∫ ∫– ∑ ∑ Association in Metallic Melts and Its Relation To

PHYSICS

Association in Metallic Melts and Its Relation to Amorphization: FeÐP Alloys A. I. Zaœtsev, N. E. Shelkova, and A. D. Litvina Presented by Academician O.A. Bannykh November 18, 1999

Received November 18, 1999

As a rule, the melts inclined to amorphization are ent sign: The first is positive, and the second is negative. ∆ ∆ characterized by an intense interparticle interaction. It Thus, decreasing mS and increasing Cp result both in is precisely the melts in which the association of their approaching the vitrification temperature Tg to the components occurs and which is accompanied by the melting temperature Tm and increasing the thermody- formation of molecule-like complexes or clusters. namic stability against crystallization of supercooled However, the relationship of these two phenomena has liquid and, consequently, glass. not been analyzed until now. This work deals with such ∆ ∆ an analysis. Keeping in mind that functions mS and Cp are vital to the vitrification thermodynamics, we will con- In the systems that can easily pass into amorphous sider the effect of association processes in liquid on state, the heat capacity for supercooled liquid, C (liq), p these characteristics. Our analysis is performed within is much higher than that for equilibrium crystals, the framework of the model of ideal associated solu- Cp(cr). Hence, the entropy of supercooled liquid tions for a binary melt AÐB because the essence of our decreases with temperature much more quickly than results is unaffected by either the introduction of an the crystal entropy, and, at a temperature TK, the entro- additional term, allowing for the interaction between pies of these two phases would become equal. In fact, the components of the associated solution, or the the vitrification occurs at the temperature Tg > TK [1]. increase in the number of these components. The Thus, the temperature TK is a lower bound of the vitri- entropy of formation for an associated solution takes fication temperature Tg, and the glass entropy at Tg is the form of two terms with different signs: larger than the crystal entropy. To consider the relative ∆ ()∆() stability of liquid and crystals, we use the Gibbs melt- f Sn=∑AiBjfSAiBj ing energy at arbitrary temperature:  T m T m ()() ∆C ÐRn∑ AiBjlnxAiBj ∆ G = ∆ ST()∆ÐT Ð CdT+ T ------pd T . (1)  (3) m m m ∫ p ∫ T T T  () () () ∆ ++n A1 lnxA1 n B1 lnx(B1). Here, mS is the entropy of melting at the temperature ∆  Tm, and Cp = Cp(liq) Ð Cp(cr). To simplify our analy- ∆ ∆ sis, we assume that Cp is independent of temperature. Here, n(AiBj) and f S(AiBj) are the number of moles This assumption does not change the essence of our and the entropy of formation, respectively, for the asso- conclusions and allows equation (1) to be rewritten in ciated complexes AiBj; x(AiBj), x(A1), and x(B1) are the the form molar fractions of the complexes and two monomer particles, respectively, in the associated solution. The T ∆ ∆ ()∆m first (negative) term in (3) is due to the arising com- m G =m STmÐT +CpT1+ ln------ÐTm.(2) T plexes, while the second (positive) term is related to the mixing of components in the associated solution. By The difference T ln T T T is negative for (1 + ( m/ )) – m lowering the temperature of the melt, the absolute value T < Tm, and the two terms in equation (2) are of differ- of first term increases and second term decreases. As a result, the entropy of formation for the solution decreases. At temperatures close to the liquidus, the Institute of the Physics of Metals and Structure Materials, first contribution dominates. In this case, the more com- Bardin Central Research Institute for the Iron plicated the empirical formula for a complex, the and Steel Industry, Vtoraya Baumanskaya ul. 9/23, greater the decrease in entropy, because the decrease in Moscow, 107005 Russia the number of particles in the solution becomes much

∆ ∆ mS, Cp, J/(mol K) with several association reactions, the extremum corre- 14 sponds to the association composition characterized by the minimum enthalpy of formation. The effect of other association reactions is less significant and increases with temperature. Thus, association processes in a liq- 12 uid result in decreasing the entropy and in increasing ∆ Cp the excessive heat capacity; i.e., they are favorable to the transition of the liquid to a glassy state. 10 We now consider the effect of association processes on the melt viscosity, that is, the most important property determining the kinetics of liquidÐglass transition. 8 To do this, we use the equation relating the viscosity to the configurational entropy Sconf [3]: ∆ mS (η) Be 6 ln = Ae + ------. (5) TSconf Here, A and B are constants. The temperature depen- 4 e e 0.70 0.74 0.78 0.82 0.86 0.90 0.94 dence of the configurational entropy can be presented in the following form: x(Fe) T conf Fig. 1. Entropy of melting and ∆C for the FeÐP melt at C p Sconf(T) = Sconf(T ) + -----p-----dT. (6) 1321 K as versus concentration. g ∫ T Tg more pronounced [2]. As a consequence, the value of Here, Cconf is the configurational heat capacity; and ∆ p mS also decreases. conf S (Tg) is the configurational entropy of the liquid at In the case of a single association reaction, with its the glass transition temperature Tg, which remains fixed enthalpy being independent of temperature, the excess when the liquid transforms into a glassy state. Accord- (with respect to the additive sum of the heat capacities conf ing to [3], S (Tg) can be presented as the difference for pure liquid components) in the heat capacity for the between the entropies of supercooled liquid and crys- associated solution is given by the expression tals at the temperature Tg. Similarly, the configurational [∆ ( )]2 heat capacity is close to a difference between the heat E f H AiB j ∆ Cp = ------. (4) capacities of liquid and crystals, i.e., to Cp.  2 2  2 1 i j The greater the liquid viscosity and the steeper its RT ----(------)- + ----(------)- + ----(------)   x AiB j x A1 x B1  increase in the course of the supercooling of a melt, the more favorable, from the standpoint of kinematics, is In the case of a large number of association reactions, the transition of the liquid into a glassy state [3]. similar expressions take a cumbersome form, but the According to (5) and (6), this transition occurs at high essence of the analysis is not changed. It follows values of the configurational heat capacity and at low from (4) that CE is always positive. In other words, the values of the configurational entropy at temperature Tg. p In essence, these conditions coincide with the require- occurrence of association processes results in increas- ment that ∆C should be large and that ∆S should be ing both the heat capacity of liquid and, consequently, p m the difference between the heat capacities of liquid and small. Thus, from the viewpoint of both kinetics and crystals. This increase is directly proportional to the thermodynamics, the association processes in a liquid enthalpy squared of formation for the associative com- are favorable to its amorphization. plex and inversely proportional to the absolute temper- We use the formulas obtained to quantitatively ature squared. When analyzing the temperature depen- describe the amorphization of the ironÐphosphorus E melt. This melt is an ideal associated solution in which dence of Cp , we have to take into account that the mole the aggregates such as FeP, Fe2P, and Fe3P are formed. fractions of the complexes and monomer particles also The thermodynamic functions of their formation are as depend on temperature. Our numerical analysis of ∆ ∆ follows (with f H in J/mol and f S in J/(mol K)) [4]: equation (4) indicated that, for arbitrary thermody- ∆ ∆ ∆ f H(FeP) = Ð81025, fS(FeP) = Ð1.4, f H(Fe2P) = namic parameters of an association reaction, the con- − ∆ ∆ 14599, f S(Fe2P) = Ð15.6, f H(Fe3P) = Ð207390, E ∆ centration dependence of Cp exhibits a peak when the and f S(Fe3P) = Ð53.5. These parameters and thermo- macroscopic composition of the solution corresponds dynamic functions for crystal phases allow us to evalu- to the composition of an associative complex. In melts ate the concentration dependences of both the entropy

DOKLADY PHYSICS Vol. 45 No. 8 2000 ASSOCIATION IN METALLIC MELTS AND ITS RELATION TO AMORPHIZATION 361

∆ ∆ ∆ ∆ of melting and Cp at the eutectic temperature of 1321 K trG, trH, trST, kJ/mol (Fig. 1) [4]. If we take into account that the entropy of melting decreases with temperature and that only the eutectic alloy with x(Fe) = 0.826 melts at the tempera- 8 ture of 1321 K, the results presented in Fig. 1 allow us to find the concentration range, 0.77 < x(Fe) < 0.86, in which the melt is most inclined to amorphization. This conclusion is very consistent with the available data 6 [5Ð7]. It is precisely the concentration range in which ∆ the biggest complexes Fe3P are intensively formed. trH Typical features of the thermodynamic functions for 4 the transition of crystal compounds into a supercooled ∆ liquid or glassy state are shown in Fig. 2 for an eutectic trG alloy. To evaluate these functions, we used the data of ∆ [8, 9] for pure components. It is clear that both the 2 trST enthalpy and entropy of the transition vary irregularly with temperature. This is due to the magnetic contributions to the thermodynamic functions of the solid solutions (with body-centered cubic lattices) of phosphorus 0 400 600 800 1000 1200 1400 in Fe and Fe3P. The separated nonmagnetic components ∆ ∆ of trH and trS monotonically decrease with tempera- T, K ture. According to the data available in literature [5Ð7], the temperatures of vitrification and crystallization for Fig. 2. Temperature dependences of (solid lines) the ther- amorphous ironÐphosphorus alloys almost coincide modynamic functions for the transition of the crystal com- and are close to 650 K. At this temperature, we have pound FeÐP with an eutectic composition into glassy state ∆ ∆ ∆ or supercooled (metastable) melt and (dotÐdash lines) the trS = 4.48 J/(mol K), trH = 6054 J/mol, and trG = differences between the corresponding thermodynamic 3141 J/mol. The crystallization enthalpy for the amor- functions of liquid and of nonmagnetic constituents of phous alloy FeÐP with an eutectic composition was crystals. found by the authors of [5] to be equal to −6.3 kJ/mol. This value nearly coincides with the result of present calculations. ln η [η in pois] To approximate the temperature dependence of vis- 30 cosity for the melts FeÐP, we assumed that the configurational entropy at the vitrification temperature was equal to the difference between the entropy of supercooled liquid (glass) and the nonmagnetic part of x(P) = 0.174 entropy of equilibrium crystalline composition. The 20 configurational heat capacity was evaluated as the difference between the heat capacity of melt and the nonmagnetic part of the heat capacity of crystals. These assumptions allowed us to describe, within the accuracy of experimental data, the temperature dependence of viscosity for ironÐphosphorus melts [5, 10Ð12] in a 10 wide range (more than 1200 K) from the vitrification temperature of 650 K to metallurgy temperatures of 1823Ð1873 K. The accuracy of the description for a melt with an eutectic composition is illustrated in Fig. 3. In this case, we found the following values of 0 ± the constants in equation (5): Ae = 44 078 540 and ± Be = –6.17 0.17. 6 8 10 12 14 16 In conclusion, it is worth emphasizing that our 4 –1 results give unambiguous evidence of a direct relation- 10 /T, K ship between the phenomena of association and amorphization. Moreover, the concept of association makes Fig. 3. Temperature dependences of the viscosity for the melt FeÐP with an eutectic composition. Solid line is the it possible to quantitatively treat thermal and transport result of our calculations based on Eqs. (5) and (6); the properties of melts in the course of their transition into points are the experimental data of [5] (circles), [10] (trian- the glassy state. gles), [11] (double triangles), and [12] (diamonds).

DOKLADY PHYSICS Vol. 45 No. 8 2000 362 ZAŒTSEV et al.

ACKNOWLEDGMENTS 5. K. Russew, L. Anestiev, L. Stojanova, and L. Sapondjiev, J. Mater. Sci. Technol. 3 (2), 3 (1995). This work was supported by the Russian Foundation 6. T. Miyazaki, Y. Xingbo, and M. Takahashi, J. Magn. for Basic Research, project no. 99-02-16465. Magn. Mater. 60, 204 (1986). We thank B.M. Mogutnov for helpful remarks and 7. M. Naka, T. Masumoto, and H. S. Chen, J. Phys. Colloq. participation in discussions of the results. 41, C8-839 (1980). 8. A. T. Dinsdale, CALPHAD: Comput. Coupling Phase Diagrams Thermochem. 15, 317 (1991). REFERENCES 9. I. Karakaya and W. T. Thompson, Bull. Alloy Phase Dia- grams 9, 232 (1988). 1. O. V. Mazurin, Structure and Stabilization of Inorganic 10. Y. Nishi, K. Watanabe, K. Suzuki, and T. Masumoto, Glasses (Nauka, Leningrad, 1978). J. Phys. Colloq. 41, C8-359 (1980). 2. A. I. Zaitsev and B. M. Mogutnov, High Temp. Mater. 11. I. Dragomir, A. F. Vishkarev, and V. I. Yavoœskiœ, Izv. Sci. 34, 155 (1995). Vyssh. Uchebn. Zaved., Chern. Metall., No. 7, 48 3. P. Richet and D. R. Neuville, Adv. Phys. Geochem. 10, (1964). 132 (1992). 12. A. A. Vostryakov, N. A. Vatolin, and O. A. Esin, Fiz. Met. Metalloved. 18, 476 (1964). 4. A. I. Zaitsev, Zh. V. Dobrokhotova, A. D. Litvina, and B. M. Mogutnov, J. Chem. Soc., Faraday Trans. 91, 703 (1995). Translated by V. Chechin

DOKLADY PHYSICS Vol. 45 No. 8 2000

Doklady Physics, Vol. 45, No. 8, 2000, pp. 363Ð366. Translated from Doklady Akademii Nauk, Vol. 373, No. 4, 2000, pp. 470Ð473. Original Russian Text Copyright © 2000 by Treﬁlov, Mil’man, Lotsko, Belous, Chugunova, Timofeeva, Bykov.

PHYSICS

Studies of Mechanical Properties of Quasicrystalline AlÐCuÐFe Phase by the Indentation Technique

Academician V. I. Trefilov, Yu. V. Mil’man, D. V. Lotsko, A. N. Belous, S. I. Chugunova, I. I. Timofeeva, and A. I. Bykov Received April 7, 2000

A special packing of atoms in quasicrystals, which in the Bürgers vectors of dislocations with the growth is characterized by the absence of translational period- of strains was observed in experiments [5]. icity and the predominantly covalent type of inter- The quasicrystalline icosahedral AlÐCuÐFe phase atomic bonds [1], causes a specific character of their (an approximate composition Al63Cu25Fe12) is very mechanical properties. Quasicrystals are brittle materi- interesting to researchers, since it has high values of als with a nearly vanishing macroscopic plasticity [2]. hardness (HV ≤ 10 GPa), elastic modulus (E ≈ At the same time, in measuring the hardness, distinct 100 GPa), and wear resistance [3]. In addition, this indentations are formed, and scratching causes stria- phase is thermodynamically stable [3]. Recently, it has tions and cuttings [3, 4]; i.e., the microplasticity is been established [7] that the deformation of the poly- revealed under high-pressure local deformation. The crystalline AlÐCuÐFe phase occurs within the charac- brittle state of quasicrystals is retained up to relatively teristic bands, where a high density of dislocations is high temperatures (600°C for AlÐCuÐFe, 700°C for − observed; the dislocations are grouped in networks or Al PdÐMn), and with a further increase of deformation walls. The analysis of the obtained results and the avail- temperature, the compression testing reveals rather able published data allowed the authors of [7] to con- high microplasticity—the deformation prior to fracture clude that under plastic deformation there occurs the can exceed 70Ð80% [2, 5, 6]. The deformation of qua- phase transformation of a quasicrystalline phase into an sicrystals at elevated temperatures is characterized by a approximant phase characterized by atomic packing pronounced work softening, which is observed neither with a translational periodicity along one of the fivefold in crystalline nor in amorphous materials. axes. At least, in AlÐPdÐMn (icosahedral quasicrystal), There is still no consensus concerning the nature of the phenomenon of work softening at elevated temper- low-temperature microplasticity of quasicrystals. It atures was explained based on the dislocation mecha- was argued that it can be caused by a phase transition nism of deformation [6]. The Bürgers vector in quasic- into a more plastic phase under the indenter [3], but rystals has two components, the phonon and phason there is no proof confirming the existence of such a ones. When the dislocations move, the phason compo- transition. nent induces a structural and chemical disorder in con- Recently, a new method based on the indentation trast to the case of the phonon component, which is technique was proposed to determine mechanical prop- similar to the Bürgers vector of dislocations in crystals. erties of low-plasticity materials [8, 9]. The calcula- It was argued in [6] that the deformation of AlÐPdÐMn tions of a mean deformation at the indenterÐsample quasicrystals is controlled by the obstacles in the form contact area allow us to plot the deformation curve sim- of Mackay’s pseudoclusters consisting of 51 atoms. ilar to the stressÐstrain curve, which is obtained under The moving dislocations disturb the ordered atomic uniaxial loading. This can be done with the help of a set arrangement in clusters, giving rise to the correspond- of indenters with different angles at the top. A new δ ing increase in the entropy. As a result, the activation characteristic of the material plasticity H is repre- energy decreases, and the deformation is facilitated. sented, which is defined as a contribution of plastic A significant enhancement of the phason components strain to the total elastoplastic strain under the indenter [8]. In this paper, the results of the first implementation Frantsevich Institute of Materials Science Problems, of this method to the studies of mechanical properties National Academy of Sciences of Ukraine, of compact AlÐCuÐFe quasicrystalline material are pre- ul. Krzhizhanovskogo 3, Kiev, 252680 Ukraine sented. The set of the used methods also includes the

364 TREFILOV et al.

(a) (b)

40 42 44 46 48 40 42 44 46 48 2ϑ

Fig. 1. Regions in X-ray diffraction patterns of the pressed samples: (a) sample 1 (annealing of the powder prior to pressing; (b) sample 2 (annealing of the pressed sample). analysis of the temperature dependence of hardness and temperature dependence of hardness, the Vickers the estimates of the elastic modulus based on the results indenter was used, and for the plotting of the deforma- of indentation with the recording of the loading diagram. tion curve, the set of trihedral pyramidal indenters with γ The powder with a quasicrystalline AlÐCuÐFe phase the angles at the top = 45°, 55°, 60°, 65°, 70°, 75°, and was obtained by sputtering the melt in the air flow. 80°. The X-ray diffraction studies were carried out Compact samples having cylindrical shape with the using the filtered CuKα radiation. The density was diameter and height equal to 5 mm were prepared by determined by hydrostatic weighing. pressing the powder with the particle size 50Ð100 µm The metallographic studies of polished surfaces of in the high-pressure chamber at 700°C under quasihy- the samples revealed the regions of homogeneous drostatic compression. The measurement of the hard- material with the size by the order of magnitude larger ness was performed at the end surfaces of the samples than the sizes of the powder particles. There were also subjected to the mechanical grinding and polishing. large pores (30Ð50 µm in diameter) at the distance of The powders and sintered samples were annealed for about 1 mm from each other. 2 h in the vacuum furnace at 700°C in order to obtain a The X-ray diffraction study (Fig. 1) revealed a pro- single-phase quasicrystalline state. Both the powder nounced difference in the linewidths for samples 1 and prior to pressing (sample 1) and the pressed cylinder 2, i.e., the pressing in the range of the high-temperature (sample 2) were annealed. plasticity led to a significant increase of the defect den- The hardness at the elevated temperatures was mea- sity in the material, and the subsequent annealing elim- sured in vacuum, and the low-temperature hardness, inated these defects. The hardness and plasticity char- under the layer of the cooling liquid. For studies of the acteristics of these samples at room temperature turned out to be also quite different (see the table). This difference cannot be attributed to the different porosity, since Mechanical properties and density of the pressed samples the densities of the samples are very close, the density of a hard sample being somewhat lower. The elastic Sample Treatment HV, GPa E, GPa δ ρ, g/cm3 modulus determined by us (see, table) corresponds to H the upper limit reported in the literature for AlÐCuÐFe quasicrystals [6]. 1 Annealing prior 4.37 113 0.612 4.709 to pressing The deformation curves obtained by the indentation method are presented in Fig. 2 for samples 1 and 2. The 2 Annealing 7.53 0.419 4.693 deformation curves plotted in the coordinates which are after pressing the mean contact pressure (hardness according to ε Meyer HM) and the total strain at the contact area t

DOKLADY PHYSICS Vol. 45 No. 8 2000

STUDIES OF MECHANICAL PROPERTIES OF QUASICRYSTALLINE AlÐCuÐFe PHASE 365 exhibit a pronounced softening, which is especially HM, GPa high for sample 2. Thus, it is shown that the deformation curves at room temperature have the same charac- 8 1 ter as that at elevated temperatures [2, 5, 6]. 2 δ Following [9], we calculated the values of H using expression 7

2 HV δ = 1 Ð 14.3(1 Ð ν Ð 2ν )------, (1) 6 H E ν where E and are the elastic modulus and the Poisson 5 coefficient for the material deformed by the diamond indenter. In the literature, we were unable to find the value of ν needed to estimate E [10] for AlÐCuÐFe. In 4 calculations, we put ν = 0.28 in agreement with the measured elastic constants of AlÐPdÐMn quasicrystal [11]. 3 We pay attention to two regions in the temperature dependence of hardness (Fig. 3): the 78Ð673 K range, where the hardness changes only slightly and the range 2 above 673 K characterized by an abrupt decrease in the hardness. In the secondary electron image of the hardness indentation for sample 2, “tongues” are seen at 1 room temperature (a thin layer pressing out around the δ indentation) (Fig. 4). The increase in H corresponds to the decrease in hardness with the growth of tempera- 0 5 10 15 20 25 30 35 ε ture. Both tetrahedral and trihedral indenters give rise t, % to radial cracks appearing sometimes at the top of Fig. 2. Hardness according to Meyer versus total strain indentation. However, it was impossible to find any reg- under the indenter. The designation of samples corresponds ularities relating the arising cracks to the load magni- to Fig. 1. tude and the shape of indenter. The surface of the indentations was smooth, without indications of fracture and δ slip of the material. HV, GPa H The athermal region on the HV(T) curve in Fig. 3 is 8 1.0 similar to that observed earlier by Gridneva, Mil’man, and Trefilov [12] for single crystals of Si and Ge semi- 0.9 conductors and then by Suzuki and Ohmura [13] at 6 nanoindentation of Si. It was shown in [12] that under 0.8 effect of the indenter, such a region can be caused by the pressure-induced transition to the more plastic 4 0.7 metallic phase. Near the low-temperature indentations in silicon, the “tongues” of the plasitified material were 0.6 found [13]. 2 0.5 In recent experiments, it was found that the icosahedral AlÐCuÐFe phase remains stable under the quasi- 0.4 hydrostatic pressure up to 35 GPa [14]. A different sit- 0 200 400 600 800 1000 1200 uation takes place upon deviation from the quasi- T, K hydrostatic conditions. It was shown in the case of − Fig. 3. Temperature dependence of the hardness HV and Al PdÐMn that, under the uniaxial load applied to the plasticity characteristic δ for sample 2 (load P = 2.34 N). sample placed in the high-pressure cell, we have the H transformation of quasicrystalline phase to the crystalline bcc phase at 20 GPa with the grain size about − pressed out in the shape of thin interlayers, area S is 3 6 nm. The grains are surrounded by amorphous increases, and stress σ decreases. The indenter stops interlayers [15]. It is obvious that under intrusion of the σ σ σ when = cr, where cr is the threshold pressure of the indenter, the stressed state is close to the latter case. phase transition. Thus, we have HV = σ in the ather- σ cr The mean stress at the contact area is = P/S, where mal region. P is the load on the indenter and S is the contact area. For P = const, at the first stages of loading, when S is These results (the work softening at room tempera- small, the pressure-induced phase transformation ture, the athermal region on the HV(T) curve, the occurs under the indenter. The arising plastic phase is “tongues” around the indentation) and the literature

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366 TREFILOV et al. It is of interest to note that, in contrast to the crystalline materials, the accumulation of distortions in the AlÐCuÐFe quasicrystal in the course of high-temperature deformation (at pressing) drastically reduced the hardness under these conditions (see table). The elimination of these distortions by annealing enhanced the hardness.

REFERENCES 1. K. Kimura, H. Matsuda, R. Tamura, et al., in Proceed- ings of the V International Conference on Quasicrystals, Avignon, 1995 (World Scientific, Singapore, 1995), 10 µm pp. 730Ð738. 2. L. Bresson and D. Gratias, J. Non-Cryst. Solids 153/154, 468 (1993). Fig. 4. Secondary electron image of the Vickers indentation 3. U. Köster, W. Liu, H. Liebertz, et al., J. Non-Cryst. Sol- at room temperature for sample 2 (P = 5 N). ids 153/154, 446 (1993). 4. E. Hornbogen and M. Schandl, Z. Metallkd. 83 (2), 128 data [2, 5] suggest that the phase transition in quasic- (1992). rystals is related to the process of plastic deformation, 5. S. S. Kang and J. M. Dubois, Philos. Mag. A 66 (1), 151 in the course of which the accumulation of phason dis- (1992). tortions results in the violation of the order in the 6. M. Feuerbacher, C. Metzmacher, M. Wollgarten, et al., atomic packing. It is clear that with the temperature Mater. Sci. Eng., A 233, 103 (1997). decrease, the dislocation density in the layer under the 7. U. Köster, X. L. Ma, J. Greiser, et al., in Proceedings of indenter increases, despite the reduction in the magni- the VI International Conference on Quasicrystals, tude of plastic strain under the indenter characterized Tokyo, 1997 (World Scientific, Singapore, 1997), by δ . This occurs owing both to the decrease in the pp. 505Ð508. H 8. Yu. V. Milman, B. A. Galanov, and S. I. Chugunova, Acta dislocation mobility and the slowing in the process of Metall. Mater. 41, 2523 (1993). polygonization and recovery. One can suggest that the observed indications of the phase transition are related 9. Yu. V. Mil’man, in Modern Materials Science of the XXI to the attainment of a certain critical value for the den- Century (Naukova Dumka, Kiev, 1998), pp. 637Ð656. sity of dislocations. 10. B. A. Galanov, O. N. Grigor’ev, Yu. V. Mil’man, et al., Probl. Prochn., No. 6, 52 (1990). Together with the decrease in hardness, the charac- δ 11. Y. Amazit, M. Fisher, B. Perrin, et al., in Proceedings of teristic of plasticity H increases with temperature the V International Conference on Quasicrystals, Avi- reaching the value of 0.95 at 1073 K (Fig. 3). According gnon, 1995 (World Scientific, Singapore, 1995), to [8], the macroscopic plasticity manifests itself at pp. 584Ð587. δ ≈ H 0.9 in the crystalline (mainly, metallic) materials. 12. I. V. Gridneva, Yu. V. Mil’man, and V. I. Trefilov, Phys. δ The comparison of the temperature dependence of H Status Solidi A 14 (1), 177 (1972). and the results of compression tests [2] demonstrates 13. T. Suzuki and T. Ohmura, Philos. Mag. A 74 (5), 1073 that in AlÐCuÐFe quasicrystals it arises at much lower (1996). δ δ ≈ δ H values, H 0.7. Note that the estimate of H in 14. J. P. Itie, S. Lefebvre, A. Sadac, et al., in Proceedings of Fig. 3 was performed without taking into consideration the V International Conference on Quasicrystals, Avi- the temperature dependence of the elastic modulus. The gnon, 1995 (World Scientific, Singapore, 1995), account taken of this dependence results in a certain pp. 168Ð171. δ decrease in the magnitude of H at elevated tempera- 15. Y. Yan, N. Baluc, J. Peyronneau, et al., in Proceedings of tures. Such an early manifestation of the microscopic the V International Conference on Quasicrystals, Avi- plasticity can be related both to the features of the gnon, 1995 (World Scientific, Singapore, 1995), deformation mechanism of quasicrystals and to the dif- pp. 668Ð671. ferent conditions of testing under compression and tension. Translated by T. Galkina

DOKLADY PHYSICS Vol. 45 No. 8 2000

PHYSICS

Parabolic Approximation in the Theory of the Sound Propagation in Three-Dimensionally Heterogeneous Media O. A. Godin Presented by Academician L.M. Brekhovskikh October 27, 1999

Received November 26, 1999

The parabolic approximation [1; 2; 3, Section 7.6] is nates and the media to be boundless or to have ideal widely used in the mathematical simulation of wave (absolutely soft or absolutely rigid) boundaries, which fields, in particular, in oceanic and atmospheric acous- are invariant with respect to the translations along the tics. In a wide scope of direct and inverse problems for Ox-axis. The differential operator Sˆ is self-conjugate finding the physically significant results in terms of this with respect to the scalar product approximation, it is necessary [4Ð6] for the reciprocity () () () () principle and the energy conservation law expressing 〈〉p 1 ,p2 =∫∫dyd zp 1 ()p2 */ρ()x, (2) the fundamental symmetries of the true wave field to remain valid for the solutions to the parabolic equation. where the integration is carried out over the waveguide Wide-angle parabolic equations providing the validity cross section. The Ox-axis is chosen along the preferen- of the reciprocity principle and the energy conservation tial direction of the wave propagation. It is assumed in two-dimensional problems related to the propagation that the medium parameters vary either weakly or ξ of sound in media with piece-wise continuous variation smoothly with x and that the propagation constants n of parameters along the trace are proposed in [7, 8]. The for local modes contributing substantially to the acous- goal of this study is the generalization of these results tic field are concentrated in the neighborhood of k0, so to the three-dimensional case. We propose a class of ε ≡ || ˆ || || || Ӷ energy-conserving and reciprocal three-dimensional that Sp / p 1. parabolic equations and investigate the properties of Similarly to [8], where the two-dimensional case their solutions. was investigated, we consider the three-dimensional parabolic equation in the form In stagnant fluid with the time-independent density ρ and the velocity of sound c, the acoustic pressure sat- []ρÐ1/2Φ ()ˆ 2 S px isfies the equation [see 3, Section 4.1] (3) ρÐ1/2Φ ()ˆ Φ()ˆ ρÐ/21 ()ˆ = ik0 1 S p 2Sp+ gSQ, ∂ 1 ∂ p 2 ------+k()1+Sˆp=Q, Φ ∂xρ ∂ x 0 where N(a) are the analytical functions satisfying the Φ Φ (1) condition N(a*) = [ N(a)]*, which serve for approxi- ρ∂ 1 ∂ ∂1 ∂ k2 1/2N ˆ------mations of the functions (1 + a) , N = 1, 2. For Q = 0 S= 2 ++ ----Ð 1 . ∂ ∂ k∂ y ρ ∂ y ∂zρ ∂ z k2 and / y = 0, equations (3) transform into the energy- 0 0 conserving two-dimensional parabolic equations pro- ω ω posed and investigated in [8]. Efficient methods for Here, k(x) = /c is the wave number; is the wave fre- solving these equations are developed for the cases quency; k = const; and Q iωρa ρ∇ ρ–1f , where a 0 = + ( ) ˆ1/2N and f are the volume densities for sources of both the when the pseudodifferential operators (1 + S ) are Φ ˆ volume velocity and extraneous force. For simplicity, approximated by the rational functions N()S [9Ð13]. we assume ρ and c to be smooth functions of coordi- We determine the function g from the condition that parabolic equation (3) approximates the solutions to the wave equation in the case of a stratified fluid, i.e., when Shirshov Institute of Oceanology, c = c(z), ρ = ρ(z), and the medium is unbounded in y. In Russian Academy of Sciences, the layered medium, the problem is reduced to the two- ul. Krasikova 23, Moscow, 117218 Russia dimensional case by the Fourier transform with respect

