Stream of asymmetric bubbles in a Hele-Shaw channel

TL;DR

This paper provides exact analytical solutions for the shape and flow of asymmetric bubbles moving steadily in a Hele-Shaw channel, using conformal mapping techniques to derive explicit formulas.

Contribution

It introduces a novel application of the generalized Schwarz-Christoffel formula to model asymmetric bubble shapes and flow in Hele-Shaw channels.

Findings

Explicit formulas for bubble shapes and velocity fields

Application of conformal mapping to asymmetric bubbles

Analytical solutions for steady bubble motion

Abstract

Exact solutions are reported for a stream of asymmetric bubbles steadily moving in a Hele-Shaw channel. From the periodicity along the streamwise direction, the flow region is reduced to a rectangular unit cell containing one bubble, which is conformally mapped to an annulus in an auxiliary complex plane. Analytic expressions for the bubble shape as well as for the velocity field are obtained in terms of the generalized Schwarz-Christoffel formula for doubly connected domains.