The problem of a gas bubble in plane ideal fluid flow

Abstract

We study the problem of two-dimensional fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall.

Two-dimensional ideal fluid flow past a gas bubble on whose boundary surface-tension forces act (or a gas bubble bounded by an elastic film) has been studied by several authors. Zhukovskii, who first studied jet flows with consideration of the capillary forces, constructed an exact solution of the problem of symmetric flow past a gas bubble in a rectilinear channel [1]. However, Zhukovskii's solution is not the general solution of the problem; in particular, we cannot obtain the flow past an isolated bubble from his solution. Slezkin [2] reduced the problem of symmetric flow of an infinite fluid stream past a bubble to the study of a nonlinear integral equation. The numerical solution of this problem has recently been found by Petrova [3]. McLeod [4] obtained an exact solution under the assumption that the gas pressure p₁ in the bubble equals the flow stagnation pressure p₀. Beyer [5] proved the existence of a solution to the problem of flow of a stream having a given velocity circulation provided p₁≥p₀.

We examine the problem of two-dimensional ideal fluid flow past a gas bubble adjacent to an infinite rectilinear solid wall. The solution depends on the value of the contact angle βπ. The existence of a solution is proved in some range of variation of the parameters, and a technique for finding this solution is given. The situation in which β=1/2 is studied in detail.