Dependence of Viscosity of Suspensions on Reynolds Number

Abstract

Calculation has been made of viscosity of concentrated suspensions of spherical particles. A term which is proportional to R_e² is estimated in the viscosity-concentration relation, where Reynolds number R_e is composed of the rate of shear, the radius of a particle and the kinematic viscosity of the medium.
Based on the cage model presented by R. Simha, a spherical enclosure was placed around a central particle. The disturbance of the original flow by the presence of the particle was considered to vanish on the surface of the cell.
The coordinate axes are taken to be unconcerned with the rotation of the particle, so that the Navier-Stokes equation is free from the effect of Coriolis force and centrifugal force. The boundary conditions, however, need to be somewhat changed.
The velocity field and the pressure are expanded into power series of the rate of shear. The Navier-Stokes equation and the equation of continuity are solved successively for several values of a/b, the ratio of the radii of the particle and the cell, up to the second approximation, a zeroth approximation being Simha's solution obtained from the Stokes approximation.
Calculations have been performed by digital computers, and the final results are as follows:
η'/η=1+17.1ξ(1-1.2 R_e²+…) for a/b=0.7,
η'/η=1+53.0ξ(1-98.3R_e²+…) for a/b=0.8,
η'/η=1+406ξ(1-5610R_e²+…) for a/b=0.9,
where η' and η are the viscosities of the suspension and the medium, respectively and ξ is the volume fraction of the particles.