On the rheology of a suspension of viscoelastic spheres in a viscous liquid

Abstract

The stress in a suspension of incompressible deformable particles exceeds that which would exist in the pure incompressible liquid undergoing the same flow by the product of the volume concentration of the particles and the tensor $\overline{p}_{ik} - 2\eta_0 e^{-(1)}_{ik}$ where η₀ is the viscosity of the pure liquid, $\overline{p}_{ik}$ is the average stress and $e^{-(1)}_{ik}$ the average rate of strain in particles in the flowing suspension. For a dilute suspension of viscoelastic spheres these tensors can be determined by using an adaptation of Jeffery's solution of the problem of an isolated rigid ellipsoid. In steady laminar flow, the material of each sphere is continuously deformed and rotates within an ellipsoidal boundary of fixed dimensions and orientation. Approximate expressions are obtained for the steady-rate viscosity and normal stress differences in terms of the dynamic viscosity and dynamic rigidity functions of the suspension. These are valid either when the rate of shear is sufficiently small or when the ratio of the dynamic viscosity of the spheres to η_o is sufficiently large. The three normal stress components are all unequal. In slow steady elongation of the suspension, the spheres suffer static deformation into prolate spheroids so the elongational viscosity depends only on their static elastic properties. It appears from the investigation of special cases, however, that this static deformation is not possible for rates of elongation above a critical value.