Global existence for a translating near-circular Hele-Shaw bubble with surface tension

TL;DR

This paper proves the global existence and stability of steady translating near-circular bubbles in a Hele-Shaw cell with nonzero surface tension, using a boundary integral approach and weighted Sobolev norms.

Contribution

It establishes the global existence and stability of translating bubble solutions for any nonzero surface tension, even in regimes where local approximations suggest instability.

Findings

Existence of a unique steady translating bubble shape.

Stability of the bubble solution under symmetric and asymmetric initial conditions.

Applicability of boundary integral methods and weighted Sobolev norms for analysis.

Abstract

This paper concerns global existence for arbitrary nonzero surface tension of bubbles in a Hele-Shaw cell that translate in the presence of a pressure gradient. When the cell width to bubble size is sufficiently large, we show that a unique steady translating near-circular bubble symmetric about the channel centerline exists, where the bubble translation speed in the laboratory frame is found as part of the solution. We prove global existence for symmetric sufficiently smooth initial conditions close to this shape and show that the steady translating bubble solution is an attractor within this class of disturbances. In the absence of side walls, we prove stability of the steady translating circular bubble without restriction on symmetry of initial conditions. These results hold for any nonzero surface tension despite the fact that a local planar approximation near the front of the bubble would suggest Saffman Taylor instability. We exploit a boundary integral approach that is particularly suitable for analysis of nonzero viscosity ratio between fluid inside and outside the bubble. An important element of the proof was the introduction of a weighted Sobolev norm that accounts for stabilization due to advection of disturbances from the front to the back of the bubble.