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Hele-Shaw flows with time-dependent free boundaries involving a multiply-connected fluid region

Consider a Hele-Shaw cell that is initially empty, and inject fluid at a number of injection points into the gap. To begin with, the plan view of the region occupied by the fluid will consist of growing circular discs, but these will then coalesce and, in general, lead to a multiply-connected geometry. Assuming a constant pressure condition to be relevant at the free boundaries, we show that the entire motion can be explicitly described analytically. When the connectivity is greater than two, the geometry is characterized by a conformal map given by a function that is automorphic with respect to a Schottky group, and we show how to construct this as a ratio of Poincaré theta series. The efficacy of our solution procedure is demonstrated by a number of examples chosen to illustrate points of both physical and mathematical interest.