Inhomogeneous viscosity fluid flow in a wide-gap Couette apparatus: Shear-induced migration in suspensions

Abstract

The first portion of this two-part paper investigates the short time evolution of a low Reynolds number flow characterized by a spatially inhomogeneous viscosity within the annular domain between two widely separated concentric circular cylinders undergoing relative rotation. The viscosity is regarded as a material property and as such is convected with the fluid. Any possible “diffusion” of this viscosity is supposed negligible, at least in the short times of interest in our calculation. The initial viscosity field, assumed to be only slightly inhomogeneous, is expanded into a Fourier series with respect to the polar angle, and the contributions of the zeroth and first harmonics are subsequently addressed. Approximate short-time analytic solutions for the velocity and viscosity fields are obtained. In the second part of this paper the results of the preceding analysis are employed in an attempt to gain insight into the experimentally observed shear-induced migration of particles in suspensions being sheared in a wide-gap Couette apparatus. The connection of the inhomogeneous viscosity problem studied in the first part to such shear-induced migration phenomena lies in the assumption that the local viscosity of a suspension of (non-Brownian) particles is functionally dependent only upon the local suspended particle volumetric fraction. In such circumstances, the local transport of suspended particles corresponds to a concomitant transport of the local suspension viscosity and vice-versa. Subject to the foregoing interpretation and limited by algebraic tractability to short times, a global radial migration is predicted. It increases with an increase in the annular gap size between the cylinders and depends upon the phase angle between the rotating outer and inner cylinders, but not upon their relative circumferential velocity—a conclusion consistent with experimental observations. Further, to leading order, particle migration is found to be independent of purely radial viscosity disturbances (the zeroth harmonic) and to arise entirely from coupling between circumferential disturbances in the velocity and viscosity (i.e., particle concentration) fields. The solution also indicates that the high shear-rate region, proximate to the inner wall, may either become less viscous on average (thereby predicting net radial migration away from the high shear rate region) or, conversely, more viscous (corresponding to migration toward the high shear rate region migration). The latter case arises in circumstances that involve a large positive radial gradient in the viscosity’s first harmonic.