Multiple finger propagation modes in Hele-Shaw channels of variable depth

摘要

We consider the propagation of an air finger into a wide fluid-filled channel with a spatially varying depth profile. Our aim is to understand the origin of the multiple coexisting families of both steady and oscillatory propagating fingers previously observed in experiments in axially uniform channels each containing a centred step-like occlusion. We find that a depth-averaged model can reproduce all the finger propagation modes observed experimentally. In addition, the model reveals new modes for symmetric finger propagation. The inclusion of a spatially variable channel depth in the depth-averaged equations leads to: (i) a variable mobility coefficient within the fluid domain due to variations in viscous resistance of the channel; and (ii) a variable transverse curvature term in the dynamic boundary condition that modifies the pressure jump over the air-liquid interface. We use our model to examine the roles of these two distinct effects and find that both contribute to the steady bifurcation structure, while the transverse curvature term is responsible for the distinctive oscillatory propagation modes.