Interface dynamics and the motion of complex singularities.

Abstract

The motion of the interface between two fluids in a quasi-two-dimensional geometry is studied via simulations. We consider the case in which a zero-viscosity fluid displaces one with finite viscosity and compare the interfaces that arise with zero surface tension with those that occur when the surface tension is not zero. The interface dynamics can be analyzed in terms of a complex analytic function that maps the unit circle into the interface between the fluids. The physical region of the domain is the exterior of the circle, which then maps into the region occupied by the more viscous fluid. In this physical region, the mapping is analytic and its derivative is never zero. This paper focuses upon the determination of the nature of the interface and the positions of the singularities of the derivative of the mapping function g. Two kinds of initial conditions are considered: case A, in which the singularities closest to the unit circle are poles; and case B, in which the t=0 interface is described by a function g with only zeros inside the unit circle. In either case, different behaviors are found for relatively smaller and larger surface tensions. In case A, when the surface tension is relatively small, the problem is qualitatively similar with and without surface tension: the singularities move outward and asymptotically approach the unit circle. For relatively large surface tension, the singularities, still polelike, move towards the center of the unit circle instead. In case B, for zero surface tension, the zeros move outward and hit the unit circle after a finite time, whereupon the solution breaks down. For finite but relatively small surface tension, each initial zero disappears and is replaced by a pair of polelike excitations that seem to approach the unit circle asymptotically, while for a relatively large surface tension, each initial zero is replaced by a polelike singularity that then moves towards the unit circle.