An analysis is presented of the deformation of a solid-like, viscoelastic sphere suspended in the infinite Stokesian flow field of a Newtonian fluid undergoing an arbitrary time-dependent homogeneous deformation far from the particle. The results of the analysis are then used to deduce the macroscopic rheological behaviour of a dilute monodisperse suspension of slightly deformable spheres.Even though inertial effects and second-order terms in the particle deformation are neglected, it is found that non-linear rheological effects can arise, because of the interaction between the deformed particle and the flow. As a consequence, the rheological relation obtained here differs from those presented earlier by Fröhlich & Sack (1946) and by Oldroyd (1955) through the appearance of certain terms which are non-linear in the deformation rate.When the suspended particles are purely elastic in their behaviour the rheological equation presented here reduces for certain flows to a special case of Oldroyd's (1958) phenomenological model, with material constants which can be directly related to suspension properties.
Non-linear anisotropic hyperelasticity for granular materials