1028-3358/00/4508-0367 $20.00 © 2000 MAIK “Nauka/Interperiodica” 368 GODIN δ ˆ to y. For the point source Q = (x – x0), the solution to and taking into account that the operator S is self-con- ≥ equation (3) for x x0 has the form jugate, we obtain ∞ ∂ + ----- 〈Φ (Sˆ ) p , Φ (Sˆ ) p*〉 ( ) [ Φ (α )( )] ∂x 2 1 2 2 p x = ∑ ∫ dqexp iqy + ik0 1 n x Ð x0 (6) n Ð∞ 1 ( 〈 〉 〈 〉) = ------p1, Q2* Ð p2, Q1* . g(α ) f (z) f (z ) 2ik0 × n n n 0 (4) ------π----Φ------(--α------)------, 2 2 n In the region free of sources, equation (6) is reduced to the formulation of the reciprocity principle considered ξ2 Ð q2 α = ----n------Ð 1, for two-dimensional parabolic equations in [7, 8]. Inte- n 2 grating (6) with respect to x, we find k0 ξ d3 x p Q /ρ = d3 x p Q /ρ where fn(z) and n are, respectively, the eigenfunctions ∫ 1 2 ∫ 2 1 . (7) and the propagation constants of normal waves in a lay- This formulation of the reciprocity principle for the ered waveguide. If the vertical wave operator has a con- solutions to parabolic equations exactly repeats the tinuous spectrum, the summation over the discrete mode well-known result (see [3, Section 4.2]) for the solu- subscript n is supplemented in (4) by integration over the tions to the wave equation. In particular, from (7) and ≥ continuous spectrum. Solution to (1) for x x0 also has the above expression for Q, it follows that in the para- the form of (4) but differs with respect to the replacement bolic approximation under consideration, as in the case 2 of true sound fields, the acoustic pressure is invariant of k Φ α by ξ q2 1/2 and of g α Φ α by 0 1( n) ( n – ) ( n)/ 2( n) with respect to a rearrangement of the receiver and the −0.5i(ξ2 – q2)–1/2. Hence, for volume-velocity point source. The vibrational-velocity n projection to the force-action direction is invariant with respect to the rearrangement of the extraneous-force ( Φ )Ð1 g = 2ik0 2 , (5) source. The validity of the energy conservation law for the the solutions to parabolic equation (3) approximate the solutions to parabolic equation (3) in the case of solutions to wave equation (1) for the waves propagat- Imk2 = 0 can be found similarly. This conclusion dif- ing toward increasing x. The degree of proximity for fers from that mentioned above by using the complex these solutions is determined by the accuracy of conjugation p* of the field under consideration instead 1/2N Φ the approximating (1 + a) by the functions N(a). of p2 and leads to the following result: Expanding the solutions in terms of two-dimensional ∂J k0 Ð1 2 modes f (y, z), it is easy to ascertain that parabolic ρ Φ ( ˆ ) ( ) n, m -∂---- = 0, J = ----ω---∫∫dydz 2 S p x . (8) equation (3), (5) also provides the approximation of the x 2 wave equation in a medium with cylindrical boundaries Here, J represents the energy flux through the and parameters depending on y and z. In the limiting waveguide cross section x = const. In the regular seg- case when the error of the approximation of (1 + a)1/2N ment of a three-dimensional waveguide with cylindri- by the functions Φ (a) tends to zero and the field cal boundaries, the contributions of normal waves into N J are additive by virtue of the orthogonality of modes. involves only the modes propagating towards increasing In the case of an isolated propagating mode, J (8) dif- x, the difference between the solutions to equation (1) fers from the true acoustic-energy flux only by the and to parabolic equation (3), (5) vanishes for an arbi- ξ replacement of the propagation constant n, m by trary Q. Φ2 ξ2 2 2 ( n, m / k0 – 1). As in the two-dimensional case [8], We show that the solutions to parabolic equation (3), parabolic equation (3) leads to a positive energy flux in (5) satisfy the reciprocity principle in the three-dimen- the nonpropagating (supercritical) modes, where sionally heterogeneous medium. Let the wave p1(x) 2 ξ , < 0, for Φ (a*) = [Φ (a)]*. The elimination of propagating towards increasing x and the wave p (x) n m N N 2 this deficiency requires using the approximations Φ (a) propagating towards decreasing x be generated by the N with complex coefficients for the radicals (1 + a)1/2N. sources Q1 and Q2, respectively. The parabolic equation The consideration of such approximations is beyond based on p2 differs from (3), (5) due to the replacement the scope of this study. We note only that, for parabolic of k0 by –k0. Multiplying the parabolic equation for p2 equation (3), (5) with complex coefficients, the above by ρ–1/2Φ (Sˆ )p and summing up the result obtained reciprocity relationships and their derivation remain 2 1 valid, including for the case of an absorbing medium. ρ–1/2Φ ˆ with the product of (3) by 2(S )p2, integrating For estimating the principal phase-error term accu- with respect to y and z over the waveguide cross section mulated with distance and introduced by the parabolic

DOKLADY PHYSICS Vol. 45 No. 8 2000 PARABOLIC APPROXIMATION IN THE THEORY OF THE SOUND PROPAGATION 369 approximation, it will suffice to compare the solutions equations used nowadays provide a high phase accu- to the wave equation and parabolic equation in a lay- racy for the grazing angles χ ≤ β, where β amounts to ered medium (see [3, Section 7.6]). In the wave zone of at least one-half of radian [9Ð12]. The usage of the Φ a source, we use the saddle-point method (see [3, same approximations 1 in three-dimensional para- Appendix A]) for calculating the integral in (4). Let bolic equation (3) guarantees the phase accuracy for the m β ≤ φ ≤ β Φ (a) = (1 + a)1/2N + O(a n ) and m ≥ 2. In this case, azimuth angles – . If the field is required to be N 1 calculated within a still wider range of azimuth angles, the exponent in (4) has a stationary point then to fulfill (10) and to cover the entire range, it may ξ ϕ[ ( m1 )] χ2 ϕ2 be necessary to solve several parabolic equations (3), qs = n sin 1 + O b , b = n + , (5) differing by the choice of the direction for the ξ y Ð y (9) Ox-axis. χ = arccos----n, ϕ = arctan------0-. n k x Ð x In the three-dimensionally heterogeneous medium, 0 0 the asymptotic errors in the parabolic approximation χ Here, n is the glancing angle formed by the wave vec- can be estimated by comparing the high-frequency tor of a mode with the xy plane; the azimuth angle φ solutions to the wave equation and to the parabolic characterizes the deviation of the direction to the equation. Restricting ourselves to consideration of the receiver from the Ox-axis, i.e., from the chosen prefer- fields far from the caustics, we present the solutions in ential direction of the wave propagation. Calculating the form of the ray (Debye) expansions p = θ the contribution of stationary point (9) into integral (4) exp(ik0 (x))A(x, k0), where k0 serves as a formal large –1 and comparing the result with the true field mode, we parameter and A(x, k0) = A0(x) + (ik0) A1(x) + …. Using find that the parabolic approximation introduces the the method of mathematical induction with respect to m m m relative error O(b 1 ) into the phase and O(b 1 + b 2 ) M = 1, 2, …, it is easy to establish the validity of the into the amplitude of the mode. For applicability of the identity φ χ parabolic approximation, and n must be small. How- θ θ  2 ˆ M( ik0 ) ik0 αM iM ∇ A ever, quantitative limitations for these angles rapidly S Ae = Ae 1 + ------α-- n⊥ ln--ρ--- weaken with the growth in the accuracy of the approx-  k0 Φ imations of N used in parabolic equation (3). The dis-  ∇ ( ) ∇ α ( Ð2) tances to a source and the frequencies for which the --+ n⊥ + M Ð 1 n⊥ ln + O k0 , parabolic equation can be used are bounded from above  (11) by the condition of correct reproduction of mode ∇θ (ν , ) ( , ν , ν ) phases: n = = 1 n⊥ , n⊥ = 0 2 3 ,

2 2 m1 2 2 2 2 2 ( )(χ φ ) Ӷ α = 1 Ð n Ð n⊥ and n = k /k . k0 x Ð x0 + 1, (10) 0 χ χ where is the characteristic value of n for the modes For an arbitrary analytical function Φ, expanding Φ(Sˆ ) contributing substantially into the ﬁeld. Note that, ˆ according to (10), in the applicability domain of the in terms of S , we obtain from (11) parabolic approximation, the ﬁeld can undergo strong θ θ ik0 ik0 i variations in the transverse directions y and z including Φ(Sˆ )Ae = Ae Φ(α) + ---- ∇(n⊥Φ'(α)) the mode-phase deviations from its value at y = 0 for x,  k0 z = const, which are large compared to unity. (12) 2  In contrast to the two-dimensional case [8], the Φ (α) ∇ A0 ( Ð2) + ' n⊥ ln----- + O k0 . amplitude error in the three-dimensional problem ρ  depends not only on the accuracy of the approximation Φ Φ provided by the function 2 but also on 1. The above Substituting the Debye expansion into (3) and using estimates of the phase error for φ = 0 coincide with ear- (11) and (12), we obtain in the major order with respect ν Φ α lier estimates for the two-dimensional waveguide [8]. to k0 that 1 = 1( ), i.e., the eikonal equation in the (It is easy to show that the last estimates are also valid parabolic approximation. Introducing the Hamiltonian for the three-dimensional waveguide with cylindrical ( , ) [Ψ(ν ) 2 2 ] boundaries.) As applied to the horizontally heteroge- H x n = 0.5 1 Ð n + 1 + n⊥ (13) neous media, the values of φ in criterion (10) are bounded from below by the angle of horizontal refrac- allows reducing the solution of the eikonal equation to tion. In the case when the sound propagates in ocean or the solution of the Hamiltonian ray equations atmosphere, this angle is, as a rule, much smaller than dx ----- = h ≡ (µ, n⊥), the grazing angles, and the horizontal refraction does dτ not practically manifest itself at the frequencies and (14) Ð1 dn 2 distances for which it is rightful to use parabolic equa- µ = [2Φ' (α)] ; ------= 0.5∇n . tion (3), (5). Two-dimensional wide-angle parabolic 1 dτ

DOKLADY PHYSICS Vol. 45 No. 8 2000 370 GODIN Here, the parameter τ describes the position of a point same eikonal equation but to different transport equa- Ψ ν under consideration at the ray, ( 1) is the function tions. It follows from (17) that inverse to Φ (a). The equations of true rays (see [3, Sec- 2 1 d  A  dx ∂ tion 5.1]) differ from (14) in the replacement of h by n. ----- ln -----0- + ∇ ----- = Ðµ----- lnρ. (18) dτ µρ  τ ∂ According to (14) in the parabolic approximation, d x the fluid turns out to be an acoustically anisotropic If the medium density varies along the propagation medium, and the directions of the normal n to the wave direction, the right-hand side in (18) differs from zero, front and of the ray tangent h do not coincide. The and the energy flux in the ray tube is not conserved. ρ ≡ anisotropy and differences between rays and eikonals When x 0, equation (18) expresses the conservation corresponding to the wave equation and the parabolic of a certain energy quantity, but its values correspond- equation tend to zero with increasing the accuracy of ing to these two parabolic equations differ. This fact the approximation of F1. It should be emphasized that leads to a difference in the amplitude errors. For sim- Φ parabolic equation (3) completely describes the three- plicity, let the error in the approximation of 1 be neg- 1/2 dimensional refraction. In the limit, F1(a) (1 + a) , ligible, so that we can ignore the difference in the H (13) transforms into the known Hamiltonian geometry of rays corresponding to the parabolic equa- H = 0.5(n2 – n2) for solutions to the wave equation tion and the wave equation. In this case, the amplitude (see [3, Section 5.1]), and, according to (14), h = n. of the solution to parabolic equation (17) differs for ρ ≡ Similarly to the derivation of the eikonal equation, x 0 from the ray amplitude of the solution to the µ τ µ τ –1/2 substituting the Debye expansion into (3) and equating wave equation by the factor [ ( 2)/ ( 1)] = 1 + O(b), τ1 τ the terms not containing k0 with the help of (12) and where and 2 correspond to the point of a position of (14), we obtain the transport equation the source and receiver, respectively. According to (15), in the case of an arbitrary ρ , the deviation of the ray 2 x d  A0 Φ2(α) ∇dx amplitudes for parabolic equation (3) amounts to ----τ- lnµ----ρ-- 2  + ----τ- = 0, (15) d d 2 Ð1/2 µ(τ )Φ (τ ) m m which allows us to calculate an amplitude of the ray ------2------2------2---- = 1 + O(b 1 + b 2 ). µ(τ )Φ2(τ ) field. This equation differs from the corresponding 1 2 2 result following from the wave equation only by the presence under the logarithm of an additional factor Therefore, the amplitude error in parabolic equation (3), Φ2 µ in contrast to (17), tends to zero with an increase in the 2 / . A physical meaning of the transport equation Φ accuracy of approximations of 1, 2. consists in the fact (see [3, Section 5.1]) that, in the ray tube, the energy flux is conserved, its density being Thus, the usage of parabolic equation (3), (5) pro- 2 Φ2 ωρµ posed in this study instead of traditional parabolic equal to k0 A0 2 / in the case under consideration. equation (17) enables us to improve the accuracy in the The local energy-conservation law for the high-fre- calculating amplitude without reducing that of the quency sound can be also found directly from the para- phase and to provide the energy conservation and the bolic equation while ignoring the consideration of the exact validity of the reciprocity principle in simulating ray geometry. Indeed, multiplying parabolic equation (3) the sound fields in three-dimensional heterogeneous ρ–1/2Φ ˆ media. by 2(S )p* and using (12) for simplifying the real part of the product, it is easy to obtain ∇I = 0, ACKNOWLEDGMENTS (16) (ωρµ)Ð1 2Φ2(α)η[ ( Ð1)] Studies underlying this paper were carried out in the I = k0 p 2 1 + O k0 . Institute of Oceanology of the RAS under the partial It is also easy to verify that expression (16) for the den- support of the Russian Foundation for Basic Research, sity of the energy-flux in high-frequency waves agrees as well as of INTAS, the INTASÐRFBR project with exact integral relationship (6). no. 95-1002, and in the University of Victoria, B.C., Canada) under the support of NSERC. We compare the high-frequency solutions to energy-conserving parabolic equation (3) and to the traditional parabolic equation REFERENCES ∂p 1. F. D. Tappert, in Propagation of the Waves and Under------= ik Φ (Sˆ ) p, (17) ∂x 0 1 water Acoustics, Ed. by R. Keller and A. Papadakis (Springer, Heidelberg, 1977; Mir, Moscow, 1980), whose coefficients involve no derivatives of medium pp. 180Ð226. parameters with respect to x. When the same approxi- Φ 2. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskiœ, Intro- mation 1 for the square root of the operator is used duction to Statistical Radio Physics (Nauka, Moscow, in (3) and (17), both parabolic equations lead to the 1978).

DOKLADY PHYSICS Vol. 45 No. 8 2000 PARABOLIC APPROXIMATION IN THE THEORY OF THE SOUND PROPAGATION 371

3. L. M. Brekhovskikh and O. A. Godin, Acoustics of Lay- 10. M. D. Collins, J. Acoust. Soc. Am. 92, 2069 (1992). ered Medium, Vol. 2: Point Sources and Bounded Beams 11. M. D. Collins, R. J. Cederberg, D. B. King, et al., (Nauka, Moscow, 1989; Springer, Berlin, 1999). J. Acoust. Soc. Am. 100 (1), 178 (1996). 4. L. Nghiem-Phu and F. D. Tappert, J. Acoust. Soc. Am. 78 12. D. Lee and A. D. Pierce, J. Comput. Acoust. 3 (2), 95 (1), 164 (1985). (1995). 5. F. B. Jensen, W. A. Kuperman, M. B. Porter, and 13. D. Yu. Mikhin, in Proceedings of the IV European Con- H. Schmidt, in Computational Ocean Acoustics (AIP, ference on Underwater Acoustics, 1998 (CNR-IDAC, New York, 1993), Chap. 6. Rome, 1998), pp. 679Ð684. 6. O. A. Godin, D. Yu. Mikhin, and A. V. Mokhov, Akust. 14. D. Yevick, C. Rolland, and B. Hermansson, Electron. Zh. 42, 501 (1996) [Acoust. Phys. 42, 441 (1996)]. Lett. 25 (18), 1254 (1989). 7. O. A. Godin, Dokl. Akad. Nauk 361, 329 (1998) [Dokl. 15. D. Lee and M. H. Schultz, Numerical Ocean Acoustic Phys. 43, 393 (1998)]. Propagation in Three Dimensions (World Scientiﬁc, 8. O. A. Godin, Wave Motion 29, 175 (1999). Singapore, 1995). 9. K. V. Avilov, in Acoustics of Ocean Medium (Nauka, Moscow, 1989), pp. 10Ð19. Translated by V. Bukhanov

DOKLADY PHYSICS Vol. 45 No. 8 2000

PHYSICS

Generalized Equation for Inverse Population in a Laser Active Medium

Academician A. L. Mikaelyan and Yu. G. Turkov Received March 28, 2000

Usually, equations describing the evolution of the either strongly bound in the continuous laser-genera- inverse population in laser theory are valid only for ide- tion mode or isolated in the energy sense while generalized energy levels of an active medium. ating short pulses. Allowing for the actual structure of energy levels in When considering processes of generation and a laser material necessitates the consideration of a sys- amplification of laser radiation, it is reasonable to iso- tem of differential equations describing the evolution of late groups of levels to which the upper and lower levels the population of all levels playing a significant role in of the laser transition belong. the process of laser generation. Such an approach requires solving a system of equations of a high order, The diagram of the energy levels for a certain active which is possible only by numerical computer simu- laser medium is presented in Fig. 1. We assume that a lation. laser transition takes place between the jth level of the level group 2 and an ith level of the level group 1 (rig- In addition, in this approach, certain difficulties orously speaking, the subscripts i and j should be appear associated with the lack of complete informa- denoted by a combination of two numbers, namely, 1, i tion on corresponding spectroscopy constants of a and 2, j, where the former implies belonging to a corre- material, which, as a rule, are known with an acceptable sponding group (1 or 2)). Thus, i = 11, 12, 13, …; and j = accuracy only for a rather limited number of energy 21, 22, 23, …). We assume also that the distribution of the states. level populations within each group always corresponds to the thermodynamic equilibrium. Then, we Nevertheless, as is shown below, under certain sim- may write out plifications, allowance for the actual structure of the energy levels of a laser material can be realized in a n ==γ n , n γ n , (1) fairly simple manner. In this case, a system of high- i i 1 j j 2 order differential equations can be reduced to only one where n1 and n2 are the total number of atoms occupy- equation. It has the meaning of an equation for the ing all energy levels belonging to the groups 1 or 2, inverse population density and generalizes well-known respectively, i.e., the populations of these groups. equations for three- and four-level active media. The simplified assumptions mentioned above are associated with the possibility to isolate separate groups of energy levels closely related to each other. 3 The energy exchange processes between these levels τ lead to the rapid establishment of thermodynamic equi- W32 = 1/ 32 librium.

In this case, it is natural that the admissible rate of j 2 establishing the thermodynamic equilibrium, for which we consider the energy levels to be closely bound with each other, is deﬁned by the speed of proceeding pro- τ τ cesses of the laser generation. For example, in different W13 A31 = 1/ 31 Wij Wji A21 = 1/ 21 situations, the same levels may be considered to be

i 1 Institute of Optoneyral Technologies, Russian Academy of Sciences, ul. Vavilova 44/2, Moscow, 117333 Russia Fig. 1.

GENERALIZED EQUATION FOR INVERSE POPULATION IN A LASER ACTIVE MEDIUM 373 γ γ The coefﬁcients i and j are easily determined from we formally arrive again at the equations for the ideal- the Boltzmann distribution for the population: ized three-level medium including the equation for inverse population density. There is only one distinc- E Ð E E Ð E Ð ----i------10-- Ð -----j------20--- kT kT tion between them. The parameter g2/g1 involved in this γ gie γ g je equation is replaced by a more general parameter i = ------E----Ð---E----, j = ------E----Ð---E----. (2) Ð ----i------10-- Ð -----j------20--- σ γ kT kT W12eff 12eff ig j ∑gie ∑g je G = ------= σ------= γ------. (8) W21eff 21eff jgi i j Indeed, assuming that active atoms are not accumu- Here, Ei and Ej are the energies of the levels i and j, Ӷ lated on the levels of the group 3, i.e., n3 n1 and respectively; E10 and E20 are the energies of lower levels dn in the groups 1 and 2, respectively; gi and gj are the ------3 = 0, we have from Eq. (5) degeneracy multiplicities for the levels i and j, respec- dt tively; k is the Boltzmann constant; and T is tempera- τ ture of an active medium. 31 τ n3 = W13-τ------τ------32n1. (9) It is worth noting that the energy scheme under con- 31 + 32 sideration generalizes the level schemes for both three- Substituting this relationship into (3) and (4), we and four-level active medium under the condition of a obtain rapid relaxation of the lower laser level. In this case, this level may be included in the group 1 of levels being d∆ ------= Ð(1 + G)∆W in the thermodynamic equilibrium with the ground dt 21eff level. (10)  G  ∆ 1  For the scheme under consideration, the system of + n0Wp Ð -τ----- Ð Wp + -τ----- , equations describing the evolution for the populations 21 21 of the level groups 1, 2, and 3 takes the form where dn1 n2 n3 ∆ ------= W γ n Ð W γ n Ð W (n Ð n ) + ------+ ------, (3) = n2 Ð Gn1 (11) dt ji j 2 ij i 1 13 1 3 τ τ 21 31 is the generalized inverse population density introduced instead of the generally accepted quantity n – dn2 γ γ n3 n2 2 ------= ÐW ji jn2 + Wij in1 + τ------Ð τ------, (4) g dt 32 21 2 ---- n1; g1 dn n n ------3 = W (n Ð n ) Ð ----3-- Ð ----3--, (5) τ 13 1 3 τ τ 31 dt 32 31 Wp = W13-τ------τ------(12) 31 + 32 where Wij and Wji are the induced-transition probabilities bound with each other by the relation is the effective value of the excitation probability for an active atom under the action of pumping; and n0 is the giWij = g jW ji. total number of active atoms per unit volume. The lifetimes and excitation probabilities of the At G = 1, equation (10) transforms into that for the active atoms under the action of pumping, which are three-level active medium; at G = 0, it transforms into involved into equations (3)Ð(5), determine the effective that for the four-level medium with the vacant low level rates of the energy exchange between the isolated level of the laser transition. groups. The gain of the active medium with the generalized As is seen, the entire difference between equa- scheme of the energy levels is tions (3)Ð(5) and usual ones applied to an ideal active α = σ n Ð σ n = σ γ n Ð σ γ n = σ ∆. (13) laser medium (see, e.g., [1]) lies in the fact that the ji j ij i ji j 2 ij i 1 21eff probabilities W21 and W12 are replaced by the quantities When the contribution to the gain of several closely γ γ jWji and iWij, respectively, which represent effective lying laser lines must be taken into account, the effec- values for the corresponding transition probabilities. tive cross sections of the induced transition should be Hence, introducing the notation introduced in the form W = g W , W = g W (6) 21eff j ji 12eff i ij σ ν σ ν γ σ ν σ ν γ 21eff( ) = ∑ ji( ) j, 12eff( ) = ∑ ij( ) i. (14) or, which is the same, a similar notation for effective ij ij values of the transition cross sections In this case, as before, the parameter G in Eq. (10) is σ σ σ σ σ σ 21eff = g j ji, 12eff = gi ij, (7) determined by the relation 12eff/ 21eff.

DOKLADY PHYSICS Vol. 45 No. 8 2000 374 MIKAELYAN, TURKOV It is worth noting that Eq. (15) is easily generalized n j 2 for the case when several lines with a close frequency lines play the role. The parameters G and Gp represent the ratios of the corresponding effective cross sections determined by summation over all lines similarly to [14]. τ At G = 0, Eq. (15) transforms into (10). Wmn Wnm Wij Wji A21 = 1/ 21 p Equations derived enable us to easily perform calculations of solid-state lasers with an allowance for the actual structure of energy levels and the active medium m i 1 temperature determining the values of the coefficients γ in accordance with formulas (2). For a more convenient application of these equations in practice, it would be Fig. 2. desirable to form information tables for the temperature dependences of the coefficients γ and parameters G and Gp of the basic spectral lines for laser crystals. If the level 3 is situated closely to the upper laser As an example, we demonstrate the advantage of the level (as this takes place for the semiconductor pump- approach proposed in calculating the output power of ing of YAG:Nd-laser), then its equilibrium Boltzmann the solid-state laser operating in the steady-state mode. population should be taken into account. With this pur- Using Eq. (10), it is not difficult to obtain the general- pose, this level may be included into the level group 2 ized formula for the output power in the form tightly bound with the upper level of the laser transition. The energy diagram depicted in Fig. 2 assumes 1 hνn V ln-- that the excitation of the active medium is attained 0 r PΣ = ------owing to the atom transitions from the level m of the (1 + G)τ (2βl Ð lnr) group 1 to the level n of the group 2, and the laser gen- 21 (17) eration corresponds to the same transition between lev- 2βl Ð lnr × k Ð G Ð (k + 1)------. els j and i. In this case, performing calculations similar 2σ n l to those used while deriving (10), we arrive at the fol- 21eff 0 lowing equation for the generalized density of the Here, inverse population: k = W τ , d∆ p 21 ------= Ð(1 + G)∆W dt 21eff hν is the energy of photons for the laser generation; V (15) and l are the volume and the length of an active sample, ( ) G ∆ ( ) 1 + n0 Wp 1 Ð GpG Ð -τ----- Ð Wp 1 + Gp + τ------. respectively; r is the reflection factor of the output mir- 21 21 ror in the optical resonator (the second mirror is β Here, ∆, G, and W are, as before, expressed by the assumed to be ideally reflecting); and is the nonre- 21eff sonance loss in the crystal. In the particular cases of same formulas (11), (8), and (6); G = 1 and G = 0, this formula transforms into well- σ γ g known expressions for the output power of three- and G = ----n---m---eff--- = ----n-----m- , (16) p σ γ g four-level lasers [1]. mn eff m n When the evolution of the inverse population obeys γ and Wp = mWmn is the effective probability for excita- Eq. (15), the formula for the output power of a contin- tion of an atom under the action of pumping radiation. uous-wave laser has the form The quantities γ γ g g have the same meaning as n, m, n, m 1 the corresponding quantities for the levels i and j. hνn V ln-- 0 r In a certain sense, we can say that Eq. (15) describes PΣ = -(------)--τ------(------β------) the evolution of the inverse population in a quasi-two- 1 + G 21 2 l Ð lnr level laser. In particular, it follows from this equation 2βl Ð lnr × k Ð G Ð (k + 1)------(18) that the inequality GpG < 1 is the necessary condition 2σ n l for the existence of the inversion (∆ > 0) in the steady- 21eff 0 state mode of such a laser. It is easy to show that this  2βl Ð lnr condition is equivalent to the well-known relation Ð kGpG + ----σ------ . 2 21effn0l E – E > E – E ; n m j i The last term in this formula determines the i.e., the frequency of pumping radiation must exceed decrease of the generation power caused by the finite the frequency of the laser radiation. population density for the upper level of the transition

DOKLADY PHYSICS Vol. 45 No. 8 2000 GENERALIZED EQUATION FOR INVERSE POPULATION IN A LASER ACTIVE MEDIUM 375 used for laser optical pumping as well as by the place from the lower of these levels E (R1 line) or from presence of several levels in a group to which the 4 the upper 2 A level (R2 line) onto the A2 ground level lower level of this transition belongs. For Gp = 0, formula (18) takes the form of (17). degenerated fourfold, which is unique in the level group 1 (as a rule, laser generation takes place for the To illustrate, we consider a ruby laser [1]. In a ruby stronger R1 line). The values of the parameter G for the crystal, two twofold degenerate metastable levels E R1 and R2 lines of the laser transition in a ruby crystal, and 2 A separated by the distance 29 cmÐ1 should be evaluated by formulas (8) and (2), are given here for the related to the level group 2. The laser transition takes temperature range between 50 and 450 K.

T, K 50 100 150 200 250 300 350 400 450

G (R1 line) 0.717 0.829 0.879 0.906 0.923 0.935 0.944 0.950 0.956 G (R2 line) 1.652 1.259 1.160 1.116 1.091 1.075 1.063 1.055 1.049

As is seen, when applied to a ruby laser, the usually output generation power at the double excess of the used three-level approximation (G = 1) is fairly accu- threshold. This product has the form rate at room and higher temperature. However, to inves- 1 tigate a laser operating at lower temperatures, as well as hνV ln-- η r in the case when we need comparative analysis of ruby Pth diff = ----τ------σ------, (19) 2 21 21effl laser characteristics for the R1 and R2 lines, it is neces- η sary to take into account actual values and temperature where Pth is the threshold pumping power and diff is dependences of the parameters G for these lines. the differential laser efficiency. In particular, for a YAG laser (assuming hν= 1.87 × In conclusion, we present a relationship useful for Ð19 σ × Ð19 2 τ µ 3 10 J, 21eff = 3.4 10 cm , 21 = 240 s, V = 1 cm , practice, which concerns continuousÐwave four-level η lasers. This relationship interrelates the differential l = 5 cm, and r = 0.9), we obtain Pth diff = 24 W. For energy efficiency (i.e., the linear part of the ratio for the example, in the case of the threshold pumping power of 1000 W, the differential energy laser efficiency is equal generation power to the increment of the pumping to 2.4%. power) and the threshold pumping power. It is evident that both characteristics are determined in the same degree by the efficiency of the pumping system; there- REFERENCE fore, their values must be interrelated. The relationship 1. A. L. Mikaelyan, M. L. Ter-Mikaelyan, and Yu. G. Tur- mentioned above is easily derived from formula (17) at kov, Solid-State Lasers (Sovetskoe Radio, Moscow, G = 0 and with an allowance for the fact that the product 1967). of the differential energy efficiency of the laser by the threshold pumping power is numerically equal to the Translated by Yu. Vishnyakov

DOKLADY PHYSICS Vol. 45 No. 8 2000

PHYSICS

Self-Organization Processes in a System of Magnetic Moments in Polycrystals V. K. Karpasyuk and E. I. Beznisko Presented by Academician Yu.V. Gulyaev December 20, 1999

Received December 21, 1999

The processes of evolution and self-organization in Due to the misorientation of the crystallographic nonlinear dissipative physical systems including those axes of neighboring grains, the normal magnetization characteristic of magnetic microstructure in solids are component has discontinuities at the grain walls. These currently attracting a considerable amount of interest. discontinuities give rise to the internal stray magnetic The self-organization phenomena in garnet ferrite films fields. The mean value of the stray field component with perpendicular anisotropy have received the most along the easy magnetization axis of a grain directed at study [1, 2]. The problem of self-organized criticality angle α with magnetization J of the polycrystal arising in polycrystals in the course of their demagneti- (parallel to applied field H) is determined by the for- zation is widely debated [3, 4]. The basic factor deter- mula [6] mining the transition between two different ordered ()α()α α states of the domain structure is the long-range magne- HM = bJÐ M cos cos ,(1) todipole interaction. It is quite difficult to take proper where b ~ 1 is the parameter depending on the configu- account of this interaction in real materials, and this α sends us to address the analysis of the simplest mod- ration of the grain surface and M( ) is the magnetiza- els [5]. The most complicated problem concerns a the- tion of the grain under study. The domain wall velocity oretical description of self-organization in a three- is proportional to the effective field α dimensional system of domains. Heff = H cos + HM Ð sHω, (2) | α | Ӷ So far, only the spatial distribution of magnetization if Hcos + HM > Hω and Heff Hcrit, where Hω is the was considered in the analysis of self-organization pro- threshold field of the wall displacement, s = cesses. In this paper, we study the evolution of angular α sgn(Hcos + HM), and Hcrit is the Walker critical field distribution of magnetic moments in polycrystalline [7]. Under the above conditions, the variation rate of ferrites under the effect of a pulsed magnetic field. We relative magnetization m in a spherical grain obeys the consider the ferrites with cubic crystal lattice having integro-differential equation the [111] axes of easy magnetization. The analysis is based on the solution of a nonlinear integro-differential dm()α, t 2 ------3= Λα(),t()1Ðx[ht()cos α equation derived here. The equation characterizes the dt dynamics of remagnetization processes in polycrystals (3) and takes into account the interaction between grains, Ð s + 2 p2()α]jt()Ð m()αα,tcos cos . hysteresis, and saturation. Here, m = M/M (M is the saturation magnetization), Following the model concepts put forward in [6], we S S j = J/M , h = H/Hω, x = –2cos((π + arccosm )/3) is the assume that magnetic moments of grains are oriented S along the easy magnetization axes closest to the direc- boundary position normalized by the grain radius and measured with respect to the grain center (–1 ≤ x ≤ 1), tion of the applied field and are uniformly distributed 1/2 within a cone with a linear opening angle 2α [α = and p = [bMS/(2Hω)] is the parameter characterizing m m the magnetodipole interaction between grains. The sat- arccos(1/) 3 ]. Each grain is characterized by a rect- uration and hysteresis are taken into account by param- angular hysteresis loop, its magnetization stems from eter Λ(α, t): the motion of 180° domain walls. Λ ∧ α ≤ ∨ = 1, if ((Heff < 0) (–1 < m( , t) 1)) ∨ ∧ ≤ α ((Heff > 0) (–1 m( , t) < 1)), (4) Astrakhan State Pedagogical University, ul. Tatishcheva 20a, Astrakhan, 414056 Russia otherwise Λ = 0.

m 0.0015

0.0010

0.0005

0 0.8 103t 1.6 2.4 0 0 0.19 0.38 0.57 0.76 0.95 α, rad

Fig. 1. Evolution of the angular distribution of grain magnetization m(α, t) under the effect of alternating-sign magnetic ﬁeld pulses. τ × Ð4 | | τ × Ð5 The ﬁrst pulse: amplitude h1 = 2, duration 1 = 8 10 ; subsequent pulses: h = 2.5, = 3 10 ; the interval between pulses is τ Ð5 p = 10 .

The relative magnetization of the polycrystal is a pulse, the grains with α = 0 have the maximum magne- functional depending on m(α, t): tization, and after the last (51th) pulse, their magnetization turns out to be minimal. If the amplitude of the α m applied field is close to threshold value Hω, then after a j(t) = (1 Ð cosα )Ð1 m(α, t)cosαsinαdα. (5) fairly large number of pulses, the distribution ceases to m ∫ respond to them and has a rather exotic form. 0 Under the effect of rather long and intense field We choose normalization of t in such a way that at h = 2 pulses (see, for example, Fig. 3), the most rapid change in the absence of intergrain interaction (p = 0), the time at the initial stage is observed in the magnetization of needed for magnetization of a grain with α = 0 from the grains with the small angles α (at not very high values | | α ≈ α state m(0, 0) = Ð1 is equal to unity. of h , the domain walls with m can initially be The integration of equation of motion (3) taking into immobile). Then, magnetization rate for grains with the account relationships (4) and (5) was performed by large α gradually increases (the larger is α, the faster is numerical methods for a discrete set of α values. the growth). Finally, the angular distribution of magnetization tends to that shown in Fig. 4. The time dependence of the angular distribution of the grain magnetic moments during magnetization of a In conclusion, the magnetization of polycrystals is polycrystal by the different series of rectangular field accompanied by self-organization related to the magne- pulses h(t) from the demagnetized state [m(α, 0) ≡ 0] at todipole interaction of grains. The self-organization p = 20 are plotted in Figs. 1Ð3. gives rise to the angular distributions of magnetic Under the effect of short alternating-sign pulses, the moments drastically different from those induced by angular distribution of magnetization can be nonmono- the applied field alone. The proposed method for the tonic with a peak and an inflection point (Figs. 1 and 2). analysis of the self-organization processes can be gen- Figure 1 demonstrates that, during the first (positive) eralized by taking into account the statistics of forma-

DOKLADY PHYSICS Vol. 45 No. 8 2000 378 KARPASYUK, BEZNISKO

m 0.0010

0.0005

0 2 4 103t 6 0 8 0.95 0 0.19 0.38 0.57 0.76 α, rad α τ × Ð4 Fig. 2. m( , t) curves under effect of the alternating-sign magnetic ﬁeld pulses. The ﬁrst pulse: h1 = 2, 1 = 6 10 ; the subsequent | | τ × Ð5 τ Ð5 pulses: h = 1, = 9 10 ; p = 10 .

m 1.0

0.5

–0.5 0.95 0.76 0.57 0.38 α, rad –1.0 0.19 0 0.2 0.4 t 0.6 0.8 0 α | | τ τ Fig. 3. m( , t) under effect of the alternating-sign magnetic ﬁeld pulses with amplitude h =10 and durations: 1 = 0.1, = 0.15, and τ p = 0.08.

