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 ϵ2ϵ2. We rigorously conclude that for bubble velocity UU relative to the fluid velocity at infinity in the interval (1,2)(1,2), analytic symmetric solutions exist in the asymptotic limit of surface tension ϵ2→0ϵ2→0 if and only if the Stokes constant for a relatively simple nonlinear differential equation is zero. This Stokes constant SS depends on the parameter α∈(0,1)α∈(0,1) corresponding to bubble size and a≡[α−(U(1+α2)−2(U(1+α2)−2α2))1/2]ϵ−43. Earlier calculations have shown SS to be zero for a discrete set of values of aa.