The effects of yield stress variation on uniaxial exchange flows of two Bingham fluids in a pipe

Buoyancy driven flows of two Bingham fluids in an inclined pipe are considered, providing a simplified model for flows occurring in the oilfield process of plug cementing. The flows studied are near-uniaxial and stratified, with the heavy fluid moving down the incline, displacing the lighter fluid upwards. Continuity and monotonicity results are derived describing general parametric dependence of the solutions on the yield stresses. Solutions are found for two specific interface configurations that are of practical interest. For these flows there exists a single curve in the dimensionless yield stress parameter space, which separates regions of zero and non-zero flow. For the example interface configurations, this curve is determined exactly. For arbitrary interface configurations a rigorous but conservative general result is proven, defining a region of zero flow. Heuristic methods are then used to improve this estimate for arbitrary interfaces.