Viscous fingering patterns in ferrofluids

Viscous fingering occurs in the now of two immiscible, viscous fluids betwe en the plates of a Hele-Shaw cell. Due to pressure gradients or gravity, th e initially planar interface separating the two fluids undergoes a Saffman- Taylor instability and develops fingerlike structures. When one of the flui ds is a ferrofluid and a perpendicular magnetic Field is applied, the labyr inthine instability supplements the usual viscous fingering instability, re sulting in visually striking, complex patterns. We consider this problem in a rectangular now geometry using a perturbative mode-coupling analysis. We deduce two general results: viscosity contrast between the fluids drives i nterface asymmetry, with no contribution from magnetic forces; magnetic rep ulsion within the ferrofluid generates finger tip-splitting, which is absen t in the rectangular geometry for ordinary fluids.