Doklady Physics, Vol. 47, No. 3, 2002, pp. 173–175. Translated from Doklady Akademii Nauk, Vol. 383, No. 1, 2002, pp. 34–36. Original Russian Text Copyright © 2002 by Rylov. PHYSICS Chaplygin Equations and an Infinite Set of Uniformly Divergent Gas-Dynamics Equations A. I. Rylov Presented by Academician G.G. ChernyÏ September 19, 2001 Received September 24, 2001 1. We consider two-dimensional potential ideal-gas (ϕ and ψ) and, as a result, a conservation law in the flows. At the hodograph plane, they are described by physical plane: the Chaplygin equations kβϕ +0,αψ ==βψ –0;αϕ (3) ϕ ψ ϕ ψ z +0,k θ ==θ –0z (1) ()αρu + βv + ()αρv – βu or the Chaplygin equations of the second order for the x y ()ρ ()ρ stream function [1–4]: = f ug+ v x +0.f v – gu x = (4) ρ 1M– 2 Remark. The solutions ϕ = f = const and ψ = g = kψθθ +0,ψ ==z ---dq, k =----------------. (2) zz ∫ q ρ2 const also allow us to obtain certain conservation laws on the (x, y)-plane. For example, substituting either Hereafter, ϕ is the potential; ψ is the stream function; q, f = 1 and g = 0 or f = 0 and g = 1 into (4), we obtain the θ, u, and v are the modulus, angle of inclination, and conservation law for either mass or circulation, respec- horizontal and vertical components of the velocity vec- tively. tor, respectively; M is the Mach number; p is pressure; 2. We consider several tentative examples of exact and ρ is the density. solutions. Eqs. (1) and (2) have an infinite set of exact solu- A flow with a source (radial motion). In this case, tions [1–4]. In this paper, we show that each solution is associated with a proper set of gas-dynamics equations, g = θ and f = −∫k dz = Ð K(z). Then, one of them being divergent (see Theorem 1). Our detailed analysis of the solutions to system (1), which kθϕ –0,Kψ ==θψ +0;Kϕ are found by the method of separation of variables led to one more algorithm for constructing an infinite set of ()Kρu – θv +0.()Kρv + θu = conservation laws (see Theorem 2). The set constructed x y includes, as a particular case, the set of the conservation A flow similar to a potential vortex. In this case, laws found in [5, 6]. They can be applied to two-dimen- θ sional steady flows (mass, momentum, and angular- f = and g = z. Then, momentum conservation laws). It is worth noting that a kzϕ +0,θψ ==zψ –0,θϕ finite set of the conservation laws was found in [5, 6] (5) ()θρ ()θρ for the cases of both steady and unsteady three-dimen- uz+ v z +0.v – zu y = sional flows. Theorem 1. For each solution ϕ = f(z, θ) ≠ const It is worth noting that system (5) is well known in gas and ψ = g(z, θ) ≠ const to Chaplygin equations (1), the dynamics [4], but irrespectively of the potential-vortex introduction of variables α and β dependent on the flow. θ θ α θ functions f(z, ) and g(z, ), respectively, = f(z, ) and 2 β θ θ = g(z, ) yields a system of uniformly divergent gas- The solution ψ = g = θz and ϕ = f = ----- Ð F(z), with dynamics equations in the plane of the potentials 2 F(z) = ∫k zdz, leads to the equations Sobolev Institute of Mathematics, Siberian Division, θ2 θ2 ()θ ()θ Russian Academy of Sciences, pr. Koptyuga 4, k z ϕ +0,----- – F ==z ψ –0;----- – F Novosibirsk, 630090 Russia 2 ψ 2 ϕ 1028-3358/02/4703-0173 $22.00 © 2002 MAIK “Nauka/Interperiodica” 174 RYLOV α β and the variables ( = fi, = gi) dependent on correspond- ing functions and an infinite set of conservation laws θ2 θ2 ρ θ ρ θ acquiring the following form: ----- – F u + zv +0.----- – F v – zu = 2 x 2 y () () f i ϕ –0,gi ψ = θ2 ()ρ ()ρ The solution ϕ = f = ÐθK(z) and ψ = g = ----- Ð G(z), f i ug+ iv x +0.f i v – giu y = 2 In the general case, the functions h(1) and h(2) are with G(z) = ∫K (z)dz, leads to the equations expressed in terms of the hypergeometric functions [1–4]. In the case of λ = 1, they take the known explicit form: θ2 θ2 ()θ ()θ k----- – G –0,K ψ ==----- – G +0;K ϕ ()1 1 ()1 1 2 ϕ 2 ψ h ()1; z = ---, h ()1; z = –,------ q z ρq and 2 ()2 p ()2 p + ρq 2 2 () () θ θ h 1; z = ---, hz 1; z = –.------------------ρ θKρu – ----- – G v +0.