i ()δ dxdy + σ ------12 + ------22 dx, () i ∫ ∂ ∂ ui' y x x yy= 1 REFERENCES which is positive definite if each velocity profile satisfied the conditions 1. V. I. Arnol’d, Prikl. Mat. Mekh. 29, 846 (1965). 2. V. I. Arnol’d, Dokl. Akad. Nauk SSSR 162, 975 (1965). u ()y Ð c ------i > 0, i = 12., 3. V. I. Sedenko and V. I. Yudovich, Prikl. Mat. Mekh. 42, () ui" y 1050 (1978). Note that the stability conditions for low surface tension are satisfied only when the velocity is continuous Translated by K. Kugel’

DOKLADY PHYSICS Vol. 47 No. 3 2002

MECHANICS

New Stationary Solutions to the Problem of Three Vortices in a Two-Layer Fluid M. A. Sokolovskiœ* and J. Verron** Presented by Academician V.V. Kozlov September 4, 2001

Received September 6, 2001

1. The universally integrable problem of three vorti- m κ1 dz 1  1 m 1 m ces [1, 4Ð6] has attracted the interest of researchers for ------2 = ------------1 []1 Ð γ z Ð z K ()γ z Ð z over one hundred years [6]. This is not associated only dt 2πi 1 m 1 2 1 1 2 z1 Ð z2 with the vortex problem in itself but also has numerous (2) analogies in the mechanics of solids, astrophysics, the κ3 Ð m  dynamics of superfluid helium, and mathematical biol- 2 []γ 3 Ð m m ()γ 3 Ð m m  + ------3 Ð m m- 1 + z2 Ð z2 K1 z2 Ð z2 . ogy [1, 5]. A new peak of interest in this problem was z2 Ð z2  stimulated by the discovery of so-called three-polar structures [13], i.e., symmetric triples of vortices m m m Here, z x iy is the complex coordinate of the (−κ, 2κ, –κ) and by later observation of their spontane- n = n + n ous origin from chaos [12]. In most studies [1, 4Ð7, 10, mth vortex in the nth layer; the overbar implies complex γ 12Ð15], the dynamics of vortices was analyzed in the conjugation; the parameter is inversely proportional framework of the homogeneous-fluid model. At the to the Rossby internal deformation radius same time, the geophysical problems (some of them g∆ρh h 1/2 were discussed in [2, 3, 8, 11]) are characterized by λ = ------1 2 - ; ρ 2() noticeable density stratification. In this paper, we ana- 0 f h1 + h2 lyze the problem of three vortices that exist in a two- layer fluid and have zero total intensity. Kk is the kth-order modified Bessel function of the second kind; g is the acceleration of gravity; and f is the 2. We take the following assumptions: (i) the upper Coriolis parameter equal to the double angular velocity and lower layers have equal thickness values (h1 = h2 = of the fluid-plane rotation about a normal to this plane. ρ ρ 1/2) and the densities 1 and 2 in the layers satisfy the According to assumption (ii) above, the variable m in condition ∆ρ = ρ – ρ > 0; and (ii) one vortex with an Eq. (2) takes the values 1 and 2. Thus, Eqs. (1), (2) are 2 1 the set of ordinary differential equations that should be κ1 intensity 1 corresponds to the upper layer, whereas complemented by initial conditions for the original κ1 κ2 coordinates of all three vortices. two vortices with intensities 2 and 2 are located in the lower layer. The complex form of equations of The set of Eqs. (1), (2) can be represented in the motion for three discrete vortices in two immiscible standard Hamiltonian form with the Hamiltonian fluid layers rotating with a constant angular velocity 1 2 1 12 12 (in the presence of a hard cap on the upper surface) is {κ κ []()γ H = Ð------π 2 2 lnd22 Ð K0 d22 2 (3) 1 1 11 11 1 2 21 21 1 2 j κ κ []κγ κ []}γ κ + 1 2 lnd21 + K0()d21 + 1 2 lnd21 + K0()d21 , dz 1 j 1 j 1 ------1 = ------∑ ------2 []1 Ð γ z Ð z K ()γ z Ð z , (1) dt 2πi j 1 2 1 1 2 1 z2 Ð z1 mn | n m | j = 1 where dkl = zl – zk . In addition to Hamiltonian (3), the original set has the first integrals for the momenta P ==κ1z1 ++κ1z1 κ2z2, P κ1z1 ++κ1z1 κ2z2 * Water Problems Institute, Russian Academy of Sciences, 1 1 2 2 2 2 1 1 2 2 2 2 Novaya Basmannaya ul. 10, Moscow, 107078 Russia and the angular momentum E-mail: [email protected] κ1 1 2 κ1 1 2 κ2 2 2 ** Laboratoire de Geophysique et Industriel Courants, M = 1 z1 ++2 z2 2 z2 , Centre National de la Recherche Scientifique, Grenoble, France whose values are evidently determined by the initial E-mail: [email protected] conditions.

e1 t2 t3 t1 {1} {2} {2}

{3} e2

(a) (b)

Fig. 1. (a) Phase portrait of the relative motion for a three-vortex system in a two-layer ﬂuid under conditions (4) and P = 1.7. Thick lines are separatrices that separate the regions of different types {1}, {2}, and {3} of interactions between the vortices. The dark region is the nonphysical region of the phase plane (axes of trilinear coordinates are also shown). (b) The same as in Fig. 1a with the singular points indicated. Squares and circles on the boundary of the physical region correspond to the coordinates of the representation points in the phase space for the original conﬁgurations of the vortices in the numerical calculations presented in Figs. 2 and 3, respectively.

It is easy to verify that the invariants H, M, and In the trilinear coordinates, the isolines of Hamilto- P á P are in involution, and therefore set (1), (2) always nian (3) coincide with those of the function has a regular solution [1, 4]. (),, Ðt1 Wt1 t2 t3 = ln ------In what follows, we assume that the total intensity of t2t3 vortices is zero and Ðt P2 t P2 t P2 1 1 2 γ 1 γ 2 γ 3 κ ==Ð2κ, κ κ =0.κ > (4) Ð2 K0------++2K0------2K0------. 1 2 2 3κ2 6κ2 6κ2 Thus, the upper-layer vortex is attributed to the anticy- In essence, these isolines are the phase trajectories for clonic vorticity compensated by the two cyclones in the the relative motion of the three-vortex system under lower layer. consideration and are shown in Fig. 1 for P = 1.7. 3. We now assume that P ≠ 0. In this case, the rela- The basic general properties of the phase curves are tive motion can be analyzed in the trilinear coordinates the following. [1, 7, 15] (I) All the phase curves start and finish at the physical-region boundary. Therefore, all relative motions of κ1κ2()21 2 κ1κ2()21 2 3 2 2 d22 3 1 2 d21 vortices are periodical, and vortices pass through two t1 ==ÐÐ------, t2 ------, P2 P2 collinear positions during a period. From this, it follows that exhaustive information on possible motions of the κ1κ1()11 2 system of vortices can be acquired from the initial con- 3 1 2 d21 dition for three vortices being located in the same t3 = Ð------, P2 straight line. (II) Possible motions for the system of three vortices which possess the obvious property can be classified into three qualitatively different types t ++t t = 3. according to the initial collinear configuration of vorti- 1 2 3 ces.

In the plane speciﬁed by the coordinates t1, t2, t3 (whose (III) The phase portrait can have singular points of meaning is illustrated in Fig. 1a), it is necessary to sep- elliptic (e) and hyperbolic (h) types (see Fig. 1b). arate so-called physical regions where the triangle ine- (IV) The phase curves are symmetric about the quality is satisﬁed. Under condition (4), this implies straight line t2 = t3, because the variables t2 and t3 are that proportional to the distances squared between the anti- cyclone of the upper layer and equivalent (with each ()2 ≤ < ≥ ≥ 12t1 + t2 Ð t3 0 for t1 0, t2 0, t3 0. other) cyclones of the lower layer.

DOKLADY PHYSICS Vol. 47 No. 3 2002 NEW STATIONARY SOLUTIONS TO THE PROBLEM OF THREE VORTICES 235

The phase portraits (we present here only one example for a particular value of the system momentum) provide total information on the relative motion of the system of vortices. However, it is known that this analysis does not reveal all characteristic properties of the absolute motion of vortices [5]. Below, the basic features of the vortices are analyzed on the basis of numerical calculations. We also present the trajectories of vortices, indicating (by markers) their synchronous collinear positions additionally marked by segments passing through these positions. Without loss of generality, the original positions of vortices are attributed to the x-axis. 1 1 We take x1 = x2 = 0 (one of the lower-layer vortices is 2 ≠ (a) (b) (c) exactly under the upper-layer vortex) and x2 0 as the simplest (reference) original positions of the vortices Fig. 2. Trajectories of absolute motion for P = 1.7. The ini- 1 1 1 1 2 tial conditions are speciﬁed by Eqs. (5) for X X ; x1 , x2 , and x2 , respectively. In this case, the momen- 1 = 2 = 0 tum P of the system of vortices is completely deter- 2 X2 = 1.7; X0 = (a) Ð0.3, (b) Ð0.2, (c) 1.1; and (a) t = 2 − mined by the initial vortex position x2 . At the same ( 1.2561, 4.0692, 0.1869), (b) (Ð1.7543, 4.6713, 0.0830), time, the sets of coordinates and (c) (Ð15.7889, 16.2768, 2.5121).

1 1 1 1 2 2 x1 ==x1, x2 x2 Ð x0, x2 =x2 + x0, (5) conserve the given P value for all x0. An arbitrary set of numerical experiments with initial conditions (5) for 2 fixed x2 corresponds to a specific phase portrait (e.g., that presented in Fig. 1). In what follows, we also use γ m γ m the notation X0 = x0 and Xn = xn along with x0 and m xn . The trajectories and vortices (for their collinear positions) are shown in Figs. 2 and 3 by solid lines and 1 triangles for the upper-layer vortex x1 , by long dashes 1 and circles for x2 , and by short dashes and squares for 2 x2 . The size of a marker is proportional to the intensity of the corresponding vortex. We imply the upward translational motion everywhere. In figure captions, the notation t = (t1, t2, t3) is used. (a) (b) (c) 4. Figure 2 shows characteristic examples of Fig. 3. The same as in Fig. 2, but for (a) X0 = Ð0.6, motions for each of three types. (b) 0.2062, and (c) 0.9611 and (a) t = (Ð0.2595, 2.5121, For type {1} (see Fig. 2a), the interaction between 0.7474), (b) (Ð4.6321, 7.5438, 0.0883), (c) (Ð13.6197, the lower-layer vortices is predominant. The system of 14.7020, 1.9177). vortices moves in the direction perpendicular to the x- axis. In this case, the anticyclonic vortex of the upper According to Fig. 2c, solutions of type {3} are fea- layer undergoes small periodic deviations from the rec- tured by anticyclonic revolutions of all three vortices. tilinear motion. The lower-layer vortices revolve in the This is caused by the defining role of the upper-layer cyclonic direction around a certain center moving vortex. translationally so that the vortices change their places The cyclicity periods for vortices in the upper and in the collinear positions for each half-period. lower layers relate to each other as 1 : 2 for motions of Motions of type {2} are characterized by dominat- types {1} and {3}, whereas these periods coincide in ing interactions between the upper-layer vortex and one the case of type {2}. This difference is explained by the of the lower-layer vortices, which was initially located fact that each phase curve of types {1} and {3} is mirror closer to the upper-layer vortex. symmetric, whereas the curve of type {2} is asymmetric.

