Uniqueness and non-uniqueness in the steady displacement of two visco-plastic fluids

We study steady miscible displacements of two visco-plastic fluids in a lon g plane channel. If the yield stress of the displacing fluid is less than t hat of the displaced fluid, uniform static residual layers can be left atta ched to the walls of the channel as the displacement front propagates stead ily. We investigate this steady finger propagation and the problem of finge r width selection. The problem is fully two-dimensional, with the two fluid s separated by a sharp interface. For a given fluid interface, chosen from a wide class of physically sensible interface shapes, we show that there ex ists a unique solution. As well as flexibility in the exact shape of the in terface, the residual; static layer thickness is also non-unique. Typically layer thickness h is an element of (h(min), h(max)) admit a physically sen sible static layer solution where h(min) and h(max) are easily computable f unctions 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 thicknesses.