Computation of Hele-Shaw Flows with Free Boundaries

A novel method for calculation of Hele-Shaw flows with receding free boundaries is presented. The method is applied to flows with suction from a point sink and to flow in a channel with parallel walls. In each case the unknown fluid region is mapped conformally onto the unit disc, the free boundary being mapped onto the unit circle. This mapping, which is a function of position and time, is calculated numerically at points on the unit circle using a version of the boundary integral method. The free boundary is thus found without explicit calculation of the pressure at internal points, and the computation times are much less than those for other numerical methods for this problem. Numerical results are compared with explicit analytic solutions for several test problems.