Apparent slip at the surface of a small rotating sphere in a dilute quiescent suspension

We consider the case of a test sphere (ball) of radius a, relating at constant angular velocity omega in an otherwise quiescent unbounded suspension of uniformly sized spheres of radii a(2) dispersed in a Newtonian fluid of viscosity mu. To the first order in the volume fraction c of suspended spheres it is shown that when the ball is small compared with the suspended spheres the suspension does not behave as regards the hydrodynamic torque L exerted on the ball like a homogeneous Newtonian fluid characterized by the usual Einstein viscosity coefficient mu(s) = mu(1+5/2c). Explicitly, the torque on the rotating sphere does not obey Kirchoff's law, L=8 pi mu(s)a(1)(3) omega for no slip. Rather, a modified form of Kirchoff's law is obtained in which the Einstein coefficient of 5/2 is multiplied by a coefficient which is less than unity in magnitude and is functionally dependent only upon the suspended-sphere/test-sphere size ratio, lambda=a(2)/a(1). In the "continuum limit," where lambda tends to zero, one recovers Kirchoff's law. Accordingly, the deviation from Kirchoff's law is interpreted in terms of an apparent Knudsen-like "slip" at the rotating ball surface since this slip vanishes in the continuum limit. The existence of an apparent slip is consistent with recent experiments performed on small rotating spheres, albeit in concentrated suspensions, in which the "viscosity" of the suspension-defined via Kirchoff's law in terms of the experimentally measured torque L as L/8 pi a(1)(3) omega-was observed to be less than the viscosity of the suspension as measured by standard viscometric methods. Similar, although quantitatively different O(c) theoretical Knudsen-like slip results were also obtained for the "inverse" case, where the torque L on the rotating ball is held constant for all time and its mean angular velocity calculated. (C) 1998 American Institute of Physics.