Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel

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Abstract
A blob of Newtonian fluid is sandwiched in the narrow gap between two plane parallel surfaces so that, a t some initial instant, its plan-view occupies a simply connected domainD0. Further fluid, with the same material properties, is injected into the gap at some fixed point withinD0, so that the blob begins to grow in size. The domainDoccupied by the fluid at some subsequent time is to be determined.It is shown that the growth is controlled by the existence of an infinite number of invariants of the motion, which are of a purely geometric character. For sufficiently simple initial domainsD0these allow the problem to be reduced to the solution of a finite system of algebraic equations. For more complex initial domains an approximation scheme leads to a similar system of equations to be solved.

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