Viscous Fingering in a Gel

Abstract

When a less viscous fluid pushes a more viscous fluid in a Hele-Shaw cell, the interface between the two fluids develops an instability leading to the formation of fingerlike patterns, called viscous fingers. This is the so-called Saffman-Taylor or viscous fingering instability [1]. The width of these viscous fingers is, for Newtonian fluids, determined by the capillary number Ca = ΔμU/γ which represents the ratio of viscous forces over capillary forces; Δe is the viscosity difference between the two fluids, U the finger velocity and γ the surface tension. The viscous forces tend to narrow the finger, whereas the capillary forces tend to widen it: the width of the finger decreases with increasing finger velocity. Due to its relative simplicity the viscous fingering instability has received much attention as an archetype of pattern forming systems, both theoretically and experimentally [1, 2] and is by now well understood.