DOKLADY PHYSICS Vol. 47 No. 3 2002 236 SOKOLOVSKIŒ, VERRON The examples of phase curves presented in Fig. 2 5. The general conditions for the existence of these correspond to the conditions when they pass far from stationary solutions for an arbitrary momentum value the separatrices and singular points. However, the sta- of the system are the following. tionary solutions corresponding to singular points are 5.1. Triangular configuration. From Eqs. (1) and of the most interest. (2), it is easy to derive the condition for the existence of the rectilinear motion of this vortex structure: In the vicinity of the singular point e1, the lower- layer vortices approach each other at an infinitesimal ϕ ()ϕ 12+ L cos K1 2L cos []() distance and should orbit a common center with a (the------= 4 1 Ð LK1 L (6) oretically) infinite angular velocity. In this case, inter- cos2 ϕ layer interaction is manifested almost exclusively as the γ linear displacement of the entire configuration. In with L = l, where l is the length of the lateral side of essence, the limiting-case structure is equivalent to a the isosceles triangle, as well as the expression for the two-layer pair of vortices of intensities –2κ and 2κ in velocity of this structure in the y direction parallel to the the upper and lower layers, respectively. The motion base of the triangle: with characteristics close to those described above is κγsin ϕ 12 []() γ V = Ð------1 Ð LK1 L . (7) exemplified in Fig. 3a, where d22 = 0.5 is not small. 8πL (The lower-layer vortices undergo 66 revolutions during the calculation time; to avoid overloading, we mark Equation (6) can be treated as a dispersion equation in the figure only the original and final collinear posi- relating the length of the triangle lateral side to the tions of the vortices.) With a decrease in the original angle ϕ adjacent the base. Equation (6) has a (unique) distance between the lower-layer vortices, the pattern solution L(ϕ) only in the interval |ϕ| < π/3; i.e., the tri- becomes less instructive. For this reason, the results of angle cannot be equilateral (more exactly, the limiting corresponding calculations are not presented. value |ϕ| = π/3 is an asymptotic one as L ∞ and |V| 0).1 The velocity of the triangular structure We now consider a point e2 belonging to the bound- coincides with that of a certain hypothetical pair con- ary of the physical region (see Fig. 1b). As is known [1], sisting of vortices located in different layers, having the collinear three-vortex configurations corresponding intensities –2κ and 2κ, and spaced by the distance to such singular points should revolve around the vor- equal to the height of the corresponding isosceles trian- ticity center. The presence of these elliptic points under gle. Figure 4a shows the dispersion curve L(ϕ) and V(ϕ) the condition of the zero total intensity of the system is ϕ (7). For = 0, we have V = 0, and L takes the value L0 = a remarkable (and surprising) property of the two-layer 0.8602 at which the degenerate (to the segment) sym- configuration. Since, in this case, the vorticity center metric three-vortex configuration is stationary. The moves to an infinitely far point, the collinear configura- extremal values of V are attained for |ϕ| = 0.82π/3 and tion of the three vortices, as a pair of vortices, should L = 1.874. move uniformly and rectilinearly in the direction nor- 5.2. Collinear configuration. We now denote as A mal to the straight line in which vortices lie (a corre- 1 2 sponding example is shown in Fig. 3b). This configura- and B, respectively, the quantities X2 and X2 that are tion, naturally referred to as a triton, can be the simplest proportional to the original distances between the example of the vortex structures referred to as a modon 1 with a raider [9]. upper-layer vortex (for X1 = 0) and its partners from the lower layer. From equations of motion (1), (2), we A hyperbolic point h apparently corresponds to the obtain the following conditions for the uniform motion unstable solution associated with the translationally of the entire configuration as a rigid body along the moving configuration in the form of an isosceles trian- y axis: gle (t2 = t3). This configuration is illustrated in Fig. 3c exhibiting trajectories of the vortex-structure motion A2 ++AB B2 ------= K ()A ++K ()B K ()AB+ . (8) for which the representation point in the phase plane is AB() A+ B 1 1 1 originally located at the boundary of the physical region near the separatrix. The markers (positions of This equality can also be treated as a dispersion equa- vortices) and segments connecting them in this figure tion relating the geometric parameters of the solutions indicate not only collinear but also other synchronous in the form of the translational collinear configurations. intermediate vortex configurations for time intervals The translational velocity is expressed as during which the representation point resides in the vicinity of the intersection point h of the separatrices. κγ 1 1 V = ------Ð K ()A Ð --- +.K ()B (9) These unstable configurations obviously cannot be 4π A 1 B 1 long-lived. As is seen in this figure, they periodically rearrange with alternation of the mutual positions of the 1 In the homogeneous fluid [10], a similar stationary state is the lower-layer vortices after passing through collinear configuration in the form of an equilateral triangle with an arbi- states. trary side length.

DOKLADY PHYSICS Vol. 47 No. 3 2002 NEW STATIONARY SOLUTIONS TO THE PROBLEM OF THREE VORTICES 237

5 (a) when the total intensity of the system is zero and the lower-layer vortices are equivalent to each other. New 4 stationary solutions are obtained, and general condi- 3 tions for their existence are found. 2 L(ϕ) ACKNOWLEDGMENTS 1 V(ϕ) We are grateful to V.M. Gryanik, J.B. Flór, V.N. Zyryanov, G.M. Reznik, and Z.I. Kizner for valu- –π/2 –π/3–1 –π/3 –π/2 able discussions of the results of this study. The work of 5 (b) one of the authors (M.A.S.) was supported by the Rus- sian Foundation for Basic Research (project no. 01-05-64646) and by the Centre National de la 4 B(A) Recherche Scientifique, Grenoble, France. 3 REFERENCES 2 1. A. V. Borisov and I. S. Mamaev, Poisson’s Structures and Lie Algebras in Hamiltonian Mechanics (Izd. Dom 1 Udmurtskiœ Univ., Izhevsk, 1999). V(A) 2. V. M. Gryanik, Izv. Akad. Nauk SSSR, Fiz. Atm. Okeana 19, 227 (1983). 012345 3. V. M. Gryanik, Izv. Akad. Nauk SSSR, Fiz. Atm. Okeana 24, 1251 (1988). –1 4. V. V. Kozlov, General Theory of Vortices (Izd. Dom Fig. 4. Geometry parameters characterizing stationary solu- Udmurtskiœ Univ., Izhevsk, 1998). tions for the (a) triangular and (b) collinear configurations. 5. V. V. Meleshko and M. Yu. Konstantinov, Dynamics of Vortex Structures (Naukova Dumka, Kiev, 1993). 6. H. Poincaré, Théorie des turbillions (Georges Carré, Figure 4b demonstrates dispersion curve (8) and Paris, 1893; Nauchno-Informatsionnyœ Tsentr: Regu- vortex-structure velocity (9). In the asymptotic limit as lyarnaya i Khaoticheskaya Dinamika, MoscowÐIzhevsk, A 0 or B 0, i.e., when the coordinates of a 2000). lower-layer vortex coincide with those of the upper-layer 7. H. Aref, Phys. Fluids 22, 393 (1979). vortex, we have, respectively, B ∞ and A ∞. 8. X. J. Carton and S. M. Correard, in Proceedings of Inter- This implies that the second vortex in the lower layer is national Union of Theoretical and Applied Mechanics, infinitely far away. The limiting velocity of the config- Symposium on Separated Flows and Jets, Lyngby, Den- uration also tends to zero but takes extremal values for mark, 1997 (Kluwer Acad. Publ., Lyngby, 1997). A/B and B/A = 0.0075. The condition A < B (A > B) leads 9. G. R. Flierl, V. D. Larichev, J. C. McWilliams, and to V < 0 (V > 0). For A = B, the velocity reverses its sign. G. M. Reznik, Dyn. Atm. Oceans 5, 1 (1980). 1 2 10. G. J. F. van Heijst, Meccanica 29, 431 (1994). In this case, the conditions X = X = L are satisfied 2 2 0 11. N. G. Hogg and H. M. Stommel, Proc. R. Soc. London, with the same value of L0, which is observed when the Ser. A 397, 1 (1985). triangular configuration with the symmetric location of 12. B. Legras, R. Santangelo, and R. Benzi, Europhys. Lett. the cyclonic vortices degenerates with respect to the 5, 37 (1988). upper-layer anticyclonic vortex (see Section 5.1). 13. C. E. Leith, Phys. Fluids 27, 1388 (1984). 6. In conclusion, we emphasize that, in the present 14. N. J. Rott, Appl. Math. Phys. (ZAMP) 40, 473 (1989). study, modes intrinsic to the problem of a system of 15. J. L. Synge, Can. J. Math. 1, 257 (1949). three vortices in a two-layer rotating fluid with layers of identical thickness values are classified for the case Translated by R. Tyapaev

DOKLADY PHYSICS Vol. 47 No. 3 2002

MECHANICS

Three-Dimensional Contact Problem for a Layer with Allowance for Friction in an Unknown Contact Area M. I. Chebakov Presented by Academician I.I. Vorovich August 30, 2001

Received September 21, 2001

σ τ τ Taking friction forces into account, we investigate z, xz, and yz are the components of the stress tensor; the interaction between an elliptic-paraboloid die and a µ is the friction coefficient; δ is the die displacement; layer. The investigation concerns the effects of the Cou- f(x, y) is the shape of the die base; and Ω is the contact lomb friction coefﬁcient, die shape, elastic constants, area. and layer thickness on contact stresses, the dependence In addition, we suppose the conditions of statics of the vertical die displacement on a pressing force, the dimensions and shape of the contact area, and the dis- σ ()Ω,, µ placement of layer-surface points outside the contact P ==∫∫ z xy0 d , T P. area. It is discovered that the contact-area shape and Ω displacements of the surface points are qualitatively Representing the components of the displacement different at small and large values of the Poisson ratio. vector in the layer as double Fourier transforms in the We investigate the case of the limit equilibrium. coordinates x and y, we obtain the following integral Quasistatic die motion along the layer surface can be equation for the unknown contact pressure q(x, y) under considered in a moving coordinate system in a similar the die: way. The integral equation derived for the problem is 2πG solved by the method of nonlinear boundary integral ∫∫q()ηξ, kx()Ð η, y Ð ξ dηξd = ------()δ Ð fxy(), , equations [1, 2]. 1 Ð ν Ω (2) Friction forces were taken into account in two- ()Ω, ∈ dimensional contact problems [3, 4] and three-dimen- xy , sional ones for a half-space [5Ð7] and a wedge [8, 9]. where G is the modulus of elasticity in shear and ν is Let a rigid die lying on the layer surface z = h be sub- the Poisson ratio. The kernel k(t, τ) can be represented jected to the normal force P and tangential force T as the difference of two terms directed along the Ox-axis of the Cartesian coordinate µ()ν (), τ ()ε, τ (), τ ε 12Ð system (x, y, z) with the origin on the lower surface of kt ==k1 t Ð k2 t , ------. (3) the layer. Under the assumptions that the friction forces 22Ð ν under the die are parallel to the force T, the layer surface Upon simplifying, these terms take the form z = 0 is rigidly fixed, and the die is in the limit-equilibrium state and does not rotate, we arrive at the boundary ∞ , τ 1 ()γ ()γγ 2 τ2 value problem presented by Lamé equations and the k1(t ) = ------+ ∫ L1()1h Ð J0 t + d , (4) boundary conditions 2 τ2 t + 0 δ (), τ µσ τ w ==Ð fxy, xz z, yz = 0, (), τ 1 k2 t = ------zh= , ()Ωxy, ∈ , t2 + τ2 (1) (5) σ τ τ ()Ω, ∉ ∞ z ===xz yz 0, zh =, xy , × ()()γ ()γγ 2 τ2 u ==ν w =0, z =0. 1 +,∫ L2 h Ð 1 J1 t + d Here, u, ν, and w are the components of the displace- 0 ment vector along the x-, y-, and z-axes, respectively; κ () 2 sinh2u Ð 4u L1 u = ------, 2κcosh2u +++ 4u2 1 κ2

Institute of Mechanics and Applied Mathematics, 2κ()cosh2u Ð 1 Ð 4u2()12Ð ν Ð1 Rostov State University, pr. Stachki 200/1, Rostov-on-Don, () L2 u = ------2 2 -, 344104 Russia 2κcosh2u +++ 4u 1 κ

