Singularities in Complex Interface Dynamics

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 which arise with zero surface tension with those which occur when the surface tension is small, but finite.

The interface dynamics can be analyzed in terms of a complex analytic function which 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 more viscous fluid. In this physical region, the mapping is analytic and its derivative is never zero.

At zero surface tension we have an integrable problem. The derivative of the mapping function, g(ω, t), then necessarily has all its zeros and poles within the unit circle. If g(ω, t) is a rational function, then the integrable dynamics simply describes the motion of these singularities. The analysis fails at a critical time at which one of the singularities hits the unit circle.

This paper focuses upon the determination of the nature of g and of the interface when the surface tension is small. Two cases are considered: In case A in which the t=0 interface is described by a g with only zeros in the unit circle; in case B in which the singularities closest to the unit circle are instead poles. In case B, the motion is qualitatively similar with and without surface tension: the singularities move outward and asymptotically approach the circle. In case A, for zero surface tension, the zeros move outward and hit the interface after a finite time, whereupon the solution breaks down. But, for finite surface tension, a zero disappears and is replaced by a pair of pole-like excitations which again seem to approach the unit circle asymptotically.