Bubble dynamics in a Hele-Shaw channel and velocity selection without surface tension

TL;DR

This paper completely solves the inviscid bubble evolution in a Hele-Shaw channel without surface tension, revealing a unique stable attractor that determines the observable bubble shape and velocity.

Contribution

It introduces a fully solvable dynamical system for bubble evolution without surface tension and identifies a new class of solutions and the stable attractor.

Findings

Unique stable attractor determines bubble shape and velocity

New rational class of solutions added to existing solutions

Numerical simulations illustrate key dynamics

Abstract

Inviscid bubble dynamics in a viscous fluid, moving with velocity $V$ far from the bubble, is considered. The Cauchy problem of recovering the bubble evolution from its initial shape is completely solved without surface tension. The well-posed (after a Tikhonov regularization) dynamical system provides an extensive list of unsteady closed form solutions due to integrability. A new (rational) class of solutions is obtained and added to the logarithmic class found earlier (Phys. Rev. E 89, 061003(R), 2014). The only attractor selects the observable bubble shape and velocity $U = 2V$ from the continuum of possible solutions. The attractor is asymptotically stable. Numerical results illustrate the most salient aspects of the bubble dynamics.