An assembly of steadily translating bubbles in a Hele–Shaw channel

This paper presents new solutions for any finite number of bubbles steadily translating along a Hele–Shaw channel. This constitutes a nonlinear free boundary problem. The solutions can be written down explicitly in terms of a special transcendental function called the Schottky–Klein prime function. This work generalizes the exact solutions for a single bubble in a channel found by Tanveer (1987 Phys. Fluids 30 651–8) as well as the solutions for streams of bubbles aligned along the channel centreline due to Vasconcelos (1994 Phys. Rev. E 50 R3306–9).