1028-3358/02/4703-0238 $22.00 © 2002 MAIK “Nauka/Interperiodica” THREE-DIMENSIONAL CONTACT PROBLEM FOR A LAYER 239 γ where Jn(x) (n = 0, 1) are the Bessel functions, = (a) y z α2 + β2 , and κ = 3 – 4ν. 2a We consider an elliptic-paraboloid die elongated along the y-axis. Then the function f(x, y) on the right- x hand side of Eq. (2) takes the form 0 2 2 –0.5 (), x y ≥ fxy = ------+ ------, R2 R1, 2R1 2R2 –1.0 –2a 2a where R1 and R2 are the radii of die curvature in the planes y = 0 and x = 0, respectively. (b) z y It should be noted that integral equation (2) involv- 2a τ ing only the kernel k1(t, ) corresponds to the contact problem of the frictionless indentation of a die into a x layer [10]. 0 Integral equation (2) with kernels (4) and (5) needs to be complemented not only by the conditions of statics but also by a condition for finding the contact area. –0.5 This condition will be formulated and realized when solving the equation. –1.0 When the layer thickness is h ∞, integral terms –2a 2a in kernels (4) and (5) vanish and Eq. (2) coincides with Fig. 1. The deformed surface z = h in the vicinity of the die the integral equation of the similar half-space problem at |x| ≤ 2a, 0 ≤ y ≤ 2a for (a) ν = 0.1 and (b) 0.4. considered in [5]. Integral equations (2) are solved here by the method According to Table 1, the die displacement δ calcu- of nonlinear boundary conditions [1, 2]. This method lated at the constant force P decreases as the Poisson enables us to simultaneously find the distribution func- ν tion of contact stresses, the contact area, and the dis- ratio increases or the layer thickness h decreases. placements of the layer-surface points outside the die in a certain region containing the contact area. Table 1 Omitting the details of the method, we note that it was previously applied to the problems where the die h displacement was given, whereas other quantities G × 10Ð10 ν including forces on the die were sought [1, 2, 8, 9, 11]. 0.5 0.2 0.1 0.05 0.02 Here, we use a more natural modification of the method in which forces applied to the die are given, whereas die 7.0 0.1 1.29 1.21 1.09 0.907 0.617 displacement is determined by solving the problem. 0.3 1.08 1.01 0.912 0.750 0.504 Certain results of the numerical calculations are pre- 0.4 0.967 0.906 0.811 0.657 0.430 sented below. The accuracy of the results was checked by comparing them for different numbers of discretiza- 1.0 0.1 4.52 4.04 3.36 2.55 1.65 tion nodes of the nonlinear integral equation that was 0.3 3.79 3.37 2.79 2.10 1.34 taken from [1, 2], equivalent to Eq. (2), and involving a condition for determining the contact area. In addition, 0.4 3.39 2.99 2.45 1.80 1.10 the results were compared with the known particular cases for µ = 0 [10, 12]. Our numerical calculations show that the die dis- Table 2 δ placement at a given force P is almost independent of P ×10Ð7 1.0 2.0 3.0 4.0 5.0 the friction coefficient µ but depends strongly on the Poisson ratio ν and other parameters. Table 1 presents δ*(h = 0.1) 0.910 1.37 1.73 2.03 2.31 the die displacement δ* = δ × 10–3 calculated at P = 107, µ δ*(h = 0.02) 0.50 0.708 0.864 1.00 1.12 = 0.9, R1 = R2 = 1.0, and indicated values of the Pois- son ratio ν, shear modulus G, and layer thickness h. We P × 10Ð7 6.0 7.0 8.0 9.0 10.0 note that variations in P and G such that P/G = const leave the results unchanged. Here and below, dimen- δ*(h = 0.1) 2.55 2.78 2.99 3.19 3.37 sional quantities are presented in the International Sys- δ*(h = 0.02) 1.22 1.32 1.41 1.49 1.58 tem of Units.

DOKLADY PHYSICS Vol. 47 No. 3 2002 240 CHEBAKOV

y surface bulges either at x > 0 (small ν) or at x < 0 (a) (ν close to 0.5). This behavior is illustrated in Figs. 1 a and 2. Figure 1 shows the deformed surfaces in the vicinity of the die and under it at P = 107, G = 7 × 1010, µ = 0.9, 0 h = 0.02, and R1 = R2 = 1. These surfaces correspond to 1 (),, 2 wxy0 | | ≤ ≤ ≤ 3 the function w*(x, y) = –------δ at x 2a, 0 y 4 ν 5 2a for (a) = 0.1 (a = 3.51) and (b) 0.4 (a = 2.93). For the same parameter values, Fig. 2 shows (curve 0) x the contact-area boundary and (1–6) contours of the –a0 a x y qxy(), R y function q*,------= ------2 describing the (b) a a π δ a G 2 dimensionless contact stresses q(x, y) at y ≥ 0. The curves numbered by n correspond to the values q* = 0.1n. Maxima of the quantity q*(x', y') in the contact 01 2 * * 3 area are qmax = q*(0.15, 0) = 0.597 and qmax = 4 q*(−0.1, 0) = 0.661, respectively. The point coordi- 5 nates x' are presented here with an absolute error less 6 than or equal to 0.05a. It is noteworthy that the point of the maximum contact stresses is displaced in the direc- x tion of the force T at ν = 0.1 and in the opposite direc- –a0 a tion at ν = 0.4, in contrast to the case of µ = 0. Fig. 2. Contact area and stress contours in it for (a) ν = 0.1 and (b) 0.4. REFERENCES 1. B. A. Galanov, Prikl. Mat. Mekh. 49, 627 (1985). Table 2 presents the δ* values at ν = 0.3, G = 2. B. A. Galanov, Dokl. Akad. Nauk SSSR 296, 812 (1987) × 10 µ [Sov. Phys. Dokl. 32, 857 (1987)]. 7 10 , = 0.9, R1 = R2 = 1.0, and indicated values of µ 3. L. A. Galin, The Contact Problems in the Theory of Elas- P and h. At other values < 1.0 and identical values of ticity and Viscoelasticity (Nauka, Moscow, 1980). the other parameters, the results are the same. 4. V. M. Aleksandrov, Prikl. Mat. Mekh. 34, 246 (1970). We numerically investigated the vertical displace- 5. L. A. Galin and I. G. Goryacheva, Prikl. Mat. Mekh. 46, ments of the points of the layer surface z = h and the 1016 (1982). contact-area shape. These characteristics also depend 6. A. S. Kravchuk, Trenie Iznos 2, 589 (1981). strongly on the layer thickness h, the coefﬁcient of fric- 7. A. A. Spektor, Prikl. Mat. Mekh. 51, 76 (1987). tion µ, and the Poisson ratio ν. The dependence on the 8. D. A. Pozharskiœ, Dokl. Akad. Nauk 372, 333 (2000) last parameter is the strongest. For Poisson ratios close [Dokl. Phys. 45, 236 (2000)]. to zero, layer-surface points in the vicinity of the die at 9. D. A. Pozharskiœ, Prikl. Mat. Mekh. 64, 151 (2000). x > 0 are less displaced in the direction of the force P 10. I. I. Vorovich, V. M. Aleksandrov, and V. A. Babeshko, than those at x < 0. For Poisson ratios close to 0.5, these Nonclassical Mixed Problems in the Theory of Elasticity displacements at x > 0 are larger than those at x < 0. (Nauka, Moscow, 1974). This difference increases as the relative layer thickness 11. I. A. Lubyagin, D. A. Pozharskiœ, and M. I. Chebakov, h* = h/D decreases or the friction coefficient µ Prikl. Mat. Mekh. 56, 286 (1992). increases. In addition, the contact area is displaced in 12. A. I. Lur’e, The Theory of Elasticity (Nauka, Moscow, the direction of the force T at small ν and in the oppo- 1970). site direction at large ν, in contrast to the case of µ = 0. The calculations also show that, at small h*, the layer Translated by Yu. Verevochkin

DOKLADY PHYSICS Vol. 47 No. 3 2002

MECHANICS

New Efficient Methods for Controlling Robots through the Internet I. R. Belousov Presented by Academician D.E. Okhotsimskiœ October 18, 2001

Received October 22, 2001

The remote control of robots in the Internet medium based on transmitting video images has certain disad- is a new promising trend in scientific research, which vantages, such as substantial delays in the feedback has important practical importance. However, the channel and the control medium that is inconvenient progress in this field is restrained by limitations on the for an operator. In addition to the considerable delays data-transmission rate intrinsic to the Internet. The while transmitting video images, their size and quality principal problem is the presence of substantial arbi- can hamper the estimate of robot positions and distances trary time delays in the communication channel. This between objects in the workspace for an operator. fact hampers the realization of Internet control and, in the majority of cases, makes it impossible. To overcome these disadvantages, the author has The author has developed several new original developed new methods for improving the efficiency of methods for the efficient control of robots through the controlling robots through the Internet [2Ð5]. These Internet. The methods proposed are based on employ- methods are based on using virtual three-dimensional ing a virtual control medium involving a three-dimen- models of a robot and its workspace in the online (real- sional model of a robot with its workspace and repre- time) regime. The idea of this approach consists in the senting the robot’s current state. The use of control fact that, instead of cumbersome video images, the models for both a robot and its operation medium pro- operator receives a minimum set of parameters unam- vides a fast response in the system being controlled to biguously defining the state of a robot and its operation the actions of an operator, which minimizes the trans- medium (the set of generalized coordinates of the robot mitted-data flow and promotes efficient operation even and of an object interacting with this robot). Then, the in the case of substantial delays in the communication operation scene is visualized by methods of computer channel. A language and a medium for the remote pro- graphics (Fig. 1). This approach makes it possible not gramming of robots through the Internet are developed. only to minimize the delays in the system response to The efficiency of the methods proposed was corrobo- the control actions (by minimizing the data transmit- rated in numerous experiments with Internet-controlled ted), but also to provide a comfortable controlling manipulation robots and mobile robots in which com- medium for the operator (with the possibility of chang- mon-use communication channels were employed. The ing the direction of view, magnifying details of the methods developed can be applied to a wide class of scene, and using half-transparent images). Application systems for the remote control of robots with delays of the computer simulation of the robot and its work- inherent in communication channels. We also present space for the Internet control provided the possibility of here certain experimental data and discuss domains for an efficient control even for low transmission rates possible application of the results of this study. (0.1Ð0.5 kbyte/s) in the case of employing commonly 1. Significant progress achieved in recent years in used communication channels. The corresponding the field of computer and Internet technologies pro- graphic module is realized in the Java3D language. vided a fast quantitative and qualitative rise in the appli- To minimize the data flow between a server and a cation of the Internet robotics. The applications of net- client, we developed a special exchange algorithm. work robotics, which became quite important in practice, Every message involves an instruction code and a set of are remote education and remote controlled automated its associated parameters (integers). For visualizing the production, in particular, in extremal media [1]. How- robot and the object in their current state, it is necessary ever, currently developed systems with robot control to transmit only 80 bytes. This provides a rate of renewal for the three-dimensional image at the operation scene on the order of 12 times per second for the Keldysh Institute of Applied Mathematics, transmission rate of 1 kbyte/s. At the same time, in the Russian Academy of Sciences, case of using video images, the scene renewal takes Miusskaya pl. 4, Moscow, 125047 Russia place with a frequency lower than 1 frame per 5 s.

242 BELOUSOV

Fig. 1. Virtual medium for the robot control.

For this reason, a unique solution for efficient Inter- two points in the robot workspace and to organize a dis- net robot control through commonly used communica- placement of the grip along the given segment: tion channels is operation with the three-dimensional Rcl > here A computer model of a robot and its workspace. This Rcl > moves 100 200 300 makes it possible to reduce time delays to an acceptable level and to provide a fast response of the system to Rcl > here B operator actions. Rcl > proc line {n} { global A; global B; 2. We have developed an Rcl language (Robot-con- for {set i 0} {$i < $n} {incr i} { trol language) for the remote programming of robots through the Internet. This language involves instruc- gos A; gos B; } tions setting motion and positions of a robot (points in } its workspace) and service commands. Rcl > line 3. Motion-setting instructions enable an operator to The use of the remote-programming medium direct the robot grip to points determined a priori, to enabled us to essentially improve the efficiency of exe- displace the grip at a desired distance from a current cuting repeated actions by an operator. We developed position, to rotate it with respect to a chosen axis, etc. an extension of the Rcl language for programming the motion of mobile robots. Robot-position instructions make it possible to 3. Currently, the methods described have no analogs determine and to change points in the robot workspace. among operating systems. On the basis of these methods, in the Keldysh Institute of Applied Mathematics Service instructions enable a user to control a grip, (KIAM), Russian Academy of Sciences, systems for to calibrate a robot, and also to store on a disk and to controlling the RM-01 manipulation robot, the CRS load programs and data needed in the course of the cur- manipulation robot, and the Nomadic XR 4000 mobile rent control session. robot through the Internet were developed (together An important feature of the Rcl language developed with De Montfort University, England; Nantes Institute for the remote programming of robots is the possibility of Cybernetics [IRCCyN], France; and the Laboratory to program operations in the on-line regime, i.e., in the of Analysis of Systems [LAAS-CNRS], Toulouse, current robot-control session. Moreover, it turns out to France, respectively). be possible to execute both individual commands and The purpose of the experiments with the RM-01 their arbitrary combinations, including those using con- robot (Fig. 2) was gripping a rod on the bifilar suspen- trol constructions in the Rcl language (cycles, condi- sion using a virtual control medium. We performed sev- tions, and procedures). For example, the following eral experimental runs when the robot established in the fragment of the program makes it possible to determine KIAM was controlled through the common-use net-

