Dynamics of viscous fingers and threshold instability.

We present a detailed weakly nonlinear analysis, along with a solvability analysis, of a new instability in viscous ngering which we term the Surface-Tension-Driven instability. This instability occurs when the surface tension is modiied in proportion to the local curvature. It is an intrinsically nonlinear instability which always requires a nite-amplitude perturbation to trigger it. It is also distinctively diierent from those arising via subcritical bifurca-tion. Numerical simulations reveal that this instability leads to spiky cellular patterns and may suggest a possibility of leading to a nite time singularity. I. INTRODUCTION The viscous ngering instability 1 which occurs when a less viscous uid pushes a uid of high viscosity has continued to play a major role in both the scientiic and industrial communities. On the practical level, it has important applications in a wide range of interdisciplinary elds, ranging through uid ow in porous media, 2?5 dendritic solidiication, 6 directional viscous ngering, 7;8 colloidal particle aggregation, 9?10 electrochemical deposition, 11 surface growth, 12 and oil recovery. 13 In addition, since the ow in a Hele-Shaw cell is simple and laminar, ngering experiments have been conducted with polymeric solutions in the hope of gaining useful insight into the role of polymers in ow phenomena, such as turbulent drag reduction. 14 Moreover , recent studies have revealed that there exist close similarities between