DOKLADY PHYSICS Vol. 45 No. 8 2000 SELF-ORGANIZATION PROCESSES IN A SYSTEM OF MAGNETIC MOMENTS 379

m 1.0

0.9 1 2 3 0.8

0.7

0.6

0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 α, rad Fig. 4. The eventual angular dependence of grain magnetization in the remanent magnetization state of a polycrystal: (1) p = 10; (2) p = 20; (3) p = 40. tion and annihilation of the domain walls, as well as the 5. F. V. Lisovskiœ and O. P. Polyakov, in Proceedings of XVI distribution in size and coercitivity of the grains. International SchoolÐSeminar (Moscow, Russia, 1998), Part 2, pp. 492Ð493. 6. V. K. Karpasyuk, V. N. Kiselev, G. N. Orlov, et al., in REFERENCES Electromagnetic Properties and Nonstoichiometry of 1. G. S. Kandaurova and A. A. Rusinov, Pis’ma Zh. Éksp. Ferrites with Rectangular Hysteresis Loops (Nauka, Teor. Fiz. 65, 60 (1997) [JETP Lett. 65, 63 (1997)]. Moscow, 1985), p. 149. 2. F. V. Lisovskiœ, E. G. Mansvetova, E. P. Nikolaeva, et al., 7. T. H. O’Dell, in Ferromagnetodynamics: The Dynamics Zh. Éksp. Teor. Fiz. 103, 213 (1993) [JETP 76, 116 of Magnetic Bubbles, Domains, and Domain Walls (1993)]. (MacMillan, New York, 1981; Mir, Moscow, 1983), 3. O. A. Chubykalo and J. M. González, J. Appl. Phys. 83, p. 256. 7228 (1998). 4. O. A. Chubykalo, J. M. González, and J. Gonzalez, Phy- sica D 113, 382 (1998). Translated by Yu. Vishnyakov

DOKLADY PHYSICS Vol. 45 No. 8 2000

Doklady Physics, Vol. 45, No. 8, 2000, pp. 380Ð383. Translated from Doklady Akademii Nauk, Vol. 373, No. 6, 2000, pp. 750Ð753. Original Russian Text Copyright © 2000 by N. Sirota, I. Sirota, Soshnina, Sokolovskiœ.

PHYSICS

Phonon Spectra for Crystal Lattices of V, Cr, Mn, Fe, Co, Ni, and Cu N. N. Sirota*, I. M. Sirota**, T. M. Soshnina*, and T. D. Sokolovskiœ*** Presented by Academician O.A. Bannykh December 8, 1999

Received December 9, 1999

Lattice phonon spectra for 3d elements with a cubic correspond to crystal structures of A1 and A2 types with structure (V, Cr, Mn, Fe, Co, Ni, and Cu) are calculated face-centered cubic (fcc) and body-centered cubic approximately using the Mie–Grüneisen potentials. (bcc) unit cells, respectively. The values of coordina- Calculations were performed based on the BornÐKar- tion numbers zi for each coordination sphere are also manÐDe LaunayÐBlackman (BKDB) method [1Ð4]. The given in Table 1. α β De Launay force constants ij and ij , as well as the For crystal structures of the A1 and A2 types, the BornÐKarman coupling constants Aij, were determined radii of the first coordination spheres are related to lat- using the Mie–Grüneisen potentials α tice constant a by expressions r () = ------and r () = a b 1 A1 1 A 2 = + ---- . 2 U Ð ----m- n (1) r r a3 β ------. Elastic moduli characterizing the rigidity of mon- The De Launay force constants ij correspond to the 2 first derivative of the binding energy U r in the crystal ( )j atomic crystals belonging to the A1 structural type are lattice of element j. The derivative is represented as a expressed in terms of the De Launay force constants [4] function of the nearest neighbor interatomic distance r1 α β α α β α β ac11 =2 1 +++2 1 4 3 12 3 +12 3 ++8 4 8 4, normalized by radius rij of the ith coordination sphere: ac =α Ð45β Ð6β+++αÐ430β α20β ,(4) 1 ∂ Ur() 12 1 1 2 3 3 4 4 β = ------. (2) α β β α β α β ij ∂ ac44 =1 ++++3 1 4 2 6 3 18 3 ++4 4 12 4. rij r rij α 1∂ 2 U The force constants ij are equal to the second deriva- − Bulk moduli B = ------2 characterizing the uniform tive of the binding energy U(r)ij as a function of r [4, 5]: V∂ V compression depend on exponents m and n in the MieÐ ∂2 () α Urj Grüneisen potential [7]: ij = ------. (3) ∂r2 rij mnU B = Ð ------m- . (5) The derivatives are taken at points rij , which corre- 9 V spond to the radius of the ith coordination sphere of ele- The De Launay force constants α and β are expressed ment j (i ≤ 4). ij ij through equilibrium energy U0 of atomization per unit r0 volume and through distance r ≡ r between the nearest The values of ratio ---- = k are presented in Table 1. 0 1 ri neighbor atoms (radius of the first coordination sphere) ≡ by the expressions as follows: Here, ri r0 is the radius of the first coordination sphere and r is the radius of the ith sphere. The data in Table 1 i mn() m + 1 U m + 2 n + 1 nmÐ α =------0 k 1 Ð,------k (6) ij nmÐ 2 m + 1 r 0 * Moscow State University of Environmental Engineering, mn U m + 2 nmÐ ul. Pryanishnikova 19, Moscow, 127550 Russia β=Ð------0 k []1 Ðk . (7) ij nmÐ 2 ** Institute of Control Sciences (Automation r 0 and Telemechanics), Russian Academy of Sciences, Profsoyuznaya ul. 65, Moscow, 117806 Russia Using expressions (6) and (7), we calculated the val- α β *** Institute of Solid State and Semiconductor Physics, ues of force constants ij and ij for the transition ele- Belarussian Academy of Sciences, ments with Z = 23–29 at fixed values of exponents m ul. Brovki 17, Minsk, 220072 Belarus and n specified in Table 2.

Table 1. Coordination numbers zi, coefﬁcients ki = r0/ri in four coordination spheres (IÐIV) for A1 and A2 structural types I II III IV Type of structure zi ki zi ki zi ki zi ki

A1 12 1 6 0.7071 24 0.57736 12 0.5

A2 8 1 6 0.8683 12 0.612 24 0.5222

3 3 Table 2. Binding energies Um (kJ/mol), U0 (kJ/cm ), atomic volumes V (cm ), lattice constants a (Å), nearest neighbor ≡ α β distances r0 r1 (Å), exponents m, n, force constants i, i of four coordination spheres (IÐIV)

Z Element a V Um U0 r1 m n 23 V 3.024 8.34 512.96 65.565 2.619 3.5 7 4 7 24 Cr 2.8846 7.23 396.64 58.48 2.5 6 8 6 10 25 Mn 3.863 7.38 279.073 38.0 2.731 5 7 26 Fe 2.8645 7.09 417.689 62.8 2.477 6 8 6 10 27 Co 3.5441 6.62 495.094 68.452 2.506 3 7 3.5 7 28 Ni 3.5238 6.59 423.84 68.56 2.492 3.5 7 29 Cu 3.6147 7.09 339.322 2.556 3.5 7 I II III IV α β α β α β α β 1 1 2 2 3 3 4 4 36318.33 0.0 1812.38 1861.16 Ð2141.39 573.60 Ð1070.80 261.27 41506.67 0.0 1404.34 2047.72 Ð2307.63 562.05 Ð1083.38 240.66 69576.03 0.0 Ð2413.87 2765.69 Ð2492.74 429.95 Ð874.68 139.95 86970.04 0.0 Ð5252.43 3031.74 Ð2344.59 369.49 Ð743.47 111.32 29180.49 0.0 Ð2579.29 644.83 Ð1040.13 208.03 Ð455.95 85.49 76139.21 0.0 Ð2641.57 3026.58 Ð2728.98 470.51 Ð957.19 153.15 95174.02 0.0 Ð5747.90 3317.72 Ð2565.76 404.35 Ð813.60 121.82 35250.89 0.0 Ð3115.80 1168.45 Ð1759.02 502.59 Ð963.89 258.19 41126.05 0.0 Ð3705.81 1227.44 Ð1907.59 489.07 Ð984.81 236.70 4216.73 0.0 Ð3795.08 1257.01 Ð1953.54 500.85 Ð1008.54 242.40 29773.50 0.0 Ð2682.85 888.62 Ð1381.01 354.06 Ð712.96 171.36

Exponents m and n in potential (1) for a large num- lations were performed by the BKDB method based on ber of elements were determined and discussed by the solution of a secular equation Fürth [8] based on available experimental data. In our paper, the phonon spectra for crystal lattices D(q) Ð Mw2(L) = 0, (8) of transition metal elements with Z = 23Ð29 are determined using the calculated De Launay force constants where M is the atomic mass, L is the unit matrix, and corresponding to four coordination spheres. The calcu- D(q) is the dynamical matrix depending on vector q.

DOKLADY PHYSICS Vol. 45 No. 8 2000 382 SIROTA et al. 5, 4, are are = = 1

2 3 10 m m Hz , K ) ) 300 T 4 3 12 8 = 7 for Co; n , 10 6 200 = 7; ( ν = 7; ( n n = 7 for Ni ( 2 4 n = 10 for Fe ( = 3.5, n 3 = 4, 100 2 = 3.5, m Cu Cu ) m m ) 2 = 6, = 3.5, ) 3 0 0 m 2 m ; ( ) 1 ) V 2 1 10 = 7; ( n = 7; ( 300 8 n = 8; ( n = 7 for = 3, 6 n = 3, 2 200 m m ) ) 1 4 = 6, 1 m : ( = 3.5, 100 ) n 2 Ni Ni 1 m ) 2 0 0 and xperimental points); ( m 1 2 10 2 = 7; ( = 10 for Fe; ( 300 n are e 1 8 n 2 alues of 6 = 2, = 6, 200 3 m m ) ) 4 1 2 wing v xperimental points); ( 100 2 = 7 for Ni ( Co Co n = 8; ( are e 0 n 0 2 1 2 2 10 = 3.5, = 6, 1 300 m m 8 ) ) 1 1 wing parameters: ( 6 200 3 = 7 for Mn ( n 4 100 2 = 5, Fe Fe m = 7 for Mn; ( 0 0 ) n 1 = 23Ð29 calculated using the follo 1 10 Z 300 = 5, xperimental points); ( 8 m 6 200 are e 3 4 2 100 2 xperimental points). = 10 for Cr; Mn Mn n xperimental points); ( 0 0 are e = 7 for Co ( 2 2 1 2 = 8, n 10 are e 1 300 m 3 ) 8 2 = 3.5, 6 200 m 7 for Cu ( ) = 8; ( 2 n 4 n = = 10 for Cr ( 3 = 7 for Cu. 100 2 n Cr Cr = 6, n = 7; ( m n = 3.5, 0 ) 0 1 = 8, m 2 ) = 3.5, ; ( 3 1 m 10 = 2, 4 1 determined based on the phonon spectra calculated using follo V m ) ) 300 m 4 3 2 T ) 8 2 ( 1 1 V C 6 200 = 8; ( = 7 for n n 4 c heat 100 = 6, = 5, V V 2 = 7 for Ni; and m m ) ) n 1 4 Phonon spectra corresponding to the crystal lattices of elements with Speciﬁ 5 0 0

25 15 20 10

3.5,

0.3 0.1 0.4 0.2

. 1. V . 2. C

(mol K) (mol / J , g ) ( = ν = 7; ( = 7; ( xperimental points); ( xperimental points); and ( e n n e Fig m Fig

DOKLADY PHYSICS Vol. 45 No. 8 2000 PHONON SPECTRA FOR CRYSTAL LATTICES OF V, Cr, Mn, Fe, Co, Ni, AND Cu 383

1 observed for copper and, below 100Ð200 K, also for the The calculations were carried out for ----- th of the 48 other elements with relatively small magnetic contribu- Brillouin zone at 1618 points. The points were chosen tion to CV(T). by the Monte Carlo method [5]. In conclusion, we demonstrated for the first time The calculated lattice phonon spectra of elements that the calculations of the phonon spectra for the ele- with Z = 23Ð29 are shown in Fig. 1. The figure also ments of cubic structure performed by the BKDB ν method using the Mie–Grüneisen potential lead us to illustrates the effect on the boundary value m of the variation in m at n = const and the same for n at m = results interesting from the viewpoint both of funda- const. At fixed exponent n, the increase in m, as well as mental science and its applications. the increase of n for m = const, is accompanied by a ν nearly linear increase of boundary frequency m. The REFERENCES determined values of m and n, which do not differ much from to those found by Fürth, correspond to boundary 1. M. Born and K. Huang, Dynamical Theory of Crystal ν ν Lattices (Clarendon Press, Oxford, 1954; Inostrannaya frequency m close to the Debye frequency D. Note Literatura, Moscow, 1958). U that at variation of m, n, and ------0 , their large deviation 2. V. Blackman, Proc. R. Soc. (London), Ser. A 148, 384 r2 (1934). 0 3. De Launay, in Solid State Physics, Ed. by F. Seits and from the equilibrium values causes an “overshoot” of D. Turnbull (Academic Press, New York, 1956), Vol. 2, the initial point of the spectrum related to “negative” pp. 219Ð348. frequencies arising in the Brillouin zone. 4. S. C. Upadhyaya, J. C. Upalhyaya, and R. Shyam, Phys. The equilibrium molar values of atomization ener- Rev. B 44, 122 (1991). gies Um [9] and interatomic distances r0 determined 5. N. N. Sirota and T. D. Sokolovskiœ, in Chemical Bond in using the equilibrium values of lattice constant a [10] Semiconductors and Solids (Consultant Bureau, New and given values of m and n are presented in Table 2. York, 1967), pp. 139Ð143. The values of exponents m and n, for which the calcu- 6. P. Debye, Ann. Phys. 39 (4), 89 (1912). lated values of CV(T) are in the best agreement with 7. N. N. Sirota, in Physics and Physicochemical Analysis experimental data, are m = 6, n = 10 for Cr, m = 6, (Moscow, 1957), Vol. 1, pp. 138Ð150. n = 10 for Fe, m = 5, n = 7 for Mn, and m = 3.5 and n = 7 8. R. Fürth, Proc. R. Soc. (London), Ser. A 193 (992), 87 for Co, Ni, and Cu. (1944). The temperature dependence of specific heat CV(T) 9. Properties of Elements, Part 1: Physical Properties, Ref- is shown in Fig. 2; it is determined based on the calcu- erence Book, Ed. by G. V. Samsonov (Metallurgiya, lated phonon spectra (see Fig. 1). The experimental Moscow, 1976). points [11, 12] are also plotted in addition to the calcu- 10. J. Emsley, The Elements (Claremdon Press, Oxford, lated curves. The experimental data CV(T)expt (points) 1989; Mir, Moscow, 1993). presented in Fig. 2 include not only the lattice contribu- 11. H. Zeise, in Termodynamik. Tabellen (S. Hirzel, Leipzig, tion to the specific heat, but the magnetic and other con- 1954), Vol. 3/1. tributions as well. The deviations of experimental 12. Handbook on Physicochemical Fundamentals of Cryo- points from the calculated CV(T) curves observed at genics, Ed. by M. P. Malkov (Énergoatomizdat, Mos- high temperatures are caused predominantly by the cow, 1985). magnetic contribution to CV(T)expt . A good agreement between calculated and experimental data for CV(T) is Translated by T. Galkina

DOKLADY PHYSICS Vol. 45 No. 8 2000

PHYSICS

Experimental Study of Inelastic Collisions of Low-Velocity Electrons with Terbium Atoms Yu. M. Smirnov Presented by Academician A.F. Andreev December 30, 1999

Received November 30, 1999

1. The terbium atom is a relatively rare object for lated. The energy matrix was parameterized by means study. Restricting ourselves by considering such fields of 11 independent parameters taking into account the of physics as atomic structure, atomic spectroscopy, electrostatic interaction, relativistic effects, and config- and electron-atomic collisions, the latest publication uration overlapping. The parameters were found on the we have found related to this scope of problems is dated basis of known experimental positions for 45 levels of 1991 [1]. In that paper, the high-lying excited and self- TbI by the method of minimization of the mean-square ionization states of TbI were investigated. The study [2] deviation ∆E, ∆E = 41 cmÐ1 being attained in this case. devoted to determining the oscillator strengths for TbI The total number of levels in the configuration under and TbII appeared almost a decade before. Studies in study is 2725. The dimension of the energy matrix which the single-charged terbium ion was investigated attains 377 (for J = 9/2). The hyperfine structure of lev- are almost equally scanty. els was calculated separately. The results of calculation It is possible that such a state of research of the ter- for 57 levels with the parental term 4f 87F, namely, posi- bium atom is one of the principal reasons for extremely tions of levels, weights of three major components, and restricted usage of this element until now. As an illus- the g factor are presented. As was indicated in the tration, we can say that in [3], none of examples for previous item, these results are included in the compi- using terbium was cited. At the same time, at the lation [12]. moment of publication of the book [3], the laser gener- Although there are rather extensive (but not always 5 7 ation on the single D4 F5 transition was already very reliable) data on the TbI spectrum, information realized for the tricharged Tb3+ ion situated in con- concerning atomic constants of the terbium atom is densed media. We imply the organometallic liquid laser almost completely absent. In the paper [14] devoted to on the basis of trifluor-acetylacetonate and the laser detailed measurements of radiative constants for atoms 3+ based on LiYF4 crystal sensitized by Gd and activated and single-charged ions of 70 elements, from several by Tb3+ [4]. However, in gas lasers, no generation using tens to several hundreds of numerical values of these terbium-atom transitions was obtained until now [5]. constants were obtained for the majority of the objects. However, only three values were obtained in this study The most complete (unpublished) tables of the ter- for TbI and none for TbII. There are neither experimen- bium spectrum are prepared by Klinkenberg on the tal nor theoretical data on the cross sections for the basis of spectrograms for an electrode-free discharge, excitation of a terbium atom by slow electrons. which were obtained in the Argonne National Labora- tory (USA). These tables involve approximately 30000 2. In this study, using the method of extended inter- spectral lines for TbI and TbII within the spectral range secting beams, we investigated inelastic collisions of from 233 to 929 nm. A further analysis of these lines low-velocity monoenergetic electrons with terbium was accomplished in the Zeeman Laboratory (Amster- atoms. The instrumentation and experimental methods dam) [6Ð11]. The results of [6Ð11] were subjected to were repeatedly described previously; the most detailed critical analysis in the compilation [12] into which the information is given elsewhere [15]. data of [13], published almost simultaneously with We used metallic terbium of the TbM-1 brand with [12], were also included. a total impurity content of less than 0.1% (the major In [13], the energy structure of the even low-level impurities were gadolinium, disprosium, and ittrium). configuration 4f 85d6s2 of the terbium atom was calcu- Terbium was evaporated from a tantalum boat in the course of electron-beam heating. The usage of a cruci- ble (of the Knudsen-cell type) with a low expense of a metal was impossible because the available electrically Moscow Power Engineering conducting constructional materials cannot hold long- Institute, ul. Krasnokazarmennaya 17, term contact with the melted terbium and fail. Temper- Moscow, 111250 Russia ature of metal at the basic stage of the experiment

EXPERIMENTAL STUDY OF INELASTIC COLLISIONS OF LOW-VELOCITY ELECTRONS 385

1.0 1.0

16 0.5 0.5 15 7

6 14 0 0 5 13 4 12

, arb. units 11

Q 10 9 3 2 8

1 2 5 10 20 50 100 200 2 5 10 20 50 100 200 E, eV Fig. 1. Optical excitation functions for terbium atom. attained 1800 K. In this case, the atomic concentration presence of such a distribution must be taken into in the region of the intersection of the atomic and elec- account when comparing the results of this study with tron beams was 7.0 × 109 cmÐ3. To minimize the reab- data of theoretical calculations (as only such results sorption while investigating the most intense transi- will appear). tions to low-lying levels, the atomic concentration was The width of the electron-beam energy distribution reduced almost by an order of magnitude. corresponded to 0.9 eV at an energy of 100 eV and to The evaporation of terbium atoms is inevitably 1.0 eV at 20 and 200 eV (for 90% of electrons). Within accompanied by the occupation of low-lying levels in the entire working energy range, the current density did 2 TbI as a result of the action of the thermal excitation not exceed 1.0 mA/cm . The spectral resolution of the mechanism. The existing technique renders it impossi- setup attained approximately 0.1 nm in the short-wave λ ≤ ble to carry out either the selection of atoms excited to range ( 600 nm) and approximately 0.2 nm in the one of the low-lying levels or the measurement of their yellowÐred spectral region. An error in measuring rela- concentration under conditions of this experiment. tive values of cross sections was within 3 to 10% Estimating the population of these levels under the depending on the line intensity. An error in determining absolute values of the cross sections was within ±16 to assumption that the Boltzmann distribution is valid, we ± find the following values for the total percent concen- 23%. Principal error sources and their contribution to the resulting error are analyzed in [15]. 9 2 6 ° tration of atoms in the beam: 27.8 for 4f 6s H15/2 (0); 8 28 8 28 3. In spectrograms detected at an exciting-electron 19.4 for 4f 5d6s G13/2 (285); 19.2 for 4f 5d6s G15/2 energies of 30 eV, we find nearly 630 spectral lines of 8 28 (462); 13.9 for 4f 5d6s G11/2 (510); 5.83 for the terbium atom. Within the electron-energy range 0 to 4f 85d6s2 (1371); 3.31 for 4f 85d6s28D (2310); 2.03 200 eV, 103 optical excitation functions (OEFs) were 9/2 11/2 detected. For almost all the transitions, the upper and 8 28 9 2 6 ° lower levels between which these transitions occur for 4f 5d6s G7/2 (2419); 2.68 for 4f 6s H13/2 (2772); 8 28 were established. However, for the very majority of the and 1.81 for 4f 5d6s G9/2 (2840). The numbers in the upper levels from the whole set of quantum character- parentheses correspond to the level energies (expressed istics, only the parity and the value of the quantum in cmÐ1) counted off from the ground level of the ter- number J are known. That is why transitions with the bium atom. Here, all TbI levels below 3000 cmÐ1 are participation of these levels are essentially inaccessible indicated. As follows from these data, the summary right now for theoretical analysis. Therefore, we dis- population of odd levels amounts to only 30.5%, cuss in this study only such transitions for which the whereas that of even levels is 65.5% (approximately upper levels are completely classified (or, at least, the 4% correspond to high-lying levels). Undoubtedly, the configurations are established), and also, the OEFs are

DOKLADY PHYSICS Vol. 45 No. 8 2000

386 SMIRNOV

Excitation cross sections of terbium atom Q , Q , λ, nm Transition J E , cmÐ1 E , cmÐ1 30 max E , eV OEF l u 10Ð18 cm2 10Ð18 cm2 max 370.311 4f85d6s28GÐ4f85d26p? 15/2Ð17/2 462 27458 12.7 13.1 18 16 371.790 4f85d6s28GÐ4f85d26p? 11/2Ð13/2 509 27399 2.59 2.70 15Ð20 15 390.133 4f85d6s28GÐ4f85d26p? 15/2Ð17/2 462 26087 117.0 123.0 18 14 431.883 4f96s26H°Ð4f96s6p (15/2, 1) 15/2Ð15/2 0 23147 248.0 261.0 16 10 432.643 4f96s26H°Ð4f96s6p (15/2, 1) 15/2Ð17/2 0 23107 331.0 348.0 16 9 433.643 4f96s26H°Ð4f96s6p (13/2, 1) 13/2Ð15/2 2771 25825 37.0 40.2 17 13 433.841 4f96s26H°Ð4f96s6p (15/2, 1) 15/2Ð13/2 0 23043 224.0 251.0 14 8 435.681 4f96s26H°Ð4f96s6p (13/2, 1) 13/2Ð13/2 2771 25717 41.4 42.6 20 12 437.202 4f96s26H°Ð4f96s6p (13/2, 1) 13/2Ð11/2 2771 25637 25.7 29.2 14 11 438.245 4f85d6s28HÐ4f85d26p? 17/2Ð17/2 4646 27458 19.7 20.3 18 16 466.279 4f85d6s28HÐ4f85d6s6p? 17/2Ð17/2 4646 26087 12.2 12.8 18 14 489.813 4f85d6s28HÐ4f85d26p? 11/2Ð13/2 6988 27399 0.65 0.68 15Ð20 15 491.523 4f85d6s26GÐ4f85d26p? 13/2Ð13/2 7059 27399 1.17 1.22 15Ð20 15 509.229 4f85d6s26HÐ4f85d26p? 15/2Ð13/2 7767 27399 0.81 0.84 15Ð20 15 518.848 4f85d26s10GÐ4f85d26p? 15/2Ð17/2 8190 27458 0.96 0.99 18 16 522.199 4f85d6s28GÐ4f85d6s6p? 15/2Ð13/2 462 19606 3.89 6.06 7.5 7 522.812 4f85d26s10GÐ4f85d26p? 13/2Ð13/2 8277 27399 3.22 3.35 15Ð20 15 523.511 4f85d6s28GÐ4f85d6s6p? 11/2Ð13/2 509 19606 4.81 7.50 7.5 7 524.871 4f85d6s28GÐ4f85d6s6p? 13/2Ð11/2 285 19332 5.85 7.21 9.5 6 531.923 4f85d6s28GÐ4f85d6s6p? 13/2Ð15/2 285 19080 12.4 15.5 13 3 533.104 4f85d26s10GÐ4f85d26p? 11/2Ð13/2 8646 27399 1.38 1.44 15Ð20 15 534.604 4f85d6s28GÐ4f85d6s6p? 15/2Ð17/2 462 19162 7.45 12.0 9.0 5 535.488 4f85d6s28GÐ4f85d6s6p? 15/2Ð17/2 462 19131 12.0 15.8 13 4 536.972 4f85d6s28GÐ4f85d6s6p? 15/2Ð15/2 462 19080 13.3 16.6 13 3 550.961 4f85d6s28GÐ4f85d6s6p? 15/2Ð13/2 462 18607 2.68 3.26 6.5 2 552.412 4f85d6s28GÐ4f85d6s6p? 11/2Ð13/2 509 18607 5.65 6.88 6.5 2 560.058 4f85d6s28GÐ4f85d6s6p? 13/2Ð13/2 285 18135 1.15 2.87 10.0 1 567.184 4f85d6s28GÐ4f85d6s6p? 11/2Ð13/2 509 18135 9.90 24.7 10.0 1 613.439 4f85d6s28DÐ4f85d6s6p? 11/2Ð13/2 2310 18607 2.09 2.55 6.5 2 688.730 4f85d6s28HÐ4f85d6s6p? 17/2Ð17/2 4646 19162 6.02 9.70 9.0 5 690.198 4f85d6s28HÐ4f85d6s6p? 17/2Ð17/2 4646 19131 6.31 8.30 13 4 831.706 4f85d6s26GÐ4f85d6s6p? 13/2Ð15/2 7059 19080 29.9 37.3 13 3 844.397 4f85d6s26HÐ4f85d6s6p? 15/2Ð13/2 7767 19606 54.4 84.8 7.5 7 848.359 4f85d6s28HÐ4f85d6s6p? 13/2Ð13/2 6351 18135 46.4 116.0 10.0 1

measured. Moreover, we omit several transitions OEF maximum, respectively, and the position Emax of blended by relatively intense unclassiﬁed lines. this maximum. The enumeration of the OEFs in the table and of the curves in Fig. 1 coincide with each The results of the measurements are shown in the other. Here, we used the logarithmic scale along the λ table. Here, we present only the wavelength , transi- abscissa axis and the linear scale along the ordinate tion notation, inner quantum number J, energies El and axis. In this case, each curve is normalized to unity in Eu of the lower and upper levels, and the cross sections its maximum and has an individual zero for counting Q30 and Qmax at an electron energy of 30 eV and in the off along the ordinate axis.

DOKLADY PHYSICS Vol. 45 No. 8 2000 EXPERIMENTAL STUDY OF INELASTIC COLLISIONS OF LOW-VELOCITY ELECTRONS 387

The state diagram for the terbium atom is presented in Fig. 2, where all known TbI conﬁgurations are E = 47295 cm–1 shown. The high-lying levels found in [1] are shown in i Fig. 2 by shaded blocks. We have determined positions for 85 even (I) and 135 odd (II) levels. However, the measurement accuracy for the level energies in [1] corresponds to ±0.3 cmÐ1, and no quantum numbers are 4 known for them by now, including J. It should be taken into account that only the two lowest terms are established deﬁnitely for odd states, I II namely, the ground term 4f 96s26H° and the excited 8 7 2 9 term 4f ( F6)6s 6p1/2(6, 1/2)° nearest to the ground one. 4f 6s7s In this case, even within the ground term, the two high- 3 est levels with J = 7/2 and 5/2 are unknown. Almost all 4f 85d 26p

the terms and conﬁgurations for the levels with an –1 energy higher than that, corresponding to 13 700 cmÐ1, cm are established only hypothetically. The mixing para- 4

meters are known only for the ground-term levels that , 10 4 involve 93 to 97% of the main component and, thus, E are very close to the pure levels in the LS-coupling 2 symbolism. 3 9 None of the odd configurations was completed. In 4f 6s6p 4f 95d6s Fig. 2, this fact is shown by dashed lines closing every configuration from the side of higher energies. The 4f 85d6s6p degree of incompleteness can be illustrated by an 4f 86s26p example contained in Section 1 of this paper: From the 2 total number 2725 levels in the even configuration 1 4f 85d26s f8 2 4 5d6s , only 57 were calculated in [13]. In the odd 1 configuration 4f 85d26p, only six levels are known; in the higher lying configuration 4f 96s7s, only two levels are known. However, they are cited in [12] as definitely established. According to the data given in [12], the 8 2 Ð1 4f 5d6s 4f 96s2 boundary of the highest odd levels is near 35000 cm , 0 the first ionization limit of TbI being 47295 cmÐ1. Here, we imply all the experimental levels independently of Fig. 2. Diagram for states of terbium atom with transitions the presence or absence of their classification (the lev- investigated. els from [1] are shown separately). In the even part of the diagram, the situation is table show that the transitions from the levels of the first somewhat more favorable. For the lowest configuration of these terms have the largest excitation cross sections 8 2 4f 5d6s , as was noted earlier, we classified 57 levels, among measured ones. The excitation cross sections for the majority of which are related to nine definitely the levels of the second of the above terms are also large established terms. Moreover, the mixing parameters are × Ð17 2 known for the levels of these terms. Only four terms and attain (3Ð4) 10 cm , whereas, for the most fre- from nine remain partially unfilled (1 to 2 levels are quently appearing level of 27 399 cmÐ1, the total exci- deficient). In the next configuration 4f 85d26s, we have tation cross section is only 0.98 × 10Ð17 cm2 for the definitely established only the lowest term 30-eV electron energy. In this case, as can be seen from 4f8(7F)5d2(3F)(9G)6s10G for which, however, the high- Fig. 1, the OEFs for the levels related to the same term est level with J = 1/2 is not found. The decuplet level are very similar in their shape (curves 8, 9, 10 and 11, 8 7 2 3 9 10 4f ( F)5d ( F)( I)6s I21/2 is also definitely established. 12, 13, respectively). The most probable reason for the Finally, in the even configuration 4f 96s6p, 11 levels marked difference between the OEF 12 and two neigh- are known. For four of them, the terms are hypotheti- boring ones is likely the spectral overlapping of the uni- cally established, and an additional six levels belong dentified line. Some other OEFs have also a significant definitely to two terms written out in denotations of the similarity in their shape (1 and 5, 3 and 4, 14 and 15, or J J -coupling: 4f 9(6H°)6s6p( 1P° ) (15/2, 1) and 14 and 16). However, their belonging to particular 1 2 1 terms remains unknown, so that this similarity can be 9 6 ° 1 ° 4f ( H13/2 )6s6p( P1 ) (13/2, 1). The data given in the caused by random reasons.

