Evolution of HeleShaw Interface for Small Surface Tension
Abstract
We consider the timeevolving displacement of a viscous fluid by another fluid of negligible viscosity in a HeleShaw 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 timedependent 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 zplane. 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 illposed initial value problem in the physical domain zeta<= 1 can be imbedded in a wellposed 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 nonzero 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 z_{0} 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  z_{0} = 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 HeleShaw interface is very sensitive to prescribed initial conditions in the physical domain.
 Publication:

Philosophical Transactions of the Royal Society of London Series A
 Pub Date:
 May 1993
 DOI:
 10.1098/rsta.1993.0049
 Bibcode:
 1993RSPTA.343..155T