Non-continuum anomalies in the apparent viscosity experienced by a test sphere moving through an otherwise quiescent suspension

A comparison is made of the ‘‘apparent viscosity’’ ~as defined by Stokes law! between two different cases of a test sphere moving slowly through an unbounded, otherwise quiescent, globally homogeneous, dilute suspension of identical, neutrally buoyant spherical particles dispersed in an incompressible Newtonian liquid. In case I the force on the test sphere is maintained constant for all time ~and the torque-free sphere allowed to rotate! — corresponding to the so-called ‘‘falling ball’’ case — and its instantaneous velocity allowed to vary with proximity to each suspended sphere encountered during its trajectory; in case II the non-rotating test sphere is towed with a uniform ~instantaneous! velocity through the suspension and the force experienced by it allowed to vary with proximity to each suspended sphere. Allowing for two-body hydrodynamic interactions between the ball and a suspended particle, the ensemble-average velocity of the test sphere is calculated in case I and ensemble-average force in case II, and Stokes law used to calculate the apparent viscosity of the suspension from the ensemble-averaged, linear force/velocity ratio obtained. In each case the ‘‘apparent suspension viscosity’’ coefficient attains, as expected, the limiting, continuum, Einstein value of 2.5 when the test sphere is much larger than the freely suspended particle. However, in the case of disparate relative sizes, the apparent viscosity is found to be significantly larger in case II than in case I. The difference arises from the locally inhomogeneous nature of the suspension and points up a fundamental non-continuum aspect of suspension behavior above and beyond the expected test/suspended-sphere size ratio ‘‘Knudsen’’ non-continuum effect. © 1997 American Institute of Physics. @S1070-6631~97!00101-3#