DOKLADY PHYSICS Vol. 45 No. 8 2000 388 SMIRNOV

The group of transitions denoted by arrow 3 in 2. V. N. Gorshkov, V. A. Komarovskiœ, A. L. Osherovich, Fig. 2 should be specially mentioned. The group of and N. P. Penkin, Opt. Spektrosk. 52, 935 (1982) [Opt. transitions 1, 2, and 4 are quite permissible by the selec- Spectrosc. 52, 562 (1982)]. tion rules in the dipole approximation because all of 3. Popular Library of Chemical Elements, Ed. by I. V. Pet- them correspond to the configuration change 6p ryanov-Sokolov (Nauka, Moscow, 1983), Vol. 2. 6s. To the contrary, the transitions of group 3 arise in 4. Laser Handbook, Ed. by A. M. Prokhorov (Sov. Radio, the case of combining the 4f 85d26p and 4f85d6s2 config- Moscow, 1978), Vol. 1. urations, i.e., in the case of a radical reconstruction 5. V. A. Gerasimov, Opt. Spektrosk. 87, 156 (1999) [Opt. 5d6p 6s2 of the electron shell. Such a transition Spectrosc. 87, 144 (1999)]. cannot be realized in the single-electron approximation 6. P. F. A. Klinkenberg, Physica (Amsterdam) 32, 1113 and is likely realized only due to the presence of a sig- (1966). nificant mixing of configurations. Actually, for low- 7. P. F. A. Klinkenberg and E. Meinders, Physica (Amster- dam) 32, 1617 (1966). lying levels (the 4f85d6s2 configuration), a significant 8. P. F. A. Klinkenberg, Physica (Amsterdam) 37, 197 mixing takes place [12]. For the transitions from high- (1967). lying levels of the group 3, the necessary data are 9. E. Meinders and P. F. A. Klinkenberg, Physica (Amster- absent. dam) 38, 253 (1968). 4. Thus, for the first time, the terbium-atom excita- 10. P. F. A. Klinkenberg and E. Meinders, Physica (Amster- tion by low-velocity monoenergetic electrons was dam) 42, 213 (1969). investigated, and the excitation cross sections were 11. P. F. A. Klinkenberg, Physica (Amsterdam) 57, 594 determined. The results obtained can be used as the ref- (1972). erence data in solving problems of physics and chemis- 12. W. C. Martin, R. Zalubas, and L. Hagan, Atomic Energy try of plasmas, astrophysics, etc. At the same time, Levels. The Rare Earths Elements (National Bureau of information obtained on the atomic constants of ter- Standards, Washington, DC, 1978). bium atom can provide for progress in the extremely 13. C. Bauche-Arnoult, J. Sinzelle, and A. Bachelier, J. Opt. complicated theoretical analysis of the TbI structure Soc. Am. 68, 368 (1978). and spectrum. 14. C. H. Corliss and W. R. Bozeman, Experimental Transi- tion Probabilities for Spectral Lines of Seventy Elements (National Bureau of Standards, Washington, DC, 1962; Mir, Moscow, 1968). REFERENCES 15. Yu. M. Smirnov, J. Phys. II (France) 4, 23 (1994). 1. V. N. Fedoseev, V. I. Mishin, D. S. Vedeneev, and A. D. Zuzikov, J. Phys. B 24, 1575 (1991). Translated by V. Bukhanov

DOKLADY PHYSICS Vol. 45 No. 8 2000

Doklady Physics, Vol. 45, No. 8, 2000, pp. 389Ð391. Translated from Doklady Akademii Nauk, Vol. 373, No. 5, 2000, pp. 615Ð617. Original Russian Text Copyright © 2000 by Isaev, Leont’ev, Baranov, Usachev.

POWER ENGINEERING

Bifurcation of Vortex Turbulent Flow and Intensification of Heat Transfer in a Hollow

S. A. Isaev*, Academician A. I. Leont’ev**, P. A. Baranov*, and A. E. Usachev*** Received April 27, 2000

Analysis of the mechanisms underlying the vortex- hollow. In general, this fact agrees with similar pro- induced intensification of heat transfer accompanying cesses that occur in the near-wall zone when a jet is the fluid flow along surfaces consisting of concave ele- injected at a finite angle with respect to the incident ments stimulates studies that reveal both the physical flow [8]. Then, the transition from a symmetric two- nature of large-scale vortex objects generated in hol- vortex flow pattern to that with one vortex is accompa- lows and their relation to the heat transfer from a wall. nied by a considerable increase in the velocity of the Experimental works concerning this problem (see, for secondary flow. example, [1]) demonstrate, first of all, that there exist non-steady-state cyclic processes involving the forma- In this paper, the main emphasis is placed on the relation between the dynamics of vortex structures in a tion of vortex flows. It is usually assumed that the latter ε are responsible for the high level of heat transfer at hollow, the extent of its deformation (spherical and comparatively low hydrodynamic losses. However, it is elliptic parts of the hollow conjoin with each other), very difficult to control these processes because their and the relative heat transfer from a surface element manifestation depends on the combination of many fac- containing the hollow. Particular attention is given to tors such as relative values of the hollow depth and a the bifurcation occurring in the vortex flow around the radius of curvature characterizing its sharp edge, the spherical hollow, which is accompanied by an abrupt arrangement of hollows at the plane (in the case of their change of the thermal regime near the wall. Similarly ensemble), the thickness of boundary layer, and the to [4], spatial jetÐvortex structures were revealed by the degree of turbulence in the incident flow. computer visualization of the flow based on tracing tagged particles in the fluid. As a basic geometrical An alternative concept is presented by the vortex- configuration, we consider a spherical hollow (0.22 in induced intensification based on steady-state asymmet- depth) having rounded edges (of radius 0.1), which ric flows around the hollows related to their asymmetric supports a steady-state turbulent flow with the devel- shapes. Numerical studies of laminar [2, 3] and turbu- oped zone of flow separation. lent [4] flows, which occur around a deep isolated hol- For the numerical simulation of turbulent flow in the low at the plane, show that one-sided transverse defor- vicinity of curved surface elements, we use an mation of the hollow can substantially change the vor- approach based on implementation of multiblock nets. tex pattern in its vicinity and intensify the fluid motion This approach was tested in calculations of two-dimen- in the direction transverse to the incident flow. In addi- sional flows with vortex cells [6]. The constructed fac- tion, the laminar flow in the asymmetric hollow does torized algorithm is based on the implicit finite-volume not change the vortex pattern with two large-scale vor- method of solving the NavierÐStokes equations with tex cells observed in spherical hollows [5Ð7], whereas the Reynolds averaging, which are closed according the in the turbulent regime arising there, the transition to a Menter zonal two-parameter model of turbulence [9]. single-vortex tornado-shaped flow pattern turns out to The algorithm involves partitioning of the calculation be possible [2]. In the latter case, the flow becomes region and generation of oblique overlapping nets of much more intense in the direction transverse to the the H and O types in selected subdomains having essentially different scales. The starting set of equations is written in the divergence form with respect to * Academy of Civil Aviation, increments of dependent variables, where the latter ul. Pilotov 38, St. Petersburg, 196210 Russia include the Cartesian components of velocity. In the ** Bauman State Technical University, source terms appearing in the equations for momen- Vtoraya Baumanskaya ul. 5, Moscow, tum, the convection-induced flows are approximated 107005 Russia using the Leonard one-dimensional counterflux *** State Research Center Zhukovskii Central Institute scheme with quadratic interpolation. In the transport of Aerohydrodynamics, equations for turbulent characteristics, these terms are ul. Radio 17, Moscow, 107005 Russia approximated by the UMIST counterflux scheme [10],

390 ISAEV et al. (the minimum near-wall step is equal to 0.0008). In the radial direction, we specify 40 cells concentrating (a) z toward the boundary of the hollow (the minimum step x is equal to 0.015). The considered subdomain of the hollow is covered by a large-scale rectangular domain, where the base of the latter coincides partly with the flat wall interacting with the flow. The origin of Cartesian coordinates coincides with the projection of the hollow center onto the plane. The domain length, height, and width are equal to 17, 5, and 10, respectively. It is partitioned by the Cartesian net containing 55 × 40 × 45 cells. Nodes of the net are concentrated in the vicinity of the hollow (the minimum step in the longitudinal and transverse directions is equal to 0.15) and near the wall (the near-wall step is equal to 0.001). To better resolve the near-wall flow in a vicinity of 0.1qΣ/q0 the axis of the cylindrical subdomain, we introduce the (b) 4 “patch” that passes through this subdomain and has a 0.15 3 shape of a curved parallelepiped. The 19 × 19 net having a uniform node distribution in the longitudinal and 0.10 5 0.05 transverse directions is constructed inside it. Steps of this net are consistent with the near-boundary step in 0 umin the neighboring cylindrical domain. In the same man- wmax ner, the arrangement of the net nodes is adjusted in the wmin ε vertical direction. –0.15 2 The velocity profile specified at the entrance bound- –0.20 1 ary of the domain corresponds to the 1/7 profile of the –0.25 0 0.05 0.10 0.15 0.20 turbulent boundary layer, where the thickness (0.175) ε of the latter is close to the hollow depth. We specify soft boundary conditions (those of continuation of the solu- Fig. 1. (a) Computer visualization of jetÐvortex objects aris- tion) at the exit boundaries and the nonslip condition at ing in the asymmetric hollow and (b) the effect of shape the heated isothermal wall. The velocity of the incident asymmetry ε on the minimum values of (1) longitudinal flow outside the boundary layer and the hollow diame- (umin) and (2) transverse (wmin) velocities, (3) on the maxi- ter are used as the normalization parameters to reduce mum value of the transverse velocity wmax, and (4) 1/10 of the problem to the dimensionless form. The Reynolds the ratio of the total heat load qΣ per circular element con- number is assumed to be equal to 2.35 × 104. While taining the hollow and the load q0 per element without the solving the heat transfer problem, we take the values hollow. Dashed line (5) corresponds to the calculations corresponding to the symmetric vortex flow pattern in the 0.7 and 0.9 for the laminar and turbulent Prandtl num- spherical hollow, which is presented in Fig. 2a. bers, respectively. Some numerical results obtained using the TECPLOT computer-based system for visualization of spatial which is a modification of the TVD scheme. In its main fields are presented in Figs. 1 and 2. features, our methodology is similar to the one used ε in [4]. For the asymmetric hollow under study ( = 0.2), a flow pattern is shown in Fig. 1a. It has a pronounced For a more accurate description of structural ele- asymmetric character, where sidewash occurs in the ments corresponding to the different scales in the turbu- direction of the deformed part of the hollow and the lent flow around a deep hollow, such as, the shear layer pressure maximum is shifted toward the opposite side. and the backward-flow zone, it is reasonable to separate As for the spherical part of this hollow, the pattern is out a near-wall domain surrounding the hollow. It is the approximately the same as in the case of the symmetric cylindrical ring of the outer radius equal to unity, the hollow in the laminar flow regime [2, 3]. It is character- inner radius of 0.1, and height 0.175, where all of the ized by a focus-type singularity, which is situated at the linear dimensions are measured in units of the hollow periphery of the hollow, and the swirling jet flow forms diameter. The domain under study is partitioned by an in the vicinity of this singularity. Inside the asymmetric oblique curved net, which conforms to the surface hollow, the large-scale vortex flow has a snail-like interacting with the flow. We place 60 uniformly dis- structure with a single-vortex tornado. This structure tributed cells along the circle, 45 cells in a vertical clearly manifests itself at the tracing of tagged particles direction with higher node concentration near the wall introduced at different spatial points. Similar to the

DOKLADY PHYSICS Vol. 45 No. 8 2000

BIFURCATION OF VORTEX TURBULENT FLOW AND INTENSIFICATION 391

exceeding the heat transfer from the flat wall by a factor of 1.7. Relative heat transfer from the element contain- (a) z ing a hollow decreases with an increase in the degree of x the hollow asymmetry, since the fraction of the hollow area in the area of the chosen element decreases. The patterns of two steady-state turbulent flows occurring in the spherical hollow are compared in Fig. 1. One of them was obtained in the absence of strong input perturbation caused by initial asymmetry of the hollow, while the other demonstrates the effect of such a perturbation. Symmetric vortex structure, characteristic of laminar flows in the spherical hollow [2] (Fig. 2a), transfers to the single-vortex tornado structure (Fig. 2b); that is, we have the vortex-flow bifurcation. A similar bifurcation arises in the separation flow that occurs in the plane-parallel symmetric channel with an abrupt widening [11]. This bifurcation leads to a significant decrease in the heat transfer in the case of the symmetric vortex pattern.

(b) z ACKNOWLEDGMENTS x This work was supported by the Russian Foundation for Basic Research, projects nos. 00-02-81045 and 99-02-16745.

REFERENCES 1. V. V. Alekseev, I. A. Gachechiladze, G. I. Kiknadze, et al., in Proceedings of the II Russia National Confer- ence on Heat Transfer, Moscow, 1998, Vol. 6, pp. 33Ð42. 2. S. A. Isaev, A. I. Leont’ev, A. E. Usachev, et al., in Pro- ceedings of the II Russia National Conference on Heat Transfer, Moscow, 1998, Vol. 6, pp. 121Ð124. 3. S. A. Isaev, A. I. Leont’ev, A. E. Usachev, et al., Izv. Akad. Nauk, Énerg., No. 2, 126 (1999). 4. S. A. Isaev, A. I. Leont’ev, and P. A. Baranov, Pis’ma Zh. Tekh. Fiz. 26 (1), 28 (2000) [Tech. Phys. Lett. 26, 15 (2000)]. Fig. 2. (a) Symmetric and (b) asymmetric patterns of jetÐ 5. S. A. Isaev, A. I. Leont’ev, D. P. Frolov, et al., Pis’ma Zh. vortex structures arising in the spherical hollow at the plane Tekh. Fiz. 24 (6), 6 (1998) [Tech. Phys. Lett. 24, 209 in the case of turbulent flow around the hollow. (1998)]. 6. S. A. Isaev, P. A. Baranov, A. E. Usachev, et al., in Pro- ceedings of the IV ECCOMAS CFD Conference, Athens, laminar flow in the asymmetric hollow [2, 3], the sepa- 1998, Vol. 1, Part 2, pp. 768Ð774. ration zone is not closed. As a result, the incident flow 7. S. A. Isaev, in Problems of Gas Dynamics and Heat and penetrates the hollow through the open lateral window Mass Transfer in Power Engineering: Proceedings of from the side of its spherical part. This effect substan- the XII School-Workshop of Young Scientists and Engi- tially intensifies the vortex motion in the fluid. neers organized by Academician A. I. Leont’ev, Moscow, When parameter ε characterizing the degree of the 1999, pp. 17Ð20. hollow deformation varies from 0 to 0.2, both the extre- 8. F. S. Henry and H. H. Pearcey, AIAA J. 32, 2415 (1994). mum values of the transverse velocity component and 9. F. R. Menter, AIAA J. 32, 1598 (1994). the minimum value of the longitudinal backward-flow 10. F. S. Lien, W. L. Chen, and M. A. Leschziner, Int. J. velocity in the hollow remain nearly unchanged. Nev- Numer. Methods Fluids 23, 567 (1996). ertheless, the ratio of the total heat load qΣ per a circular 11. F. Battaglia, S. J. Tavener, A. K. Kulkarni, et al., AIAA element of unit radius containing the asymmetric hol- J. 35, 99 (1997). low and load q0 applied to a similar element without a hollow exhibits a pronounced peak with the height Translated by Yu. Verevochkin

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

A Method of Transferring Boundary Conditions by CauchyÐKrylov Functions for Rigid Linear Ordinary Differential Equations A. Yu. Vinogradov and Yu. A. Vinogradov Presented by Academician G.G. Chernyœ November 12, 1999

Received November 22, 1999

The problem of stable numerical transferring of the We divide the interval [0, 1] into segments of the sta- boundary conditions for rigid linear ordinary dif- ble evaluation by the points ferential equations was first considered in 1961 by x = 0, Abramov [1] and Godunov [2]. After separation of vari- 0 ,,,,,,,,,… … ables, partial differential equations in deformation x1 x2 xi Ð 1 xi xi + 1 xn Ð 2 xn Ð 1 xn = 1. mechanics for plates and shells are reduced to these It is well known [3] that the general solution to dif- equations. In numerically integrating differential equa- ferential equation (1) for an arbitrary stable-evaluation tions by step methods, e.g., in the Godunov method, the segment can be written in two forms. When integrating orthonormalization procedure is applied for stable cal- from the left to the right, it is culations. Y =Kx(),x Y+,Y()x,x (3) In this study, we demonstrate a radically new i iÐ1 i iÐ10iÐ1i method for improving the efficiency of transferring the and when integrating from the right to the left, it is boundary conditions to a given point x* within the inte- Y=Kx(),x Y+Y().x,x gration interval [0, 1]. This method is not time consum- ii+1 i i+10i+1i (4) ing, decreases requirements to the computer RAM, and Here, K(xi – 1, xi) and K(xi + 1, xi) are the matrices of the provides a given error of calculations. numerical values for of fundamental CauchyÐKrylov We consider a boundary value problem represented functions for homogenous differential equation (1), i.e., by the system of first-order differential equations in the when Q = 0, and Y0(xi – 1, xi) and Y0(xi + 1, xi) are columns matrix form for the partial solutions to equation (1). It is worth noting that solutions (3) and (4) to differ- Y' =FYQ+, (1) ential equation (1) are derived without allowing for the T boundary conditions since the CauchyÐKrylov func- where (*)' = d(*)/dx; Y = {y1(x), …, ym(x)} is the tions satisfy arbitrary initial conditions of the problem. unknown vector function with the dimension m, F = || || It is evident that solutions (3) and (4) using matrices K fij(x) is the matrix of variable coefficients fij(x) with for the values of the CauchyÐKrylov function and the × T the dimension m m, and Q = {q1(x), …, qm(x)} is the partial solutions Y0 derived on their basis link the val- vector function with the dimension m standing on the ues of the unknown quantities of the columns Yi – 1, Yi right-hand side of equation (1) and describing an exter- and Yi + 1 at edges of an arbitrary chosen stable-evalua- nal load. The boundary conditions at the edges of the tion segment. This property is used for successive inde- interval [0, 1] are given by the relations pendent transferring the boundary conditions in each of H()0Y()0 ==r(),0 H()1Y()1 r(),1 (2) adjacent segments of the stable evaluation, starting L L R R from the edge ones to a given point x* within the inter- where HL and HR are the matrices of the boundary con- val [0, 1]. ditions having dimensions (m – r) × m and r × m and the The values of the fundamental CauchyÐKrylov ranks m – r and r, respectively. rL and rR are vectors functions for a homogeneous differential equation with with the dimensions m – r and r, respectively. the constant coefficients F = const, i.e., the matrix K for the stable-evaluation segment, are determined with the help of the matrix Taylor series mk= Institute of Applied Mechanics, , ()∆m Russian Academy of Sciences, Kx()iÐ1 xi =∑Fx/m! Leninskiœ pr. 32a, Moscow, 117312 Russia m=0

1028-3358/00/4508-0392 $20.00 © 2000 MAIK “Nauka/Interperiodica” A METHOD OF TRANSFERRING BOUNDARY CONDITIONS 393 ∆ ( x = xi – xi – 1 = const is the length of the stable-evalu- From the second equation of the same system, we have 2 ation segment), which is derived by the successive W = {w , …, w } and ω , where w = w' /(w' + … + Picard approximation or on the basis of Newton matrix 2 21 2m 2 2k 2k 11 2 1/2 ω ω 2 2 1/2 binomial formula w1' m ) , k = 1, …, m, and 2 = 2' /(w21' + … + w2' m ) , ω , ( ∆ )m where w2' k = h2k – (h2, w1)w1k and 2' = r2 – (h2, w1)r1. K(xi Ð 1 xi) = E + F xm Henceforth, (hi, wj) = (hi1wj1 + … + h1mwjm). (∆x = ∆x/m = const, m is the number of sections, into m From the ith equation, we ﬁnd Wi = {wi1, …, wim} which the stable-evaluation interval ∆x is divided, and ω 2 2 1/2 E is the identity matrix), which is, by definition, the and i , where wik = wi'k /(wi'1 + … + wi'm ) , k = 1, …, solution to the Volterra integral. ω ω 2 2 1/2 m, and i = i' /(wi'1 + … + wi'm ) , where wi'k = hik – For differential equations with variable coefﬁcients, ω the values of the fundamental CauchyÐKrylov func- (hi, w1)w1k – (hi, w2)w2k – … – (hi, wi – 1)wi – 1, k and i' = tions are calculated, with the multiplicativity of the Vol- ri – (hi, w1)r1 – (hi , w2)r2 – … – (hi, wi – 1)ri – 1. terra integral taken into account, provided that the inte- We now denote the orthonormalization procedure gration direction coincides with the positive direction by the operator M. Then, of the x-axis. This is obtained by multiplying the matrix ∆ {WY = w} = M{HY = r}. Taylor series in the stable-evaluation segment x = xi – xi – 1 by the formula Transferring the boundary conditions from the left

j = 1 m = k edge over adjacent segments of the stable evaluation to , ( ∆ )m a given point x* of the interval [0, 1] consists in the fol- K(xi Ð 1 xi) = ∏ ∑ F j x j /m!. lowing. j = e m = 0 The orthonormalized conditions HL(0)Y(0) = rL(0) ∆ ∆ Here, xj = x/e = const is the length of the stable-eval- at the left edge or those already transferred to the point uation segment, xi , positioned to the left of the point x* within the interval [0, 1], are transferred to the point x . This is per- , , , …, , , , …, , , ∈ ∆ i + 1 x0 x1 x2 x j Ð 1 x j x j + 1 xe Ð 2 xe Ð 1 xe x j; formed using the values of the fundamental CauchyÐ Krylov functions by excluding from the conditions j = 1 , ( ∆ ) W (x )Y(x ) = w (x ) K(xi Ð 1 xi) = ∏ E + F j x j , L i i L i j = e the column Y(xi). In this case, solution (4) is used, ∆ which was obtained by integrating equation (1) in the where the quantity xj is chosen from the condition for admission of averaging the variable elements of the direction from the right to the left, i.e., in the direction matrix F in equation (1). not coincident with the positive direction of the x-axis. As a result, we have the new vector-matrix equation In the case of the opposite direction of integrating, the order of multiplying in the given formulas is HL(xi + 1)Y(xi + 1) = rL(xi + 1), reversed. where The values for partial solutions Y of equation (1) 0 H (x ) = W (x )K(x , x ), are calculated on the basis of the deﬁnition of the so- L i + 1 L i i + 1 i called Cauchy matrix [3]. and The iteration process of transferring the boundary r (x ) = w (x ) Ð W (x )Y (x , x ). conditions to a given point x* of the integration interval L i + 1 L i L i 0 i + 1 i [0, 1] is the following. At each edge, boundary condi- Using the operator M, we transform the latter equations (2) in the form HY = r are transformed to the tion to the form equivalent ones written out in the form WY = w in W L(xi + 1)Y(xi + 1) = wL(xi + 1). which the rows wi = {wi1, …, wim} of the rectangular matrices W are derived by orthonormalizing the rows By repeating the process described above, the con- hi = {hi1, …, him} of the rectangular matrix H. ditions for the left edge are transferred to a given point x* of the integration interval [0, 1] The rows hi = {hi1, …, him} are orthonormalized in ∗ ∗ ∗ the following way [4]: W L(x )Y(x ) = wL(x ). (5) From the ﬁrst equation of the system HY = r for the Similarly, the conditions H (1)Y(1) = r (1) for the boundary conditions, we obtain w w w and R R 1 = { 11, …, 1m} right edge are transferred to a given point x* within the ω 2 2 1/2 1, where w1k = h1k/(h11 + … + h1m ) , k = 1, … , m interval [0, 1] by using solution (3) ω 2 2 1/2 2 2 ∗ ∗ ∗ and 1 = r1/(h11 + … + h1m ) , (w11 , …, w1m ) = 1. W R(x )Y(x ) = wR(x ). (6)

DOKLADY PHYSICS Vol. 45 No. 8 2000 394 A. YU. VINOGRADOV, YU. A. VINOGRADOV Taking into account the fact that expressions (5) and desired solution rather than for the rows of the bound- (6) contain the same column Y(x*) and introducing the ary conditions. The advantage is the absence of the notation necessity to memorize in calculations the matrices of ∗ ∗ the orthogonal transformation of the rows for boundary ∗ W L(x ) ∗ wL(x ) conditions. At the same time, in the Godunov method, W = ------, w = ------, W (x∗) w (x∗) we need to conserve the transformation matrices in R R order to use them again in determining the desired val- we obtain the system of algebraic equations ues of the problem. It is evident that this property of the method constructed above makes it possible to signifi- W∗Y(x∗) = w∗ , cantly reduce the requirements to both the volume of the computer RAM and computer-time expenditures. where W* and w* are the square matrix and the column with the dimension m, respectively. Abandoning step methods of integrating differential Finally, we find the solution to the boundary value equation (1) within segments of the stable evaluation problem at an arbitrary point in the form and finding the integral on the basis of the convergent matrix Taylor series or according to the Volterra Ð1 Y(x∗) = (W∗) w∗. method additionally and significantly reduce computer time expenditures compared to other known methods. The significant features of principal novelty for the The error in evaluating the integral on the basis of method described are the convergent matrix Taylor series is determined by (i) transferring the boundary conditions to a given comparison of its partial sums. In the case of taking the point within the integration interval with their succes- integral by the Volterra method, the error is determined sive orthogonalization at the ends of each stable evalu- by comparison of products for different divisions into ation segment with preservation of the vector-column parts of the stable-evaluation segment. of the unknown quantities; Also taking into account a possibility of estimating (ii) independent transferring of the boundary condi- the round-off errors inherent in calculations [5] clarifies tions from each edge of the stable-evaluation segments that the method constructed makes it possible to obtain to a given point of the integration interval, which is the solutions to the boundary value problem with the based on using the fundamental CauchyÐKrylov func- given error. tions of differential equations; The efficiency of the method constructed is con- (iii) a possibility to provide a given error in calcula- firmed by numerical simulations while solving the tion results. boundary value problems in mechanics of deformation The method proposed is characterized by a simple of shells with various parameters under the action of algorithm. After orthogonalization by the operator M at concentrated and local loads. the edge of each section for the stable evaluation, boundary conditions (2) are transferred to a given point x* of the interval [0, 1] with the help of solutions (3) REFERENCES and (4) of differential equation (1). The boundary con- 1. A. A. Abramov, Zh. Vychisl. Mat. Mat. Fiz. 1, 542 ditions transformed in the equivalent way with the pre- (1961). served column of the desired quantities are combined at the given point x* into a system of algebraic equations 2. S. K. Godunov, Usp. Mat. Nauk 16 (3), 171 (1961). for determining unknown values. 3. F. R. Gantmakher, Matrix Theory (Nauka, Moscow, Among all known methods of transferring boundary 1967). conditions, the method described above can be consid- 4. I. S. Berezin and N. P. Zhidkov, in Calculation Methods ered as the most adequate since its algorithm using (Fizmatgiz, Moscow, 1962), Vol. 2. equivalent transformation transfers given boundary 5. A. P. Filin, Matrices in Statics of Rod Systems (Stroœiz- conditions to a given point of the integration interval. dat, Moscow/Leningrad, 1966). Such an algorithm has a principal advantage compared, for example, to the Godunov method, in which the orthogonalizion is carried out for columns of the Translated by Yu. Vishnyakov

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

Amplification of a Shock Wave in a Saturated Porous Medium

Academician V. E. Nakoryakov and V. E. Dontsov Received March 16, 2000

On the basis of numerical calculations describing a shock wave from a rigid wall. The role of the skeleton collapse of a cavitation-bubble layer near a rigid wall [1], is only technological and consists in retaining gas bub- it was found that inertia effects of the collective bubble bles in liquid. collapsing can cause a series of high-amplitude pressure However, the situation becomes qualitatively differ- pulses. Experimental investigations [2Ð4] have shown ent when a shock wave in the porous medium com- that the shock-wave amplification occurs in a liquid sat- posed of closely packed polyethylene particles is urated with bubbles of vapors or of an easily soluble gas. reflected from a rigid wall. For such a shock wave, The amplification can occur both for a direct wave and experimental values of its amplitude P (dots 1 and 2) after its reflection from a rigid boundary. 2 and velocity U2 (dots 3) are shown in the figure as a In this paper, we experimentally investigate the pro- function of the amplitude P1 of the shock wave imping- cess of amplification of shock waves in a porous ing a wall (P0 is the initial static pressure in the liquid). medium saturated with a liquid containing bubbles of a The values of P2 (curves 4 and 5) and U2 (curve 6) are soluble gas, which occurs when the waves are reflected calculated on the basis of the isothermal model [4, 5]. from a rigid boundary. A mechanism governing the The values of P2 (curves 7 and 8) are calculated for the shock-wave amplification in saturated porous media is reflection from a rigid wall of a complete-condensation proposed. A setup of the shock-tube type [4] was used (gas dissolving) shock wave [4, 5], Cm representing the to carry out the experiments. Its working space was low-frequency sonic velocity in a porous medium satu- filled with a porous medium saturated with water con- rated with a liquid without gas bubbles. taining air bubbles that had a characteristic radius of ~50 µm. As a porous medium, we used either chaotic It is seen that an increase in the ratio P1/P0 for the dumped packing of polyethylene particles having a wave amplitude causes considerable amplification of characteristic size of 3.5 mm (porosity m0 = 0.37) or P2/P1 U2/Cm soft foam plastic (m0 = 0.98). Step pressure waves were 10 1.0 generated by breaking off a membrane separating a 8 7 high-pressure chamber and the working space. Pres- sure-wave profiles were recorded by piezoelectric pres- 6 0.8 sure sensors situated along the working space. The sen- 8 sors were not in contact with the porous-medium skeleton and measured pressure in the liquid phase. From 4 0.6 the sensors, the signals were transferred to an analog- 6 to-digital converter and then processed by a computer.

As a result of our experiments, it was shown that in 5 0.4 the soft foam plastic saturated with water containing air 4 bubbles, the amplitude and velocity of a shock wave reflected from a rigid wall are well described in the iso- 0.2 thermal approximation [4, 5]. This is caused by the fact 1 2 that the thermal-relaxation time of gas in the bubbles is 2 3 much shorter than the duration of the wave-front time. 0 Due to its low rigidity and high porosity, the porous 4 8 12 16 20 skeleton does not appreciably affect the reflection of a P1/P0 Amplitudes and velocities of a shock wave reflected from a Kutateladze Institute of Thermal Physics, Siberian Division, rigid wall in a porous medium (m0 = 0.37) saturated with Russian Academy of Sciences, water containing air bubbles with the initial volume gas con- ϕ ϕ pr. Akademika Lavrent’eva 1, Novosibirsk, tent 0 = 0.105 (P0 = 0.103 MPa): (1, 3, 4, 6, 7); and 0 = 630090 Russia 0.095 (P0 = 0.203 MPa): (2, 5, 8).

