Multiple bubbles in a Hele-Shaw cell

What is it about?

A complete set of exact solutions is presented for multiple bubbles steadily propagating in a Hele-Shaw cell. The solutions are written in explicit form in terms of a conformal mapping involving a novel class of special transcendetal functions, the so-called secondary Schottky-Klein prime functions, recently introduced by the author and collaborators. To motivate the introduction of these special functions, a generalized method of images is first employed, which (we hope) makes the paper accessible to a larger audience. It is also hoped that the mathematical formalism described in the paper may find application in other areas of potential theory and fluid mechanics.

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Why is it important?

Because it solves the long-standing problem of the steady motion of multiple bubbles (including the special case of multiple fingers and bubbles) in a Hele-Shaw channel. All previous solutions found for this problem are special cases of the general solutions presented in our paper. Our formalism also paves the way for constructing exact time-dependent solutions for the evolution of multiple bubbles, starting from an arbitrary initial configuration.

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