Variational principle and variational inequality for a yield stress fluid in the presence of slip

Minimum and maximum principles, which have been derived for the creeping flows of a yield stress fluid, are extended to flows when wall slip is present. In addition, the minimum principle is shown to lead to a variational inequality when wall slip is absent, and the latter is generalised to include inertial effects in specific cases, including flows in pipes and the flow past a body at rest due to a uniform flow at infinity. Moreover, the variational inequality is extended to deal with problems where wall slip may be present. Finally, the squeezing flow between two co-axial and parallel disks is re-examined as an application of the variational principles obtained here.