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

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.

arXiv comment17 pages, 8 figures

References in corpus (7)

• Selection of the Saffman-Taylor Finger Width in the Absence of Surface Tension: an Exact Result
• Corner and finger formation in Hele--Shaw flow with kinetic undercooling regularisation
• Saffman-Taylor fingers with kinetic undercooling
• Multiple bubbles and fingers in a Hele-Shaw channel: complete set of steady solutions
• Selection of the Taylor-Saffman Bubble does not Require Surface Tension
• Stream of asymmetric bubbles in a Hele-Shaw channel
• Multiple steadily translating bubbles in a Hele-Shaw channel

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• Modelling finger propagation in elasto-rigid channels
• The interaction of multiple bubbles in a Hele-Shaw channel
• Interfacial dynamics and pinch-off singularities for axially symmetric Darcy flow