DOKLADY PHYSICS Vol. 47 No. 3 2002

NEW EFFICIENT METHODS FOR CONTROLLING ROBOTS THROUGH THE INTERNET 243 work from various institutions in Moscow, England, France, and South Korea [2Ð4]. Potentialities of the Internet-control system for the RM-01 robot were shown during the Day of the Faculty of Mechanics and Mathematics in Moscow State Uni- versity (MSU). The robot was controlled by students from the MSU main building (at a distance of approximately 25 km). The remote control of the RM-01 robot through the common-use network was successfully demonstrated at the IEEE International Conference on Robotics and Automation ICRA’2001 (Seoul, South Korea, May, 2001). The robot was controlled from the conference hall at a distance of over 10 000 km. Using the virtual controlling medium, an object was gripped in the real- time scale, although a delay in the transmission of video images reached more than 30 s. Fig. 2. RM-01 manipulation robot. Similar experiments were performed with the CRS manipulation robot that gripped parallelepiped-shaped objects [4]. The remote-control system for the CRS robot was demonstrated at the Exposition of digital technologies in Vandee province (Montegue, France). Access to the robot situated at a distance of 50 km from the exposition hall was realized through a conventional telephone line (34800 kbyte/s) and using a mobile phone (9 600 kbyte/ s). Experiments with the Nomadic robot (Fig. 3) were devoted to developing a platform for mobile telecom- munication conferences [5]. The robot was controlled by choosing target points on two-dimensional or three- dimensional maps of its workspace, while the naviga- tion between the chosen positions was realized locally in the automatic regime. The robot was equipped with a set of sensors, a stereoscopic pair of TV cameras, and devices for vocal communication. 4. We have developed methods for the efficient Internet control of robots. The methods are based on using virtual three-dimensional models of a robot and its workspace in the online regime, as well as a medium for the remote programming of robots. This enabled us to overcome a principal problem of the Internet control, namely, the presence of substantial arbitrary time delays in communication channels, to provide a fast response of the system on controlling actions of an operator, and to provide a convenient and efficient control medium. Our further investigations are focused on developing methods of reconstruction of the three-dimensional model of the robot workspace including processing Fig. 3. Nomadic XR 4000 mobile robot. mobile objects. We intend to develop various technologies for the remote Internet control of robots, which will provide efficient functioning of an operator in the These algorithms are based on both the application of a case of various network-connection rates. In particular, system of technical vision and the prediction of the it is assumed to analyze events and the workspace state object motion on the basis of an adequate dynamic of a robot and to transmit a small data set determining model [6, 7]. this state to an operator. We develop algorithms for gripping a complicated dynamic object (e.g., a rod at Of special interest is the creation of a medium for the bifilar suspension) by an Internet-controlled robot. remote education in the field of robotics and mecha-

DOKLADY PHYSICS Vol. 47 No. 3 2002

244 BELOUSOV tronics on the basis of developing systems for Internet- 2. G. Clapworthy, I. Belousov, A. Savenko, et al., in Con- controlled robots. The principal features of these sys- ﬂuence of Computer Vision and Computer Graphics tems is the possibility to carry out experiments with an (Kluwer Academic Publisher, Amsterdam, 2000), actual robot and actual equipment, which is of special pp. 215Ð228. interest for universities and other educational institutions 3. I. R. Belousov, in Proc. 8th Congress on Theoretical and having no such device and equipment. The methods Applied Mechanics, Perm’, Russia, 2001 (Perm’, 2001), developed make it possible to realize the joint use of p. 91. expensive robotics equipment through the Internet. 4. I. R. Belousov, R. Chellali, and G. Clapworthy, in Proc. IEEE Intern. Conf. on Robotics and Automation, Seoul, ACKNOWLEDGMENTS Korea, 2001 (Seoul, 2001), p. 1878. This work was supported by the Federal Purposeful 5. R. Alami, I. R. Belousov, S. Fleury, et al., Proc. IEEE Program “Integration of Higher Education and Funda- Intern. Conf. on Intelligent Robots and Systems, Taka- mental Science”, the Complex Program of Scientiﬁc matsu, Japan, 2000 (Takamatsu, 2000). Research of the RAN Presidium, and the RANÐCNRS 6. I. R. Belousov, D. E. Okhotsimskiœ, V. V. Sazonov, et al., Agreement. Izv. Akad. Nauk, Mekh. Tverd. Tela, No. 4, 102 (1998). 7. I. R. Belousov, D. E. Okhotsimskiœ, V. V. Sazonov, et al., REFERENCES Izv. Akad. Nauk, Mekh. Tverd. Tela, No. 1, 194 (2001). 1. P. Backes, K. Tso, J. Norris, et al., in Proc. IEEE Intern. Conf. on Robotics and Automation, San Francisco, USA, 2000 (San Francisko, 2000), p. 2025. Translated by V. Bukhanov

DOKLADY PHYSICS Vol. 47 No. 3 2002

MECHANICS

Self-Similar Solutions to Problems on Compaction and Water Recovery of Clays while Extracting Water from Strata A. G. Egorov and A. V. Kosterin Presented by Academician V.P. Myasnikov August 25, 2001

Received October 26, 2001

INTRODUCTION 1. STATEMENT OF THE PROBLEM Infiltration of liquids in a stratum system and the We consider a porous stratum lying on an imperme- deformation of surrounding rocks are known to be able rigid base and separated from overlying rocks by a interrelated processes (see, i.e., [1]). In certain cases, soft clay layer. such an interrelation is manifested especially clearly Using the conventional infiltration equation for liq- and, as a rule, is caused by the presence of insufficiently uid in a layer and considering, as in [5, 6], the superposed compacted clay layers contacting a header in use. rocks as an impermeable elastic plate, we arrive at While boring and exploiting a stratum, water flows in ∂ p q from clay layers compressed by overlying rocks. The mβ------Ð k∆p = ---, (1) softer the clay layer, the greater its strain and, therefore, dt h the larger the ground shrinkage and the water yield from the stratum to the header. These effects were D∆2w = Γ. (2) repeatedly observed in the practice of oil production β and hydrogeology. Reiterated many-meter shrinkages Here, m, , k, and h are the porosity, compressibility, fil- were observed in the Wilmington [2] and Eckofisk [3] tration coefficient, and thickness of the stratum, respec- oil fields, in the process of the underground-water tively; q is the overflow of the liquid from the clay layer into the stratum; p is the pressure variation with respect extraction in Mexico [2], etc. Balance calculations per- to the initial pressure in the stratum; D is the rigidity formed for the Belozerskiœ water-bearing complex [4] modulus for the plate of overlying rocks; Γ is the nor- can serve as one more example. These calculations mal load (per unit area) applied to the lower plate-base showed that the water inflow into a water-bearing stra- surface, i.e., to the boundary with the clay layer; and w tum, which had been caused by compressing slightly is the plate’s bending equal to the shrinkage of the clay permeable rocks surrounding the stratum, provides the layer. It is necessary to supplement Eqs. (1) and (2) by predominant contribution to the water discharge rate. relations describing macroscopic rheology of the clay layer. The corresponding conditions should relate the Theoretical description of the effects indicated Γ above requires setting and solving combined problems quantities q and entering into the right-hand sides of of rock geomechanics and water infiltration in deform- Eqs. (1) and (2) to pressure p at the base surface of the clay layer and to the layer’s shrinkage w. These rela- able strata. Some of the results of studies of this kind tions are found by solving the problem on the unidi- were described in [5Ð7]. However, the specific nature of mensional consolidation of a clay layer. One of them, the object under consideration, which is presented by a soft clay layer contacting a header, does not allow the qw= ú , (3) results of these papers to be used immediately. In this paper, we propose a new formulation of the problem, is a consequence of the assumption conventional for the which takes into account the key role of the clay layer theory of infiltration consolidation [8]. According to in the process under study. Additionally, we find an ana- this theory, the compressibility of both the liquid and lytical solution to this problem. the skeleton nuclei is lower than that of the skeleton by itself. The second relation significantly depends on the deformation model accepted for describing the behavior of the clay-layer porous matrix. Considering insuf- Research Institute of Mathematics and Mechanics, ficiently compacted clays, we accept the ultimate rheo- Kazan State University, logical scheme in which the porous matrix is assumed ul. Universitetskaya 17, Kazan, 420008 Russia to be force-free deformable only until its strain ⑀ has

246 EGOROV, KOSTERIN ⑀ attained a certain value 0. The model allows for two 2. ASYMPTOTIC METHOD main features of the deformation process in insuffi- FOR SOLVING THE PROBLEM ciently compacted clays: (a) at the initial-compaction ε stage, the strain resistance is primarily associated with The smallness of the quantity implies that studying the asymptotic behavior of solutions to Eqs. (5) and (6) infiltrating a liquid rather than with stresses in the ε matrix; and (b) the secondary compaction is small com- as 0 is the main task to be considered. Indeed, pared to the initial one. this is the asymptotic solution that describes the process for long intervals (months or years) and, because of Within the framework of this scheme, solving the this, is of great practical interest. On the other hand, the unidimensional problem on the consolidation of a clay method proposed is applicable only in this asymptotic layer is not difficult and yields the following macro- region. As we show below, for time intervals on the scopic relation: order of or shorter than a day (i.e., outside the asymp- ⑀ 0 totic region), the size of the disturbed-pressure region wú = K---- ()Γ Ð p , ww ≤ = ⑀ H. (4) w 0 0 in a stratum is smaller than its depth of occurrence. Because of this, the treatment of superposed rocks as an Here, K and H are the filtration coefficient and the elastic plate becomes inadequate. thickness of the clay layer, respectively. The physical It is natural to take the solution to Eqs. (5) and (6) in meaning of relation (4) is evident: the filtration flow from ε the layer into the stratum [the left-hand side of (4)] is the case of = 0 as a principal term of the corresponding asymptotic solution. From the physical standpoint, w the condition ε = 0 implies the following assumptions: determined by the current penetration depth ----⑀ and the 0 (i) the compressibility of the stratum is negligible com- pressure drop (Γ – p) in the compression wave. pared to that of the clay layer and (ii) the rock pressure Eliminating Γ and q from (1)Ð(4), we arrive at the is constant [1]. In the case of ε = 0, Eqs. (5) and (6) are system of two nonlinear parabolic equations with reduced to a nonlinear steady-state equation with respect to the pressure p and the shrinkage w. After respect to the auxiliary function introducing the dimensionless coordinates (p0 is a char- t acteristic pressure) ᏼ()xt, = Ð∫ pxt(), dt. (7) w p t x w ------, p ----- , t --- , x ---- - ; 0 w0 p0 t0 x0 With regard to the equality 2 w0 2 khw0 t0 ==------⑀-, x0 ------⑀ , ᏼú (), ᏼ p0K 0 K 0 p ==Ðmin12with w , (8) the system can be written out in the form which follows from (6) and (7), we write out the equa- ∂ ∂ tion mentioned above in the form ε p w ∆ b ------∂ Ð ------∂ - = p, (5) t t min()∆ 1, 2ᏼ Ð0.ᏼ = (9) ∂w 2 pw+ ------= ε∆ w, w ≤ 1. (6) The boundary conditions for ᏼ are related to those for ∂t pressure. The latter are specified by the exploitation The dimensionless parameters ε and bε are determined regime for the stratum and determine the parametric by the relationships dependence of the desired function ᏼ on time. mβhp Dw K 2 In many cases, the solution to Eq. (9), with p and w bε ==------0, ε ------0 ------. w p khH subsequently found from (8), describes both the stra- 0 0 tum pressure and the clay-layer shrinkage with an accu- The typical values of these parameters are racy sufficient for practical applications. As a rule, an 2 2 improvement of the description is needed near the Ð9 m Ð13 m k ∼ 10 ------, K ∼ 10 ------, hH∼∼10 m, boundaries (boreholes, isobars), where the correspond- Pa s Pa s ing shrinkage field, in general, does not satisfy natural boundary conditions. To do this, a boundary layer ∼ ⑀ ∼ Ð1 ∼ 6 ∼ 15 2 m 1, 0 10 , p0 10 Pa, D 10 Pa m . approximation should be constructed and then sewed In this case, with the solution to Eq. (9). t ~ 3 years, x ~ 1 km, ε ~ 10–3, and b ~ 1. In order to illustrate and estimate capabilities of the 0 0 asymptotic method proposed, we now consider a stra- The value of D corresponds to the stratum depth of tum being exploited by a gallery of boreholes that may occurrence and to the Young’s modulus on the order of operate either under constant-pressure conditions or 100 m and of 1010 Pa, respectively. constant-yield conditions.

