FINITE-TIME SINGULARITY FORMATION IN HELE-SHAW SYSTEMS

We investigate the behavior of the interface between two fluids in a t wo-dimensional flow driven by surface tension. The geometry is chosen so that one can apply a variant of the lubrication approximation and s o that the more-viscous fluid will have a tendency to change its topol ogy by separating into two masses. Simulations are used to show that, with appropriate initial and boundary conditions, this separation can occur in a finite time. We particularly focus our attention at the pin ch point, i.e., the space-time point at which the width of the viscous fluid first goes to zero. The lubrication approximation used contains a parameter rho which measures the strength of the inertial forces. S ince the fluid velocity diverges as the pinch is approached, the behav ior is qualitatively different for small rho and for rho = 0. Simulati ons and asymptotic analyses are used to isolate this difference. For r ho = 0, at the pinch time there is a region of space in which the widt h grows quadratically as one moves away from the pinch. The curvatures , however, are different on the two sides of the pinch. In contrast, w hen rho is different from zero, the width rises nearly linearly with d istance from the pinch.