396 NAKORYAKOV, DONTSOV the reflected shock wave (dots 1 and 2) compared to the content behind the shock wave for time intervals under isothermal calculations (curves 4 and 5), which do not investigation t ~ 10 ms. At appropriate parameters of take into account gas dissolving in the liquid behind the both the dense porous medium and the wave, a consid- wave. With an increase in the amplitude P1/P0 of the erable increase in the mass exchange is observed wave, the experimental data (dots 1 and 2) approach the behind the wave front. For example, for the wave corresponding calculated curves (7 and 8) taking into amplitude P1/P0 = 16.5 at the initial stage of bubble col- account complete gas dissolving in the liquid behind lapsing, the mass-transfer coefficient exceeds by two the shock wave that impinges the wall. For higher ini- orders of magnitude that calculated in the boundary- tial static pressures P0 in the medium, amplification of layer approximation for the diffusion regime of bubble the amplitude of the reflected shock wave occurs at dissolving [6, 7]. The convective mass transfer associ- lower wave amplitudes P1/P0. When amplitudes of the ated with the relative motion of air bubbles with respect waves grow, experimental values for the velocity of a to water is insignificant, because velocities of both shock wave reflected from the rigid wall (dots 3) devi- phases equalize in tens of microseconds, i.e., much ate from the calculated curve 6. Consequently, gas dis- more rapidly than the times under investigation. Such a solving occurs in the liquid behind the shock wave high increase in the mass exchange can be caused by impinging the wall. This leads to a decrease in the vol- turbulent pulsations of the velocity of the liquid, which ume gas content behind the wave and, therefore, to an arise in porous medium behind the shock wave. Indeed, increase in the velocity of the reflected shock wave. being defined by the diameter of solid particles consti- When the shock-wave amplitude is P1/P0 > 10, experi- tuting the porous medium and also by the relative mental values for the velocity of the reflected shock wave velocity of the liquid and solid phases behind the shock deviate considerably from the calculated curve. Thus, the wave, the Reynolds number intrinsic to the process Re process of dissolving gas in the liquid essentially deter- is rather high (Re ӷ 100). Hence, in the porous mines the gas behavior behind the shock wave. medium, the liquid flows in the turbulent regime [8]. It is noteworthy that the time of equalizing the velocities Hence, the availability of a dense porous medium of the solid and liquid phases behind the shock wave leads to intense dissolving gas behind a shock wave due to interphase friction is much longer time than the and, therefore, to the amplification of the reflected shock-wave durations under investigation. shock wave. The essence of the amplification mecha- Thus, we may conclude that, for sufficiently high nism is associated with converting the kinetic energy of amplitudes (P /P0 > 30) that propagate in a dense the liquid (i.e., the energy of its radial motion arising as 1 a result of collapsing bubbles) into the potential energy porous medium saturated with a liquid containing gas of its pressure [1Ð4]. In order to realize this mechanism, bubbles, the amplification effect caused by the turbuli- it is necessary that complete dissolving (due to diffu- zation of the motion of the liquid behind the shock- sion-caused processes) of gas contained in the bubbles wave front occurs not only due to the reflection of the would occur for a time on the order of the shock-wave shock wave from a rigid wall but also behind the front leading-edge time duration. Otherwise, the regime of of the direct shock wave. accelerating collapse of the bubbles, which leads to the occurrence of pressure spikes in the liquid and, conse- REFERENCES quently, to the amplification of the reflected shock wave, cannot be observed [2]. We note that the process 1. V. K. Kedrinskiœ, Fiz. Goreniya Vzryva, No. 5, 14 (1980). of amplification of the reflected shock waves is not 2. A. A. Borisov, B. E. Gel’fand, R. I. Nigmatulin, et al., associated with splitting the bubbles behind the shock Dokl. Akad. Nauk SSSR 263, 594 (1982) [Sov. Phys. waves, as was observed when the waves evolved in a Dokl. 27, 237 (1982)]. liquid with sufficiently large bubbles [4]. In our exper- 3. V. E. Nakoryakov, E. S. Vasserman, B. G. Pokusaev, and iments, the Weber number, determining the instability N. A. Pribaturin, Teplofiz. Vys. Temp. 32, 411 (1994). and splitting of bubbles in a shock wave, is substan- 4. V. E. Dontsov and B. G. Pokusaev, Teor. Osn. Khim. tially smaller than its critical value. Tekhnol. 33, 485 (1999). 5. R. I. Nigmatulin, in Dynamics of Multiphase Media In the experiments under consideration, an intense (Nauka, Moscow, 1987), Vol. 2. mass exchange behind the shock wave is caused, seem- 6. P. S. Epstein and M. S. Plesset, J. Chem. Phys. 18, 1505 ingly, by the turbulent liquid motion occurring behind (1950). the shock wave in the dense porous medium. Using both the experimental values for the amplitude and the 7. S. S. Kutateladze and V. E. Nakoryakov, Heat-and-Mass velocity of a reflected shock wave and the model [4, 5], Exchange and Waves in GasÐLiquid Systems (Nauka, Novosibirsk, 1984). we have succeeded in calculating the content of the dis- solved gas behind a shock wave impinging the wall 8. M. É. Aérov and O. M. Todes, Hydraulic and Thermal and, consequently, the mass-transfer coefficient at the Principles for Operation of Apparatuses with a Steady gasÐliquid interface. It is shown that, in the soft foam and Boiling Granular Layer (Khimiya, Moscow, 1968). plastic saturated with water containing air bubbles, diffusion processes do not noticeably vary the volume gas Translated by Yu. Verevochkin

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

Effect of Oriented Elastic and Strength Characteristics on the Impact Fracture of Anisotropic Materials A. V. Radchenko and S. V. Kobenko Presented by Academician V.E. Panin November 15, 1999

Received November 15, 1999

The small quantity of papers concerning the behav- ing to the generalized Hook law for the orthotropic ior of anisotropic materials under impact loads is an material [6, 8] expressed through the increments and indication of the fact that this problem is far from well involving the components of deformation rate tensor understood. At the same time, the anisotropic materials eij. To describe the failure in the orthotropic material of are finding ever-widening application, especially in the plate, we use the fourth-order tensorÐpolynomial connection with advanced modern technologies that strength criterion [4, 5]. It is assumed that the fracture make it possible to produce materials with specified of the anisotropic material under intense dynamic load- properties. To ensure an optimum fitting of material ing occurs as follows [6, 7]: properties to various structures, it is necessary to take If the strength criterion [5] is violated under com- into account the conditions (loads) under which this pression (e ≤ 0), the material loses its anisotropic material operates and, based on that, to choose the nec- kk essary parameters of the material and their dependence properties, and its behavior can be described by the on the orientation. Taking into account the directional- hydrodynamic model and the material retains only the ity of material properties is of special importance for compression strength. the structures working under extreme conditions and If criterion [5] is violated under tension (ekk > 0), the subjected to intense dynamic loads. These are the air- material is considered to be destructed, and the stress- craft and spacecraft equipment, as well as containers tensor components are assumed to be zero. for storage and transportation of explosive and toxic 2. Discussion of the results. The compact cylinder substances. In this paper, we present the model of of steel with dimensions d0 = l0 = 15 mm (d0 is the behavior for anisotropic materials undergoing dynamic diameter and l is the length) runs symmetrically loads and study the effect of orientation-dependent prop- 0 against the orthotropic organoplastic plate of thickness erties on the impact fracture of orthotropic materials. h = 15 mm. Cylinder velocity v0 ranges from 700 to 1. Formulation of the problem. We consider the 1500 m/s. To find a solution, we used the finite element interaction between a steel isotropic indenter in the method [9]. The mechanical properties of the original form of a compact cylinder and an orthotropic plate material (material 1) of the plate obey the following made of the organoplastic material. We solve the three- relationships: Ex > Ey > Ez, Ex/Ey = 2.28, Ex/Ez = 6.8, dimensional problem in the Cartesian XYZ coordinate σ σ σ σ σ σ σ bx > by > bz, bx/ by = 2.26, bx/ bz = 6.8, where Ex, system. The OZ-axis coincides with the indenter axis, σ σ σ Ey, and Ez are the elastic modules and bx, by, and bz are and it is opposite to the impact direction. The symmetry the yield strengths in the corresponding directions [5]. axes of the orthotropic material of the plate coincide with the coordinate-system axes, while the face surface We obtain material 2 from the original material by of the plate coincides with the XOY plane at the initial rotation about the OY-axis by 90° assuming that there moment of time. The indenter material characterized by exists an elastic potential for the anisotropic material ν the shear modulus, the dynamic yield strength, and the under consideration, and, thus, the relationships Ek ik = ν ν state-equation constants is modeled by an elastoplastic Ei ki, where ki are the Poisson coefficients; i, k = x, y, medium [1, 2]. The behavior of the orthotropic plate z, are met. For material 2, in this case, we obtain the fol- material is considered in the framework of the phenom- lowing relationships between the characteristics in the enological approach [3Ð7]. The components of the initial system of coordinates: Ex < Ey < Ez, Ex/Ey = 0.34, σ σ σ σ σ stress tensor before the fracture are determined accord- Ex/Ez = 0.15, bx < by < bz, bx/ by = 0.33, and σ σ bx/ bz = 0.15. In Fig. 1, we show the calculated configurations of Tomsk Branch of the Institute of Structural Macrokinetics the indenter and the plates made of materials 1 and 2 at and Materials Science Problems, various moments of time in the ZOX cross section. The Russian Academy of Sciences, Tomsk, Russia regions, in which the material of the plate became

398 RADCHENKO, KOBENKO

t = 10 µs

t = 20 µs v0 = 700 m/s

t = 10 µs

t = 20 µs v0 = 1000 m/s

Fig. 1. Conﬁguration of the indentor in the ZOX cross section. Material 1 is to the left, and material 2 is to the right.

≤ destructed at ekk 0 are completely shaded, while the In material 2, owing to the fact that cx < cz, the rarefac- regions where the fracture takes place at ekk > 0 are tion waves have a higher component of velocity along hatched. the z direction and, catching up with the compression The fracture in the material of the plate starts in the wave, reduce the level of compressing stresses. The fur- compression wave initiated at the time moment corre- ther propagation of the weakened compression wave sponding to the impact. In material 1, the failure taking still does not induce any failures in the material. place in the compression wave propagates across the The analysis of wave processes in these materials plate through its whole thickness. In material 2, the fail- can be performed in more detail using the plots (Fig. 2) ure in the compression wave occurs only in the upper σ of the z-stress distribution over the plate thickness half of the plate, but the width of this region in the ZOX along the OZ-axis at various moments of time. The cross section exceeds by a factor of 1.4Ð1.7 (depending cross section z = 0 corresponds to the position of the on the impact rate) the corresponding dimension in plateÐindenter contact surface at the moment of time material 1. Such a difference can be related not only to t = 0. To 1.5 µs, the compression-wave front propagated the different strength parameters, but also by the differ- along the z direction by 2h/3 in material 2 and by h/3 ent velocities ci of the wave propagation in materials 1 in material 1. At the same time, in material 2, the rar- and 2 along the corresponding directions: i = x, y, z. efaction wave formed at the lateral surface of the In material 1, cx/cz = 2.61; this results in the fact that indenter and the face surface of the plate begins to catch the rarefaction waves propagating from the lateral sur- up with the compression wave. At 3 µs after the impact, face of the indenter and the face surface of the plate the rarefaction wave already weakens considerably the reduce the width of the compression-wave front in compression wave in material 2; the highest stress in material 1 to a greater extent than in material 2, in this material decreases from Ð3 GPa to Ð2.2 GPa as which cx < cz. For material 1, the evolution of a narrow compared with 1.5 µs. Up to this moment in time, the fracture zone (crack) propagating from the face surface regions beyond the compression-wave front in materi- of the plate at an angle of 45° to the OZ-axis is charac- als 1 and 2 are fractured. The further propagation of the teristic for the case of increasing impact rate. For the compression wave in material 2 does not cause any impact velocities to 1000 m/s, this zone does not fracture. On the contrary, in material 1, the fracture in emerge at the rear surface of the plate. With an increase the compression wave occurs across the whole thick- in the interaction rate, the crack attains the rear surface. ness of the plate. Up to 4.5 µs, the compression wave in

DOKLADY PHYSICS Vol. 45 No. 8 2000 EFFECT OF ORIENTED ELASTIC AND STRENGTH CHARACTERISTICS 399

σ z, GPa Vp/V0 1.0 0.30 0.5 6 v0 0.25 0 0.20 3 –0.5 4 5 6 –1.0 5 0.15 3 1 –1.5 0.10 4 1 –2.0 2 0.05 –2.5 2 –3.0 –14 –12 –10 –8 –6 –4 –2 0 0 5 10 15 20 25 30 z, mm t, µs σ Fig. 2. Proﬁle of the stress z. Curves 1, 3, and 5 correspond Fig. 3. Time variation of a relative volume of the fractured to material 1, and curves 2, 4, and 6, to material 2 at the material for ekk > 0. Curves 1, 3, and 5 correspond to mate- µ moments of time 1.5, 3, and 4.5 s, respectively, for v0 = rial 1, and curves 2, 4, and 6, to material 2 for v0 = 700, 700 m/s. 1000, and 1500 m/s, respectively.

material 1 only reaches the rear surface, while the rar- It is 5% for v0 = 1500 m/s, 12% for v0 = 1000 m/s, and efaction wave formed as a result of reflection of the 20% for v0 = 700 m/s. compression wave from the rear surface of the plate already propagates along material 2. The level of com- Thus, the fracture of the plate made of material 1 pressing stresses in material 2 remaining in the upper takes place owing to and during the wave processes; it half of the plate after the passage of the compression starts in the compression wave and terminates in the wave is already considerably lower (by a factor of 2Ð3) reflected rarefaction wave. In the plate of the material than that in material 1. An approximately equal level of with reoriented properties (material 2), the wave pro- compressing stresses in the materials is retained only cesses do not cause any macroscopic fracture. The close to the indenterÐplate contact surface. The rarefac- compression wave destroys the material in the upper tion wave reflected from the rear surface propagates in half of the plate, while the rarefaction wave induces the material 1 across the weakened substance fractured by shear fractures near the rear surface. The further devel- the compression wave across the whole thickness of the opment of the fracture in material 2 is caused by the plate and having only the compression strength that penetration of the indenter. The fracture in material 2 leads to the complete destruction of the material in demands a higher energy; i.e., only owing to a change front of the penetrating indenter. In material 2, the in the orientation of properties is it possible to provide reflected rarefaction wave propagates across the unde- more efficient protection against the impact. structed substance inducing the failures of the shear- Our studies show that the anisotropy of properties is fracture type near the rear surface of the plate. The fur- a significant factor that must be taken into account to ther failure in material 2 develops already not owing to describe and adequately predict the evolution of the the wave processes, but as a result of developing the shock-wave processes and the fracture in materials sub- tensile stresses in the course of penetration of the jected to dynamic loadings. It was found that a change indenter. in the orientation of properties results in qualitative The curves in Fig. 3 make it possible to estimate a changes in the mechanisms of macroscopic fracture of fraction of the completely fractured material in the anisotropic materials. The proposed model of fracture plate, which exhibits no resistance to the destruction. using the fourth-order tensorÐpolynomial criterion pro- During the process of the indenterÐplate interaction, a vides an opportunity to model the behavior of a wide fraction of the complete failures in material 1 exceeds class of anisotropic materials with various degrees of the corresponding value in material 2. A value of resid- anisotropy. ual velocity of the indenter after piercing the plates also demonstrates that material 2 as a higher resistance to the shock fracture and penetration: For all initial impact ACKNOWLEDGMENTS velocities under study, a more intense deceleration of the indenter is observed during its interaction with the This work was supported by the Ministry of Educa- plate of material 2. With a decrease in the interaction tion of Russian Federation according to the grant on rate, the difference in the indenter velocity after pierc- Research in the Field of Fundamental Natural Science ing the plate from material 1 and material 2 increases: (item 4.3: Mechanics of Deformable Solids).

DOKLADY PHYSICS Vol. 45 No. 8 2000 400 RADCHENKO, KOBENKO

REFERENCES 5. E. K. Ashkenazi and É. V. Ganov, Anisotropy of Struc- tural Materials: Reference Book (Mashinostroenie, Len- 1. M. L. Wilkins, in Fundamental Methods in Hydrody- ingrad, 1980). namics, Ed. by B. Alder, S. Fernbach, et al. (New York, 6. A. V. Radchenko, Mekh. Kompoz. Mater. Konstr. 4 (4), 1964; Mir, Moscow, 1967). 51 (1998). 2. V. A. Gorel’skiœ, A. V. Radchenko, and I. E. Khorev, 7. A. V. Radchenko, S. V. Kobenko, I. N. Marzenyuk, et al., Prikl. Mekh. 23 (11), 77 (1987). Int. J. Impact Eng. 23, 745 (1999). 3. E. M. Wu, in Mechanics of Composite Materias, Ed. by 8. S. G. Lekhnitskiœ, Theory of Elasticity for Anisotropic G. P. Sendeckyj (Academic, New York, 1974; Mir Mos- Materials (Nauka, Moscow, 1977). cow, 1978). 9. G. R. Johnson, J. Appl. Mech. 44 (1), 95 (1977). 4. E. K. Ashkenazi, Anisotropic Materials in Mechanical Engineering (Mashinostroenie, Leningrad, 1969). Translated by V. Bukhanov

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

Solving the Boltzmann Equation in the Case of Passing to the Hydrodynamic Flow Regime F. G. Cheremisin Presented by Academician V.V. Rumyantsev October 19, 1999

Received October 21, 1999

As the hydrodynamic regime is approached, the gas I(fM, fM) = 0 taken into account, eliminates the term on flow is usually accompanied by the formation of nar- the order of K–1 in the right-hand side of equation (2). row highly nonequilibrium zones (Knudsen layers) The idea of the method proposed is to provide for the with the characteristic size on the order of the mean free fulfillment of the indicated properties in the discrete path λ for molecules. The structure of these zones is approximation of the Boltzmann equation. In accor- determined by fast kinetic processes. In unsteady flows, dance with the general theory developed in [4, 5] for an initial layer with the time scale on the order of the solving rigid equations of form (1), the solution to the τ λ mean free time 0 = /vT (here, vT is the molecular ther- difference equations obtained is constructed in such a mal velocity) occurs as well. In the macroscopic scale manner that the asymptotic solution to the difference ӷ λ l0 , the flow parameters vary smoothly beyond these problem is of form (3) as τ ∞. zones [1]. From a computational standpoint, solving We write out the Boltzmann collision integral for λ τ τ the Boltzmann equation with steps hx < and < 0 is monoatomic gas in the form inefficient everywhere over the calculation domain. ∞ π Moreover, it can result in the early cessation of the iter- 2 bm (), ()ε ξ ative process as soon as the error of the numerical If f = ∫ ∫ ∫ f'f1'Ðff1 gbd b d d 1, method becomes equal to the small difference of two Ð∞ 0 0 successive approximations. When passing to the mac- λ Ӷ τ Ӷ τ v where the conventional notation of [1] is used. The roscopic steps hx < l0, 0 < t0 , where t0 = l0/ T, ξ ξ the problem of a large factor standing ahead the colli- velocity arguments and 1 of the functions f and f1 are ξ ξ sion integral arises [2]. In terms of the dimensionless related to the arguments ' and 1' of the functions f ' ξ ξ variables t = t*/t0, x = x*/l0, = */vT (here, the asterisk and f' by the relations denotes dimensional variables), the Boltzmann equa- 1 tion takes the form ξ ξ ()ξ ξ () ' ==+nng, 1' 1Ðnng. (4) ∂ f ∂ f Ð1 +Kξ = If(), f. (1) Here, n is the unit vector aligned with the centerline of ∂t ∂x ξ ξ the colliding molecules, and g = – 1 is their relative For the Knudsen numbers K Ӷ 1, the equation velocity. acquires a rigid nature, and specific methods [3–5] We consider the integral operator should be used for its numerical solving. ∞∞2πbm According to the EnskogÐChapman theory, for 1 Ӷ Q()φ=--- ()φξ()+ φξ()φξÐ()'Ðφξ()' K 1, the solution to equation (1) takes the form 4∫∫∫∫1 1 ()1 … Ð∞Ð∞00 ff=M++Kf , (2) ×()ε ξ ξ (5) where fM is the Maxwell function determined by the gas f'f1'Ðff1 gbdbd d d 1. density n(x, t), temperature T(x, t), and velocity u(x, t), δ δ ξ ξ while Taking the three-dimensional -function ( – β) for φ(ξ), we obtain 2 Ð3/2 ()ξ Ð u ()δξ() ξ f= n()2πTexp Ð ------.(3) Iβ = Q Ð β . (6) M 2T The laws of conservation for mass (density), momen- The substitution of (3) into (1), with the property tum, and energy Q(ψ) = 0 follow from (5) for the additive collision invariants ψ = (1, ξ, ξ2). Computer Center, Russian Academy of Sciences, We bound the space of the ξ-variable by the region ul. Vavilova 30/6, Moscow, 117900 Russia Ω with a volume V and introduce in this region a net

1028-3358/00/4508-0401 $20.00 © 2000 MAIK “Nauka/Interperiodica” 402 CHEREMISIN ξ consisting of N0 equidistant nodes β with a step h = we obtain (h , h , h ). Furthermore, the following notation is used: 1 2 3 Nν   fβ ≡ f(ξβ, x, t), Iβ ≡ I(ξβ, x, t). Here, only the node of the (1) (2) (3) (4) Iβ = B∑Ð jν + jν + ∑ pν, ( jν + jν ) , (9) velocity net is indicated at which a given value is calcu-  s  lated. Equation (1) will be approximated by a system of ν s N0 equations in terms of the functions fβ. From (6) and (1) where B = Vπb /4Nβ, Nβ = Nν/N , jν = f α f β gνbν, (5), a cubature formula for calculating the integral Iβ is m 0 ν ν (2) (3) constructed, which satisﬁes, as in the method [6, 7], the jν = f α' f β' gνbν for αν = β or βν = β, and jν = conservation laws ν ν (4) f α f β gνbν, jν = f α' f β' gνbν for λν + s = β or ψ ψ ( , ξ , ξ2 ) ν ν ν ν ∑ β Iβ = 0, β = 1 β β , (7) µν − s = β. β We determine the coefﬁcients ps from conservation as well as the condition laws (7). To this end, we substitute (9) into (7), change the order of summation, and require each term with the ν Iβ( f , β, f , α) = 0 (8) index in the external sum to vanish. Thus, we arrive M M at the same system of equations that has been obtained ξ in [6] when constructing a conservative method for the at any node β of the velocity net. Here, fM, β is the magnitude of function (3) at the node ξβ. collision integral with “inverse collisions” ignored, i.e., without the term f α' f β' in (5), namely, ξ ξ ε ν ν We employ a uniform cubature net αν , βν , bν, ν

ν ξα ξβ 1 Ð ∑ pν, s = 0, with N nodes, such that ν and ν belong to the (10a) velocity net. Then, we exclude those values of the inte- s ε ξ' gration variables b, that remove the velocities αν and ξα + ξβ Ð pν, (ξλ + ξµ ) = 0, ν ν ∑ s ν + s ν Ð s (10b) ξ' Ω × Ω s βν after the collision beyond the region . Since 2 2 2 2 ξ' ξ' δ ξα + ξβ Ð pν, (ξλ + ξµ ) = 0. the arguments αν and βν of the -functions appearing ν ν ∑ s ν + s ν Ð s (10c) in (5) with negative sign, in general, do not coincide s with the net nodes, these functions will be approxi- System (10) of equations has a particular solution mated by the sum of δ-functions with arguments taken containing only two nonzero coefﬁcients: The ﬁrst is at nodes nearest to ξα' and ξβ' . 1 Ð pν and corresponds to s = 0. The second is pν and ν ν corresponds to a certain value s ≠ 0, which depends on We assume that ξλ and ξµ are the nearest nodes to a combination of parameters entering into the energy ν ν equation. Thus, we arrive at the approximation ξα' and ξβ' , respectively; ∆λ = ξα' – ξλ and ∆µ = ν ν ν ν ν ν Nν (2) (1) (3) (4) ξβ' – ξµ . From (4), it follows that the points ξα' and ( ( ) ν ν ν Iβ = B∑ Ð jν + jν + pν jν, 0 Ð jν, 0 ξ' ν (11) βν are situated antisymmetrically in the corresponding (3) (4) ∆ ∆ ∆ ∆ ∆ ∆ ∆ + (1 Ð pν)( jν, Ð jν, ) ). cells: λν = – µν . We denote = λν /h = ( 1, 2, 3), s s ∆ ≤ ( ) ( ) | i| 1/2, i = 1, 2, 3, and introduce a displacement vec- 2 4 The value of f α' f β' entering into jν , jν, , and tor on the net s = (s , s , s ), s = 0, or s = sgn(∆ ), i = ν ν 0 1 2 3 i i i (4) 1, 2, 3. Then, the vertices of the cells, inside of which jν, s should be found from interpolation of the nearest ξ' ξ' ξ the points αν and βν fall, can be represented as λν + s values f λν + s , f µν Ð s . From (10c), it follows that the for- ξ mula and µν Ð s .

ln( f α' f β' ) = (1 Ð pν)ln( f λ f µ ) We replace the off-net δ-functions in (5) by the ν ν ν ν (12) expansions ( ) + pν ln f λν + s f µν Ð s δ(ξ ξ ) δ(ξ ξ ) β Ð α' = ∑ ps β Ð λ + s , is exact for a net function of the type (3), and it makes ν ν each term of sum (11) vanish. s Formulas (11) and (12) specify the method to calculate the collision integral ensuring the rigorous fulﬁll- δ(ξβ Ð ξβ' ) = ∑ p δ(ξβ Ð ξµ ). ν s ν Ð s ment of conditions (7) and (8). s Equation (1) is solved on the basis of the decompo- Substituting these expansions into (5) and using (6), sition method. In the interval [t j, t j + 1], the following

DOKLADY PHYSICS Vol. 45 No. 8 2000 SOLVING THE BOLTZMANN EQUATION 403

p n 1.0 0.5

0.8 0.4

0.6 0.3 0.4

0.2 0.2

0 0.1 –400 –200 0 200 400 600 0 100 200 300 400 500

–8000 –4000 0 4000 8000 12000 0 2000 4000 6000 8000 10000 x x Fig. 1. Fig. 2. equations are sequentially solved discontinuities of hydrodynamic parameters, which are spread by the scheme viscosity. ∂ * ∂ * , f β ξ f β * j j ---∂------+ β--∂------= 0, f β = f β, (13) As the ﬁrst example, the well-known one-dimen- t x sional nonstationary problem of propagation of an arbitrary initial-data discontinuity was considered, which is ∂ f β Ð1 j , j + 1 ------= K Iβ, f β = f β* . (14) often used as a test for numerical methods of gas ∂t dynamics. The initial data (at t = 0) are the following: The system of equations (13) is approximated by the To the left of the discontinuity point x = x0, gas is characterized by the parameters n = 1, T = 1, U = 0. To scheme [8] having the second order of accuracy in hx, 1 1 1 which is well adapted to calculating discontinuous ﬂows. the right of the discontinuity point, it has the parame- For solving the system of nonlinear equations (14), we ters n2 = 0.125, T2 = 0.8, U2 = 0. The Boltzmann equa- j + 1 j tion was solved for the hard-sphere molecule model introduce the discrete variable tν = τν/Nν, τ = t – t ν with the following values for discretization parameters j + /Nν ξ and the intermediate values of the solution f β ; in the -variable: N0 = 3604, Nν = 36000, h1 = h2 = h3 = τ then, we employ the explicit Euler method 0.3vT, and with various values of hx and .

j + ν/Nν j + (ν Ð 1)/Nν Ð1 (1) (2) Figure 1 shows the pressure plots for the cases hx = f β = f β + τK B(Ð jν + jν λ τ τ λ τ τ λ Ð 1 Ð 1 (15) 5 1, = 0.2 1 and hx = 100 1, = 3 1. Here, 1 is the (3) (4) (3) (4) mean free path and τ is the mean free time to the left + (1 Ð pν, )( jν , Ð jν , ) + pν, ( jν , Ð jν , ) ). 1 0 Ð 1 0 Ð 1 0 s Ð 1 s Ð 1 s of the initial discontinuity. The number of nodes for ( ) ( ) 1 2 both cases is equal to kx = 200. The initial discontinuity Here, the values jν , jν , … are calculated from the Ð 1 Ð 1 is located at x . The solution corresponds to t = 200τ (ν ) 0 1 j + Ð 1 /Nν τ functions f β . Method (15) is conservative and and t = 4000 1 for the first and second cases, respec- enables the verification of the positive value of the solu- tively. Since the solution to the gas-dynamic problem is tion. The solution to system (14) is attained for ν = Nν. invariant with respect to the transformation x' = cx, t' = It is correct at arbitrary values of τ and tends to (3) as ct, the plots are rather close. However, in the first case, τ ∞. On substituting the net function in form (3), a shock-wave structure is visible, whereas it is a discon- equation (13) becomes equivalent to the kinetic tinuity for the second case. The solutions in the case of approach for solving equations of gas dynamics [5]. the rarefaction wave are close, and they essentially coincide in the contact region. Figure 2 shows the den- The method proposed allows us to solve the Boltz- sity plots to the right of the rarefaction wave in an mann equation with both the separation of the Knudsen enlarged scale. In the first calculation (solid line), the τ –1 layers (that must be calculated with steps < K , hx < contact discontinuity and the shock wave are spread out K–1) and without such a separation (with steps τ > K–1, by molecular viscosity. The shock-wave thickness is –1 λ λ hx > K ) over the entire calculation domain. In the lat- approximately equal to 5 2, where 2 is the mean free ter case, as for calculating hydrodynamic equations, path to the right of the initial discontinuity; this magni- highly nonequilibrium zones manifest themselves as tude corresponds to the actual value for the given Mach

DOKLADY PHYSICS Vol. 45 No. 8 2000 404 CHEREMISIN

–Fx an unperturbed flow. In the second calculation, the 1.0 Knudsen layers were not separated, and a net with a λ constant step hx = hy = 5 , kx = 43, ky = 22 was employed, the other parameters being the same as 0.8 before. In both cases, the calculations were carried out τ up to t = 200 0, and the attainment of the steady-state solution was controlled. Figure 3 shows the plots for 0.6 the friction along the plate obtained in the first case (solid line) and in the second one (dashed line). The 0.4 separation of the Knudsen layers is seen to play a significant role. The third example shows the efficiency of the 0.2 method for calculating slow flows. A flow similar to that of the second example is considered but with M = 0 0.2 0.4 0.6 0.8 1.0 0.001 and K = 0.1. The parameters of the calculation x λ λ are the following: hx = 0.5 , hy = 0.333 , kx = 40, ky = τ Fig. 3. 20, = 0.1, N0 = 3604, Nν = 36000, and h1 = h2 = h3 = 0.26vT. In this case, the deviation from the equilibrium × Ð4 y is determined by the parameter Keff = M K = 10 , and 0 an especially high accuracy is required. Isolines of the 0.8 0.04 0.08 0.04 gas-velocity component normal to the plate (in the – 0.08 0.6 0.12 – 1000-fold scale) are shown in Fig 4. The plate is situated between x1 = 0.7 and x2 = 1.7. In the case of using 0.16 0.4 0.12 0.12 the method of [6, 7], attempts to calculate these small 0.08 0.16 – – 0.08 deviations from the equilibrium even for M = 0.01 and 0.2 – with a tenfold increase in the parameter Nν turned out 0 unsuccessful. 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 x The examples given above show that the method proposed allows the gas flows with the Knudsen num- Fig. 4. bers K Ӷ 1 to be efficiently calculated. number M = 1.62. In the second case (shown by cir- ACKNOWLEDGMENTS cles), gas-dynamic discontinuities are spread by the This study was supported by the Russian Founda- viscosity over a few cells. tion for Basic Research, project no. 98-01-00443. As for the second example, we consider a longitudinal flow around a plane-parallel plate of the infinite REFERENCES swing for M = 2 and K = 0.01 in the case of the surface temperature equal to that of the approach stream and 1. M. D. Kogan, Dynamics of Rarefied Gas (Nauka, Mos- diffuse reflection of molecules from the surface at the cow, 1967). condition of total accommodation. For such flow 2. V. A. Rykov, in Modern Problems in Computational parameters, the Knudsen layer with the thickness on the Aerodynamics (Mir, Moscow, 1991), pp. 377Ð395. order of λ is known [1] to be formed near the plate sur- 3. F. Coron and B. Perthame, SIAM (Soc. Ind. Appl. Math.) face. At the same time, a region with abrupt longitudi- J. Numer. Anal. 28, 26 (1991). λ nal flow changes at a distance on the order of is 4. S. Jin, J. Comput. Phys. 122, 51 (1995). formed near the leading edge. Accurate calculation of the flow requires that the steps of a three-dimensional 5. R. Gaflisch, S. Jin, and G. Russo, SIAM (Soc. Ind. Appl. net in the regions indicated would be smaller than λ. Math.) J. Numer. Anal. 34 (1), 246 (1997). These conditions are met in the first calculation, where 6. F. G. Cheremisin, Dokl. Akad. Nauk 357, 53 (1997) λ [Phys.ÐDokl. 42, 607 (1997)]. the varied net in x and y with the steps hx = 0.6 and λ 7. F. G. Tcheremissine, in Proceedings of the XX Interna- hy = 0.6 was used near the leading edge and surface, respectively. The numbers of nodes in x and y were k = tional Symposium on Rarefied-Gas Dynamics (Peking x Univ. Press, Beijing, 1997), pp. 297Ð302. 54 and ky = 32, respectively; the steps hx and hy attained 20λ at the periphery of the calculation region. The other 8. J. P. Boris and D. L. Book, J. Comput. Phys. 11, 38 τ τ (1973). discretization parameters are the following: = 0.35 0, N0 = 3604, Nν = 36000, and h1 = h2 = h3 = 0.32vT. Here, λ τ is the mean free path, and 0 is the mean free time in Translated by V. Tsarev

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

An IsobaricÐIsothermal Ensemble in Statistical Mechanics K. M. Magomedov Presented by Academician I.I. Novikov November 10, 1999

Received July 6, 1999

The isobaricÐisothermal ensemble is rarely used in tion, we derive an expression for the Gibbs free energy statistical mechanics and thermodynamics due to diffi- depending on the number of particles, temperature, culties with integration and some other computational pressure, and on several typical values of enthalpy. The problems [1Ð4]. Nevertheless, this ensemble provides limiting cases correspond to the Debye theory for heat certain advantages related to rapid convergence with capacity of solids and to the chemical potential of ideal respect to the number of particles [5]. The canonical gas. Note that in the latter case, the mean volume per and grand canonical distributions are the most popular, particle tends at T 0 to its value in the condensed since they ensure exact solutions for ideal gas and har- state. This equation of state is in good agreement with monic oscillators. the experimental data. At first glance, it seems that the problem could be 1. The modified constant pressure ensemble can solved by means of the expansion of thermodynamic be obtained directly from the Meyer definition of prob- potential Ω = –pV in terms of chemical potential µ and ability [1] the group integrals for interacting particles [1, 2]. How- β[]᏷ ,,᏷ ever, the mathematical formalism of such a type does W()᏷ =eGÐ NpV()ÐEN() V ,(1) not provide an opportunity to explicitly obtain a final result. First, it is possible to calculate only about ten where β = (kT)–1; E and V are the energy and the vol- first terms in the series expansion without any proofs of ume, respectively; G = G(β, p, N) is the Gibbs free its convergence. Second, this approach is based on energy; and ᏷ are the characteristic quantum numbers model potentials for two-particle interactions with the of the system. parameters refined by fitting to experimental data. Third, we also need to derive the inverse function by The conventional approach involving the summa- expressing µ through measured temperature T and pres- tion over possible ᏷ and the integration over volume sure p. leads to the relationship

In this paper, we demonstrate that precisely the ∞ ensemble corresponding to constant pressure underly- V ÐβG= ln Z()β,,NVd------, (2) ing the Gibbs free-energy distribution turns out to be ∫ v free of many disadvantages inherent in other ensem- 0 * bles. Our approach starts from clear basic principles: The mathematical models considered in statistical where Z is the partition function of the canonical ensemble. The physical meaning of small volume v is mechanics serve as approximations to real physical * systems [1], and the actual accuracy of the models is not quite clear, and it is possible to eliminate it using determined by phenomenological thermodynamics in the following expression (see [1]): the broad understanding of the term [6]. The latter ∞ enables us, in particular, to extrapolate the available experimental data using the theory of similarity. ÐβG = ln∫Z()β,,NVd()βpV . (3) Here, we suggest a new approach for interpreting 0 the constant-pressure ensemble based on the quantum- Nevertheless, expressions (2) and (3) imply a phys- mechanical concepts. In the one-particle approxima- ical paradox: We need to know Z(β, N, V) for the case of N ~ 1024 particles conﬁned within a small volume of the order of 10Ð24 cm3. On the other hand, even for ideal Institute of Geothermal Problems, gas occupying a macroscopic volume, partition func- Dagestan Scientiﬁc Center, Russian Academy of Sciences, tion Z calculated in the usual way is valid under condi- pr. Kalinina 39a, Makhachkala, 367003 Russia tions

406 MAGOMEDOV

2/3 1/3 1 V V 16 ∂βG v Ӷ l = ------= ------, N Ӷ ------∼ 10 , and enthalpy. Note that H = ------, and the similar for- nσ Nσ σ ∂β σ mula relates the Helmholtz free energy and internal where l is the mean free path and is the cross section energy. for collisions between particles. Note that N ~ 1019 for ideal gas under normal conditions. However, with Relationship (4) refines (1) by eliminating physi- account taken for particle collisions, we obtain quite a cally improbable and nonessential states. To simplify different picture. Let us perform a Gedanken experi- the problem further on, we assume ment by tracing the motion of a chosen particle in the Nν Nν Z(β, Nν, V ν) = zν , Y(β, p, Nν) = yν , gas of hard spheres of the radius σ1/2 during a macroscopic time interval. Let n1 be the number of particles where zν = zν(β, vν) and yν = yν(β, p). Actually, this σ1/2 with the free path , n2 being that with a free path means that we consider only the normal systems [8]. twice as large, etc. Similar results could be obtained if Below, we present a method allowing us to calculate zν we use the ergodicity postulate to calculate the number and yν. Here, we use the one-particle approximation to of particles with free paths equal to σ1/2, 2σ1/2, 3σ1/2, …. determine statistical weights for identical particles dis- This implies that, even in classical mechanics, we can tributed over volumes Vν. Let Mν be the maximum pass to the statistical description of a system with dis- number of particles occupying the νth spatial subset in crete values of volume treating n1, n2, n3, … as mean such a way that Nν cells contain one particle and the 3/2 3/2 3/2 numbers of particles in volumes σ , 2σ , 3σ , …. remaining Mν – Nν cells are empty. As a whole, there The above discussion becomes more evident and exist Mν! ways to distribute these particles, but the clearer if we start a priori from the quantum-mechani- exchange of particles between the occupied cells and cal approach, where the state of a system (subsystem) the rearrangement of empty cells do not affect the value is described by a wave function parametrically depen- of energy (enthalpy) determined by the summation dent on the volume. For example, for a free particle in (4). Hence, the weight factor for the νth layer is Wν = confined within the volume vν, we have Mν!(Nν!(Mν – Nν)!). To determine Mν and the common weight factor, we can imagine the following experi- π2ប2n2 ment. Let us assume that all N particles are in the Ᏹ = ------. n 2/3 ground state at T = 0 and each of them occupies a vol- 2mv ν ume v0 . The system is heated at p = const (under a Now, let us consider a set of weakly interacting sub- movable piston). Then, some of the particles would systems, with a large number of particles in each of leave the ground state and occupy other levels. It is pos- them. If the long-range forces are absent and the sur- sible to consider the state where N particles leave the face effects are negligible, the system as a whole can be ground state and occupy a single νth layer. Then, it is evident that Mν = N. While additional conditions are not described by wave function Ψ = ψν and by the cor- ∏ specified, all layers are independent and the total ν weight is W = ∏ W ν . The special role of the ground responding relationships N = ∑ Nν , V = ∑ Nν vν, ν = 1 Hν = ∑[ pNνvν + E(Nν, Vν, ᏷)]. Performing the sum- state is taken into account by condition W0 = 1. In the mation over ᏷ in (1), we get case of N2 = N3 = … = 0, this corresponds to the well- known examples of solving similar problems in statis-   tical physics [8]. The physical meaning of this condi- β G Ð p ∑ N v  ν ν ν  tion is the absence of additional contribution to entropy W(Nν) = e Z(β, Nν, v ν). related to N0 particles. Then, summing over Nν = 0, 1, 2, etc., and taking into Thus, it is necessary to calculate the sum account that the sum over all probabilities is equal to β N y Mν! Nν lnyν unity, we obtain eÐ G = ∑…∑e 0 0 ∏ ------e . Nν!(Mν Ð Nν) (5) β Ðβp N v Ð G ∑ ν ν ∑ Nν = N ν = 1 e = ∑…∑e Z(β, Nν, v ν) Let us employ the well-known method of statistical ∑ Nν = N (4) physics that allows us to calculate similar sums based … β, , = ∑ ∑Y( p Nν). on the maximum term [1]. We add condition C(N – δ It should be emphasized that Nν is the mean number of ∑ Nν ) to (5) and find variations Nν using the Stirling particles occupying volume Vν = Nνvν; Z (β, Nν, Vν) is formula for factorials. Then, we find the partition function of canonical ensemble, whereas lny0 + C = 0, Y(β, p, Nν) is the constant-pressure partition function. The latter is evidently related to the Gibbs free energy lnyν Ð ln Nν + (N Ð Nν)ln(N Ð Nν) + C = 0.

