Hele-Shaw flows with time-dependent free boundaries involving a concentric annulus

We consider the classical Hele-Shaw situation with two parallel planes separated by a narrow gap. A blob of Newtonian fluid is sandwiched between the planes and, initially, its plan-view occupies a doubly connected region, so that we physically have a fluid surrounding what we will refer to as an air hole. We seek to predict the evolution of the plan-view of the blob as fluid is added to or removed from the blob at specific injection or suction points, or as air is added to or removed from the air hole. We suppose the relevant free boundary condition to be one of constant pressure, but allow different constant pressures to act along each free boundary. Lf we begin with a blob in the form of a concentric circular annulus, we obtain an analytic solution to describe its subsequent evolution when subjected to any pattern of injection or suction of fluid, or addition or removal of air, provided only that the plan-view of the blob either remains doubly connected or becomes simply connected as the air hole disappears. Since each state we can reach by this process may itself be regarded as an initial state for some subsequent motion, the class of problems we are thus able to solve is quite extensive. The solutions are expressed in terms of conformal maps that have an explicit analytic form incorporating a finite number of parameters; in general, these parameters must be determined numerically by solving a finite set of transcendental equations. The efficacy of the procedure is illustrated by means of specific examples, chosen to exhibit a number of features of physical interest. The development has an independent mathematical interest, for it involves the solution of a nonlinear functional equation; we here deal only with the particular form of equation that arises in the present application, but generalizations can be handled by similar techniques. Also, the solutions are expressed in terms of functions which themselves have interesting properties. However, we do not pursue these mathematical questions in this paper.