# The effect of surface tension on steadily translating bubbles in an unbounded Hele-Shaw cell

## TL;DR

This paper presents new numerical solutions for steady bubble translation in an unbounded Hele-Shaw cell, revealing multiple solutions influenced by surface tension and differences from channel flow behaviors.

## Contribution

It introduces a conformal mapping approach using the Schottky-Klein prime function to solve the selection problem for one and two bubbles, uncovering multiple solutions and their dependence on surface tension.

## Key findings

- Multiple solutions exist for each fixed surface tension value.
- Bubble shapes become more complex with higher solution branch numbers.
- A single solution is selected as surface tension approaches zero.

## Abstract

New numerical solutions to the so-called selection problem for one and two steadily translating bubbles in an unbounded Hele-Shaw cell are presented. Our approach relies on conformal mapping which, for the two-bubble problem, involves the Schottky-Klein prime function associated with an annulus. We show that a countably infinite number of solutions exist for each fixed value of dimensionless surface tension, with the bubble shapes becoming more exotic as the solution branch number increases. Our numerical results suggest that a single solution is selected in the limit that surface tension vanishes, with the scaling between the bubble velocity and surface tension being different to the well-studied problems for a bubble or a finger propagating in a channel geometry.