DOKLADY PHYSICS Vol. 45 No. 8 2000 AN ISOBARICÐISOTHERMAL ENSEMBLE IN STATISTICAL MECHANICS 407

ν βᏱ Unfortunately, in the literature, insufficient attention is ν  Ð i  0 Ð ∑g ln 1 Ð e Ðβhν i   payed to the ground state, i.e., to the first condition. As i a result, parameter C corresponding to the particle yν = e zν, zν = e , ν (9) number conservation is not eliminated from the distri- Ᏹ Ᏹ ν Ᏹ ν bution. This makes the problem much more compli- i = i Ð 0 . cated and changes the distribution type. For example, if we retain C (actually, the chemical potential), we would Since each particle moves inside its own volume vν , the deal with the so-called generalized ensemble [2]. phase volume can be calculated for one particle and ν ν Eliminating C, we obtain the desired distribution multiplied by Nν: gi = Nν gi . For free particles, we have by definition [1, 9] Nν Nν yν ------≡ ------= ------, ν = 1, 2, 3, …, Mν N y0 + yν ν 4π 2 (v v ) (6) gi = ∑ 1 = ν + 1 Ð ν -----3-P dP . N0 yν h v ≤ v ≤ v Ᏹ ≤ Ᏹ ≤ Ᏹ ------= 1 Ð ∑ ------. ν ν + 1; i i + 1 N y0 + yν ν = 1 Correspondingly, keeping the linear terms in zν, we Taking (6) into account, the maximum exponent in (5) obtain has the form ν ν Ðgi ν ÐβᏱ ν ÐβᏱ yν N Ð Nν yν  i  i N   zν = ∏ 1 Ð e = ∑g e Nν ln---- + N ln------+ Nν ln------= N ln1 + ---- .   i y0 N Ð Nν Nν y0 i i ∞ Thus, the desired expression for the Gibbs potential is (v ν Ð v ν) β 2 ⁄ v ν Ð v ν = ------+---1------Yπ∫P2eÐ P 2mdP = ------+---1------, (10) h3 Λ3  yν 0 ÐβG = N lny + ln ∏ 1 + ---- . (7) 1/2 0  y  2 ν 0  h  = 1 Λ = ------, 2πmkT 2. Partial statistical sums in (7). If we take ψν as a sum of symmetrized products of one-particle wave where Λ is the thermal de Broglie wavelength. In the functions, then the enthalpy of the system with particles general case, when a particle moves in effective poten- occupying volumes vν can be written in the form tial field u(r), and P = 2m(Ᏹ Ð u) , we find

ν ν ν v ( Ᏹ ) ν + 1 Hν = ∑ pv i + i ni , ∑ni = Nν. π ∆Ᏹ ν 4 m 2 (Ᏹ ) i i gi = ------∫ r 2m Ð u dr. (11) h3 Then, for bosons, we have [9] v ν

ν ν 2 2 ( ) β( Ᏹν) ν For the harmonic oscillator (u = mω r /2, Ᏹi = ωបi), ni + gi Ð 1 ! Ð pv ν + i ni … ν Y ν = ∑ ∑∏------ν------ν------e , (8) 2 ν n !(g Ð 1)! simple calculations give gi = i /2. At large i, it coin- M i i i i cides with the exact quantum-mechanical result ν (i + 1)(i + 2)/2. This example demonstrates that, in the Ᏹ where i are the one-particle energy levels, numbers general case, we should use expression (11) to deter- ν ν mine zν. ni specify the quantum state of the system, and gi is ν Ᏹ ν Ᏹ 0 the number of states at the ith level, with g0 = Nν. We Measuring energies i from 0 and the volume estimate the sum based on the maximum term approxi- from v0, we divide (7) by N and ﬁnd the expression for mation. Repeating exactly the same procedure as in chemical potential taking into account (9), Section 1, we obtain

0 ν ν β ⁄ ν 0 Ð hν zν z0 ÐNν ∑ g ln(1 Ð e Ð βᏱi ) Ðβ( pv ν + Ᏹ )Nν i µ = h (p) Ð kT lnz + ln (1 + e ) , 0 i 0 0 ∏ Y ν = e e ν = 1 (12) 0 Ðβh N ν ν β, 0 ( ) Ᏹ 0 = e Zν( Nν), hν = p v ν Ð v 0 + ν . or, in terms of yν (zν), Differentiating (12) with respect to p, we obtain the

DOKLADY PHYSICS Vol. 45 No. 8 2000 408 MAGOMEDOV

z = pv/kT lowing expression for the heat capacity of a solid with 6 v = v0:

n * 4 bn k 2 n e dn បω ϑ -----D- c p = --b ∫ ------2 , b = ------= . (15) 2 ( bn ) kT T 4 0 e Ð 1 1/3 At n* = (24) = 2.884, expression (15) coincides exactly with the well-known Debye formula [1], but not with the Einstein formula, as one could expect. This can 2 be simply explained: Einstein considered the vibrations in the whole Nv 0 volume, rather than vibrations within each elementary volume v0 occupied by one particle. Thus, we take into account the interaction of energy levels in the framework of statistical analysis. 0 2 4 6 The second example is an ideal gas with a character- 0 q = pv0/kT ӷ Ᏹ β Ӷ istic volume vν v0 meeting condition ν 1. In Figure. this case, we can retain only the ﬁrst-order terms in the product entering the logarithm argument. Then, assum- ν ing that vν + 1 – vν = v∞, vν = v∞, we obtain the expres- µ ӷ mean volume per particle, sion for (zν z0):

v ν Ð v v Ðβpv ∞ν v = v + ------0------µ = h 0 Ð kT ln ∑ -----0 e 0 ∑ β 0 ( ⁄ ) 0 3 hν Ð ln zν z0 Λ ν = 1 e + 1 ν = 0 (13) (16) v Ðβpv Ð1 0 0 ( ∞ ) Nν = h0 Ð kT ln----- 1 Ð e . = v + ∑------(v ν Ð v ). 3 0 N 0 Λ ν At β 0, we find the well-known expression for the As should be expected, this expression demon- 0 chemical potential of ideal gas: µ – h = kTln(pΛ3/kT). strates that v = v (p) at β ∞. Thus, we get the 0 0 At the same time, expression (16) provides an opportu- equation for a zero isothermal curve, which can be nity to obtain a more general equation of state by differ- derived based on the method of differential virial rela- entiating it with respect to p: tionships [10]. In addition, a stepwise change of v occurs at temperatures corresponding to a zero argu- v ∞ v Ð v = ------. ment of exponentials in the denominator of (13) (this 0 βpv ∞ recalls the behavior of the well-known FermiÐDirac e Ð 1 distribution) [1, 8]. This is an indication of phase tran- Assuming that v ≡ v∞, we can write the following sitions, and the summation in (12) and (13) demands 0 certain care and calls for further analysis. expression for compressibility factor z: ρ pv q v 0 Distribution (12) cannot be treated as closed while z = ------= ------, q = ------. (17) 0 kT Ðq kT hν and vν are unspecified. To begin with, we consider 1 Ð e three examples. In Fig. 1, the curve calculated according to (17) is com- The first example is the heat capacity of solids. Let pared with the experimental data for argon at T = 300 K (dots). In the same figure, we present the calculations in us assume that Nν/N Ӷ 1, ν = 1, 2, 3, …. Then, according to (9) and (12), we have the framework of the Kirkwood method [2], using the pair interaction function and the model Lennard-Jones

2 potential (crosses). The dashed line corresponds to the 0 n Ðβωបn free volume theory [2]. It is clear that our approach pro- µ = h (p) + kT ∑ ---- ln(1 Ð e ). (14) 0 2 vides a rather simple way to obtain results that agree n = 0 well with experimental data. Differentiating (14) with respect to T, we obtain s. After The third example is sublimation. Keeping in (12) subsequent differentiation with respect to lnT and the z0 value for the solid phase, we relate zν summed replacing the summation by integration, we ﬁnd the fol- according to (16) with the gas phase. Then, measuring

DOKLADY PHYSICS Vol. 45 No. 8 2000 AN ISOBARICÐISOTHERMAL ENSEMBLE IN STATISTICAL MECHANICS 409 the energy from the lowest level z0 and introducing However, a direct approach is expected to provide Ᏹ 0 the most accurate results. Let us specify an appropriate notation 1 = u, we obtain effective potential for a particle. Then, we can calculate Ᏹ β v Ð1 n by using, e.g., the BohrÐSommerfeld quantization µ 0 Ð u 0 ( Ðq) = h0 Ð kT ln z0 + e ----- 1 Ð e . (18) conditions [7] Λ3 rν Let us deﬁne the phase transition point as that where the ប2(l + 1 ⁄ 2)2 2m(Ᏹ Ð u(r)) Ð ------dr contributions from both phases to µ become equal: ∫ n 2 r r0 β v Ð1 Ð u 0 ( Ðq) z0 = e ----- 1 Ð e . (19) = បπ(M + 1 ⁄ 2); Λ3 r 0 Solving (19) with respect to q, we get hence, we can choose vν and Ᏹν .

 v Ðβu In any case, the distributions over T, p, and N corre- q ≡ βpv = Ðln 1 Ð ------0---e . (20) sponding to the Gibbs potential (12) provide an oppor- 0  Λ3  z0 tunity to construct rather good models for the thermal and caloriﬁc equations of state. In the case of v0 0, this expression coincides with the well-known expression for equilibrium vapor [8] REFERENCES kT Ðβu P(T) = ------e . (21) Λ3 1. J. E. Mayer and M. G. Mayer, Statistical Mechanics z0 (Wiley, New York, 1977; Mir, Moscow, 1980).

In these expressions, z0 is given by the Debye formula (9), 2. T. L. Hill, Statistical Mechanics: Principles and Selected (14). It is clear that more general expression (20) Applications (McGraw-Hill, New York, 1956; Inostran- remains valid up to the triple point. When we use the naya Literatura, Moscow, 1960). Gibbs function, the phase transition turns out to be con- 3. P. Lloyd and J. J. O’Dwyer, Mol. Phys. 6, 573 (1963). tinuous, but it manifests itself in the steep volume 4. D. R. Cruise, J. Phys. Rev. 74, 405 (1970). change from v0 to V. At the phase transition point itself, 5. W. Kohn and J. R. Onffroy, J. Phys. Chem. B 8, 2485 we have v = V/2. This could be demonstrated by dif- (1973). ferentiating (18) with respect to p and using (19). 6. M. P. Vukalovich and I. I. Novikov, Thermodynamics Note that a number of problems considered in [8] (Mashinostroenie, Moscow, 1972). based on other distributions can be solved rather simply 7. L. D. Landau and E. M. Lifshitz, Quantum Mechanics: by means of the Gibbs potential (12). Non-Relativistic Theory (Nauka, Moscow, 1989, 4th ed.; The last examples demonstrate that we can use rela- Pergamon, Oxford, 1977, 3rd ed.). v νv 8. R. Kubo, Statistical Mechanics: An Advanced Course tionship ν = for the gas phase. The clearest with Problems and Solutions (North-Holland, Amster- description can be attained if v1 = 2v0, i.e., when the dam, 1965; Mir, Moscow, 1967). particle hopping to neighboring sites (corresponding to 9. Yu. B. Rumer and M. Sh. Ryvkin, Thermodynamics, Sta- the Shottky defects) is taken into account. In this case, tistical Physics, and Kinetics (Nauka, Moscow, 1977). we can choose two characteristic values of the activa- 10. K. M. Magomedov, Dokl. Akad. Nauk 368, 189 (1999) Ᏹ 0 Ᏹ 0 [Dokl. Phys. 44, 615 (1999)]. tion energy, 1 and 2 . A more accurate partition can be performed assuming that vν = v0/2 for both solid and liquid phases (the Frenkel’ defects). Translated by O. Chernavskaya

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

On Singularities of Boundaries for Parametric Resonance A. A. Maœlybaev and A. P. Seœranyan Presented by Academician V.V. Rumyantsev November 23, 1999

Received November 30, 1999

The theory of parametric resonance has broad appli- j = 1, …, m, system (1) is asymptotically stable. If at |ρ | cations in science and technology [1]. The basic issue least one multiplier lies outside the unit circle j > 1, of this theory is constructing stability domains in the system (1) will be unstable (parametric resonance). space of parameters. The boundary of the stability The matrix G in (1) and period T will be considered domain may possess singularities, whose occurrence to depend smoothly on the vector p = (p , …, p ) of real causes computational problems and affects the physical 1 n properties of a system. parameters. Then, the monodromy matrix is a smooth function of parameters F(p), and its first derivatives are In this study, a classification of generic-position sin- of the form [5] gularities of stability-domain boundaries is performed T for systems of general-form linear differential equa- ∂ ∂ ∂ tions with coefficients periodic in time and dependent F T G () T ∂ =FY∫∂ Xdt+GTF∂ , (2) on two or three parameters. A constructive approach is pk pk pk proposed allowing the stability domain in the vicinity 0 of a point of its boundary to be found in the first approx- where Y(t) is the matriciant of the conjugate system imation. This approach is based on information avail- Yú = −GTY, Y(0) = I. The matriciants X(t) and Y(t) are able at that point, namely, the values of multipliers, the bound by the relation X t TY t I. eigenvectors and adjoint vectors of the monodromy ( ) ( ) = matrix, and the first derivatives of the system operator 2. The asymptotic-stability condition divides the with respect to parameters. It is worth noting that, space of parameters Rn into a stability region and insta- unlike some previous studies [1], the closeness of a bility one (the region of parametric resonance). The periodic system to a stationary one was not assumed in transition from the stability region to the instability one the present paper. Singularities of stability-domain is accompanied by the emergence of certain multipliers boundaries for stationary systems have been studied in from the unit circle. In particular, when real multipliers [2Ð4]. go through the points 1 and Ð1, it is said to be the primary resonance; the emergence of a complex-conju- 1. We consider a system of linear homogeneous dif- ± ω ferential equations with periodic coefficients gate pair through the unit circle at the points exp( i ) (ω ≠ πk, k ∈ Z) is spoken of as a combinative resonance. xú = Gx, (1) Thus, the stability-domain boundary is determined by the presence of multipliers in the unit circle, providing where x is the real vector of the dimension m, and G = the rest of the multipliers to lie inside the unit circle. G(t) is a real-valued matrix function of the dimension × Generally, the stability-domain boundary is a smooth m m, which is continuous in time t and periodic with hypersurface in a parameter space with certain singu- minimum period T, i.e., G(t + T) = G(t). larities (points where the smoothness is lost). Singular- A matrix-valued function X(t) obeying the equation ities of the generic position (typical singularities) are of Xú = GX, X(0) = I, where I is the unit matrix, is referred particular interest. The emergence of such singularities to as a matriciant, with its value F = X(T) being the is always expected in studies of particular systems. As monodromy matrix [1]. The eigenvalues of the mono- for the singularities of the nongeneric position, they are dromy matrix F are said to be multipliers. a consequence of a certain degeneracy or a symmetry of a system and disappear if the system is subjected to a The stability of system (1) is determined by the fol- ρ perturbation as small as is wished [2]. lowing conditions imposed on the multipliers 1, …, ρ [1]: If all multipliers lie inside the unit circle |ρ | < 1, We denote the types of the boundary points using m j the product of multipliers located in the unit circle, with exponents equal to the dimensions of the corresponding 2 ± ω ± ω Jordan boxes. For example, 1 exp( i 1)exp( i 2) Institute of Mechanics, Moscow State University, implies that the monodromy matrix F has twofold ρ = 1 Michurinskiœ pr. 1, Moscow, 117192 Russia with the Jordan box of the second order and two pairs

1028-3358/00/4508-0410 $20.00 © 2000 MAIK “Nauka/Interperiodica” ON SINGULARITIES OF BOUNDARIES FOR PARAMETRIC RESONANCE 411 ρ ± ω ± ω of simple multipliers = exp( i 1), exp( i 2), such C B ω ω ∈ π ω ≠ ω i jk that 1, 2 (0, ), 1 2 . For convenience, we introduce a concise notation B B B B for certain types of boundary points i 3 j k ( ) ( ) ( (± ω)) S S B1 1 , B2 Ð1 , B3 exp i , (3) ( 2) (( )2) C1 1 , C2 Ð1 , (4) Fig. 1. ( 3) (( )3) (( (± ω))2) D1 1 , D2 Ð1 , D3 exp i , (5) C B i jk B and for their combinations as well i3 B3 (± ω) ( ) (± ω) Bi B3 Bj Bk Bi S B12(1(Ð1)), B13(1exp i ), B23( Ð1 exp i ), S S ( (± ω ) (± ω )) Di Ci B33 exp i 1 exp i 2 , (6) ( ( ) (± ω)) CiBj Bijk B123 1 Ð1 exp i , B33 S B B ( (± ω ) (± ω )) Ci B 3 ij Bij B jk B B133 1exp i 1 exp i 2 , j ik 3 (( ) (± ω ) (± ω )) S S B233 Ð1 exp i 1 exp i 2 , D3 ( (± ω ) (± ω ) (± ω )) B333 exp i 1 exp i 2 exp i 3 , Fig. 2. (7) ( 2( )) C1 B2 1 Ð1 , We introduce the real n-dimensional vectors r and k (( )2 ) ( 2 (± ω)) C2 B1 Ð1 1 , C1 B3 1 exp i , with the components C B ((Ð1)2 exp(±iω)). 2 3 T ∂F v0 ∂ u0 Qualitative analysis of the stability domain in the s s ps r + ik = ------, s = 1, …, n, (9) vicinity of its boundary point is carried out on the basis vT u of the theory of versal deformations [2, 6]. The result 0 0 obtained is stated in the form of a theorem. ρ where i is the imaginary unit. If 0 is a real number, then Theorem 1. In the generic-position case, the stabil- k = 0. ity-domain boundary for system (1) consists of (a) iso- (b) We now consider the case of the twofold multi- lated points of B1-, B2-, and B3-types [see (3)] corre- ρ sponding to the primary resonance and combinative plier 0 with the second-order Jordan box. Jordan one in the case of one parameter; (b) smooth curves of chains for right and left eigenvectors and adjoint vec- type (3) intersecting each other transversally (at non- tors are of the form zero angle) at break points of types (4), (6) in the case ρ ρ of two parameters; (c) smooth surfaces of type (3), F0u0 = 0u0, F0u1 = 0u1 + u0; (10) whose singularities are curves of types (4), (6) (i.e., the T ρ T T ρ T T v0 F0 = 0v0 , v1 F0 = 0v1 + v0 . dihedral angle) and separate points of type (5) D1, D2 (i.e., the edge break), D (i.e., dead end in the edge), 3 We consider the vectors u0 and u1 as fixed and intro- and (7), (i.e., trihedral angle) in the case of three T T parameters. duce the normalization v0 u1 = 1, v1 u1 = 0 unambigu- The stability domains in the vicinity of singular ously defining the vectors v0, v1 . We define the real points of the above-listed types, up to the nondegener- vectors f1, f2, q1, and q2 with the components ate smooth change of parameters (diffeomorphism), are of the form presented in Figs. 1 and 2 (the stability s s T ∂F f 1 + i f 2 = v0 ∂ u0, domain is denoted by the letter S). ps (11) 3. Consider a point of the stability-domain boundary ∂ ∂ p = p . s s T F T F , …, 0 q1 + iq2 = v0 ∂ u1 + v1 ∂ u0, s = 1 n. ρ ps ps (a) Let 0 be a simple multiplier of the matrix F0 = F(p0) and u0, v0 be the right and left eigenvectors, If ρ is a real number, then f = q = 0. respectively, corresponding to the multiplier 0 2 2 (c) Next, we consider the case of the threefold real ρ T ρ T ρ F0u0 = 0u0, v0 F0 = 0v0 . (8) multiplier 0 with the third-order Jordan box. The Jor-

DOKLADY PHYSICS Vol. 45 No. 8 2000 412 MAŒLYBAEV, SEŒRANYAN

r T ρ T T T ρ T T f1 f1 v1 F0 = 0v1 + v0 , v2 F0 = 0v2 + v1 . C1 q1 – f1 q1 – f1 B1 C B 1 S B1 S 3 Considering the vectors u0, u1, and u2 as ﬁxed, we S T T impose the normalization conditions v0 u2 = 1, v1 u2 = 0, Fig. 3. T ρ and v2 u2 = 0 on v0, v1, and v2. Since 0 is a real number, the vectors ui and vi can be chosen to be real. We dan chains for 1 and left eigenvectors and adjoint vec- introduce the n-dimensional vectors g, h, and t and a tors are of the form matrix R with the dimension n × n of the form ρ ρ F0u0 = 0u0, F0u1 = 0u1 + u0, s T ∂ s T ∂ T ∂ ρ T ρ T g = v0 ∂ (F)u0, h = v0 ∂ (F)u1 + v1 ∂ (F)u0, F0u2 = 0u2 + u1, v0 F0 = 0v0 , (12) ps ps ps

∂ ∂ ∂ s T T T , …, t = v0 ∂ (F)u2 + v1 ∂ (F)u1 + v2 ∂ (F)u0, s = 1 n, ps ps ps (13) ∂ ∂ ∂2 T [ ρ T ]Ð1 F T F R = v0 ∂ (F) F0 Ð 0I Ð v0v2 ∂ u0 Ð v0 ∂------∂------u0 , pi p j pi p j i, j = 1, …, n.

Expressions for the second derivatives of the matrix F where vectors are calculated for the multiplier deter- are presented in [5]. mining the type of a boundary point. The tangent cones 4. We draw a smooth curve p = p(ε) from the point for the combined types (6), (7) are produced by the intersection of tangent cones found for each of the sub- of the boundary of a parametric resonance [p0 = p(0), ε ≥ 0]. In this case, some curves will lie (for small types. For example, in the case C1B2, we have KC B = ε 1 2 > 0) in the stability domain and others, in the instabil- K ∩ K . C1 B2 dp ity one. We select a direction e = of the curves In the cases B , B , B , C , C , and D , all smooth dε 1 2 3 1 2 3 ε = 0 curves drawn in e-direction satisfying rigorous inequal- lying in the stability domain. The set of such directions ities (14) lie in the stability domain for sufﬁciently ε forms a cone tangent to the stability domain at the point small > 0. In the cases D1 and D2, curves satisfying of its boundary p = p0. The tangent cone is an approxi- additional conditions mation for the stability domain in the vicinity of point of its boundary. (Re, e) Ð (t Ð h, e)(h, e) < (g, d) < (Re, e) Theorem 2. Cones tangent to the stability domain and at its boundary points of the type (3)Ð(5) are determined by the relations (Re, e) < (g, d) < (Re, e) Ð (t + h, e)(h, e)

K = {e: (r, e) ≤ 0}, K = {e: (r, e) ≥ 0}, 2 B1 B2   1 d p respectively, where d = - - ------  lie in the stabil- K = {e: (rcosω + ksinω, e) ≤ 0},  2 dε2  B3 ε = 0 { ( , ) ≤ , ( , ) ≤ } ity domain. It is worth noting that the tangent cones KC = e: f 1 e 0 q1 Ð f 1 e 0 , 1 K , K , and K are degenerate (are two-dimen- D1 D2 D3 K = {e: (f , e) ≤ 0, (q + f , e) ≥ 0}, C2 1 1 1 sional angles in the three-dimensional parameter space). K = {e: (g, e) = 0, (h, e) ≤ 0, (t Ð h, e) ≤ 0}, (14) D1 Theorem 2 enables us to find, in the first approxima- K = {e: (g, e) = 0, (h, e) ≤ 0, (t + h, e) ≥ 0}, tion, the stability domain in the vicinity of a point of the D2 stability-domain boundary from information at this { ( ω ω, ) K D = e: f 2 cos2 Ð f 1 sin2 e = 0, point [using first derivatives of the operator G from (1) 3 with respect to parameters and values of multipliers and ( ω ω, ) ≤ f 1 cos2 + f 2 sin2 e 0, corresponding eigenvectors and adjoint vectors calculated for p = p ]. Relations (14) provide a clear idea of ( ω ω ω ω, ) ≤ } 0 q1 cos + q2 sin Ð f 1 cos2 Ð f 2 sin2 e 0 , the stability domain. For example, in the case of non-

DOKLADY PHYSICS Vol. 45 No. 8 2000 ON SINGULARITIES OF BOUNDARIES FOR PARAMETRIC RESONANCE 413

V 3.0

(a) (b) 2.8 n V V 3.0 3.0 2.6 ν 0.6 0.4 6 2.6 p0 2.6 p0 ν 0.5 ν 0.5 0 0 S 2.2 2.2 8 0 0 ω S ω 5 ω 5 S 10 10 10

Fig. 4.

singular point B1, a tangent cone is determined by the is denoted by a dot. The following notation is used: ≤ inequality (r, e) 0. Therefore, a plane tangent to the 2 stability-domain boundary is specified by the equation τ = αt, f (τ) = vú (τ) + v (τ), (r, e) = 0, and the vector r is a normal to the boundary ω lying in the region of a parametric resonance (see (τ) ( ν τ) v = V 1 + sinw , w = -α--, Fig. 3). For the case of the singular point C1, the tangent cone is the intersection of the half planes (half U 2 c spaces) f e ≤ and q f e ≤ . These inequalities V = -----, and α = ------. ( 1, ) 0 ( 1 – 1, ) 0 α 3 determine a two-dimensional (dihedral) angle in a l ml space of two (three) parameters, which is the first The matrix operator G(τ) with the period T = 2π/w approximation to the stability domain (Fig. 3). smoothly depends on the dimensionless parameters p = 5. As an example, we consider two-dimensional (w, ν, V). vibrations of a two-link pipe, along which a fluid with We consider the point p0 = (8, 0.737, 2.8) in the the linear mass m and pulsating velocity parameter space. We calculate the monodromy matrix F0 at this point and find the multipliers of the matrix: u(t) = U(1 + νsinωt) ρ (± ) ρ ρ 1, 2 = exp 0.882i , 3 = 0.535, 4 = 0.152. flows. The pipe parts are connected by elastic hinges having the rigidity coefficients equal to c; they have the Since ordinary complex-conjugate multipliers lie in the |ρ | length l and linear mass M = 2m. The right end is free. unit circumference 1, 2 = 1, and the other multipliers We choose the angles ϕ and ψ of the deviation of the are inside the unit circle, the point p0 is the regular point pipe parts from a horizontal axis as generalized coordi- of the stability-domain boundary of the B3-type. nates. The linearized equations for the motion of the According to (14), the K cone tangent to the stability B3 system in terms of dimensionless variables are of the domain at the point p0 is described by the expression form [7] ω ω ≤ ω ρ (rcos + ksin , e) 0, where = arg 1; the vectors r T and k are calculated according to (2), (9). The vector (ϕ, ψ, ϕú , ψú ) xú = Gx, x = , n = rcosω + ksinω = (0.03, Ð2.05, 0.51) is the normal to the stability-domain boundary and lies in the para-     O I 4 1.5 metric-resonance region. The numerically calculated G(τ) =   , M =   , (15) boundary of the stability domain and the vector n are  Ð1 Ð1    ÐM C ÐM B 1.5 1 shown in Fig 4a.

