Dynamic equilibria of a nonisothermal fluid

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Introduction. The study of the motion of a rotating self-gravitating fluid was started as long ago as by Isaak Newton [1, 2], and it is currently far from being completed. Newton proved to be the first to endeavour to obtain, from the motion equations, a solution simulating the shape of the Earth. Newton’s pioneering study gave rise to numerous investigations in this field of mechanics and physics and a great impetus to the development of the mathematics used practically in all the sections of mathematics. The development of the potential theory, the creation of the theory of special functions and the formation of the rapidly developing theory of integrable systems should be mentioned first. P r o s v i r y a k o v E. Yu. Practically all the most prominent scientists, such as Appel, Basset, Betti, Gagen, Helmholtz, Dedekind, Dirichlet, Kirchhoff, Lipschitz, Lame, McLoren, Poincare, Riemann, Chandrasekhar, Jacoby, took part in the construction of the theory [3-13]. This list of discoverers of exact solutions can be extended [3- 13]. The history of discovering the solutions can be found in the monographs and surveys [3-13]. It should be noted that the scientific breakthrough in the simulation of rotating equilibrium figures was made by the Russian scientists Steklov, Lyapunov, Ovsyannikov and Zel’dovich. Note that the development of the theory of motion stability described by systems of ordinary differential equations was initiated by studying the stability of equilibrium figures and the motion of an elastic body partially filled with a fluid [4, 5]. It is the work in these fields that offered a formulation to the wellknown Lyapunov theorems of stability and instability, among other things, from the first approximation [4, 5]. Original research has currently been done on the stability of Dirichlet and Jacobi ellipsoids [8]. Ovsyannikov’s exact gas-hydrodynamic solution became the first solution simulating gas cloud motion in the class of linear velocities [8]. Afterwards models of Dyson and Fujimoto were proposed [8]. There is another substantial model, which influenced the development of the theory of rotating fluids. It is the generalized solid proposed for research by Arnold [14]. In the survey made by Dolzhansky [15] there is a detailed analysis of this approach with some generalizations for different classes of fluids. As a rule, dynamic fluid equilibria are simulated with the application of two mathematical models. One model is Lagrangian fluid simulation, i.e. the use of the ordinary differential equations of Lagrange, Hamilton and Jacoby to describe the relative equilibrium (solid-state) of the continuum. The other model consists in using the Euler equations with potential forces to analyze the structures of a relative fluid equilibrium. Thus, in the simulation of dynamic fluid equilibria no account is taken of dissipation and temperature effect. In other words, an isothermal perfect incompressible fluid is dealt with. It is apparent that this deficiency in studying fluid flows can be compensated if this problem is analyzed with the application of the Oberbeck-Boussinesq system of equations. Note that it is not only the Boussinesq approximation that has not been used before to describe dynamic fluid equilibria, but also the characteristic scale of solutions. This statement needs to be clarified. Despite Chandrasekhar [6] used the Boussinesq approximation to simulate astrophysical objects, he used the virial method of physical field expansion for solving his problems. The application of this body of mathematics enabled one to reduce the Oberbeck-Boussinesq system to a system of ordinary differential equations and to use the classical problem statements when interpreting results. Similar reasoning holds for [14, 15]. There are studies dealing with large-scale fluid or gas motions [6], when it is necessary to take into account the inhomogeneity of the gravitational field, and with scales for which surface tension forces are principally important, e. g. the Hadamard-Rybczynski problem [16, 17]. In both extreme cases, the principal equilibrium shape is spherical. The consideration of rotation was always viewed 736 Dynamic equilibria of a nonisothermal fluid as some disturbance of the principal flow, however it was not viewed as a forming factor for equilibrium structures. Note that the very possibility of the existence of equilibrium figures in the region of intermediate scales, when neither the inhomogeneity of the gravitational field nor the surface tension forces are essential, does not seem to have been studied. At any rate, the authors are unfamiliar with works viewing the problem this way. Thus, we intend to ascertain the possible existence of equilibrium structures in the region of the so-called intermediate scales, when the Boussinesq approximation is obviously applicable to the simulation of a heat-conducting viscous incompressible fluid. 1. Problem statement. In the Boussinesq approximation [6, 18] in the Cartesian coordinates (the

About the authors

Evgeny Yu Prosviryakov

Institute of Engineering Science, Urals Branch, Russian Academy of Sciences; Ural Federal University named after the First President of Russia B. N. Yeltsin

Email: evgen_pros@mail.ru
Dr. Phys. & Math. Sci., Professor; Head of Sector; Sector of Nonlinear Vortex Hydrodynamics 34, Komsomolskaya st., Ekaterinburg, 620049, Russian Federation; 19, Mira st., Ekaterinburg, 620002, Russian Federation

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