Effective viscosity of suspensions of spheres

Modeling the effective viscosity of suspensions of spheres beyond the dilute limit is a difficult problem that has challenged researchers for nearly a century since Einstein derived an equation for dilute suspensions. Hydrodynamics modeling requires complex mathematical treatments of integral equations describing interactions between spheres, and its solutions have been limited to semidilute dispersion of monosized hard spheres. Mechanics modeling treats suspensions in a statistical sense and generally provides upper and lower bounds solutions for polydispersed suspensions. A closed-form solution for the effective viscosity of concentrated suspensions of monosized spheres using both the elastic-viscous analogy and the analogy between the effective dielectric constant of a polarizable medium and the effective viscosity of two-phase flow is derived in the present study. Existing measurements and numerical simulations support present analytical results. A closed-form solution for polydispersed suspensions using a differential model is also derived, and the results are within existing bounds solutions.