The effective viscosity of a dilute suspension of dipolar fluid droplets in a viscous fluid

A classical Newtonian viscous incompressible fluid in which small spherical particles of a linear homogeneous incompressible fluid are suspended is analyzed under the assumption that the Reynolds number of the motion disturbed by the particles is small as compared to unity and that the effects of inertia and gravity on particle motion are negligible, so that particles move locally with the ambient fluid. This means that the disturbance flow is caused by the background pure straining motion only. It is shown how, using certain integral identities derived by Saffman (1965) and modified by Hills (1967), the effective viscosity of the system can be computed without calculating explicitly the velocity and pressure distributions in the droplets and in the fluid.