Viscoplastic fingering in rectangular channels

We experimentally study the viscous fingering problem of viscoplastic fluids in channels of rectangular cross section. We find that a yield stress-dependent capillary number (Ca^{*}) and an aspect ratio-dependent Bond number (Bo^{*}) can classify the finger shape into ramified and unified fingering patterns, and the finger flow regime into yield stress, viscosity, and aspect ratio-buoyancy-dominated regimes. For these regimes, we provide the transition boundaries using Ca^{*} and Bo^{*} and propose simple relations to predict the finger width, for a wide range of flow parameters, versus the capillary number, the channel aspect ratio, and the rheology of the viscoplastic fluid.