DOKLADY PHYSICS Vol. 47 No. 3 2002 SELF-SIMILAR SOLUTIONS TO PROBLEMS ON COMPACTION AND WATER RECOVERY 247 ᏼ 3. THE PRINCIPAL TERM In the region x*(t) < x < x*(t), the function is given, OF THE ASYMPTOTIC SOLUTION as before, by formula (10), where x*(t) = x*(t) + 6 . In the cases under consideration, the problem becomes unidimensional and Eq. (9) should be solved The most important characteristics of the solutions in the region x > 0 under the boundary condition presented above are listed in the table. ᏼ(∞) = 0. The boundary condition at the borehole gallery has a form of either ᏼ(0) = t (constant pressure) or In order to estimate the range of applicability of the ᏼ'(0) = –t (constant yield). In both the cases, the distur- asymptotic method under consideration, we compared ᏼ ≡ orders of the terms omitted in Eqs. (5), (6) and of the bance propagation velocity is ﬁnite, with (x, t) 0 for basic terms entering into solution (10). Simple calcula- x > x*(t), and there are two time ranges characteristic for tions showed that the ratio of these terms for the two the process. For 0 < t < t*, the shrinkage is everywhere ε –1/2 ε –1 ᏼ regimes are on the order of t and t , respectively. smaller than the ultimate value, and the function is Therefore, the lower bound for the range of applicabil- given by the formula ity of the asymptotic method is on the order of ε2 (minutes) and of ε (days), respectively. It is worth noting that ()4 ≤ ᏼ xxÐ * in such time intervals the shrinkage-crater size is only t t , x < x (t): = ------. (10) ε1/2 ε1/3 * * 72 on the order of x0 (about 30 m) and of x0 (about 100 m), respectively (see table). As a rule, these values In the two regimes under consideration, respectively, do not exceed the stratum depth of occurrence. 1 2 t = --- , with x (t) = (72t)1/4, and t = --- , with x (t) = * 2 * * 3 * 4. THE BOUNDARY LAYER CORRECTION (18t)1/3. At the second stage of the process, an ultimate- shrinkage region 0 < x < x*(t) appears near the gallery It is evident that at the ﬁrst stage of the process the and then begins growing. For the two regimes, x*(t) = principal term of the asymptotic solution found does 1 2 2 not satisfy the natural symmetry conditions 2t Ð --- Ð --- and x*(t) = t Ð --- , respectively. In the 3 3 3 ᏼ ∂w ∂3w region mentioned above, the function is given by for- x = 0: ------==------0. mulas ∂x ∂x3

xx()Ð x* 2 This discrepancy is explained by the presence of a t ≥ t*, x < x*(t): ᏼ = t + ------– --- x 2 3 boundary layer with the thickness on the order of ε1/4 near the point x = 0. Introducing the boundary layer (constant pressure), coordinate y = xε–1/4, we seek the boundary layer corrections to the principal term (w = 2ᏼ , and p = ()txÐ 2 1 0 0 t ≥ t*, x < x*(t): ᏼ= ------– --- ú 2 6 Ðᏼ ) of the asymptotic solution in the form

ε1/4 ε3/4 (constant yield). w = w0 + w1(t, y) + …, p = p0 + p1(t, y) + … . (11)

Basic characteristics of processes when exploiting strata with the help of a borehole gallery Pressure Yield Shrinkage Maximum Exploitation regime Time interval p(0, t) Ðp'(0, t) crater x*(t) shrinkage w(0, t)

1 8t Ð1/4 Constant pressure t < --- Ð1 ----- ()1/4 2 9 72t 2t

1 1 Ð1/2 1 2 t > ---2Ð1t Ð --- 2t Ð2--- + --- 1 2 3 3 3

Ð1/3 1/3 2 2t 3 2/3 Constant yield t < --- Ð ----- 1 ()18t 1/3 --- t 3 3 2

2 2 t > --- Ðt 1 t + 2 --- 1 3 3

DOKLADY PHYSICS Vol. 47 No. 3 2002 248 EGOROV, KOSTERIN

As a result, we find that the function w1 satisfies the lin- ACKNOWLEDGMENTS ear equation This work was supported by the Russian Foundation for Basic Research, project no. 99-01-00466. ∂ ∂4 ()(), w1 ----- w0 0 t w1 Ð0------= ∂t ∂y4 REFERENCES 1. V. N. Nikolaevskiœ, Mechanics of Saturated and Cracked under the following boundary and initial conditions: Media (Nedra, Moscow, 1984). 2. V. N. Shchelkachev, Foundations and Applications of ∂w ∂w ∂3w Unsteady Filtering Theory, Part 2 (Neft’ i Gaz, Moscow, ------1 =,------0 ------1 = 0, 1995). ∂y ∂x ∂ 3 y = 0 x = 0 y y = 0 3. V. Mori, in Mecanique des roches appliquee aux prob- w = 0, w = 0. lemes d’exploration et de production petroliers (Bous- 1 t = 0 1 y = ∞ sens, 1993; Mir, Moscow, 1994). 4. V. A. Mironenko, in Proceedings of International Sym- Thus, the problem formulated above has the self-sim- posium on Engineering and Geological Properties of 3/8 –1/8 5/12 –1/12 ilar solutions w1 = t W(yt ) and w1 = t W(yt ) in Clay Rocks and Processes in Them, Moscow, Russia, the two regimes, respectively. The solution to the corre- 1971 (Mosk. Gos. Univ., Moscow, 1972), p. 97. sponding ordinary differential equations with respect to 5. V. N. Nikolaevskiœ, Prikl. Mat. Teor. Fiz., No. 4, 35 W can be found in an explicit form. We here write out (1968). only corrections to the maximum shrinkage w(0, t). For 6. E. F. Afanas’ev and V. N. Nikolaevskiœ, Prikl. Mat. Teor. Fiz., No. 5, 113 (1969). − ε1/4 3/8 the two regimes, they are equal to 0.5204 t and 7. V. M. Entov and T. A. Malakhova, Izv. Akad. Nauk 1/4 5/12 −0.4786ε t , respectively. The corrections to pres- SSSR, Mekh. Tverd. Tela, No. 61, 53 (1974). sure and the yields are significantly smaller than those 8. A. G. Egorov, A. V. Kosterin, and É. V. Skvortsov, Con- presented above. According to (11), they are on the solidation and Acoustic Waves in Saturated Porous order of ε3/4 and ε1/2, respectively. Therefore, for typical Media (Izd. Kazan. Univ., Kazan, 1990). values of the problem parameters, these corrections can be ignored in contrast to the correction to the shrinkage. Translated by V. Chechin

DOKLADY PHYSICS Vol. 47 No. 3 2002

MECHANICS

Structure of Isobaric Flows in a Gas and in an Ideal Incompressible Fluid V. E. Shemarulin Presented by Academician Yu.A. Trutnev April 9, 2001

Received June 4, 2001

Gas or fluid flows in which pressure p is identically u = u(x, t) [1]: constant (p ≡ const) are referred to as isobaric (inertial) ()⋅ ∇ flows [1]. ut +0,u u = (1) In this paper, we study the structural characteristics divu = 0. of isobaric flows in a gas and an ideal incompressible v fluid (below, for brevity, isobaric flows). Equations Here, u = (u, , w), x = (x, y, z) [for two-dimensional flows, we have u = (u, v) and x = (x, y)]. Let x = (ξ, η, describing isobaric flows are completely integrable. For ζ the first time, formulas for the general solution to these ) be the Lagrangian coordinates of gas (fluid) parti- equations were derived in [2, 3] (see also [4], where a cles: x = x(0). From the first equation in (1), it follows more general set of equations was integrated). Chrono- that u = u(x), and the condition of divergence-free flow logically, the first but less successful attempt to study implies that the function u(x) satisfies a set of three the integrability of equations for isobaric flows was pre- equations obtained by putting to zero the invariants of ∂u sented in [5]. The general solution found in [2, 3] was the Jacobian matrix J = ------[1Ð3]. In particular, we have written in an implicit form as a set of functional equa- ∂x tions for the components of the velocity vector, which involve a certain number of arbitrary functions. In this uξ uη uζ connection, the problem arises to present a simpler and more constructive description of the entire variety of detJ ==v ξ v η v ζ 0. (2) flows being determined by implicit formulas concern- wξ wη wζ ing the general solution. This problem is partially solved in the present paper. Namely, we propose here We restrict our analysis to the consideration of the an explicit geometrical description of the structure of local structure of isobaric flows, assuming that every- three-dimensional steady-state and two-dimensional where in the domain under study the flow is smooth and unsteady-state flows. The case of two-dimensional the rank rkJ of the Jacobian matrix has a constant value. steady-state flows is rather trivial and is not considered Based on equation (2), we can classify the isobaric here. Indeed, it is easy to show that all such isobaric flows according to their rank: flows (even in the assumption that they are determined only locally) are of a shear type in any connected (i) rkJ ≡ 0. In this case, u = const and the flow is uni- domain. This implies that the flow paths are parallel. form. The problem concerning the explicit description of (ii) rkJ ≡ 1. In this case, two components of the three-dimensional unsteady-state flows is still open. velocity vector are functions of the third component. The flow is referred to as a simple wave [2] or a flow of rank 1. 1. LOCAL CLASSIFICATION OF ISOBARIC FLOWS ACCORDING (iii) rkJ ≡ 2. For such a flow, one component of the TO THEIR RANK [2, 3] velocity vector is a function of two other components. The flow is referred to as a double wave [2] or as a flow In Euler variables, the isobaric flows are described of rank 2. by the following set of equations for the velocity field 2. BASIC CLASSES OF THREE-DIMENSIONAL ISOBARIC STEADY-STATE FLOWS All-Russia Research Institute of Experimental Physics, pr. Mira 17, Sarov, Nizhni Novgorod oblast, In our terminology, basic classes are three classes 607190 Russia composed of the following isobaric flows:

250 SHEMARULIN

(I) shear flows, i.e., flows occurring in parallel application point of the vector u and the straight line lt. planes and along parallel straight lines in each plane; Imposing the requirement that u smoothly depends on t (II) conic (conic-type) flows, i.e., flows occurring and d, we obtain a smooth vector field u = u(x, y, z) along families of half-planes tangential to arbitrary π+ γ defined in domain G = ∪ t . The convexity of curve convex conic surfaces, and, in each half-plane, moving t along straight lines parallel to the generatrix of the ensures the absence of intersections of the half-planes conic surface belonging to this half-plane; from the chosen family and, hence, the uniqueness of (III) tangential (tangential-type) flows, i.e., flows the field u in G. For the field u constructed in such a occurring along families of half-planes tangential to the manner, Eq. (4) is satisfied automatically and the valid- so-called tangential surfaces (the surfaces formed by ity of equation (5) is verified by direct calculations. tangents to arbitrary three-dimensional curves). These This implies that such an arbitrary field u is the velocity surfaces obey certain conditions of convexity (ensuring field of the eventual steady-state isobaric flow. We refer | the absence of intersections between the tangential to each pair (u G', G'), where G' is a subdomain of G, as half-planes). In each half-plane, the flows move along a conic-type flow. In the most general situation for straight lines parallel to the generatrix of the tangential domain G', we can admit flows with discontinuities surface belonging to this half-plane. along the lined surfaces consisting of paths (stream- In the general case, for all aforementioned flows, lines), as well as to start and terminate the paths, gas (fluid) flows with its own velocity along each thereby allowing the existence of surface sources and straight line (streamline). sinks. The shear flows are well known [6]. In the steady- In a similar manner, we can construct isobaric flows state case, they are specified by the following explicit of the tangential type determined by the tangential sur- γ formulas in the coordinate system with the Oz-axis face T with the directrix three-dimensional curve . In orthogonal to the flow planes this case, generatrices lt of the surface T are tangents to γ π+ πÐ u = Ðβϕ() αx + βy, z , , whereas the tangent half-planes t and t of the sur- αϕ() α β , face T coincide with the half-planes into which lt v = x + y z , π γ (3) divides the osculating plane t of the curve . Half- w = 0, π+ πÐ planes t and t differ from each other, because one 2 2 αα==(),z ββ(),z α ()z + β ()z ≡ 1. of them contains vector n(t) of the main normal to the Here, α and ϕ can be arbitrary functions. In the case of directrix curve γ and the other half-plane contains the vector fields u of form (3), equations (1) for the steady- opposite vector Ðn(t). Similar to the case of conic flows, state flows, t is the parameter of motion along the curve γ. ()u ⋅ ∇ u = 0, (4) divu = 0 , (5) 3. LOCAL STRUCTURE OF THREE-DIMENSIONAL ISOBARIC are satisfied in a trivial way. STEADY-STATE SIMPLE WAVES We now describe in more detail the structure of Without the loss of generality, we can write for the conic-type isobaric flows. Let K be an arbitrary convex simple wave conic surface in space R3(x, y, z) with the vertex O and a directing curve γ, r = r(t) be the vectorial parametric u ==ϕ(),w v fw(). (6) γ dr()t γ π ≠ ∇ ≠ equation for , t(t) = ------be a vector tangent to , t be Here, w 0 and w 0 everywhere within the range of dt the definition of the flow. To analyze the structure of a plane tangent to the surface K at the point r(t), lt = steady-state simple waves, we can use the general π ∩ π t K be the generatrix of the surface K lying in t, and expression implicitly describing the class of three- π+ πÐ dimensional isobaric flows of rank 1 [2, 6]. If relation- t and t be half-planes for which the straight line lt π ships (6) are met, then this expression has the form divides the t plane and which contain vectors t(t) and –t(t), respectively. A conic flow of the general form ξϕ= '()w ζλ+ (),w, η Ð f '()w ζ (7) related to the surface K (and determined by it or, more where ϕ(w), f(w), and λ(α, β) are arbitrary smooth π+ πÐ precisely, by families {t } and {t }) is constructed in functions. The passage from representation (7) to the the following manner. We choose one from two fami- representation in Eulerian variables is performed using lies of tangent half-planes. For definiteness, let it be the equations π+ π+ {t }. In each half-plane t , we define a vector field u x ===ξ + ut, y η + vt, z ζ + wt. (8) ⊂ π+ parallel to the generatrix lt t and depending on a On the other hand, in such an analysis, we can use rela- single parameter d, which is the distance between the tionships (6) directly substituting them into equations (4)

