Uniaxial exchange of two bingham fluids in a cylindrical duct

Buoyancy driven flows of two Bingham fluids in an inclined duct are considered, providing a simplified model for many oilfield cementing processes. The flows studied are near-uniaxial and stratified, with the heavy fluid moving down the incline, displacing the lighter fluid upwards. Existence and uniqueness results are obtained for quite general flows and for those which satisfy an axial flow rate constraint. Parametric dependence of the solutions on the axial pressure gradient is studied. Flows which satisfy a zero net axial flow constraint result from an axial pressure gradient which minimises the viscous dissipation, but not the plastic dissipation. A regularisation method is used to compute solutions to these problems for (more or less) arbitrary fluid-fluid interfaces and duct cross-sections. Examples related to a number of practical applications are presented.