θKρv + ----- – G u = q q 2 2 x y The functions h(1) correspond to the Ringleb flow [1, It should be noted that the angle θ of inclination of 3, 8]. The streamline pattern for the flow corresponding the velocity vector enters into the conservation laws to the function h(2) is very similar to that for the Ringleb under consideration as both an explicit argument and flow [1]. We now present the relations that describe the argument of the trigonometric functions. Moreover, such flows, the systems of gas-dynamics equations in the two last solutions depend on the variable θ2. This the form of Eqs. (3), and the corresponding conserva- fact can be used to carry out certain estimates. For tion laws in the physical plane. example, using the known asymptotic form of the sym- Ringleb flow. (a): metric flow around a body [7] and the conservation law cosθ sinθ presented above, we can prove that the following rela- ϕ ==f ------------, ψ ==q -----------; tionships are satisfied at an arbitrary streamline over its 1 ρq 1 q entire length: θ θ θ θ +∞ +∞ sin cos sin cos 2 k----------- +0,------------ = ----------- –0;------------ = θ q ϕ ρq ψ q ψ ρq ϕ ()Gz()– Gz()∞ qld = -----qld > 0. ∫ ∫ 2 ∞ ∞ θ θ θ θ – – cos ρ sin cos ρ sin -----------ρ - u + -----------v + ------------ v – -----------u 3. In order to construct new systems of equations q q x pq q y and conservation laws, we now consider known solu- tions to Eqs. (1), which are found by the method of sep- = 1x +0.0y = aration of variables. The four independent solutions are Ringleb flow. (b): of the form sinθ cosθ ϕ ==f –-----------, ψ ==g ------------; 1 ()1 λθ ()1 λθ 2 ρq 2 q f 1 ==–λ---hz cos , g1 h sin ; cosθ sinθ cosθ sinθ 1 ()1 λθ ()1 λθ k------------ –0,----------ρ - ==------------ +0;----------ρ- f 2 ==λ---hz sin , g2 h cos ; q ϕ q ψ q ψ p ϕ (6) () () sinθ cosθ sinθ cosθ 1 2 λθ 2 λθ –-----------ρu + ------------v – -----------ρv + ------------u f 3 ==–---hz cos , g3 h sin ; ρ ρ λ q q x q q y () () = 0 –0.1 = 1 2 λθ 2 λθ x y f 4 ==λ---hz sin , g4 h cos . Modified Ringleb flow. (a): Here, λ is an arbitrary constant, and the functions 2 p + ρq p h(1) λ z and h(2) λ z are two independent solutions to ϕ θ ψ θ ( ; ) ( ; ) ==f 3 ------------------ρ cos , ==g3 --- sin ; the ordinary differential equation of the second order: q q λ2 hzz(z) = kh(z). The subscript z stands for differentia- ρ 2 p θ p + q θ tion with respect to z. k--- sin +0,------------------ cos = q ρq ψ Since the solutions depend on the arbitrary constant ϕ λ , it is possible to construct an infinite set of the func- ρ 2 tions (f , g ). Therefore, according to Theorem 1, we can p θ p + q θ i i --- sin –0;------------------ρ cos = find an infinite set of gas-dynamics equations (3) with q ψ q ϕ DOKLADY PHYSICS Vol. 47 No. 3 2002 CHAPLYGIN EQUATIONS AND AN INFINITE SET 175 ()ρ 2 ()ρ (), p + u x +0.uv y = R 41 ==Ypx∫ d , Modified Ringleb flow. (b): with the integrals taken along streamlines. In the case of λ = 1, in the potential plane and the p + ρq2 p ϕ ==f –------------------ sinθ, ψ ==g --- cosθ; physical plane, conservation law (7) takes the following 4 ρq 4 q forms, respectively: 2 2 2 ρ ρ ρ p + q θ p + q θ p θ p + q θ ------------------ cos y – ------------------ sin x k--- cos –0,------------------ sin = ρq ρq ϕ q ϕ ρq ψ p p – --- sinθy + --- cosθx = 0, ρ 2 p θ p + q θ q q ψ --- cos +0;------------------ sin = q ψ ρq ϕ 2 2 ()()p + ρu y – ρuvx x +0.()ρuvypp– ()+ v x y = ()()ρ 2 puv x +0.p + v y = This is simply the angular-momentum conservation law. λ The Ringleb flow was comprehensively discussed in For = 1, the functions f and g entering into (7) cor- the literature [1, 3, 8]. It represents one of a few exact respond to the modified Ringleb flow, while the func- solutions to the problem on a flow with a local super- tion R corresponds to the Ringleb flow. At the same sonic area. However, the Ringleb flow leads to no time, these flows interchange in conservation law (8). instructive conservation law in the (x, y)-plane. At the As a result, we have same time, the modified Ringleb flow was only briefly θ θ θ θ mentioned in the literature [1], not being discussed in cos sin sin cos -----------ρ -X – ----------ρ -Y –0,-----------X + ------------Y = detail. However, as was shown above, this is the flow q q ϕ q q ψ that is associated with the momentum conservation law.