Analytical solutions of the boundary-value problem of nonstationary flow of viscoplastic medium between two plates

Summary

Nonstationary flow of a viscoplastic medium between two parallel plates is considered for the case of a varying pressure gradient. The problem is reduced to the Stephan problem, with the condition on the boundary separating the flow domain from the quasi-rigid domain. Four multiparameter families of exact solutions are found. The first family describes the flow decelerations up to a full stop. The second family determines the development of the flow from the state of rest as the pressure gradient increases. The third family describes the development of the flow for the case where (1) the pressure gradient is constant and exceeds the threshold value related to the yield stress, (2) the upper plate does not move, and (3) the lower plate moves with a constant acceleration. Finally, the fourth family determines the flow retardation, when the pressure gradient is constant and is less than the threshold value. The decrease in the flow of the viscoplastic medium can be achieved for certain values of parameters by increasing the quasi-rigid domain, whereas the viscoplastic flow remains unchanged.