Solutions for Shearing Flows

Abstract

It is common physical experience that when a nondilute multicomponent fluid is placed in a viscometric testing device, the flow that is produced is not a viscometric flow, that is, the velocity gradient is not necessarily constant, and the concentration is not uniform. Instead, relatively thin layers of fluid with relatively few suspended particles accumulate near the boundaries. These layers exhibit high shear rates. The suspended particles accumulate at some distance from the boundaries, giving a high local viscosity and a low shear rate. These phenomena are so mathematically robust that they are exhibited by a large number of theories for suspensions. Here, we consider two properly invariant theories of multicomponent fluids. We investigate the simplest type of viscometric test, steady flow between parallel plates with one plate stationary and the other plate moving parallel to it at constant speed. We also consider steady flow in a channel, with the flow forced by a pressure gradient. For one theory, it is possible to demonstrate exact solutions to the field equations. For the other, plausible approximate solutions are found. Both types of solutions exhibit the phenomenon of a core of concentrated suspension surrounded by a layer of clear fluid.