DOKLADY PHYSICS Vol. 47 No. 3 2002 STRUCTURE OF ISOBARIC FLOWS IN A GAS AND IN AN IDEAL INCOMPRESSIBLE FLUID 251 and (5) and then find an equivalent (in the case under in the Eulerian coordinates have the following form study) system of two scalar quasilinear equations [2, 3]: ϕ()w w ++0,fw()w ww = (9) ,, ∇ ≠ x y z Gu()0,1 u2 u3 = G 0, ϕ 3 '()w wx ++0.f '()w wy wz = (10) ∂ ()G ,, ∑ xi Ð tui ------∂ = Hu(),1 u2 u3 These equations describe simple waves in terms of ui the velocity-vector third component. i = 1 3 ∂2 (11) Let ()()G ∑ xi Ð tui x j Ð tuj ∂------∂ π {}(),, ∈ 3 ,, ui u j w = xyz R : wx() yz ==w const ij, = 1 be the level surface of the velocity field u. 3 ∂ ()H Σ ,, Proposition 1. For isobaric three-dimensional Ð2∑ xi Ð tui ------∂ = (),u1 u2 u3 ui π i = 1 steady-state simple waves (6), surfaces w are cylindri- cal (in particular, they can be planar and even degen- where u = (u1, u2, u3), x = (x1, x2, x3), u = u(x, t); G, H, erate to straight lines). Generatrices of these surfaces and Σ are arbitrary smooth functions for which set of are streamlines of the field u, which are determined by equations (11) is solvable with respect to u1, u2, and u3. equations (8) [characteristics of equation (9)]. The Without the loss of generality, we can assume that range of definition for a simple wave can be divided everywhere in the definition range of the solution u into subdomains such that for each of them one of the under study the following conditions are met following properties takes place. (A) Everywhere, characteristics of equations (9) ∂G u ≠ 0, ------≠ 0. and (10) coincide. In this case, these characteristics are 3 ∂u parallel and the simple wave corresponds to the flow 3 along parallel straight lines. For the steady-state solutions u = u(x), set (11) (B) Everywhere, characteristics of equations (9) becomes “split” with respect to t. As a result of such a π and (10) are different; in this case, all surfaces w are splitting, we obtain a system of functional equations planar (and, hence, form a one-parameter family); implicitly describing isobaric three-dimensional π steady-state double waves each surface w is a combination (family) of streamlines passing through various points of a certain char- Gu()0,,,u u = ∇G ≠ 0, acteristic of equation (10). Below, the simple wave is 1 2 3 referred to as the flow in a one-parameter family of π 3 ∂ planes { w}. G ,, ∑ xi------∂ = Hu(),1 u2 u3 Based on Proposition 1 and using the well-known ui classification of one-parameter families of planes [7, 8], i = 1 we can prove the following theorem. 3 ∂ Theorem 1. An arbitrary isobaric three-dimen- G ∑ ui------∂ = 0, ui sional steady-state simple wave is locally a flow of one i = 1 of the following four types: flows along parallel (12) 3 ∂2 3 ∂ straight lines (special shear motion) with different (in G H Σ ,, the general case) velocity values for each streamline, ∑ xi x j∂------∂ Ð 2∑ xi------∂ = (),u1 u2 u3 ui u j ui as well as shear, conic, and tangential flows. For shear ij, = 1 i = 1 π flows, the flow is constant in each plane p of the corre- 3 3 sponding parallel bundle, whereas for conic and tan- ∂2G ∂H ∑ xiu j∂------∂ Ð0,∑ ui------∂ = gential flows the flow is constant in each tangent half- ui u j ui π± ij, = 1 i = 1 plane t corresponding to the conic or tangential sur- π 3 2 face. In the general case, the velocity in each plane p ∂ G and the absolute value of the velocity in each half-plane ∑ uiu j∂------∂ = 0. ui u j π± ij, = 1 t are different. The following set of equations plays an important role in the analysis of the structure characteristic of the 4. LOCAL STRUCTURE double waves OF ISOBARIC THREE-DIMENSIONAL λ λ STEADY-STATE DOUBLE WAVES 2 = f (),1 The formulas implicitly specifying the general solu- λ ()λ ()λ λ , λ tion of rank 2 for equations that describe isobaric flows Ð f '()1 x1 Ð 1 x3 + x2 Ð 2 x3 = h(),1 2

DOKLADY PHYSICS Vol. 47 No. 3 2002 252 SHEMARULIN ∂ 5. LOCAL STRUCTURE λ ()λ 2 ()λ h Ð f ''()1 x1 Ð 1 x3 Ð 2 x1 Ð 1 x3 ∂λ------(13) OF ISOBARIC TWO-DIMENSIONAL 1 UNSTEADY-STATE FLOWS ∂ We now put in correspondence an unsteady-state ()λ h Σ ,, Ð2 x2 Ð 2 x3 ∂λ------= 1()u1 u2 u3 , vector field u = u(x, y, t) in R2(x, y) to the steady-state 2 vector field u1 = u1(x, y, t) = u + k, where k is the unit vector of the time axis Ot in space R3(x, y, t). The field λ u1 λ u2 1 ==-----, 2 -----, u satisfies the set of equations (1) if and only if the field u3 u3 u1 satisfies equations (4) and (5), where Σ ∂ ∂ ∂ where f, h, and 1 can be arbitrary smooth functions for ∇ ==i ++j k , divu u ++v w . ∂x ∂y ∂t x y t which Eq. (13) is solvable with respect to u1, u2, and u3. Σ All such functions f, h, and 1 are referred to as admis- By virtue of this fact, a new theorem immediately sible. follows from Theorem 4. Theorem 2. Arbitrary set (12) as a set of equations Theorem 5. An arbitrary two-dimensional unsteady-state isobaric flow considered as a steady- with respect to variables u , u , u is equivalent to a cer- 1 2 3 state flow in R3(x, y, t) is a combination of domains for tain set of form (13). The entire variety of isobaric shear, conic, and tangential flows of rank 1, for which three-dimensional steady-state double waves is the third component w (along the Ot-axis) of the veloc- described by solutions of rank 2 to sets (13) with vari- v Σ ity vector u1 = (u, , w) is identically equal to unity ous admissible functions f, h, and 1. ≡ (w 1). In an arbitrary given streamline of the field u1, v From Theorem 2, it follows that level surfaces πλ of both components, u and , are uniquely determined ≡ the vector function l = (λ , λ ) play the main role in the from the condition w 1. The case of “flow” in 1 2 R3(x, y, t) along parallel straight lines is excluded, as it local classification of isobaric steady-state double 2 waves. corresponds to the constant flow in R (x, y). Proposition 2. For any isobaric three-dimensional ≠ 6. EXAMPLES steady-state flow of rank 2 in a domain, where u3 0, Theorems 4 and 5 yield a constructive method for the surface πλ is a part of the plane πλ* determined by determining all three-dimensional steady-state and first two equations of set (13). If a certain streamline l two-dimensional unsteady-state isobaric flows. Below, of the velocity field u has a common point with a certain we present two examples of isobaric flows constructed surface πλ , then l ⊂ πλ . All streamlines (even if they according to this method. 0 0 are defined only locally) lying in any fixed surface πλ Example 1. Let K be a conic surface specified by the 0 equation d ≡ x2 + y2 – r2z2 = 0, r > 0, G = {(x, y, z): d > 0}. λ0 λ0 * λ0 γ l0 = (1 , 2 ) are parallel, and the vector l0 = (1 , As the directrix of surface K, we take circle = {(x, y, 1): 2 2 2 0 x + y = r }. If the point P = (x, y, z) ∈ G, then half- λ , 1) plays the role of a directrix. 2 π± γ planes P tangent to K and passing through P contact Theorem 3. If in the set of equations (13), function − ± rxz+ y d ryz± x d f ≡ f(λ ) is linear, then an isobaric three-dimensional at points ξ = r------, η± = r------. Conse- 1 2 2 2 2 steady-state flow determined by this set is of the shear x + y x + y type. In the case of nonlinear function f [it is assumed quently, vector fields u± = (ξ±, η±, 1) are velocity fields λ ≠ that f ''( 1) 0 everywhere within the definition range of for two conic axisymmetric flows of rank 1, which are defined in G and are associated with K. the solution], the one-parameter family of planes πλ* (see formulation of Proposition 2) containing surfaces Example 2. Flows with singularities. Let K be a conic surface specified by the equation d ≡ z2 – 4xy = 0. πλ has an envelope which is either a conic or tangential As the directrix γ of the surface K, we take the ellipse surface. Hence, in this case, the isobaric flow deter- obtained by intersecting K by the plane x + y = 1. If P = mined by the set of equations (13) is a flow of either the (x, y, z) is a point belonging to the outer part of the conic conic or tangential type. surface K (i.e., to the domain d > 0), then coordinates ξ±, η±, ζ± of points contacting the curve γ for the half- The following classification theorem is a conse- ± quence of Theorems 1Ð3. planes π passing through P satisfy the following relationships Theorem 4. An arbitrary isobaric three-dimen- − ± ± zd+ ± ± zd± ± ± sional steady-state flow is a combination of domains for ξ ==ζ ------, η ζ ------, ξ +1.η = shear flows, conic flows, and tangential flows. 4y 4x

DOKLADY PHYSICS Vol. 47 No. 3 2002 STRUCTURE OF ISOBARIC FLOWS IN A GAS AND IN AN IDEAL INCOMPRESSIBLE FLUID 253

Hence, the formulas Note 2. Using the constructions suggested in the present paper, it is possible to construct flows of ideal − ± zd+ ± zd± ± Ð1 incompressible fluid even with nonzero pressure gradi- u ===------, v ------, w d (14) 4yd 4xd ent as a superposition of isobaric flows. By a method similar to that under discussion here, it is also possible specify velocity fields for two conic-type flows of to construct unsteady-state three-dimensional isobaric rank 2 associated with K. Domain G = {(x, y, z): d > 0, flows. In this case, it is necessary to consider single- ≠ xy 0}, which is outer with respect to K, is a common 3 λ ∈ 1 part of the definition range for these flows. We assume parameter families {Rλ : R } of hyperplanes in that R4(x, y, z, t). π {}(),, , < 1 = xyz: y = 0 z 0 , REFERENCES π = {}()xyz,, : y = 0, z > 0 , 2 1. L. V. Ovsyannikov, Lectures on Gas-Dynamics Founda- πd πd tions (Nauka, Moscow, 1981). where 1 and 2 are the half-planes supplementary to 2. L. V. Ovsyannikov, Differ. Uravn. Ikh Primen. 30, 1792 π π 1 and 2, respectively. The first flow [upper signs (1994). πd πd 3. Yu. A. Bondarenko, Voprosy Atomnoœ Nauki i Tekhniki, in (14)] is extended by continuity to 1 and 2 π π Ser. Mat. Simulation of Physical Processes, No. 3, 41 (excluding the Oz-axis), whereas at 1 and 2 it has sec- (1994). ond-order discontinuities. The second flow [lower signs 4. A. P. Chupakhin, Dokl. Akad. Nauk 352, 624 (1997) π π in (14)] is extended by continuity to 1 and 2 (exclud- [Phys. Dokl. 42, 101 (1997)]. ing the Oz-axis), whereas it has the second-order dis- 5. A. S. Zil’bergleœt, Dokl. Akad. Nauk 328, 564 (1993) [Phys. Dokl. 38, 61 (1993)]. continuities at πd and πd . 1 2 6. V. V. Aleksandrov and Yu. D. Shmyglevskiœ, Dokl. Akad. Note 1. It can be shown that the steady-state axi- Nauk SSSR 274, 280 (1984) [Sov. Phys. Dokl. 29, 22 symmetric isobaric flows described in [3, 9, 10] are (1984)]. conic-type flows of rank 2 (in the general case) in 7. A. V. Pogorelov, Differential Geometry (Nauka, Mos- R3(x, y, z), whereas the two-dimensional (planar) cow, 1969). unsteady-state isobaric flow described in [2] is the axi- 8. M. Ya. Vygodskiœ, Differential Geometry (Gostekhteor- symmetric conic-type flow of rank 1 in R3(x, y, t), izdat, Moscow, 1949). which is obtained from the flow corresponding to the 9. Yu. D. Shmyglevskiœ, Zh. Vych. Mat. Mat. Fiz. 30, 1833 velocity field u+ from the Example 1, if we put in it (1990). 10. Yu. D. Shmyglevskiœ, Analytical Study of Gas and Fluid r = 1 and z = t Ð 1. Hence, the cone K1 characteristic of the flow described in [2] is obtained from the cone K of Dynamics (Editorial URSS, Moscow, 1999). Example 1 (at r = 1) by shifting this cone upward by 1 along the z axis. Translated by K. Kugel’

