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Abstract
The local volume averages of the equations of motion as well as the appropriate boundary conditions are developed for a flowing suspension of non-neutrally buoyant, uniform spheres in an incompressible, weak, second-order fluid under conditions such that inertial effects can be neglected. These equations do not represent an asymptotic theory with respect to the volume fraction of solids. Higher order terms have been retained everywhere, except where it has been necessary to estimate the stress distribution on the surface of each sphere. In estimating the stress distribution on the surface of each sphere, we have assumed that the suspension is sufficiently dilute that the disturbance neighborhoods associated with any two spheres do not significantly overlap. The resulting local volume-averaged equations of motion and their boundary conditions involve no free or undetermined parameters.
For the special case of a very dilute suspension of neutrally buoyant spheres and for a special class of flows of neutrally buoyant spheres that includes the cone-plate viscometer and the parallel-plate viscometer, the total intrinsic average of Cauchy's first law reduces to the form appropriate to a single-phase equation with the material parameters replaced by effective quantities. However, in general the complete boundary value problem to be solved is not the same as that for a single-phase second-order fluid.
