Uniqueness and Non-uniqueness in the Steady Displacement of Two Visco-plastic Fluids

Abstract

We study steady miscible displacements of two visco-plastic fluids in a long plane channel. If the yield stress of the displacing fluid is less than that of the displaced fluid, uniform static residual layers can be left attached to the walls of the channel as the displacement front propagates steadily. We investigate this steady finger propagation and the problem of finger width selection. The problem is fully two-dimensional, with the two fluids separated by a sharp interface. For a given fixed interface, chosen from a wide class of physically sensible interface shapes, we show that there exists a unique solution. As well as flexibility in the exact shape of the interface, the residual static layer thickness is also non-unique. Typically layer thicknesses h ∈ (h min , h max ) admit a physically sensible static layer solution, where h min and h max are easily computable functions of the dimensionless problem parameters. The dependency of h min and h max on the dimensionless problem parameters is explained and example solutions are computed for different static residual thick-nesses.