Single waves in a gas-liquid bubble mixture

V. Sh. Shagapov ; M. N. Galimzyanov ; U. O. Agisheva

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 2, pp. 232-240

Résumé

Nonlinear wave processes in a two-phase medium (bubbly liquid) do not lose their relevance as an object of study due to their wide use in various fields of physics, engineering, chemical and petroleum industries. Last decades the jump in the development of computing has expanded the possibilities for the study of significantly nonlinear problems. The aim of this work was to obtain a stationary solution of equations describing the motion of a solitary wave in a gas-liquid mixture without taking into account dissipative processes. A one-dimensional stationary flow of a liquid with gas bubbles was considered under the following assumptions: the mixture is monodisperse, i.e. in each elemental volume all bubbles are spherical and of the same radius; viscosity and thermal conductivity are considerable only in the process of interfacial interaction and during bubble pulsations. Moreover, it is assumed that there is no mass transfer between the phases, and the liquid temperature is constant unlike the gas temperature in a bubble. This is always fulfilled under not very high pressures due to the bigger mass content of the liquid (therefore it can be considered as a thermostat). It greatly simplifies the task since there is no need to consider the equation of the energy in the liquid. The pressure in the bubble was assumed to be uniform. It is ensured if the radial velocity of the bubble walls is significantly less than the speed of sound in the gas. Phase pressure and bubble size were bound by the condition of combined deformation. The Rayleigh equation corresponding to the pulsations of a single spherical bubble in an infinite incompressible fluid was taken as the condition in this case. Properties of the gas in bubbles were described by the polytropic law. Based on one-dimensional stationary equations of fluid flow with gas bubbles, a solution of the “solitary wave” type is constructed. This solution in a special case of weak solitons is equal to the results taken on the basis of the Korteweg – de Vries equation for bubble media.