Kinetic model of bubbly flow

Abstract

A kinetic approach based on the approximate calculation of the fluid flow potential and formulation of Hamilton's equations for generalized coordinates and momenta of bubbles is employed to describe processes of collective interaction of gas bubbles moving in an inviscid incompressible fluid. Kinetic equations governing the evolution of the distribution function of bubbles are derived. These equations are similar to Vlasov equations.Kinetic approaches for the description of fluid flows with gas bubbles have been developed in a number of recent papers [1][2][3][4]. Some of the systems of equations obtained are similar in structure to Vlasov equations which are used to describe plasma flows. In the derivation of these equations, Hamilton's ordinary differential equations that describe the motion of individual particles are employed. If gas bubbles moving in an inviscid incompressible fluid are treated as particles, then for the derivation of the above-mentioned ordinary differential equations, one needs to "know the fluid flow potential in the region between the particles. For the simplified situation where the bubbles are considered incompressible, an approximate Hamiltonian describing the motion of the bubbles for a rarefied bubbly medium was obtained by Russo and Smereka [4] and used to derive a system of kinetic equations governing the evolution of the one-particle distribution function. The assumption on the incompressibility of bubbles can be used to describe real flows with bubbles of sufficiently small size where the surface tension, which maintains the shape of the bubbles, is considerably greater than the variations of the hydrodynamic pressure. This model can be used for description of concentration waves for small pressure differences.The motion of a system of compressible bubbles in a fluid is often modeled using averaged equations, supplemented with the Rayleigh-Lamb equation for a single bubble. Various models of this type differing in additional terms of equations describing real effects are discussed in [5,6]. Certain hydrodynamic effects related to motion of bubbles in a fluid were considered in a monograph by Lavrent'ev and Shabat [7]. A kinetic approach for modeling bubbly flows in which the evolution of the bubble distribution function is governed by equations similar to Boltzmann equations or Vlasov equations, allows one to describe the motion more thoroughly and to derive average equations using a regular procedure. In particular, this approach can, in principle, yield certain basic relations that are postulated in a hydrodynamic description.In the present paper, we derive a system of kinetic equations for bubbly flow that describes the motion of compressible gas bubbles in an inviscid incompressible fluid. We first formulate the system of Hamilton's equations for generalized coordinates of spherical bubbles (spatial coordinates of the centers and radii) and the corresponding momenta. This system is easily written if the potential of the irrotational fluid flow in the region between the bubb...

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

“…Solutions of the form f (t, x, p) =f (k(t, x), p) and p i (t, x) =p i (k(t, x)), where k(t, x) is an arbitrary smooth function, will be called the simple waves of the system of integrodifferential equations (3) and (4). Let us construct a simple wave for the special class of the solutions.…”

“…A kinetic approach has been developed [1][2][3][4][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.…”

Shock Waves in Bubbly Media

“…In conclusion, note the recent experimental and theoretical investigations of wave processes in passive and reactive bubbly media [92][93][94][95]. The refraction of a bubble detonation wave from a chemically active bubble media into a liquid, its structure and evolution as well as reflection from the butt-end of a shock tube were studied experimentally by Sychev [92,93] to evaluate the effects of poly-dispersion of bubble media, gas-bubble size and to understand the energy-dissipation mechanisms.…”

“…Mathematical and numerical modeling of two-phase compressible flows with micro-inertial were carried out by Gavrilyuk and Saurel [94] which using the Hamilton principle of stationary action have developed the new model for compressible multiphase mixtures with full coupling between micro-and macroscale motion and validated for computation of oscillating shock waves as well as for the multidimensional interaction of a shock wave with a large bubble. Teshukov [95] has considered a kinetic approach based on the approximate calculation of the fluid flow potential and formulation of Hamilton's equations for generalized coordinates and momenta of bubbles to describe processes of collective interaction of gas bubbles moving in an inviscous incompressible fluid. Kinetic equations governing the evolution of the distribution function of bubbles (similar to Vlasov equation) were derived.…”