Evolution of Hele Shaw Interface for Small Surface Tension
Abstract
For time evolution of a Hele-Shaw interface described by a conformal map z(zeta,t) that maps a unit circle (or semi-circle) in the zera plane into the viscous fluid flow region in the physical z-plane, we present results on the motion of singularities outside the unit circle. For zero surface tension, we extend earlier results to show that for any initial condition, each singularity of z(zeta,t) present initially in the absolute value of zeta > 1 continually approaches the interfacial boundary the absolute value of zeta = 1 without any change of form. However, depending on the singularity type, it may or may not impinge the absolute value of zeta = 1 in finite time. Under some assumptions, we give analytical evidence to suggest that the ill-posed problem in the physical domain the absolute value of zeta < or = 1 can be imbedded in a well- posed problem in the absolute value of zeta > or = 1. We present a numerical scheme to calculate such solutions. For each initial singularity of certain type, which in the absence of surface tension would have merely moved to a new location zeta sub s (t) at time t from an initial zeta sub s (O), we find an immediate transformation of the singularity structure for nonzero surface tension Beta; however, for O < Beta << 1, surface tension effects on this singularity are limited to a small 'inner' neighborhood of zeta sub s (t) when t << 1/Beta.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1991
- Accession Number
- ADA242984
Entities
People
- S. Tanveer