On the effective viscosity of a dilute suspension of rigid spherical particles

The problem of the motion of a spherical bubble of arbitrary size with a specified velocity in a dilute suspension of solid spherical particles is considered in the Stokes approximation for a carrying continuous medium. An expression is obtained for the drag of the bubble in the first approximation in the volume concentration of the dispersed phase. Using the results obtained an expression is derived for the effective viscosity of the suspension, “as seen” by the spherical bubble as it moves through the dispersed medium. It is shown that the coefficient for the volume concentration in the formula for the effective viscosity of the suspension depends on the ratio of the dimensions of the dispersed particles and of the bubble. In the limit, when this ratio approaches zero, the coefficient obtained is identical with Einstein's result for the effective viscosity of the suspension. However, this coefficient may differ considerably from Einstein's result for “non-point” dispersed particles. By comparing this result with the similar result obtained earlier for the case of the motion of a rigid sphere in a viscous suspension it is shown that the coefficient for the volume concentration in the formula for the effective viscosity of the suspension depends not only on the size of the body, but also on the form of the boundary conditions on its surface.