Evolution of Hele-Shaw interface for small surface tension

Abstract

We consider the time-evolving displacement of a viscous fluid by another fluid of negligible viscosity in a Hele-Shaw cell, either in a channel or a radial geometry, for idealized boundary conditions developed by McLean & Saffman. The interfacial evolution is conveniently described by a time-dependent conformal map z(£, t) that maps a unit circle (or a semicircle) in the £ plane into the viscous fluid flow region in the physical z-plane. Our paper is concerned with the singularities of the analytically continued z((,,t) in |£| > 1, which, on approaching |£| = 1, correspond to localized distortions of the actual interface. For zero surface tension, we extend earlier results to show that for any initial condition, each singularity, initially present in |£| > 1, continually approaches |£| = 1, the boundary of the physical domain, without any change in the singularity form. However, depending on the singularity type, it may or may not impinge on |£| = 1 in finite time. Under some assumptions, we give analytical evidence to suggest that the ill-posed initial value problem in the physical domain |£| ≤ 1 can be imbedded in a well-posed problem in |£| ≤ 1. We present a numerical scheme to calculate such solutions. For each initial singularity of a certain type, which in the absence of surface tension would have merely moved to a new location £ s ( t ) at time t from an initial £ s (0), we find an instantaneous transformation of the singularity structure for non-zero surface tension B; however, for 0 < B << 1, surface tension effects are limited to a small ‘inner’ neighbourhood of £ s ( t ) when t << B -1 Outside the inner region, but for ( — £ s (t)1, the singular behaviour of the zero surface tension solution z 0 is reflected in On the other hand, for each initial zero of z £ , which for B = 0 remains a zero of z 0 £ at a location £ 0 ( t ) that is generally different from £ 0 ( 0 ), surface tension spawns new singularities that move away from £ 0 ( t ) and approach the physical domain |£| = 1. We find that even for 0 < B << 1, it is possible for z — z 0 — O (1) or larger in some neighbourhood where z 0 £ is neither singular nor zero. Our findings imply that for a small enough B, the evolution of a Hele-Shaw interface is very sensitive to prescribed initial conditions in the physical domain.