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

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.