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

This paper concerns the existence of a steadily translating bubble in a Hele–Shaw cell for small but non-zero surface tension 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 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 α(0,1) corresponding to bubble size and . Earlier calculations have shown S to be zero for a discrete set of values of a.