Finally, we consider the point p0 = (3.643, 0.5555,    (τ) (τ )  v (τ) 1 2   2 Ð f Ð 1 + f  2.6), at which the monodromy matrix F 0 has the multi- B = , C = , ρ ρ ρ ρ  0 1   Ð1 1  pliers 1 = 2 = –1, 3 = 0.225, 4 = 0.026, second-order Jordan chains (10) being relevant to a twofold multi- ρ where O is the zero matrix of the dimension 2 × 2, and plier (we denote it by 0 = Ð1). Since the twofold mul- τ ρ differentiation with respect to the dimensionless time tiplier 0 = Ð1 belongs to the boundary, and two others

DOKLADY PHYSICS Vol. 45 No. 8 2000 414 MAŒLYBAEV, SEŒRANYAN lie inside the unit circle, from Theorem 1, a singularity Our results can be employed in solving various of the stability-domain boundary of the C2 type (i.e., of problems of stabilization of parameter-dependent peri- the dihedral-angle type) is realized at the point p0 . odic systems with the use of gradient methods. According to (2), (11), we find the vectors f1 = (Ð5.15, 45.2, Ð7.77), q1 = (4.49, Ð31.1, 3.16). From Theorem 2, REFERENCES these vectors define the cone tangent to the stability 1. V. A. Yakubovich and V. M. Starzhinskiœ, Parametric domain Resonance in Linear Systems (Nauka, Moscow, 1987). 2. V. I. Arnol’d, Complementary Chapters in the Theory of K = {e: (f , e) ≤ 0, (q + f , e) ≥ 0}. C2 1 1 1 Ordinary Linear Equations (Nauka, Moscow, 1978). 3. A. A. Mailybaev and A. P. Seyranian, SIAM J. Matrix The vectors f1 and –(q1 + f1) are normals to the sides of Anal. Appl. 20 (4) (1999). the dihedral angle, and they lie in the region of the para- 4. A. A. Maœlybaev and A. P. Seœranyan, Prikl. Mat. Mekh. metric resonance. The vector eτ tangent to the edge of 63, 568 (1999). × the dihedral angle is equal to eτ = (q1 + f1) f1. In 5. A. P. Seyranian, F. Solem, and P. Pedersen, Arch. Appl. Fig. 4b (to the left), a dihedral angle is shown, which is Mech. 69, 160 (1999). an approximation for the stability-domain boundary in 6. D. M. Galin, Usp. Mat. Nauk 27, 241 (1972). the vicinity of the point p0 . For comparison, at the right 7. Z. Szabo, S. C. Sinha, and G. Stepan, in Proceedings of of Fig. 4b, the numerically calculated stability-domain the European Nonlinear Oscillation Conference, boundary is shown confirming the presence of the sin- EUROMECH II, Prague, 1996, Vol. 1, pp. 439Ð442. gularity and reasonable agreement of the results obtained. Translated by V. Tsarev

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

Structure of Solutions to Linear Unsteady Boundary Value Problems in Mechanics and Mathematical Physics A. D. Polyanin Presented by Academician D.M. Klimov December 16, 1999

Received December 17, 1999

We consider unidimensional and multidimensional can be represented as the sum linear unsteady nonhomogeneous boundary value x problems for equations of the parabolic and hyperbolic t2 types with coefficients being arbitrary functions of spa- wx(), t =∫∫Φ()y,τGx(),,, yt τdydτ tial coordinates and time. We derive general formulas 0x1 allowing solutions to these problems to be expressed in x terms of the Green’s functions in the case of boundary 2 conditions of all basic types. These results can be used +∫fy()Gx(),,, yt 0dy in the theory of heat transfer and mass transfer, in wave x theory, and in other branches of mechanics and theoret- 1 (6) t ical physics. τ,τΛ,,τ τ The formulas obtained generalize results of a large +∫g1()ax()1 1()xt d number of studies (e.g., [1Ð7]) in which special 0 unsteady boundary value problems were considered. t 1. Parabolic equations with one spatial variable. τ,τΛ,,τ τ +∫g2()ax()2 2()xt d, We consider a linear nonhomogeneous differential parabolic equation of the general form with variable coef- 0 ∂ ficients where xw is the partial derivative with respect to x. τ ∂w Here, G(x, y, t, ) is the Green’s function that, for arbi------ÐL ,[]w =Φ(),xt, (1) trary t > τ ≥ 0, satisfies the homogeneous equation ∂t xt ∂G where ------Ð0L ,[]G = (7) ∂txt ∂2 ∂ []≡, w , w , Lxt, w ax() t------++bx() t------cx() tw, with the special nonhomogeneous initial condition and ∂x2 ∂x (2) homogeneous boundary conditions ax()0., t > G = δ(x – y) for t = τ, (8) ∂ Hereafter, subscripts in the operator L indicate that its s1xGk+01G= for x = x1, (9) coefficients depend on x and t. ≥ ≤ ≤ s∂Gk+0G= for x = x . (10) In the domain {t 0, x1 x x2}, the solution to the 2x 2 2 boundary value problem for equation (1) under arbi- Here, the quantities y and τ enter into problem (7)Ð(10) trary nonhomogeneous initial conditions and boundary ≤ ≤ δ as free parameters (x1 y x2), and (x) is the Dirac conditions delta function. Λ τ Λ τ wfx= () for t = 0, (3) The functions 1(x, t, ) and 2(x, t, ) in the inte- s∂wk+w= g()t for x = x , (4) grands of two last terms in solution (6) are expressed in 1 x 1 1 1 terms of the Green’s function G(x, y, t, τ). For the s∂wk+w= g()t for x = x , (5) boundary value problems of basic types, the corre- 2 x 2 2 2 Λ τ sponding formulas for m(x, t, ) (m = 1, 2) are listed in Institute of Problems in Mechanics, Table 1. Russian Academy of Sciences, It is important to emphasize that both the Green’s Λ pr. Vernadskogo 101, Moscow, 117526 Russia function G and functions m are independent of the

Table 1 Λ τ Type of the boundary value problem Form of the boundary conditions Functions m(x, t, ) Λ τ ∂ ( , , , τ) First (s1 = s2 = 0, k1 = k2 = 1) w = g1(t) at x = x1 1(x, t, ) = yG x y t y = x1 Λ τ ∂ ( , , , τ) w = g2(t) at x = x2 2(x, t, ) = Ð yG x y t y = x2 ∂ Λ τ τ Second (s1 = s2 = 1, k1 = k2 = 0) xw = g1(t) at x = x1 1(x, t, ) = ÐG(x, x1, t, ) ∂ Λ τ τ xw = g2(t) at x = x2 2(x, t, ) = G(x, x2, t, ) ∂ Λ τ τ Third (s1 = s2 = 1, k1 < 0, k2 > 0) xw + k1w = g1(t) at x = x1 1(x, t, ) = ÐG(x, x1, t, ) ∂ Λ τ τ xw + k2w = g2(t) at x = x2 2(x, t, ) = G(x, x2, t, ) Λ τ ∂ ( , , , τ) Mixed (s1 = k2 = 0, s2 = k1 = 1) w = g1(t) at x = x1 1(x, t, ) = yG x y t y = x1 ∂ Λ τ τ xw = g2(t) at x = x2 2(x, t, ) = G(x, x2, t, ) ∂ Λ τ τ Mixed (s1 = k2 = 1, s2 = k1 = 0) xw = g1(t) at x = x1 1(x, t, ) = ÐG(x, x1, t, ) Λ τ ∂ ( , , , τ) w = g2(t) at x = x2 2(x, t, ) = Ð yG x y t y = x2

Φ functions , f, g1, and g2 that deﬁne nonhomogeneous and arbitrary nonhomogeneous boundary conditions (4) terms in the boundary value problem. If the coefﬁcients and (5), can be represented as the sum of operator (2) are independent of time t, then the t x Green’s function depends only on three arguments: 2 G(x, y, t, τ) = G(x, y, t – τ). w(x, t) = ∫∫ Φ(y, τ)G(x, y, t, τ)dydτ

Formula (6) also holds true for this problem with the 0 x1 boundary conditions of the third kind provided that k1 = x2 k1(t) and k2 = k2(t). In this case, the relations between ∂ Λ Ð f (y) G(x, y, t, τ) dy the functions m (m = 1, 2) and the Green’s function G ∫ 0 ∂τ τ = 0 are the same as in the case of the constant coefficients x1 k1 and k2 (although the Green’s function in itself is dif- x ferent). 2 [ ϕ , ] , , , The solutions to the first, second, and third + ∫ f 1(y) + f 0(y) (y 0) G(x y t 0)dy (14) boundary value problems in the semi-infinite interval x1 ≤ ∞ x1 x < are often required to vanish at infinity ∞ t (w 0 as x ). In this case, the solution can be τ , τ Λ , , τ τ Λ + g1( )a(x1 ) 1(x t )d found by formula (6) with 2 = 0 and with the expres- ∫ Λ sion for the function 1 given in Table 1. 0 It is worth noting that the Green’s function for vari- t ous unsteady boundary value problems can be found, τ , τ Λ , , τ τ + ∫g2( )a(x2 ) 2(x t )d . for example, in the books [1, 2, 5Ð8]. 0 2. Hyperbolic equations with one spatial variable. We consider a linear nonhomogeneous hyperbolic Here, G(x, y, t, τ) is the Green’s function that satisfies equation of the general form with variable coefficients the homogeneous equation 2 ∂2G ∂G ∂ w ∂w ϕ , ------[ ] ------+ ϕ(x, t)------Ð L , [w] = Φ(x, t), (11) ------+ (x t) Ð Lx, t G = 0 (15) 2 x t ∂ 2 ∂t ∂t ∂t t with the semi-homogeneous initial conditions where the form of the operator Lx, t[w] is given by formula (2). G = 0 at t = τ, (16) ∂ δ τ The solution to equation (11), which satisfies the tG = (x – y) at t = (17) nonhomogeneous initial conditions and homogeneous boundary conditions (9) and (10). w = f0(x) at t = 0, (12) Λ τ Λ τ The functions 1(x, t, ) and 2(x, t, ) in the inte- ∂ tw = f1(x) at t = 0 (13) grands of two last terms in solution (14) are expressed

DOKLADY PHYSICS Vol. 45 No. 8 2000 STRUCTURE OF SOLUTIONS TO LINEAR UNSTEADY BOUNDARY VALUE PROBLEMS 417

Table 2 Type of the boundary value problem Form of the boundary conditions (21) Function U(x, y, t, τ) ∂ ∈ τ G τ First w = g(x, t) at x S U(x, y, t, ) = Ð∂------(x, y, t, ) My ∂ w ∈ τ τ Second -∂------= g(x, t) at x S U(x, y, t, ) = G(x, y, t, ) Mx ∂ w ∈ τ τ Third -∂------+ kw = g(x, t) at x S U(x, y, t, ) = G(x, y, t, ) Mx in terms of the Green’s function G(x, y, t, τ). For the can be represented as a sum boundary value problems of basic types, the corre- t sponding formulas for Λ (x, t, τ) are listed in Table 1. m , Φ , τ , , , τ τ w(x t) = ∫∫ (y )G(x y t )dV yd Formula (14) also holds true for this problem for the 0 V boundary conditions of the third kind with k1 = k1(t) and k2 = k2(t). , , , + ∫ f (y)G(x y t 0)dV y (22) 3. Parabolic equations with many variables. We V consider a linear nonhomogeneous differential para- t bolic equation in n spatial variables, which takes the , τ , , , τ τ general form + ∫∫g(y )U(x y t )dSyd . 0 S ∂w Here, G(x, y, t, τ) is the Green’s function that, for all t > ------Ð L , [w] = Φ(x, t), (18) ∂t x t τ ≥ 0, satisfies the homogeneous equation ∂G ------Ð L , [G] = 0 (23) where ∂t x t

n with the special nonhomogeneous initial conditions ∂2 [ ] ≡ , w and homogeneous boundary condition Lx, t w ∑ aij(x t)∂------∂------xi x j i, j = 1 G = δ(x – y) at t = τ, (24) n ∂ Γ [ ] ∈ , w , x, t G = 0 at x S. (25) + ∑ bi(x t)∂------+ c(x t)w, (19) xi i = 1 The quantity y = {y1, …, yn} enters into problem (23)Ð (25) as an n-dimensional free parameter (y ∈ V), and n n 2 δ δ δ …δ { , …, } , ξ ξ ≥ σ ξ σ > (x Ð y) = (x1 Ð y1) (x2 Ð y2) (xn Ð yn). x = x1 xn , ∑ aij(x t) i j ∑ i , 0. i, j = 1 i = 1 The Green’s function is independent of the functions Φ, f, and g defining nonhomogeneous terms in the Equations of this type, with n = 1, 2, 3, are the basis of boundary value problem. The integration in solution (22) the mathematical theory for the mass-and-heat transfer is carried out over the parameter y, with dVy = [1, 7, 8]. dy1dy2…dyn. Let V be a closed domain in the space ᏾n having a The function U(x, y, t, τ) in the integrand of the last sufficiently smooth indivisible surface S = ∂V. The solu- term of solution (22) is expressed in terms of the tion to the linear boundary value problem in the domain Green’s function G(x, y, t, τ). For the boundary value V for equation (18) with arbitrary nonhomogeneous ini- problems of three basic types, the corresponding for- τ tial and boundary conditions mulas for U(x, y, t, ) are listed in Table 2 (in the case of the third boundary value problem, the coefficient k can depend on x and t). The operators of differentiation x w = f( ) at t = 0, (20) along the conormal of operator (19), which enter into the boundary conditions of the second and third kinds Γ [ ] , ∈ x, t w = g(x t) at x S (21) and in the solution to the first boundary value problem,

DOKLADY PHYSICS Vol. 45 No. 8 2000 418 POLYANIN are defined by the relations [4] Here, G(x, y, t, τ) is the Green’s function satisfying n both the homogeneous equation ∂G ∂G ≡ , 2 ------∑ aij(x t)N j------, ∂ ∂ ∂M ∂x G ϕ , G [ ] x , i ------+ (x t)------Ð Lx, t G = 0 (31) i j = 1 (26) ∂t2 ∂t ∂ n ∂ G ≡ , τ G with the semihomogeneous initial conditions -∂------∑ aij(y )N j-∂------, τ My yi G = 0 at t = , i, j = 1 ∂ G = δ(x Ð y) at t = τ where N = {N1, …, Nn} is the unit outward normal to t the surface S. In the special case when ai,i(x, t) = 1 and and homogeneous boundary condition (25). ≠ ai,j(x, t) = 0 for i j, operator (26) coincides with the If the coefficients entering into equations (31) and usual operator of differentiation along the normal to the boundary condition (25) are independent of time t, then surface S. the Green’s function takes the form G(x, y, t, τ) = If the coefficients of equation (23) and boundary G(x, y, t – τ). condition (25) are independent of time t, then the Green’s The function U = U(x, y, t, τ) in the integrand of the function takes the form G(x, y, t, τ) = G(x, y, t – τ). last term in solution (30) is expressed in terms of the Remark. If boundary conditions of different types are Green’s function G(x, y, t, τ). For the boundary value problems of principal types, the corresponding formulas imposed in certain domains of the surface S = ∑Si , i.e., for U are listed in Table 2 (in the case of the third bound- i ary value problem, the coefficient can depend on x and (i) t). It is worth noting that the relation between the func- Γ , [w] = g (x, t) for x ∈ S , x t i i tion U and the Green’s function in the case of the bound- then the last term in solution (22) should be replaced by ary value problem for parabolic equation (18) is the same the sum as that for hyperbolic equation (27) provided that the t boundary conditions imposed are identical (naturally, the , τ , , , τ τ Green’s functions for these problems are different). ∑∫∫gi(y )Ui(x y t )dSyd . i 0 Si 4. Hyperbolic equations with many variables. We ACKNOWLEDGMENTS consider a linear nonhomogeneous hyperbolic equation This work was supported by the Russian Foundation in n spatial variables which takes the general form for Basic Research, projects nos. 00-02-18033 and ∂2 ∂ 00-03-32055. w ϕ , w [ ] Φ , ------+ (x t)------Ð Lx, t w = (x t), (27) ∂t2 ∂t REFERENCES where the operator Lx, t[w] is explicitly given by formula (19). 1. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Clarendon Press, Oxford, 1959; Nauka, Moscow, The solution to the boundary value problem in the 1964). domain V for equation (27) under the arbitrary nonho- 2. Linear Equations of Mathematical Physics, Ed. by mogeneous initial conditions S. G. Mikhlin (Nauka, Moscow, 1964; Holt, Rinehart w = f0(x) at t = 0, (28) and Winston, New York, 1967). ∂ w = f (x) t 3. G. E. Shilov, Mathematical Analysis: The Second Spe- t 1 at = 0 (29) cial Course (Nauka, Moscow, 1965). and arbitrary linear boundary condition (21) takes the 4. Mathematical Éncyclopedia (Sov. Éntsiklopediya, Mos- form cow, 1977), Vol. 1, p. 1130. t 5. A. G. Butkovskiœ, Green’s Functions and Transfer Func- w(x, t) = Φ(y, τ)G(x, y, t, τ)dV dτ tions Handbook (Nauka, Moscow, 1979; Horwood, ∫∫ y Chichester, 1982). 0 V 6. V. F. Zaœtsev and A. D. Polyanin, Handbook on Partial ∂ ( , , , τ) Differential Equations: Exact Solutions (Moscow, 1996). Ð ∫ f 0(y) ∂τG x y t dV y τ = 0 7. A. D. Polyanin, A. V. Vyaz’min, A. I. Zhurov, and V (30) D. A. Kazenin, Handbook on Exact Solutions to Equa- tions of Heat-and-Mass Transfer (Faktorial, Moscow, + [ f (y) + f (y)ϕ(y, 0)]G(x, y, t, 0)dV 1996). ∫ 1 0 y 8. A. N. Tikhonov and A. A. Samarskii, Equations of Math- V ematical Physics (Nauka, Moscow, 1972; Pergamon, t Oxford, 1964). , τ , , , τ τ + ∫∫g(y )U(x y t )dSyd . 0 S Translated by V. Chechin

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

On Invariants of the StressÐStrain State in Mathematical Models for Mechanics of Continua Academician E. I. Shemyakin Received March 14, 2000

In the mechanics of continua, it is usual practice to where adopt the following standard set of invariants for a sym- σ Ð σ σ + σ metric second-rank stress-state tensor regardless of the T ==------1 -3, σ ------1 3 , medium type (solid body, viscous fluid, etc.): 2 n 2 I ==σ ++σ σ , I σσ++σσσσ, σσσ σ σ 1 1 2 3 2 12 23 31 2Ðn2 2 Ð 1 Ð 3 µσ==------()T ≠ 0 . σσσ (1) I3=123. TT Here, the principal values of the stress tensor It is rather probable that the LodeÐNadai parameter µσ characterizing a type of the stress state was introduced σ σ σ 1, 2, 3 (2) in exactly the same manner. and the trihedron for the principal directions of tensor We now make two comments concerning the new σ ij (i, j = x, y, z) determine the stress state in a small vol- invariant set σ σ ume of a continuous medium; and ij = ji [1, 2]. σσσ σ 1Ð3 1 + 3 T==------, σ ------, µ σ , (4) It is generally believed in models that a change only 2n 2 in the second invariant is essential; effects of the first one for incompressible media and the third one associ- which completely specifies the stress state in an ele- ated with the type of a stress state are often ignored. As mentary volume of the medium. In other words, the fol- a result, the information on the principal-direction tri- lowing statement is valid: Three invariants (4) are nec- hedron is lost. Thus, choosing the first and second essary and sufficient to specify the stress state in an ele- invariants (for the stress deviator) does not call for mentary volume of a continuum. information on principal directions. µp A representation of invariants, which is similar dε to (1), is also used for the symmetric strain tensor or the 0 strain rate tensor. In this case, a similarity of the stress tensor and strain (strain rate) one is adopted to relate invariants of a stress state to those of a strain (strain rate) state. Note that deviations from the similarity are –0.2 both possible and natural in actual situations [3Ð5] (Fig. 1). –0.4 Taking into account these facts and proceeding from physical interpretation of the energy dissipation processes in models of plastic and viscous media, we propose to introduce another set of invariants related to –0.6 areas of the maximum tangential stresses T and principal shears Γ (shear-strain rate). Data from solid-state physics and recent results obtained in mesomechanics –0.8 [6] indicate these directions of investigation [7, 8]. Copper Aluminum In this case, in performing identity transformations, Steel it is easy to pass from (2) to substantiating this new set of invariants: –1.0 –0.8 –0.6 –0.4 –0.2 0 µσ σ≥≥σσσ≥≥σσ 123,n+T2nÐT, (3) Fig. 1. Lode parameter of the plastic-strain increment tensor ≥σσ≥ ≥≥µ T 2Ðn ÐT,1 σ Ð1, as a function of the same parameter for the stress tensor.

420 SHEMYAKIN

σ σ 1 – 3 It is well known that for a uniaxial tension (biaxial σ z compression) 1.2 Fe, Cu, Ni µσ = 1, 1.1 while in the case of a torsion, µ = 0. 1.0 σ –1.0 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1.0 µ This implies that the role of areas with tangential stresses (5) increases at µσ ≠ 1 compared to the case Fig. 2. Relative value of the maximum tangential stress as a µσ = 0. Considering the data of [4] (Fig. 2) in this σ function of the Lode parameter ( z is the ultimate strength context, we emphasize that the necessity of introducing at the uniaxial tension). the smooth plasticity conditions of the HuberÐMises type does not follow from these LodeÐNadai data. However, allowance for the LodeÐNadai parameter, T i.e., for the type of the stress state, turns out to be necessary (1 ≥ µσ ≥ –1). τ max I Thus, it is stated that introducing these three invari- σ µ ants, T, n, and σ enables us to describe anisotropy of II III the resistance to shears, which arises at irreversible strains and failure. Moreover, owing to this, we are able τ to construct a mathematical model for the deformation residual of solid bodies, which is adequate to physical processes [6Ð8]. In the case under consideration, principal facts used as the basis for the construction of this model are the experimental relationship between the maximum µ 0 tangential stress and principal torsional shear (for µσ = 0) and attraction to the consideration (in accor- ΓÂ Γ dance with the data of Fig. 2) of other area elements with extreme tangential stresses in accordance with the Fig. 3. Illustration of the descending branch of the maximum tangential stress on the principal strain diagram (at the values of the LodeÐNadai parameter. Note, that the uniaxial tension). same experimental law T = T(Γ) valid for principal elementary areas manifests itself for these areas (Fig. 3). In recent years, wide attention was attracted in geo- The first comment is associated with the fact that the physics and geomechanics to studying ordered block condition of isotropy for set (1), which is understood structures in massifs, and the hierarchy of these struc- that it does not depend on directions of the principal tures was revealed [7, 8]. Thus, they establish the exist- axes, was never discussed while introducing invariants ence and development of blocks that are closely related (except for rare cases when the third invariant should be with areas of maximum tangential stresses and princi- taken into account). Actually, allowing for the first and pal shears, which were mentioned above. In this con- second invariants did not require an account for the ori- nection, it should be pointed out that, from the physical entation of the principal-direction trihedron of the standpoint, the set of invariants proposed describes stress tensor. most closely the deformation of a massif under the In contrast to this, introducing set (4) definitely indi- action of technogenic and tectonic loads. Apparently, as cates such a dependence, because the invariant T is is shown by investigations in the strength of solid bodies related to area elements for maximum tangential conducted by the School of Academician V.E. Panin, it is stresses, which bisect angles between the first and the also reasonable to follow this route in the mechanics of σ third principal directions. The invariant n is indepen- deformable solid body. µ dent of principal directions, while the invariant σ, as The proposed model allows us to confidently trace a well as T, is closely related to the orientation of the certain important phenomenon, namely, the behavior of principal axis and indicates the influence of other (two) a material in the supercritical state. In fact, after the extreme tangential stresses maximum value of the tangential stress has been σ σ σ σ attained for a material, a practically new material 2 Ð 3 1 Ð 2 T 23 = ------, T12 = ------, (5) arises, whose behavior calls for different description 2 2 [10, 11, 13]. Nevertheless, the laws of mechanics (laws because of mass, momentum, and energy conservation) remain valid. It is these laws that determine the material prop- T 23 Ð T 12 erties in the post-ultimate (post-peak) state. When the µσ = ------. T extreme (for a given body) value of the principal shear

DOKLADY PHYSICS Vol. 45 No. 8 2000 ON INVARIANTS OF THE STRESSÐSTRAIN STATE IN MATHEMATICAL MODELS 421 Γ Γ ( = e) is attained, this shear value is preserved in an cated by Academician S.A. Christianovich in his elementary volume, and the growth of the domain of famous paper [12]. irreversible deformations (and failures) occurs only at From the practical standpoint, the results presented, the expense of an increase in an amount of such ele- including new equations for the plane problem (6) and ments. The fact that other extreme area elements come (7), (8), allow us to consider from a new standpoint the into play, in accordance with Fig. 2 and in correspon- boundary value problem for the brittle failure [10, 11, dence to the same law [T = T(Γ)], determines the resid- 13] and for a press tool on a compliant (with allowance ual strength of the material. for irreversible strains) base. In order to illustrate the proposed synthetic model, we present the system of basic equations for the problem of the plane deformation of a solid body [10, 13]. REFERENCES Papers [10, 11] contain the preliminary results concern- 1. A. A. Il’yushin, Uch. Zap. Mos. Gos. Univ. 39 (1940). ing the statement of new boundary value problems on 2. V. V. Novozhilov, Prikl. Mat. Mekh. 15 (2) (1951). the basis of the models developed. For example, new 3. L. M. Kachanov, Principles of Plasticity Theory (Nauka, equations describing irreversible deformation can be Moscow, 1969). derived on the basis of the model proposed for the case of the plane deformation of an elastoplastic body (here, 4. A. Nadai, Plasticity; Mechanics of the Plastic State of Materials (McGraw-Hill, New York, 1931; ONTI SSSR, the usual notation is used): Moscow, 1936). ∂ωˆ ∂ψ ∂ψ 5. S. A. Khristianovich and E. I. Shemyakin, Izv. Akad. ------Ð cos2ψ------+ sin2ψ------= 0, ∂x ∂x ∂y Nauk SSSR, Mekh. Tverd. Tela, No. 4, 87 (1967); No. 5, (6) 138 (1969). ∂ωˆ ∂ψ ∂ψ ------+ sin2ψ------+ cos2ψ------= 0, 6. Physical Mesomechanics and Computer Designing ∂y ∂x ∂y Materials, Ed. by V. E. Panin (Nauka, Novosibirsk, 1995). where 7. A. F. Revuzhenko, S. B. Stazhevskiœ, and E. I. Shemy- ∂ ∂ ε ω akin, Fiz. Tekh. Probl. Razrab. Polezn. Iskop., No. 3, 130 ω 1 v u ψ xy 2 ωˆ = ---∂------Ð -∂---- , tan2 = -ε------ε---, -Γ------= . (1974). 2 x y y Ð x e 8. A. F. Revuzhenko, Fiz. Tekh. Probl. Razrab. Polezn. This system is of the hyperbolic type and has two fam- Iskop., No. 2, 9 (1974). ilies of real characteristics 9. Yu. I. Yagn, Vestn. Inzh. Tekh., No. 6 (1931). dy dy 10. E. I. Shemyakin, Izv. Akad. Nauk, Mekh. Tverd. Tela 32 ----- = Ðtanψ, ----- = cotψ (7) dx dx (2), 145 (1997). 11. A. F. Revuzhenko and E. I. Shemyakin, Prikl. Mekh. with the following relationships for them: Tekh. Fiz., No. 2, 128 (1979). ω + ψ = α = const, ω Ð ψ = β = const. (8) 12. S. A. Khristianovich, Mat. Sb. 1 (4), 512 (1936). 13. E. I. Shemyakin, Fiz. Mezomekh. 2 (6) (1999). It is easy to see a close analogy between the new system for strains and the classical Saint-Venant system for stresses. Apparently, it is this analogy that was indi- Translated by V. Devitsyn

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

Undular Bore in the Counterflow V. I. Bukreev Presented by Academician V.M. Titov December 5, 1999

Received December 20, 1999

In this paper, we call an undular bore a free plane exposition time per one frame at photographing, and (as a rule, nonlinear) gravity wave, whose free surface V = Q/Bh. In the examples presented below, the flow level passes from its lower to higher value executing was subcritical, h decreased monotonically, and V gradually convergent oscillations, rather than monoton- increased downstream. ically or by a jump. The term undular wave has a At t = 0, the flow was stopped in the cross section broader meaning and includes the case of stimulated x = 0 by a rapidly dropping gate. Trajectories of labeled waves. An undular wave may be smooth, with the col- micron-size particles (i.e., particles of aluminum pow- lapsing first crest or several collapsing first crests. An der) were recorded with a photographic camera and example of recording a smooth undular bore, which video camera. In contrast to the PIV-method [5] that was performed by a stationary wavemeter, is presented has recently become popular, the time of exposure for in Fig. 1. each frame was taken on purpose greater: It ranged We have not managed to obtain adequate analytical from 1/60 to 1/2 s. This made it possible to trace trajec- representations for undular waves due to their nonlin- tories of individual particles for a sufficiently long time, earity and unsteadiness. Incomplete available informa- although the shape of the free surface was strongly dis- tion on undular waves is based on physical and numer- torted. Information on the shape of the free surface was ical experiments. The majority of investigations are obtained by the video recording, which determined devoted to the undulating hydraulic jump [1]. Detailed ∆z(x) at a fixed t, and also by stationary wavemeters experiments with free undular waves in a channel were recording ∆z(t) for fixed x. The velocity of the longitu- conducted in [2]. An example of numerical experiments dinal displacement c for the height-averaged point of based on the NavierÐStokes equations is represented by the wave leading edge was calculated according to sig- the paper [3]. Mathematical models yielding solutions nals of two wavemeters spaced by ∆x = 30 cm apart. of the undular-bore type with the collapsing leading Henceforth, this quantity is referred to as the bore prop- edge were proposed in [4]. agation velocity. Below, we present some results of a physical exper- One new interesting result of measurements with iment, which are useful to qualitively compare the inner wavemeters is the following. Previously, in experi- structure of the undular bore with the results of numer- ments with initially stagnant fluid, it was found [6] that, ical calculations based on different mathematical mod- in contrast to predictions of the first approximation of els. Here, by the term inner structure, we imply a ran- the shallow-water theory, undular waves retained their dom ensemble of trajectories (limited in time) of smoothness, while passing by the first critical velocity labeled particles. The statement of the problem is clarified in Fig. 1. ∆ A steady nonuniform turbulent flow with the constant z, cm volumetric rate Q was created in a rectangular channel with the length of 3.8 m and the width B = 6 cm with 1 1 the plane horizontal bottom. The following notation is 5 used therewith: h(x, t) is the depth of the undisturbed 0 2 4 t, s flow, h0 is the depth ahead of the leading edge, x is the Q z, cm longitudinal coordinate, ∆z is the deviation of the free 2 3 h(x) 1 surface from the equilibrium position, t is time, ∆t is the 4 x, m 2 1 0

Lavrent’ev Institute of Hydrodynamics, Fig. 1. Unperturbed ﬂow and undular bore for Q = 0.323 l/s: Siberian Division, Russian Academy of Sciences, (1) ∆z(t) at x = 210 cm; (2) wavemeter; (3) unperturbed free pr. Akademika Lavrent’eva 15, Novosibirsk, surface; (4) channel bottom; (5) gate; for (1), h0 = 2.8 cm; 630090 Russia c1* = 33.2 cm/s; c = 40 cm/s; and c2* = 48.9 cm/s.

UNDULAR BORE IN THE COUNTERFLOW 423

Fig. 2. Bore structure for Q = 0.648 l/s (the bore propagates leftwards). In Figs. 2Ð4, 1 is the channel bottom. For scale lines x = 230 Ӎ ∆ ± and 240 cm, h0 3.55 cm; t = 1/15 s; V = 30.5 cm/s; c = 44 1.5 cm/s; c1* = 28.5 cm/s; c2* = 46.2 cm/s.

