Modeling pressure-driven flow between adjacent surfaces of viscoplastic and seemingly viscoplastic materials

A simplified model is derived for pressure-driven flow between adjacent surfaces of materials modeled as seemingly viscoplastic or truly viscoplastic. The material response to external forces is traditionally described by constitutive relations in which the extra stress tensor $S$ is expressed as a function of the symmetric part of the velocity gradient $D$. However, for viscoplastic materials, $S$ cannot, in general, be written as a function of $D$, whereas $D$ can be expressed in terms of $S$. Motivated by this observation, a model based on constitutive relations of the form $D = f(S)$ is proposed, leading to a system of first-order partial differential equations. A local Poiseuille law is also formulated, and a reduced-dimensional equation for the pressure is derived. Explicit velocity profiles are obtained for selected cases.