## 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($\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|\leq $ 1 can be imbedded in a well-posed problem in $|\zeta|\geq $ 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 $\zeta _{\text{s}}$ (t) at time t from an initial $\zeta _{\text{s}}$(0), we find an instantaneous transformation of the singularity structure for non-zero surface tension $\beta $; however, for 0 < $\beta \ll $ 1, surface tension effects are limited to a small `inner' neighbourhood of $\zeta _{\text{s}}$(t) when t $\ll \beta ^{-1}$. Outside the inner region, but for $|\zeta -\zeta _{\text{s}}$(t)$|\ll $ 1, the singular behaviour of the zero surface tension solution z$_{0}$ is reflected in z($\zeta $, t). On the other hand, for each initial zero of z$_{\zeta}$, which for $\beta $ = 0 remains a zero of z$_{0\zeta}$ 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 < $\beta \ll $ 1, it is possible for z - z$_{0}$ = O(1) or larger in some neighbourhood where z$_{0\zeta}$ is neither singular nor zero. Our findings imply that for a small enough $\beta $, the evolution of a Hele-Shaw interface is very sensitive to prescribed initial conditions in the physical domain.

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