DOKLADY PHYSICS Vol. 47 No. 3 2002

MECHANICS

On the Flutter of a Plate S. D. Algazin* and I. A. Kiœko** Presented by Academician D.M. Klimov November 20, 2001

Received November 22, 2001

kp 1. INTRODUCTION Here, D is the torsional rigidity of the plate, β = ------0 , c The first systematic investigation of the flutter of a 0 plate dates back to Movchan [1] in 1956. He studied the 2 kp0 c0 = ------ρ , k is the polytropic index, and L is the differ- vibrations and stability of a rectangular plate under the 0 condition that the velocity vector of a flow be parallel ential operator known from the theory of plates. The to a plate side. For aerodynamic-interaction pressure, parameter λ is related to the vibration frequency by the the “plunger-theory” formula was used. This special expression ρhω2 + βω + λ = 0, where ρ and h are the formulation was used in almost all the studies pub- density of the plate and its thickness, respectively. lished before the survey [2]. The conventional methods of solving particular problems, for example, the finite- The plate vibrations are stable and unstable when element method or the difference method, being the Reω > 0 and Reω < 0, respectively. The boundary methods with saturation [3] turned out to be inefficient between these regions in the complex plane λ is the sta- [4]. The reliability of the frequently used BubnovÐ bility parabola Galerkin method is not definitively clarified as to the plate configuration and the type of boundary condi- β2Reλ = ρh(Imλ)2. (2.4) tions; this fact was pointed out in [2]. In the generalized formulation [5, 6], this paper discusses the flutter of a If all λ values are inside the parabola, the motion is plate with an arbitrary contour given in a parametric stable; if at least one eigenvalue is outside the parabola, form. For this case, we develop the saturation-free the motion is unstable. It is proven [7] that Reλ > 0 for method previously proposed in [4]. We present the the boundary conditions of fixing and simply support- results of calculations for the rectangular and elliptic ing, all λ are real for v = 0, and certain eigenvalues plates in the case of an arbitrary direction of the flow- become complex with increasing v. Therefore, the velocity vector. problem is to find such a v value at a given θ for which a certain the eigennumber first falls on parabola (2.4). 2. FORMULATION OF THE PROBLEM This flow velocity vcr is taken as the critical velocity of flutter. Let a plate occupy the region K with the piecewise smooth contour ∂K in the plane xy. A gas flow with the To solve problem (2.1)Ð(2.3) under condition (2.4), θ θ | | ρ velocity vector v = {vcos , vsin }, v = v , density 0, we developed a numericalÐanalytical method without and pressure p0 (parameters of a unperturbed state) pass saturation in two modifications. The first modification over one side of this plate. As usual, the plate deflec- [4] is used when the analytical (relatively simple for- tions w(x, y, t) are assumed to be w(x, y, t) = mula-specified) conformal mapping of the region K ϕ(x, y)exp(ωt). In this case, we obtain the eigenvalue onto the unity circle is known. Here, this restriction is problem in the generalized formulation [5, 6]: removed (the second modification): when the contour ∂ D∆2ϕ – β(v, gradϕ) = λϕ, (2.1) K of the region is given by parametrical equations, the conformal mapping is constructed numerically accord- ϕ| ∂K = 0, (2.2) ing to the procedure described in [8]. ϕ | L( ) ∂K = 0. (2.3) The algorithm of solving boundary value problem (2.1)Ð(2.3) is as follows. Upon the double conversion of the Laplace operator, Eq. (2.1) changes to an integral * Institute for Problems in Mechanics, equation and boundary conditions (2.2) and (2.3) are sat- Russian Academy of Sciences, isfied exactly and approximately, respectively. For pr. Vernadskogo 101, Moscow, 117526 Russia numerical solution, we use the interpolation formula [3] ** Moscow State University, providing the method without saturation. The software Vorob’evy gory, Moscow, 119899 Russia package realizing this calculation was reported in [9].

ON THE FLUTTER OF A PLATE 255

3. RESULTS OF PARTICULAR CALCULATIONS Table In the calculations, we used the following parameter Rectangular BubnovÐGa- × 5 ρ 3 µ θ Elliptic plate values: p0 = 1.0133 10 Pa, 0 = 1.2928 kg/m , = plate lerkin method 0.33, k = 1.4, E = 6.867 × 1010 Pa, ρ = 2.7 × 103 kg/m3, and relative plate thickness h = 0.003 (the ratio of plate 0 0.4029 (1) 0.3546 (1) 0.3042 π thickness to a half-length along the x-axis). The other --- 0.4221 (1) 0.3737 (1) 0.3307 parameters were made dimensionless, as in [4]. 8 3.1. Rectangular simply-supported plate occu- π | | ≤ | | ≤ --- 0.4868 (1) 0.4346 (1) Ð pying the region K: { x 1, y b}. Boundary condi- 4 tions (2.3) have the form 5π ------0.5404 (1) 0.4801 (1) 0.4207 ∂2ϕ ∂2ϕ 16 ------= 0, ------= 0. (3.1) ∂x2 ∂y2 3π x = 1 yb= ------0.6012 (1) 0.5235 (1) Ð 8 Let H be the matrix of the discrete Laplace operator 15π with boundary condition (2.2). In this case, the matrix ------0.6124 (2) 0.5275 (2) 0.4022 of the discrete biharmonic operator with boundary con- 32 dition (3.1) is H2. This fact enables us to relatively sim- π --- 0.6108 (2) 0.5257 (2) 0.4121 ply perform the discretization of spectral problem (2.1), 2 (2.2), (3.1). This procedure is described in detail in [10]. The results of calculations are presented in the third column of the table (eigenvalue numbers are indi- π cated in parentheses). θ 15 The presence of maxvcr near the point = ------is the 3.2. Elliptic simply-supported plate. The contour 32 of the region K is given parametrically by the equations so-called effect of stabilization of the plate vibrations x = cost and y = bsint; the ellipse is inscribed into the against the fluctuations of the flow-velocity direction rectangle from the above example. The simply-support- π near the point θ = --- . It should be noted that this effect ing condition requires that the bending moment van- 2 ishes; therefore, we have was previously discovered in the problem of the flutter of an infinite strip. ∂2ϕ ∂2ϕ ∂ϕ L()ϕ ≡ ------+ µ ------+ κ------= 0 ∂K ∂ 2 ∂ 2 ∂n n s ∂K REFERENCES instead of Eq. (2.3). Here, µ is the Poisson ratio and κ 1. A. A. Movchan, Prikl. Mat. Mekh. 20, 231 (1956). is the curvature. The calculations were carried out using 2. Yu. N. Novichkov, Itogi Nauki Tekh., Ser. Mekh. numerical conformal mapping according to the proce- Deform. Tverd. Tela 11, 67 (1978). dure from [8]; the results are presented in the second 3. K. I. Babenko, Principles of Numerical Analysis (Nauka, column of the table. For comparison, the fourth column Moscow, 1986). presents the results calculated for the rectangular plate 4. S. D. Algazin and I. A. Kiœko, Prikl. Mat. Mekh. 60, 171 by the BubnovÐGalerkin method. (1997). 5. A. A. Il’yushin and I. A. Kiœko, Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., No. 4, 40 (1994). 4. DISCUSSION OF THE RESULTS 6. A. A. Il’yushin and I. A. Kiœko, Prikl. Mat. Mekh. 58, θ 167 (1994). The table demonstrates that the dependence vcr( ) is nonmonotonic. First, v increases with θ, and this 7. S. D. Algazin and I. A. Kiœko, Izv. Akad. Nauk, Mekh. cr Tverd. Tela, No. 1, 170 (1999). π 3π increase is more rapid in the region θ ∈ ---, ------. Near 8. É. P. Kazandzhan, Preprint No. 82, IPM AN SSSR (Inst. 4 8 of Applied Mathematics, Academy of Sciences of π USSR, 1979). θ 15 θ the point = ------, vcr( ) has a maximum. Finally, vcr- 9. S. D. Algazin and I. A. Kiœko, Preprint No. 684, IPMekh 32 RAN (Inst. of Applied Mechanics, Russian Academy of π Sciences, 2001). decreases slightly when θ approaches --- . It is character- 2 10. S. D. Algazin, Zh. Vychisl. Mat. Mat. Fiz. 35, 400 istic and very interesting that the plate-motion form is (1995). modified near the point maxvcr so that the second eigenvalue first falls onto the stabilization parabola. Translated by V. Bukhanov

DOKLADY PHYSICS Vol. 47 No. 3 2002

MECHANICS

On Rolling and Sliding of a Cylinder along a Perfectly Plastic Half-Space with Allowance for Contact Friction R. I. Nepershin Presented by Academician A.Yu. Ishlinskiœ November 28, 2001

Received November 29, 2001

Rolling of a rigid cylinder along a viscoelastic half- OA. Stresses and velocities in the plastic domain satisfy space was considered in [1, 2]. When a contact arc hyperbolic equations of the plane plastic flow in the between the cylinder and a plastic domain is small, the sliding lines ξ and η [4]: problem of rolling a smooth cylinder along the rigid- dy dy plastic half-space was solved in [3] by the small param------==tanϕ for ξ and ------Ðcotϕ for η , (1) eter method. In this study, we consider the problem of dx dx rolling and sliding a cylinder along the boundary of a dσ Ð0indϕ == ξ , dσ +0indϕ η , (2) perfectly plastic half-space with allowance for contact friction. The limiting values of forces and a contact arc dVξ Ð0inV ηdϕ = ξ , for steady plastic flow are obtained. (3) dVη +0inV ξdϕ = η , Figure 1a shows a steady plastic domain arising in the process of rolling and sliding a cylinder along the where σ is the mean stress, ϕ is the slope angle of the boundary of an incompressible perfectly plastic half- space [3]. We assume that the cylinder axis is fixed and Q the half-space moves with a unit velocity V = 1. Along the C (a) M boundary OEDB, the velocity is continuous. In the case of F a stationary plastic flow, the boundaries AB and OA are ω streamlines, the tangents to which at the points B and O are directed along the half-space boundary. Therefore, the 2.571 σ 1.508 lower point O lies on the half-space boundary y = 0. – n For a cylinder of a radius R rolling with an angular A α velocity ω without slipping, we have a relationship R c ωR = 1 at its contact point O with the rigid domain. At y β the other points of the contact arc, the plastic material 0 ψ B slides from the point O to the point A and tangential η x stresses of friction appear on the contact surface and η ξ form a positive moment M. This is the case for rolling ξ V = 1 the cylinder with the sliding it forward along the con- D tact arc. If the angular velocity ω is equal to zero or is E so small that the plastic material slides from the point A (b) Α to the point O in the entire contact arc, the contact-fric- Vy tion stresses and the moment M reverse their signs. This β 0 is the case for rolling while sliding at the point O or O–B α sliding the cylinder backwards without rolling in the –1 c Vx contact arc OA. If the cylinder is smooth, the plastic domain is independent of ω and is the same for both Α forward and backward slide. ωR = 1 We measure stresses in the units of double the shear- flow stress, and lengths, in the units of the contact arc Fig. 1. (a) Field of sliding lines and the normal-pressure dis- Moscow State Academy of Instrumentation Engineering tribution at the contact boundary and (b) hodograph of and Informatics, ul. Stromynka 20, velocities in the process of rolling a cylinder with forward α τ Moscow, 107846 Russia sliding for c = 0.5236 and c = 0.2 at the point A.

