Variational principles and variational inequalities for the unsteady flows of a yield stress fluid

Abstract

A minimum principle, which has been derived for the steady, creeping flows of a yield stress fluid with shear-dependent viscosity, is extended to flows when the yield stress is also shear dependent, and the flow may be unsteady. As an application of the minimum principle, the unsteady squeezing flow between two co-axial and parallel disks is examined. Next, the variational principle is extended to a variational inequality, and situations where inertia may be incorporated into the latter are discussed. Using this, the specific forms of the variational inequalities are derived for five flows: unsteady pipe flows, flow past a solid at rest, the reservoir problem, the cavity driven flow, and, finally, for a class of problems with free surfaces. Further, the variational principle and the inequality are extended to deal with those problems where wall slip may be present. In a manner similar to the way the minimum principle has been extended, a maximum principle for the stress in the above class of yield stress fluids is established, and is easily reworded to include the case of wall slip as well. In addition, this principle is converted to a variational inequality for the stress. Finally, it is shown that the mimimum velocity functional and the maximum stress functional are identical when the velocity and stress fields satisfy the equations of motion and the relevant boundary conditions.