Models for the dynamics of dust-like matter in the self-gravity field: The method of hydrodynamic substitutions
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, PARTICLES, FIELDS, GRAVITATION, AND ASTROPHYSICS
Models for the Dynamics of Dust-Like Matter in the Self-Gravity Field: The Method of Hydrodynamic Substitutions V. M. Zhuravlev* Ulyanovsk State University, Kapitza Research Technological Institute, ul. L. Tolstogo 42, Ulyanovsk, 432017 Russia *e-mail: [email protected] Received March 2, 2017
Abstract—Models for the dynamics of a dust-like medium in the self-gravity field are investigated. Solutions of the corresponding problems are constructed by the method of hydrodynamic substitutions generalizing the Cole–Hopf substitutions. The method is extended to multidimensional ideal and viscous fluid flows with cylindrical and spherical symmetries for which exact solutions are constructed. Solutions for the dynamics of self-gravitating dust with arbitrary initial distributions of both fluid density and velocity are constructed using special coordinate transformations. In particular, the problem of cosmological expansion is considered in terms of Newton’s gravity theory. Models of a one-dimensional viscous dust fluid flow and some problems of gas hydrodynamics are considered. Examples of exact solutions and their brief analysis are provided. DOI: 10.1134/S1063776117090102
1. INTRODUCTION The problems of research on self-gravitating systems are among the main problems of astrophysics and cosmology. The problem of the dynamics of self-gravitating dust and gas in a self-consistent statement was quantitatively investigated by Jeans [1] and is known as the Jeans instability problem. This problem is considered in various statements, in particular, as the problem of the initial formation stage of compact objects (protostars) [2] and the problem of the cosmological expansion of the Universe [3, 4]. In all these cases, the main problem is to reveal the dynamics of the medium and the structures forming in it during the collapse of a local perturbation or its expansion, depending on the initial conditions in the medium [5]. In particular, an interesting process is the formation of shock waves in such systems, which can be of particular interest for astrophysical problems. Such processes are strongly nonlinear. This requires using the methods of constructing exact solutions of the equations of dynamics without their linearization by means of the perturbation theory. Only this approach allows the essentially nonlinear formation stages of structures in a self-gravitating medium to be revealed. The problems being investigated can be subdivided into two classes. The first, simplest, class describes a self-gravitating dust-like medium with the equation of state p = 0, i.e., the pressure in the medium is zero. The second class describes a medium in which the pressure is nonzero and that is defined by an equation of state with a pressure dependent on the density of the medium and its temperature (entropy). In all these
cases, the system of equations describing the dynamics of a self-gravitating medium is essentially nonlinear and generally cannot be integrated completely. Therefore, app
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