Coupling of the interfacial and bulk flow in knife-edge viscometers

The operation of the knife-edge viscometer requires knowledge of the interfacial velocity profile in order to determine the viscous traction between the surface film and the knife edge and hence measure the surface shear viscosity of the film. The interfacial velocity profile can be obtained analytically in two limiting regimes. One is the limit of the surface shear viscosity going to infinity, in which case the interfacial velocity profile is independent of the bulk flow and a simple analytic expression is available. The other limit corresponds to vanishing bulk flow inertia, allowing one to reduce the Navier–Stokes equations to the Stokes equation, and the resulting linear system can be solved analytically. For finite inertia and finite surface shear viscosity, the knife-edge viscometer hydrodynamics is governed by the coupled nonlinear set of equations. Here, we study these numerically, explore the coupling between the interfacial and bulk flow, and delineate the ranges of surface shear viscosity and knife-edge rotation rates where the analytic approximations are appropriate. We also examine a variant of the knife-edge viscometer, known as the double-wall ring viscometer, which is essentially the same geometry but with the addition of a stationary inner cylinder so that the bulk fluid is contained in an annular channel rather than a cylinder.