2 1

Fig. 3. Initial stage of developing the bore presented in Fig. 2: (1) channel bottom; (2) the gate. For the middle of the leading edge, x = 18 cm and ∆t = 1/15 s.

c1 = gh and began to collapse at the greater second In Fig. 2, a wave is shown in the transition state critical velocity. A good quantitative measure of the lat- between the smooth and collapsing forms. The wave ter is the ultimate propagation velocity of solitary leading edge has collapsed at x < 210 cm. Then, it became smooth but the corner point, characteristic for waves, c2 = 1.294 gh , which was derived in [7] from the critical state, retained on its frontal slope rather than the complete system of equations for the potential on the wave crest, as is the case in the theory of stand- motion. These experiments confirmed the result for ing or solitary waves with the ultimate amplitude. The waves across the counterflow as well. Note that, in the comparison of the values of c, c1* , c2* placed under- first approximation, the first and second critical veloci- neath the photograph confirms the aforesaid about the ties can be defined as c1* = gh – V and c2* = conditions for the transition from collapsing waves to smooth undular ones. 1.3 gh – V, respectively. For example, the condition c < c < c is valid for the plot 1 in Fig. 1, and a wave At the initial stage of the bore evolution (Fig. 3), 1* 2* apart from the clearly defined stratification into the jet retained its smoothness. and the stagnant zone, the formation of vortices near The most important result obtained by photograph- the free surface and collapsing of the leading edge, ing consists in the following. In the counterflow, the which are accompanied by an intense involvement of severe vertical stratification of the particle-motion air into water, do occur. velocity occurs in the fluid under the undular bore. A A photograph of the head part of the undular bore sufficiently thick boundary layer of the damped fluid is propagating in the fluid stagnant ahead of it is presented formed near the bottom. Above this layer, particles con- for comparison in Fig. 4. A perturbance was introduced tinue to move in the previous direction with velocities by removing a partition creating the level difference ∆ on the order of V forming a twisting jet. In a collapsing h. In this example, the condition c1 < c < c2 was ful- wave, damping of the incoming flow also takes place in filled, and the wave was smooth. Its inner structure is the near-surface layer. The unsteadiness and collapsing considerably simpler than in the case of the propaga- result in the fact that the separation into domains of tion across the counterflow. In particular, particle tra- damped and moving fluid has an alternating character jectories under the wave first crest are quite similar to in both time and space. Photographs (Figs. 2, 3) are pre- those for a solitary wave [8]. The structure of a collaps- sented as illustrations. ing undular wave propagating in stagnant fluid is con-

DOKLADY PHYSICS Vol. 45 No. 8 2000 424 BUKREEV

Fig. 4. Bore structure for the stagnant ﬂuid (the bore propagates rightwards): 1 is the channel bottom. For the scale lines x = 210 and ∆ ∆ 220 cm, t = 1/30 s; h0 = 3.5 cm; h = 3.2 cm; c = 71.5 cm/s; c1 = 58.6 cm/s; and c2 = 76.2 cm/s. siderably simpler than in the presence of the counter- 3. C. M. Lemos, Int. J. Numer. Meth. Fluids 23, 545 ﬂow. (1996). 4. V. Yu. Lyapidevskiœ, Prikl. Mekh. Tekh. Fiz. 39 (2), 40 ACKNOWLEDGMENTS (1998). 5. R. J. Adrian, Ann. Rev. Fluid Mech. 23, 261 (1991). The author is grateful to A.V. Gusev for his assis- tance in photographing. 6. V. I. Bukreev and A. V. Gusev, Izv. Akad. Nauk, Mekh. This study was supported by the Russian Founda- Zhidk. Gaza, No. 1, 82 (1999). tion for Basic Research, project no.98-01-00750. 7. M. S. Longuet-Higgins and J. D. Fenton, Proc. R. Soc. London, Ser. A 340, 471 (1974). 8. J. W. Daily and S. C. Stephan, Jr., in Proceedings of the REFERENCES III Conference on Coastial Engineering (Univ. of Cali- 1. Ven Te Chow, Open-Channel Hydraulics (McGraw Hill, fornia, Berkley, 1952), p. 13. New York, 1959). 2. H. Favre, Ondes de translation dans les canaux deco- verts (Dunod, Paris, 1935). Translated by V. Devitsyn

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

Integration of Equations for Long Waves in the DrozdovaÐKulikovskiœ Form A. G. Petrov Presented by Academician G.G. Chernyœ December 23, 1999

Received December 27, 1999

1. The DrozdovaÐKulikovskiœ equations for long- tion Sl = V = const. The pressure is calculated by con- ρ δ δ wave approximation. While deriving the Boussinesq sidering the work P l = –QS S under the transverse equations, two small parameters are usually assumed to tension of the element filled with incompressible fluid exist: the depth-to-wavelength ratio and the amplitude- to-depth one [2]. The derivation of the Boussinesq SQS S δL P ==------Ð ------. (1.2) equations for fluid flow in channels [1] does not assume ρl ρlδS the smallness of the second parameter and is as follows. The mass and momentum conservation laws for the Equations (1.1) and (1.2) with the additional condition potential flow of perfect incompressible fluid in a hori- Sl = V = const have been obtained in [1]. In the case of zontal channel of an arbitrary cross section are written the calculated function L(,Súú Sú , S), they form the out as desired closed system of the Boussinesq equations for determining u and S. ∂S ∂ Su ∂u∂ u ∂P ------+0,------==S------+ u ------Ð.------(1.1) ∂t ∂x ∂t∂ x ∂x Taking into account the constant value of the volume V, it is more convenient to express the total pres- Here, t is time; x is the coordinate along the bed axis; u sure in terms of the variational derivative for the func- is the velocity along the bed axis, which is averaged tion L/(ρV), the layer width l not entering into this over the cross section; P is the sum (divided by the den- derivative. As a result, we obtain the closed system of sity) of the pressure forces in the cross section; and S is two equations the flow cross-section area. ∂ S ∂ Su The function P is determined in [1] by application of -----∂ - +0,------∂ = the Lagrange generalized coordinates to a certain fluid t x ≤ ≤ (1.3) volume. We isolate a fluid element x0 – l/2 x x0 + l/2 ∂u∂ u ∂2 δ L S------+ u ------=------S ------. of a small width l and with the cross section S (see ∂t∂ x ∂xδ S ρ V Fig. 1). This fluid element moves in the flow being deformed. The element deformation is ignored, so that Approximation (1.3) can be shown to be the first term the lateral faces of the element remain parallel. S is úúú taken as the generalized coordinate of the element, and of the asymptotic series in the derivatives S, S, S, …. the Lagrange equation is written out: The second term of the series is obtained by introduc- δδ ∂ ∂ 2 ∂ LL L d L d L … ------==Ð Q S , ------Ð ------+ ------Ð . z δSδ S ∂ S dt∂ Sú dt2∂ Súú On the left-hand side, there is the variational derivative of the Lagrange function. This function L = T – U is equal to the difference between the kinetic and poten- S(t, x) tial energies of the fluid element and depends on S and l time derivatives Sú, Súú. The function L depends also on the size l, which is expressed in terms of S by the rela- x x0 Institute of Problems in Mechanics, y Russian Academy of Sciences, pr. Vernadskogo 101, Moscow, 117526 Russia Fig. 1.

1028-3358/00/4508-0425 $20.00 © 2000 MAIK “Nauka/Interperiodica” 426 PETROV ing the second generalized coordinate determining the Examples of exact solutions to steady-state equa- fluid-element deformation, etc. tions (1.3) obtained with general integral (2.2) are pre- 2. An integral of steady motion. If the function sented below. L/(ρV) does not explicitly depend on t and x, three inte- 3. Solitary waves in channels. Methods for calcu- grals of system (1.3) can be found for a steady motion. lating the kinetic energy Two of them are the constant flow rate and the Ber- noulli integral: 1 2 2 T (v v ) ρ---- = --∫ y + z dy dz 2 l 2 Q 2 δ  L  S Su = Q, Ð------+ S ------= A . S δSρV 1 in a channel cross section of an arbitrary form are presented in [1]. Here, y and z are the Cartesian coordi- The Bernoulli integral can be written out in the form of nates in the channel cross section; the z-coordinate the Lagrange equations or of the Hamilton principle, being directed vertically upwards; and vy, vz are the which is equivalent to these equations: corresponding components of the velocity for fluid particles. The integration is carried out over the wetted 2 δ  L Q A1  area of the cross section. -----------+ ------+ ----- + A2 δS ρV 2S2 S The velocity field has a potential ϕ, which is deter- ∞ mined from the solution of the boundary value problem  L Q2 A  for the Poisson equation = 0 ⇒ δ ------+ ------+ -----1 + A dt = 0. ∫ ρ 2 2 V 2S S 2 2 Ð∞ ∂ ϕ ∂ ϕ Sú QS'(x) ------+ ------= -- = ------. ∂ 2 ∂ 2 S 2 Since the Lagrange function depends only on the gen- z y S eralized coordinate S and its derivatives and does not depend explicitly on time, the Lagrange equations The kinetic energy was calculated exactly in [1] for obtained have a third energy integral. It is more conve- channels of the rectangular and triangular shapes. For nient to express this integral in terms of x-dependent shallow and narrow channels, approximate asymptotic functions. To do this, we make use of the substitution formulas were obtained. The potential energy U includes the energy U of the dt = (S/Q)dx, Sú = QS'/S (the prime denotes the deriva- g gravity force and may contain, in addition, the surface tive with respect to x) in the action integral. Then, the tension energy Uσ or the energy U of elastic film at the Hamilton principle transforms to the form E fluid surface: ∞ 2 Q Ug δ Λdx = 0, Λ = Λ + ------+ A + A S, U = U + Uσ + U , ------= Sgz , ∫ 0 2S 1 2 g E ρl c Ð∞ (2.1) L 2 Λ (S'', S', S) = ----. Uσ bσ( 1 + h' Ð 1) bσ 2 0 ρl ------= ------≈ ------(h') , (3.1) ρl ρ 2ρ The EulerÐLagrange equations corresponding to prin- U bEI 2 ciple (2.1) have the energy integral -----E- = ------(h'') . ρl 2ρ d ∂Λ ∂Λ ∂Λ ÐS '------0 + S''------0 + S'------0 Ð Λ Here, z is the vertical coordinate of the geometric cen- dx ∂S'' ∂S'' ∂S' 0 c (2.2) ter for the wetted cross section; S is the cross-section Q2 area, which is a function of the depth S(h) for a channel = ------+ A + A S, 2S 1 2 of a given shape; b = Sh is the width of the free surface; σ is the coefficient of surface tension; E, I are the in which all three arbitrary constants Q, A , and A are Young’s modulus and moment of inertia of the elastic 1 2 film, respectively. transferred to the right-hand side. For periodic waves, these three constants are expressed in terms of wave Thus, the method developed in the present paper parameters, such that the amplitude a, wavelength λ, allows us to find exact solutions to the Boussinesq and mean depth H. Furthermore, the dependence of the equations for capillary-gravitation waves in channels wave-propagation velocity on the wave parameters is and for waves with an elastic film. found from the relation between the flow rate and a, λ, We consider solutions for solitary waves with the and H. condition imposed at infinity: h h∞ as x ±∞.

DOKLADY PHYSICS Vol. 45 No. 8 2000 INTEGRATION OF EQUATIONS FOR LONG WAVES 427

We introduce the dimensionless coordinates of the nel is equal to S = h2/p. The exact expressions for the wave proﬁle X, Y and the wave velocity c: kinetic and potential energies are written out [1] as

2 v 2 T Q2(h')2 U 2h3g T Ð U x h 2 Q ∞ ---- = ------, ---- = ------, Λ = ------. X = -----, Y = -----, c = ------= ------. (3.2) ρ 2 ρ 0 ρ h∞ h∞ gh∞S∞ gh∞ l 12h p l 3 p l Here, the flow rate is expressed in terms of the cross- Energy integral (2.2) is reduced, in terms of dimension- section area S∞ and the fluid velocity at the infinity (Q = less variables (3.2), to the form S∞v∞). 2 5 For a rectangular channel with a unit width and free 1 + 3 p ( )2 2 4 4Y ------Y' = 1 + A1Y + A2Y Ð ------surface, the cross-section area is S = h. The kinetic and 6 p2 3c2 potential energies are calculated exactly: ( )  Y  ( )2 4 1 + a 2 2 2 = 1 Ð ------ Y Ð 1 1 + AY + ------2------Y  . T 1 2 Q (h') U 1 2 1 + a 3c ---- = --hhú = ------, ---- = --gh , ρl 6 6h ρl 2 (3.3) In the expansion in terms of powers of Y, the coeffi- T Ð U 3 Λ = ------. cients of Y and Y are zeros. Thus, we find the constant 0 ρl A and the velocity c: Λ Substituting 0 in energy integral (2.2), we arrive at the 3 + 2a A = ------, equation 1 + a

2 2 2 Q (h') 1 2 Q 1 + a 2a ------+ --gh = ------+ A + A h, c 2 = ------1 + ----- 6h 2 2h 1 2 1 + a/2 3

5 5 2 v ∞ 2 whose integration is carried out completely by the = 1 + --a Ð -----a + … = ------. method of separation of variables. For periodic waves, 6 36 gh∞ we obtain the well-known solution expressed in terms of elliptic functions [3]. 4. Capillary-gravitational solitary waves. In the For solitary waves, the energy integral can be rectangular channel, the wave under consideration can reduced in terms of dimensionless variables (3.2) to the be described by the Froude and Weber dimensionless form numbers

3 gh∞ σ 2  2 Y  (Y') = 3 1 + A Y + A Y Ð ----- G = ------, W = ------.  1 2 2 2 ρ 2 c v ∞ h∞v ∞ (3.4)   2 Y ( ) We determine the Lagrange function L = T – Ug – Uσ = 31 Ð ---- Y Ð 1 . c2 from (3.1) and (3.3) and substitute it into (2.1). Then, in terms of dimensionless variables (3.2), variational prin- From this expression, the constants A1, A2 are deter- ciple (2.1) and energy integral (2.2) take the form mined and the relation between the wave velocity and ∞ the dimensionless amplitude a related to the depth h∞ is found: δ∫Λ(Y', Y)dx = 0, ∞ (4.1) Q 1 1 2 c = ------= 1 + a = 1 + ----- Ð ------+ …. (3.5) (Y') 3 2 3 2a 8a2 Λ = ------(1 Ð 3WY) + ---(1 Ð GY)(Y Ð 1) , gh∞ 6Y Y

The exact trinomial expansion 2 2 1 Ð GY 1 (Y') = 3(Y Ð 1) ------, G = ----. (4.2) 1 Ð 3WY c2 1 1 1 … c = 1 + ----- Ð ------2 Ð ------2 + 2a 8a 40a The energy equation for solitary wave (3.4) follows from (4.2) for W = 0. The employment of variational [4, 6] differs insigniﬁcantly from approximation (3.5). principle (4.1) for a capillary-gravitation wave is an The problem of a solitary wave in a triangular chan- essential adjunct, because it allows for the discovery of nel with the cross-section shape z = p|y| is solved new solutions with a discontinuous derivative. In this equally easily. The cross-section area for such a chan- case, it is insufﬁcient to solve equation (4.2). It is essen-

DOKLADY PHYSICS Vol. 45 No. 8 2000 428 PETROV

Y – 1 the wave velocity is β-independent and equals c = a > 0 (a) a 1 + a . The solitons are symmetric with respect to the β < 1 point X = 0 and have a smooth profile. In the vicinity of |X| Ӷ 1, the profile is parabolic: β > 1 3a2 X2 Y = 1 + a Ð ------2------. (4.4) X 4(1 Ð β ) ( 1 + a) Ò β 1 + a β (b) For < 1, the solitons have the positive amplitude a > 0 (elevation); for β > 1, the amplitude is negative, i.e., 1 + a a < 0 (inverse soliton). The effect of the soliton inversion for the Weber number exceeding a certain threshold value was noted in the numerical studies [7]. For 1 β periodic waves, this effect was mentioned in [8]. In the 1 analytic investigation [9], the threshold value for the Weber number W = 1/3G was found, wherein the sign Fig. 2. of the soliton amplitude changes. The analytic solution for the inverse soliton in the case of a large value of the surface-tension coefficient was obtained in [10]. tial that, at the discontinuity point of solution (4.2), the Weierstrass–Erdman conditions should be satisfied, Solitons with discontinuity of a tangent, a(1 – b) < 0. The parameter t varies within the limits 1/β < t < i.e., the conditions of the continuity of the momentum ∞ Λ Λ Λ ; the numbers G, W, and B are determined by the for- Y' and energy Y' Y' – . Solution (4.2) satisfies the mulas latter condition automatically. When these conditions 1 1 are fulfilled, the solution realizes the extremum of func- G = ------, W = ------, Λ ≥ 2 ( ) tional (4.1), i.e., the minimum for Y'Y' 0 and the β (1 + a) 3 1 + a maximum for Λ ≤ 0. Y'Y' a B The general solution to equation (4.2), which satis- = ------2 , fies the conditions Y → 1 for X → ±∞, has the paramet- 1 + a Ð 1/β ric representation the wave velocity depends linearly on the Bond number 1 + βt 1 + Bt β ± 3X = Ðβln ------+ Bln ------, c = 1 + a . Solitons are also symmetric with respect 1 Ð βt 1 Ð Bt to the point X = 0, but the derivative Y'(X) has a second- kind discontinuity. For |X| Ӷ 1, the velocity profile is of 2 σ another asymptotic form: 2 1 Ð t β2 3W 3 Y = c ------2-----2 , = ------= ------2-, (4.3) 1 Ð β t G ρgh∞ 2 1/3 (β Ð 1)(1 + a) 2 Y = 1 + a Ð 3a ------2------X . (4.5) 1 Ð 3W 4β a B = ------. 1 Ð G The fulfillment of the WeierstrassÐErdman conditions The point X = ∞ corresponds to the parameter value t = is easily verified. Therefore, the solution obtained with 1/β; the values t = 0 or t = ∞ correspond to the point a discontinuity of the tangent at X = 0 is an extremal for X = 0. Therefore, the range of values for the parameter the functional in (4.1). In the extremal, for a > 0, the ∈ β Λ ≥ t is confined by the intervals t [0, 1/ (smooth soli- condition Y'Y' 0 is met and the minimum of the action ∈ β ∞ Λ ≥ tons) or t (1/ , ) (discontinuous solitons). functional in (4.1) is realized; for a < 0 and Y'Y' 0, the It is more convenient to introduce the wave ampli- maximum of the functional is realized. tude a and the Bond number β > 0 instead of the wave In Fig. 2a, the elevation-soliton profiles for constant parameters G and W. The range of values for the param- positive amplitude and various Bond numbers are eters a and β breaks down into two regions correspond- shown. As β increases, the soliton area decreases, and ing to a(1 – β) > 0 and a(1 – β) < 0 in which waves have the curvature at the vertex monotonically increases to quantitative distinctions. infinity as β 1. For β > 1, solitons have vertical tan- Smooth solitons, a(1 – b) > 0. The parameter t var- gent at the vertex and the derivative Y' becomes infinite. ies within the limits 0 ≤ t < 1/B; the numbers G, W, and At a constant positive amplitude, the soliton dimen- B depend on a and β as sionless velocity c is β-independent for β < 1 and is proportional to β for β > 1 (see Fig. 2b). 1 β2 1 + a Ð β2 G = ------, W = ------, B = ------, Figure 3a shows the profiles of the inverse solitons 1 + a 3(1 + a) a at the constant negative amplitude and various Bond

DOKLADY PHYSICS Vol. 45 No. 8 2000 INTEGRATION OF EQUATIONS FOR LONG WAVES 429

Y – 1 ACKNOWLEDGMENTS a < 0 (a) X The author is grateful to A.G. Kulikovskiœ for his β < 1 fruitful remarks and to G.G. Chernyœ for his attention to this study. β > 1

– a REFERENCES 1. Yu. A. Drozdova and A. G. Kulikovskiœ, Izv. Akad. Nauk, c Mekh. Zhidk. Gaza, No. 5, 136 (1996). (b) 2. Yu. A. Madsen and H. A. Shaffer, Philos. Trans. R. Soc. 1 Ð a London, Ser. A 356, 3123 (1998). 3. L. N. Sretenskiœ, Theory of Wave Motion of a Fluid β 1 Ð a (Nauka, Moscow, 1977). 4. E. V. Laitone, J. Fluid Mech. 9, 430 (1960). β 5. V. I. Smirnov, A Course of Higher Mathematics (Nauka, 1 Moscow, 1974; Addison-Wesley, Reading, 1964), Vol. 4, Part 1. Fig. 3. 6. Li Zhi and N. R. Sibgatullin, Prikl. Mat. Mekh. 61, 184 (1997). β 7. B. T. Brook, Quart. J. Appl. Mech. 40, 231 (1982). numbers. As increases from zero to infinity, the soli- 8. A. G. Petrov and V. G. Smolyanin, Vestn. Mosk. Univ., ton area increases from zero to infinity. At the lower Ser. 1: Mat., Mekh., No. 3, 92 (1991). point, the curvature is infinite for 0 < β < 1 and mono- 9. G. Yoss and K. Kirchgassner, Proc. R. Soc. Edinburgh, tonically decreases from infinity to zero as β increases Sect. A 122, 267 (1992). from unity to infinity. At a constant negative amplitude, 10. F. L. Chernous’ko, Prikl. Mat. Mekh. 29 (5), 856 (1995). the soliton velocity is proportional to β for β < 1 and is β-independent for β > 1 (Fig. 3b). Translated by V. Tsarev

DOKLADY PHYSICS Vol. 45 No. 8 2000

MECHANICS

Macrostructure and Microstructure of a Cocurrent Stratified Flow behind a Cylinder Yu. D. Chashechkin and V. V. Mitkin Presented by Academician A.Yu. Ishlinskiœ December 1, 1999

Received December 7, 1999

High-resolution optical and probe methods are used problem are presented by the thicknesses of the veloc- δ ν to isolate not only large-scale elements (internal waves ity boundary layer ( u = /U) and of the density bound- δ κ immersed into a wake and soaring vortices) in a contin- ary layer ( ρ = s/U), both formed on the obstacle, and λ uously stratified fluid flow around a two-dimensional the length of the attached internal wave = UTb. Both obstacle but also internal boundary flows. The latter internal boundary flows and unsteady flows induced by represent thin high-gradient interlayers inside and, diffusion on an impermeable surface [7] are character- sometimes, outside a cocurrent flow [1]. An advancing δ ν disturbance and both internal waves and vortices in a ized by the scales of velocity ν = /N and density δ κ lagging wake behind the obstacle, which were investi- s = s/N . The basic dimensionless parameters are gated in detail by experimental and theoretical meth- presented by ratios between the internal scales. They ods, are presented in [2Ð4]. Parameters of high-gradient ν δ are the Reynolds number Re = UD/ = D/ u, the internal shells that belong to the density wake and contact with Froude number Fr = U/ND = λ/2πD, the scale ratio C = the boundary layer on the obstacle are determined Λ ν κ δ δ ≈ /D, and the Schmidt number Sc = / s = u/ ρ 700 in [5]. Patterns of internal boundary flows in the field of (the Schmidt number was constant during the experi- attached internal waves, which are obtained by differ- ments). The necessity of recording all elements of the ent shadow methods, can be found in [6]. In the present flow pattern puts forward strict requirements to experi- paper, we, for the first time, have experimentally deter- mental equipment. It is the highest resolution shadow mined boundaries for a domain of existence (in the instruments that actually satisfy this necessity [8]. coordinate plane given by the Froude and Reynolds numbers) of isolated discontinuities in a wave wake The experiments were carried out in a tank with vol- behind a cylinder. We have also shown that the structure ume of 220 × 40 × 60 cm3. It had transparent windows of the extended regime diagram depends on a stratifica- and was filled with a stratified solution of common salt tion level. The usually assumed universal character of with the buoyancy period Tb from 5.3 to 25 s. Monitor- the extended diagram [3, 4] violates due to an allowing of both the homogeneity and the stratification level ance for the entire complex of steadily observed ele- was carried out by the method of a density marker [9]. ments, which include fine-structure components of Plastic horizontal cylinders were towed in the tank with both the wave wake and density wake. the constant speed U = 0.01Ð6.5 cm/s. They had the diameter D = 1.5Ð7.6 cm and a length equal to the tank Dimensional parameters of the problem are the den- width. Monitoring of motion uniformity was carried dρ sity ρ(z) and its vertical gradient ------, the free-fall accel- out by an optical method. The experiments were con- dz ducted at the following values of the dimensionless eration g, the diameter D and the velocity U of the cyl- problem parameters: 10 < Re < 5000, 0.008 < Fr < 8, inder, the kinematic fluid viscosity ν, and the diffusion and 100 < C < 15 000. κ coefficient s of a component causing stratification. The We performed the observations by the IAB-451 and exponential stratification is characterized by the scale IAB-458 shadow instruments using Maksutov methods d lnρ Ð1 [10]. The most detailed flow-pattern structure in the Λ = ------, the buoyancy frequency N = g/Λ, and dz vertical plane is obtained by the method called “Verti- the buoyancy period T = 2π/N. Internal scales of the cal slit–thread in focus” (Fig. 1). It is characterized by b the complete set of scales intrinsic to the problem. The obstacle dimension D determines the vertical size of a region occupied by the fluid held in front of the body. Institute for Problems in Mechanics, The length λ characterizes unsteady internal waves, Russian Academy of Sciences, which smoothly turn into attached waves behind the δ δ pr. Vernadskogo 101, Moscow, 117526 Russia obstacle. The scales u and ρ determine the thickness

MACROSTRUCTURE AND MICROSTRUCTURE OF A COCURRENT STRATIFIED FLOW 431 of a shell for the velocity wake and density wake, respectively, at points of flow separation from the δ δ obstacle. The scales ν and s characterize isolated interlayers that are formed in the field of internal waves and are separated from both the wake and the body by a fluid layer free from fine-structure features. At small values of the Reynolds number (Re < 200) and moderate values of the Froude number (Fr < 1.2), the density wake splits into separate extended interlayers, whose deformation reflects distribution of wave- induced displacements behind the body. Darker lines visualize crests of unsteady waves and attached internal waves, where the former arise in front of the body and the latter, behind it. Binary light gray lines show wave troughs. The wave field is asymmetric with respect to the central horizontal plane. Wave surfaces behind the body in the quiescent fluid represent semicircles [11]. Actual shapes of the crests and the troughs differ from the calculated ones due to the Doppler effect in the flow Fig. 1. Shadow pattern of a flow with isolated soaring dis- layer with a velocity shift. A wavelength measured continuities in the field of attached internal waves. A cylin- λ der with the diameter D = 2.5 cm moves from the left to the along a normal to the phase surface is equal to u = λ right at a speed of U = 0.29 cm/s. The other parameters are 4 cm in the upper half-space and to b = 3.1 cm in the T = 15 s, Fr = 0.28, Re = 74, and C = 5500. bottom half-space. The difference is caused by the b dependence of the buoyancy period on the depth (Tb = 15.0 s above the line drawn by the moving center of the Re body and Tb = 11.3 s below this line). Here, the high-gradient interlayers do not form an entire exterior high-gradient shell of the density wake, 400 6 as is observed in many other regimes [5]. Therefore, the attached internal waves pass through the density wake. 5 Crests and troughs from the upper and lower half- spaces join consecutively, forming a single wave sys- 200 tem. The shape of the connecting line inside the density wake varies with increasing distance from the obstacle, 8 which reflects the complex vortex structure of the wake and the dependence of the buoyancy period on the depth. 0 Horizontal high-gradient interlayers formed in the 0 0.2 0.4 Fr field of attached internal waves are situated at a distance h = 3.1 cm from the wake axis. The leading edge Fig. 2. Domain of existence of isolated soaring discontinui- of the interlayers is situated in the region of maximum ties arising in the field of attached internal waves in the amplitudes of the internal waves and lies at a distance coordinate plane composed of the Froude and Reynolds numbers. Flow regimes are classified according to Figs. 3 of r = 4 cm from the cylinder center. The thickness of and 4: (5) the regime of soaring vortices, (6) vortex fila- these interlayers depends weakly on parameters of the ments in the wake, and (8) the density wake split into wave- motion, while their length grows with increasing the and-vortex interlayers. Froude number and, under the conditions of this experiment, substantially exceeds the dimension of the window. In the case of slower motion, when these interlay- Similarly to papers [3, 4], a domain of existence of this ers are not long [1, 6], their back and front edges are flow regime is shown in the plane given by the Froude free from singularities. and Reynolds numbers (Fig. 2). The regime of curls [3] Analysis of the experimental data shows that, in or vortices [4] soaring in the field of attached internal waves is realized to the left of this domain at small val- both strongly (Tb = 6.0 s) and weakly (Tb = 12.5 and 20.5 s) stratified fluid, isolated discontinuities are ues of the Froude number (Fr < 0.3). Then, a multipole observed in the field of attached internal waves near the vortex system [4] forms at the front interlayer edge in cylinders having the diameters D from 1.5 to 7.6 cm the region of maximum amplitudes of the internal and towed with the speed U = 0.04Ð0.5 cm/s if values of waves. This system is a source of its own density wake the internal Froude number are small (0.02 < Fr < 0.5) presented by a rectilinear or wavelike high-gradient and the Reynolds number is moderate (30 < Re < 500). interlayer. With increasing the Reynolds number, a flow

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432 CHASHECHKIN, MITKIN

Re Allowing for steadily reproducible small-scale components in a stratified-flow pattern behind the cylinder, 1000 the regime map loses its universality and becomes dependent on the scale ratio C. For the strong stratification of the fluid (C < 2000, see Fig. 3), within the parameter ranges under investigation (5.3 < Tb < 25 s, 1.5 < D < 7.6 cm, and 0.01 < U <6.5 cm/s), we observe 100 different flow regimes (according to the classification performed towards increasing values of the Froude and Reynolds numbers). They are a wake (1) with a split high-gradient shell; a wake (2) with a central high-gra- 1 2 3 4 5 10 dient interlayer in which a microscale instability (3) 6 7 8 9 10 develops [12]; isolated discontinuities (4) in the field of attached internal waves with an adjacent regime of 0.01 0.1 1 Fr soaring vortices (5) arising at the front edges of the discontinuities. At 0.1 < Fr < 1 and 80 < Re < 1000, the Fig. 3. Diagram of flow regimes at strong stratification internal waves strongly deform the cocurrent flow, and (C < 2000). isolated vortex filaments (6) are observed in regions of wake widening. Near regions of wake narrowing, trains Re of soaring vortex systems, which are separated from the 1000 wake by a fluid layer (7) free from fine-structure distur- bances, are formed in the field of internal waves. Points corresponding to a regime of cusped wave-and-vortex objects (8), in which there is no whole high-gradient shell of the density wake, are situated at the lower boundary of the given parameter range. At Re > 200 100 and 0.45 < Fr < 1.4, a narrow turbulent wake (9) is observed behind the cylinder. At Re > 350 and Fr > 1, turbulent vortex objects (10) are formed inside the density wake. Overlapping the regimes occurs at small and moderate values of the Froude number (0.02 < Fr < 1 10 1 2 4 5 6 8 9 10 11 12 13 and 20 < Re < 1000). The regime map has a clearer structure in a weakly 0.1 1 Fr stratified fluid (C > 2000). It is shown in Fig. 4, where the notation used in Fig. 3 is preserved for repeated Fig. 4. Diagram of flow regimes at weak stratification flow types. Similarly to the previous case, the laminar (C > 2000). cocurrent flow with the split high-gradient shell (1), which occurs at Fr < 0.02 and Re < 40, is changed by pattern at the front edge of the interlayer becomes more the wake (2) having the central high-gradient interlayer as the Reynolds number increases. At Re > 70 and Fr < complicated; i.e., a larger number of vortices are being 0.1, the isolated discontinuities (4) and the soaring vor- identified in it [4]. A line that separates the regimes of tices (5) arise in the field of attached internal waves. At soaring vortices from those of isolated discontinuities Fr > 0.1, the separate vortex filaments (6) or the oblique is given by the relation Re = 1200Fr + 15. wave-and-vortex interlayers (8) are observed in the Two flow types exist to the right of the region of the density wake. In the last case, at small values of the isolated discontinuities. At small values of the Rey- Reynolds number and moderate values of the Froude number, there is no entire high-gradient shell of the nolds number (Re < 200), the flow pattern is the sim- density wake (8) split into separate layers and interlay- plest. It includes only an advancing disturbance, inter- ers. When the speed increases, an attached vortex forms nal waves, and a density wake split into separate both in the bottom part of the body and in the uniform oblique interlayers [4]. A boundary between these fluid. Dynamics of its intrinsic motion generates char- regimes is given by the relation Re = 960Fr – 215. At acteristic wake elements with longitudinal scales dif- large values of the Reynolds number (Re > 200 and fering from a wavelength of the attached internal wave. Fr > 0.5), the vertical wake dimension grows and At Re > 200, Fr > 0.7, as in strongly stratified fluid, the decreases cyclically with the distance from the body. narrow turbulent wake (9) is observed rather than the Horizontal vortex filaments (vortex bubbles [3]) appear wider turbulent flow with separate vortex elements (10) in the regions of wake widening, their positions being at Re > 350 and Fr > 1. determined by the phase structure of the attached inter- Furthermore, we discuss flow regimes that are nal waves. present in the diagram and are absent in strongly strat-

DOKLADY PHYSICS Vol. 45 No. 8 2000 MACROSTRUCTURE AND MICROSTRUCTURE OF A COCURRENT STRATIFIED FLOW 433 ified fluid. Within a wide range of the parameters, there gram for supporting unique instrumentation) and the is an undular wake (11) for which the scale of longitu- Russian Foundation for Basic Research (project dinal variation is determined by the period of natural no. 99Ð05Ð64980). oscillations of the bottom vortex. An increase in the wake speed leads to the growth of the amplitude of the wake transverse oscillations. In this case, laminar vorti- REFERENCES ces forming the StrouhalÐKarman vortex street (12) 1. Yu. D. Chashechkin and V. V. Mitkin, Dokl. Akad. Nauk enter the wake sequentially. At a still higher speed, 362, 625 (1998) [Dokl. Phys. 43, 636 (1998)]. when the flow inside the bottom vortex becomes irreg- 2. P. G. Baines, Topographic Effects in Stratified Flows ular, an array of turbulent vortices (13) forms in the (Cambridge Univ. Press, Cambridge, 1995). wake. 3. D. L. Boyer, P. A. Davies, H. J. S. Fernando, et al., Phi- Qualitative features of the regime maps presented in los. Trans. R. Soc. London, Ser. A 328, 501 (1989). Figs. 3 and 4 differ essentially from each other. For 4. Yu. D. Chashechkin and I. V. Voeœkov, Izv. Akad. Nauk, weak stratification, domains of existence of the men- Fiz. Atmos. Okeana 29, 821 (1993). tioned flow regimes are compact and, at relatively large values of the Froude and Reynolds numbers, have clear 5. I. V. Voeœkov and Yu. D. Chashechkin, Izv. Akad. Nauk, Mekh. Zhidk. Gaza, No. 1, 20 (1993). boundaries. For strong stratification, domains for the existence of the given regimes in the parameter space 6. V. V. Mitkin, Cand. Sci. (Phys.ÐMath.) Dissertation are being extended. Moreover, at close values of the (Moscow, Inst. Probl. Mech., Russ. Acad. Sci., 1998), parameters, three and more flow types can exist around p. 111. the bodies differing in their diameters. The observed 7. V. G. Baœdulov and Yu. D. Chashechkin, Izv. Akad. ambiguity of the flow regimes (each of them actually Nauk, Fiz. Atmos. Okeana 29, 666 (1993). forms a compact domain in the three-dimensional 8. L. A. Vasil’ev, Shadow Methods (Nauka, Moscow, parameter space given by the Froude and Reynolds 1968). numbers and the scale ratio) testify to the incomplete- 9. S. A. Smirnov, Yu. D. Chashechkin, and Yu. S. Il’inykh, ness of the plane regime map [3, 4] and the necessity of Izmer. Tekh., No. 6, 15 (1998). constructing the three-dimensional map in the C-, Fr-, 10. D. D. Maksutov, Shadow Methods for Researching Opti- and Re-axes. In general, the regimes should be classical Systems (Gostekhteorizdat, Moscow, 1934). fied in the four-dimensional space given by the Froude, 11. M. J. Lighthill, Waves in Fluids (Cambridge Univ. Press, Reynolds, and Peclet numbers as well as the scale ratio. Cambridge, 1978; Mir, Moscow, 1981). 12. V. V. Voeœkov, V. E. Prokhorov, and Yu. D. Chashechkin, ACKNOWLEDGMENTS Izv. Akad. Nauk, Mekh. Zhidk. Gaza, No. 3, 3 (1995). This work was supported by the Ministry of Sci- ence and Technology of Russian Federation (the Pro- Translated by Yu. Verevochkin

DOKLADY PHYSICS Vol. 45 No. 8 2000