Special class of solutions of the kinetic equation of a bubbly fluid

Abstract

A new class of solutions is constructed for the kinetic model of bubble motion in a perfect fluid proposed by Russo and Smereka. These solutions are characterized by a linear relationship between the Riemann integral invariants. Using the expressions following from this relationship, the construction of solutions in the special class is reduced to the integration of a hyperbolic system of two differential equations with two independent variables. Exact solutions in the class of simple waves are obtained, and their physical interpretation is given. Introduction.A kinetic approach has been developed [1-5] to simulate the motion of gas bubbles in a perfect fluid taking into account the effects of their collective interaction. The construction of the equations of motion is based on the calculation of the fluid kinetic energy, which is represented as the quadratic form of the bubble velocity [6], whose coefficients are determined by the fluid flow potential in the region between bubbles. Assuming that all bubbles are identical and are solid imponderable spheres, Russo and Smereka [3] approximately calculated the kinetic energy and Hamiltonian of bubble motion by asymptotic expansion of the solution of the Laplace equation in a small parameter (the ratio of the bubble radius to the average separation between the bubbles). In [3], the kinetic equation describing the evolution of the one-particle distribution was obtained using a system of Hamilton's ordinary differential equations for the bubble coordinates and momenta and a method of deriving Vlasov's equation.The characteristic properties of the Russo-Smereka kinetic model [3] for the case of one space variable are studied in [7], where hyperbolicity conditions are formulated and Riemann integral invariants and infinite series of conservation laws are found. The exact solutions of the kinetic model in the class of simple waves propagating over a specified, spatially homogeneous, stationary background are obtained in [7,8].In the present paper, we propose a new method for constructing solutions of integrodifferential equations that admit a formulation in terms of Riemann invariants. This method is used to find a wide class of solutions of the Russo-Smereka kinetic equation that is described by a system of first-order hyperbolic differential equations.1. Mathematical Model. In the one-dimensional case, the Russo-Smereka kinetic equation in dimensionless variables is written as follows [3,7]: