Solution for the oscillation of a newtonian fluid in a co-axial viscometer

Abstract

When one cup of a co-axial viscometer oscillates, the measured moment on the other (stationary) cup shows a phase lag, partly due to the inertial effect of the fluid within the gap between the cups. In this paper such an effect is illustrated by a new exact solution of the Navier-Stokes equation, which is derived herein by a scheme of reducing it to a two-point boundary value problem for ODEs. Our numerical results indicate that, as the Womersley number a or the dimensionless gap width δ increases. the fluid velocity profile within the gap gradually deviates from the linear one and transits to that of the boundary layer type, with the result that the moment decreases in the magnitude and lags behind in the phase. With the advantage of high accuracy and excellent stability, the scheme proposed herein can readily be extended to solve other linear periodic problems.