Selection of the Taylor-Saffman Bubble does not Require Surface Tension

TL;DR

This paper introduces exact solutions for bubble evolution in Hele-Shaw flows that show bubble selection occurs without surface tension, challenging previous theories that deemed surface tension necessary for pattern selection.

Contribution

It provides a new class of solutions demonstrating bubble selection without surface tension and extends previous models to doubly-connected geometries.

Findings

Selection corresponds to a stable fixed point in the dynamics.

Surface tension is not required for bubble pattern selection.

Solutions extend to arbitrary connectivity in Hele-Shaw flows.

Abstract

A new general class of exact solutions is presented for the time evolution of a bubble of arbitrary initial shape in a Hele-Shaw cell when surface tension effects are neglected. These solutions are obtained by conformal mapping the viscous flow domain to an annulus in an auxiliary complex-plane. It is then demonstrated that the only stable fixed point (attractor) of the non-singular bubble dynamics corresponds precisely to the selected pattern. This thus shows that, contrary to the established theory, bubble selection in a Hele-Shaw cell does not require surface tension. The solutions reported here significantly extend previous results for a simply-connected geometry (finger) to a doubly-connected one (bubble). We conjecture that the same selection rule without surface tension holds for Hele-Shaw flows of arbitrary connectivity. We also believe that this mechanism can be found in other, similarly described, selection problems.