Hele–Shaw flows with time-dependent free boundaries in which the fluid occupies a multiply-connected region

Abstract

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 we suppose its plan-view to occupy a bounded, multiply-connected domain; physically, we have a viscous fluid with the holes giving rise to the multiple connectivity occupied by relatively inviscid air. The relevant free boundary condition is taken to be one of constant pressure, but we allow different pressures to act within the different holes, and at the outer boundary. The motion is driven either by injection of further fluid into the blob at certain points, or by injection of air into the holes to change their area, or by a combination of these; suction, instead of injection, is also contemplated. A general mathematical theory of the above class of problems is developed, and applied to the particular situation that arises when fluid is injected into an initially empty gap bounded by two straight, semi-infinite barriers meeting at right-angles: injection into a quarterplane. For a range of positions of the injection point, air is trapped in the corner and, invoking images, the problem is equivalent to one involving a doubly-connected blob. When there is an air vent in the corner, so that the pressure is the same on the two free boundaries in these circumstances, the air hole rapidly disappears, as might be expected. If, however, there is no air vent and we suppose the air to be incompressible, so that the area of the region occupied by the air in the plan-view remains constant, we find there to be no solution within the framework of our model. Other scenarios within this same geometry, involving both suction and injection of fluid at the injection point, and air at the corner, are also examined.