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Bubble dynamics and velocity selection in a Hele-Shaw cell

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CAT - Complex analysis: techniques, applications and computations

The unsteady motion of a finite assembly of bubbles in a Hele-Shaw channel is studied in the case when surface tension is neglected. A general exact solution is obtained in terms of a conformal map from a multiply connected circular domain to the fluid region exterior to the bubbles. The correspond- ing mapping function is given explicitly in terms of certain special transcen- dental functions, known as the secondary Schottky-Klein prime functions. Exploring the properties of these solutions, we show that steady configura- tions where the bubbles move with a velocity, U, which is twice greater than the velocity, V , of the background flow, i.e., U = 2V , are the only attrac- tor of the dynamics; whereas solutions with U ≠ 2V act as repellors. This demonstrates that the special nature of the solutions with U = 2V is already built-in in the zero-surface-tension dynamics, which is confirmed by the in- clusion of regularization effects. In particular, the case of a single bubble will be discussed in detail and several numerical examples of bubble evolution and bubble selection will be presented.




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