Motion stability of a periodic system of bubbles in a liquid

Abstract

Wave-like motion in a periodic structure of bubbles that steadily moves through ideal incompressible liquid is considered. The wavelength is microscopically short. Some general local properties containing general information about two-phase flow are found. The dynamics of small-amplitude disturbances is studied in linear systems (called trains) and in spatial structures (such as a cubic lattice). The behaviour of one-dimensional waves in various structures is shown to differ widely: one-dimensional waves in the train do not magnify, whereas in the three-dimensional structure there may be stability and instability of one-dimensional waves. In the continuum limit the one-dimensional instability is demonstrated not to be related to the mean parameters of two-phase flow. The long-wave dynamics is shown to depend significantly on the relative velocity vector orientation in the lattice, but orientation is not included in the usual equations for the two-phase continuum. One result of this study is the relation between the short-wave-type instability of the periodic structure, on the one hand, and the instability of one-dimensional flow of inviscid bubbly liquid discovered by van Wijngaarden on the other. Long microscopic waves are analysed to determine the coefficients of one-dimensional equations for a two-phase continuum model. The velocity orientation at which the coefficients of the traditional one-dimensional model are obtained is found. Short waves in a stationary structure are studied by using the system of equations based on the equation of motion of a small sphere in a general potential flow. A refined equation for the force applied on a sphere in a non-uniform potential flow is derived.