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

Abstract

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. C 2015 AIP Publishing LLC. [http://dx.

Bulk flow driven by a viscous monolayer

“…The numerical technique used is the same as that implemented in Lopez & Hirsa (2015), and only a brief overview is presented here. The Navier-Stokes equations, (2.4) and (2.5), for the bulk flow are solved using a second-order finite difference scheme for space together with a second-order predictor-corrector scheme for time evolution.…”

“…Geometric parameters governing this flow are the cylinder-to-knife-edge radius ratio R = r o /a, the cylinder-height-to-knife-edge radius ratio H = h/a, and the knife-edgethickness-to-radius ratio . How the flow depends on these has been described in the numerical study in Lopez & Hirsa (2015), and in this paper, we shall use values of these that correspond to the specific experiments that are reported below.…”

“…Using the knife-edge thickness instead of its radius to define the Boussinesq number becomes problematic in the limit of the thickness going to zero. Retaining the Bo using the radius as the length scale, one can examine effects of variations in Bo with a zero thickness knife edge, as was done numerically in Lopez & Hirsa (2015).…”

“…There are only a few studies that discuss or take into account the coupling to the bulk flow that is driven by a knife edge or a needle (Reynaert et al 2008;Vandebril et al 2010;Verwijlen et al 2011;Lopez & Hirsa 2015), and we are not aware of any detailed measurements of such bulk flows. Here, we utilize the knife edge for several reasons.…”

“…In the experiment, the knife edge can be carefully treated so that there is no meniscus and the interface can be made essentially flat down to the mesoscale (order of a micron, which is essentially the limit of optical microscopy). Also, we have a numerical code based on a Newtonian surface model that is nonlinearly coupled to the bulk flow and includes the effects of inertia via the Navier-Stokes equations (Lopez & Hirsa 2015). The model predictions are compared to measurements in the bulk flow using particle image velocimetry.…”