Existence and selection of steady bubbles in a Hele-Shaw cell

摘要

This paper concerns the existence of a steadily translating bubble in a Hele-Shaw cell for small but non-zero surface tension epsilon(2). We rigorously conclude that for bubble velocity U relative to the fluid velocity at infinity in the interval (1, 2), analytic symmetric solutions exist in the asymptotic limit of surface tension epsilon(2) -> 0 if and only if the Stokes constant for a relatively simple nonlinear differential equation is zero. This Stokes constant S depends on the parameter alpha epsilon (0, 1) corresponding to bubble 4 size and a = [alpha-(U(1+alpha(2))-2/(U(1+alpha(2))-2 alpha(2)))(1/2)] epsilon(-4/3) Earlier calculations have shown S to be zero for a discrete Set Of Values of a. (c) 2007 Elsevier Ltd. All rights reserved.