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arXiv · 1310.4861

Multiple steadily translating bubbles in a Hele-Shaw channel

Christopher C. Green·Giovani L. Vasconcelos
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Abstract

Analytical solutions are constructed for an assembly of any finite number of bubbles in steady motion in a Hele-Shaw channel. The solutions are given in the form of a conformal mapping from a bounded multiply connected circular domain to the flow region exterior to the bubbles. The mapping is written as the sum of two analytic functions---corresponding to the complex potentials in the laboratory and co-moving frames---that map the circular domain onto respective degenerate polygonal domains. These functions are obtained using the generalised Schwarz-Christoffel formula for multiply connected domains in terms of the Schottky-Klein prime function. Our solutions are very general in that no symmetry assumption concerning the geometrical disposition of the bubbles is made. Several examples for various bubble configurations are discussed.

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Christopher C. Green, Giovani L. Vasconcelos. 2013-10-17. Multiple steadily translating bubbles in a Hele-Shaw channel. https://doi.org/10.1098/rspa.2013.0698

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