1028-3358/02/4703-0256 $22.00 © 2002 MAIK “Nauka/Interperiodica” ON ROLLING AND SLIDING OF A CYLINDER 257 tangent to the sliding line ξ and Vξ and Vη are the com- direction of the material sliding along the contact ponents of the velocity vector in the directions ξ and η. boundary OA. In this case, the fields of stresses and For a steady plastic flow, the boundary AB, which is velocities in the plastic region are independent of the free of external stresses and coincides with the direc- rotation of the cylinder and are the same for forward tion of the second principal stress, is a streamline: and backward slides. ψ π The angle of the centered fan of the sliding-line ϕ V y σ 1 tanÐ --- ==------, Ð--- in AB , (4) field at the point A is determined by the expressions 4 V x 2 3π ψ = ------Ð ()α ++βθif ωR > V (11) where Vx and Vy are the velocity-vector components 4 c c related to Vξ and Vη as ϕ ϕ ϕ ϕ for forward sliding or V x = V ξ cos Ð V η sin , V y = V ξ sin + V η cos . (5) π Along the rigid plastic boundary O–B, velocities are ψ = --- + θαÐ ()+ β if ωR < V (12) 4 c c continuous: Vx = –1 and Vy = 0. From Eqs. (5), we find

V ξ ==Ðcosϕ, V η sinϕ in O ÐB. (6) for backward sliding, where Since the contact arc OA is taken as the characteris- V β = Ðarctan ------y (13) tic dimension, the cylinder radius R and the contact V α α x A angle c are related as R c = 1. The surface velocity of the cylinder can vary in the range 0 ≤ ωR ≤ 1. The shear is the slope angle of the tangent to the boundary AB at τ stresses c of contact friction appear at the boundary the point A. OA, and the sliding lines intersect this boundary at the The mean stress at the point O is found from the ﬁrst angle: of Eqs. (2) for ξ of the sliding line O–B and boundary conditions (8) and (9) at α = 0: θ 1 τ ≤≤τ 1 = --- arccos2 c,0 c ---. (7) 2 2 π σ 13 θ ω 0 = Ð---1 + ------+ (14) For forward sliding, R > Vc, where Vc is the veloc- 2 2 ity of the material at the contact boundary and θ is the angle between the sliding line η and the tangent to OA. for forward sliding or ω τ For backward sliding, R < Vc, the direction of c θ ξ 1π changes, and is the angle between the sliding line σ = Ð--- 1 + --- Ð θ (15) and the tangent to OA. Therefore, the angle ϕ specify- 0 22 ing the direction of sliding lines with respect to OA is determined as for backward sliding, where θ is determined by Eq. (7). π Expression (14) shows that, for forward sliding, the ϕαθ= + Ð --- if ω RV > (8) bearing capacity of a rigid wedge at the point O is valid 2 c  π if τ = 0 θ = --- at this point. Therefore, the plastic for forward sliding or as c  4 ϕαθ ω < = Ð if RV c (9) domain shown in Fig. 1a can be formed only if the con- for backward sliding. For a smooth cylinder, τ = 0, it tact-friction stress varies and equals zero at the point O. c Experimental data on rolling corroborate that stress π follows from Eq. (7) that θ = --- and Eqs. (8) and (9) vanishes at the point of the outlet of a workpiece from 4 the contact with the roll [5]. In this study, we assume π that τ varies linearly and is maximal at the point A. For ϕ α c yield the same = – --- value at the boundary OA. backward sliding, the bearing capacity of a rigid wedge 4 τ at the point O is valid for each value of c that is Since the velocity normal to the cylinder is equal to assumed to be constant in OA. zero, kinematic boundary conditions at OA have the form The mean value of σ decreases in magnitude along OA and, at the point A, takes the following value θ θ V ξ ==V η tan , V ξ V η cot (10) depending on the angle ψ, for forward and backward sliding cases, respectively. 1 For a smooth cylinder, Eqs. (10) both yield the equality σ = Ð---()12+ ψ . (16) A 2 Vξ = Vη. Thus, for a smooth cylinder, the boundary condi- If the distribution of σ along the contact arc OA is tions for stresses and velocities are independent of the known, it is possible to determine the normal pressure

DOKLADY PHYSICS Vol. 47 No. 3 2002 258 NEPERSHIN on the cylinder field dependent on the distribution of σ over OA, Eq. (4) is the determining equation for the unknown distribu- σ σ 1 θ tion of σ over OA. Ð n = ÐÐ --- sin2 , (17) 2 The fields of sliding lines and velocities for a given σ and the forces and moment with allowance for the rela- distribution of over the boundary OA were calculated tionship Rα = 1 are found in the form numerically by using a finite-difference approximation c of differential equations (1)Ð(3). The boundary AB free α c of external stresses is obtained by solving the sequence 1 []α()ασ ± τ α of the elementary inverse Cauchy problems starting Q = α----- ∫ Ð n cos c sin d , (18) c worth the an initial point A. The coordinates of these 0 boundary points are found from the differential equa- α c dy  π tion of the AB contour ------= tan ϕ- – --- and differen- 1 []α()ασ +− τ α dx  4 F = -----α ∫ Ð n sin c cos d , (19) c tial equation (1) for the ξ-line. The velocities at the 0 boundary OA are obtained by solving the elementary α c mixed problems for differential equation (3) along the ± 1 τ α η-line with allowance for boundary conditions (10). M = α----- ∫ cd , (20) c Since differential equations (2) and (3) are linear, the 0 finite-difference equations are also linear. For this rea- where the upper and lower signs correspond to the for- son, the calculation of a detailed network of sliding ward and backward sliding cases, respectively, and lines and the field of velocities for a σ distribution given at 20 nodes at the boundary OA takes a fraction of a sec- σ σ 1 τ ond on a Pentium-133 computer. Ð n ===ÐÐ --- and M c 0 2 Let s be the vector of unknown σ values at the N for a smooth cylinder. nodes at the boundary OA and f be the vector of the dif- τ ferences between the slope angles of tangents to the For a rough cylinder and linearly varying c, boundary AB and velocity vector at the N nodes of this 1 Eq. (20) yields M = --- (τ ) and Ðτ for forward and boundary. These differences are the errors of stationary 2 c A c condition (4) for given s. The procedure of computing backward sliding cases, respectively. the fields of sliding lines and velocities provides a con- The above equations show that the problem of roll- tinuous fÐs dependence, and condition (4) takes the ing and sliding a cylinder calls for the joint consider- form of the N-dimensional nonlinear vector equation ation of stress and velocity fields. The solution to the f()s = 0. (21) problem can be found by the following way. Equation (21) is solved by the Broyden method [6], in Taking into account known values (14) and (15) at σ which the iterative process does not require the evalua- the point O, we set a continuous distribution of at the tion of derivatives. Taking the initial approximation for boundary OA and an initial approximation for the angle β 0 β. Then, σ and the boundary conditions (8) and (9) the angle , we specify the initial approximation s by define the Cauchy data for Eqs. (1) and (2) and the field a linear distribution over OA from the value specified of sliding lines in the domain OAE. In the domain AED, by Eqs. (14) and (15) at the point O to value (16) at the ∂f the field of sliding lines is found by solving the Goursat point A. The functional matrix ------i at the initial point problem with given σ and ϕ in AE and at the singular ∂s ψ j point A with the angle specified by Eqs. (11) and s0 is found via solving N problems for the variations of (12). In the domain ABD, we solve the inverse Cauchy s0 by the finite-difference method. Equation (21) is problem with given σ and ϕ in AD and conditions | | ≤ solved in several iterations with an accuracy of fi max 1 dy π –4 σ ==Ð---, ------tan ϕ Ð --- 10 , i = 1, 2, …, N. The condition y = 0 at the point B 2 dx 4 is satisfied with an accuracy of 10Ð6. at the unknown boundary AB. As a result, we find the Figure 1 shows (a) the field of sliding lines with dis- free boundary AB and the rigid plastic boundary O–B. tribution of the contact pressure and (b) the hodograph Next, we find the field of velocities in the plastic of velocities for rolling a rough cylinder without slip- ω α domain by solving the mixed boundary value problem page at the point O ( R = 1) for the contact angle c = for Eqs. (3) with boundary conditions (6) and (10). If 0.5236 for forward sliding with the maximum value τ β steady-flow condition (4) is satisfied at the boundary c = 0.2 at the point A. For this case, we obtain = AB, the distribution of σ over OA is the solution to the 0.703, ψ = 0.55, Q = 2.03, F = 0.394, and M = 0.1. With α problem. Since the sequence of solving the boundary increasing the contact angle c, the centered-fan angle value problems for Eqs. (1)–(3) determines the field of ψ vanishes, the domain AED in the physical plane con- sliding lines with the boundary AB and the velocity tracts to a line, the velocity at the singular point A

DOKLADY PHYSICS Vol. 47 No. 3 2002 ON ROLLING AND SLIDING OF A CYLINDER 259

σ becomes single-valued, and the corresponding arc A–A 2.323 – n 1.759 at the hodograph of velocities tends to the fixed point O. (a) Thus, along the free boundary AB, the velocity of a material particle decreases from 1 at the point B to 0 at y A α ψ c the point A when 0 Then, the velocity increases 0 β from 0 to 1 when the particle moves along the cylinder B τ x boundary from the point A to the point O. For c = 0.2, α* we find the ultimate contact angle c = 0.65 at which V = 1 a steady plastic flow with the forces Q = 1.828 and F = D 0.427 is possible. An increase in friction for forward E sliding leads to increasing both the angle ψ and the ulti- (b) α ψ A Vy mate contact angle c* as 0. F The ratio ---- can be treated as the coefficient of roll- β 0 O–B α Q c –1 Vx ing friction caused by the asymmetry of the plastic ωR = 0.59 domain about the y-axis coinciding with the axis of the A τ cylinder. In the case of forward sliding for given c val- Fig. 2. The same as in Fig. 1, but for the process of rolling α α τ ues, it is possible to find the contact angles c for which a cylinder with backward sliding for c = 0.2 and c = 0.2. the force F = 0. This is the limiting case of rolling a thick workpiece without penetrating plastic deforma- tions into its depth; i.e., when only the surface layer is and the domain OAE degenerates into a shear line with π plastically deformed. σ the uniform pressure – n = 0.5 + --- at the boundary OA. Figure 2 shows (a) the field of sliding lines with the 4 contact-pressure distribution and (b) the hodograph of This is the case of sliding an absolutely rough plane die velocities for rolling and sliding a cylinder with back- along the boundary of a plastic half-space. α ward sliding at the contact boundary OA for c = 0.2 ≤ ω τ and 0 R < 0.59 and the contact-friction stress c = REFERENCES 0.2. In this case, we have β = 0.364, ψ = 0.801, Q = 2.03, F = 0.39, and M = Ð0.2. An increase in friction for 1. A. Yu. Ishlinskiœ, Prikl. Mat. Mekh. 3, 245 (1939). backward sliding leads to decreasing the angle ψ, a lim- 2. A. Yu. Ishlinskiœ, Applied Problems of Mechanics, Vol. 1: α* Mechanics of Viscoplastic and Elastic Solids (Nauka, iting contact angle c , and normal pressure on the cyl- Moscow, 1986). inder. Moreover, this pressure is distributed more uni- 3. E. A. Marshall, J. Mech. Phys. Solids 16, 243 (1968). formly than pressure for rolling the cylinder with for- 4. D. D. Ivlev, Theory of Ideal Plasticity (Nauka, Moscow, ward sliding. 1966). At ω = 0, we obtain the sliding of a round die ahead 5. A. I. Tselikov, Foundations of Rolling Theory (Metal- of which a steady plastic domain depending on the ver- lurgiya, Moscow, 1965). τ tical force Q and the contact friction c is formed. This 6. J. E. Dennis and R. B. Shnabel, Numerical Methods for problem also describes the process of drawing a thick Unconstrained Optimization and Nonlinear Equations workpiece between circular matrices when only the (Prentice-Hall, Englewood Cliffs, N. J., 1983; Mir, Mos- surface layer undergoes plastic deformation. As cow, 1988). τ α c 0.5 (for backward sliding), c 0 vanishes, the plastic region ABDE degenerates into the point A, Translated by Yu. Vishnyakov

DOKLADY PHYSICS Vol. 47 No. 3 2002