Theoretical basis for falling-ball rheometry in suspensions of neutrally buoyant spheres

The effective viscosity μ of a dilute suspension of neutrally buoyant, polydisperse, rigid spheres of macroscopic size ( characteristic radius = c ) randomly distributed throughout a Newtonian liquid of viscosity μ 0 is derived theoretically. This result is obtained by two closely related, but nevertheless independent, schemes: (i) from the properties of the fundamental Stokeslet solution viewed at the suspension scale; and (ii) from the settling velocity U of a single, nonneutrally buoyant falling ball ( radius = b ) instantaneously (and quasistatically) settling through the unbounded suspension. To terms of leading order, both yield the classical Einstein result, μ = μ 0(1 + 1 2 φ) , apparently independently of the ratio of c/b, as well as of the size distribution of the suspended spheres. Also studied are wall effects for the special case where the falling ball is instantaneously situated at the center of a hollow sphere ( radius = r o ) that bounds the suspension externally. For circumstances where b/r o ⪡ 1 and c/r o ⪡ 1 , it is demonstrated that classical falling-ball wall effects for a homogeneous Newtonian fluid of viscosity μ = μ 0(1 + 1 2 φ) apply equally well to the present suspension case. This example strongly suggests that the apparent viscosity of dilute suspensions can be experimentally measured via falling-ball rheometry using the well-known (circular cylindrical) wall-effect corrections developed for Newtonian liquids. This observation is in agreement with existing experimental data.