Evolution of Hele-Shaw Interface for Small Surface Tension

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(zeta , t) that maps a unit circle (or a semicircle) in the zeta plane into the viscous fluid flow region in the physical z-plane. Our paper is concerned with the singularities of the analytically continued z(zeta , t) in |zeta| > 1, which, on approaching |zeta| = 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 |zeta| > 1, continually approaches |zeta| = 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 |zeta| = 1 in finite time. Under some assumptions, we give analytical evidence to suggest that the ill-posed initial value problem in the physical domain |zeta|<= 1 can be imbedded in a well-posed problem in |zeta|>= 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 zeta _s (t) at time t from an initial zeta _s(0), we find an instantaneous transformation of the singularity structure for non-zero surface tension β ; however, for 0 < β << 1, surface tension effects are limited to a small `inner' neighbourhood of zeta _s(t) when t << β ^-1. Outside the inner region, but for |zeta -zeta _s(t)|<< 1, the singular behaviour of the zero surface tension solution z₀ is reflected in z(zeta , t). On the other hand, for each initial zero of z_zeta, which for β = 0 remains a zero of z_0zeta at a location zeta ₀(t) that is generally different from zeta ₀(0), surface tension spawns new singularities that move away from zeta ₀(t) and approach the physical domain |zeta| = 1. We find that even for 0 < β << 1, it is possible for z - z₀ = O(1) or larger in some neighbourhood where z_0zeta is neither singular nor zero. Our findings imply that for a small enough β , the evolution of a Hele-Shaw interface is very sensitive to prescribed initial conditions in the physical domain.