The viscosity of a concentrated suspension of spherical particles

Einstein's viscosity equation for an infinitely dilute suspension of spheres is extended to apply to a suspension of finite concentration. The argument makes use of a functional equation which must be satisfied if the final viscosity is to be independent of the sequence of stepwise additions of partial volume fractions of the spheres to the suspension. For a monodisperse system the solution of the functional equation is ƞτ=exp2.5φ1 − kφ role=presentation style=font-size: 90%; display: inline-block; position: relative;>$\text{ƞ}{}_{\mathrm{\tau }}\text{=}\text{exp}\text{2.5φ}\text{1 − kφ}$$\text{ƞ}{}_{\mathrm{\tau }}\text{=}\text{exp}\text{2.5φ}\text{1 − kφ}$ where η_r is the relative viscosity, φ the volume fraction of the suspended spheres, and k is a constant, the self-crowding factor, predicted only approximately by the theory. The solution for a polydisperse system involves a variable factor, λ_ij, which measures the crowding of spheres of radius r_j by spheres of radius r_i. The variation of λ_ij with rirj role=presentation style=font-size: 90%; display: inline-block; position: relative;>$\text{r}{}_{\mathrm{i}}\text{r}{}_{\mathrm{j}}$$\text{r}{}_{\mathrm{i}}\text{r}{}_{\mathrm{j}}$ is roughly indicated. There is good agreement of the theory with published experimental data.