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| Many free boundary problems naturally arise from the consideration of different types of Hele-Shaw systems, where two viscous fluids (typically with one fluid much less viscous than the other) are sandwiched between two closely spaced parallel plates. If the flow domain is simply (or at most doubly) connected, a plethora of analytical solutions can be found for both steady and time-dependent Hele-Shaw flows by using conformal mapping techniques. In the case of higher connectivity, however, the situation is much more complicated because conformal mappings for such domains are notoriously difficult to obtain. In this work, I will describe a large class of conformal mappings from a bounded circular domain to radial and circular slit domains. The slit maps are written in terms of functions appropriately chosen from a family of Schottky-Klein prime functions associated with the circular domain which we have recently obtained. Using these slit maps, exact solutions for an arbitrary number (finite or infinite) of steady bubbles in a Hele-Shaw cell can be constructed in closed form. Nonsingular time-dependent solutions for multiple Hele-Shaw bubbles can also be obtained. In particular, it will be shown that the steady solutions for which the bubbles move twice as fast as the fluid at infinity are the only attractor for the nonsingular solutions, thus showing that the velocity selection is inherently determined by the zero surface tension dynamics. |
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