Suspensions of rigid spherical particles in polymeric solutions

Suspensions of solid particles in polymeric solutions are regarded as suspensions of solid particles and polymer molecules in a Newtonian fluid. Disturbances caused by the particles propagate in the Newtonian fluid and influence the distribution of the particles and conformation of the molecules. States of the particles are characterized by a distribution function (two particle distribution function in the dilute limit) and states of the polymer molecules by one molecule distribution function depending on the end-to-end vector of the molecule and the position coordinates of the particles. The rheological model consists of an expression for the extra stress tensor and time evolution equations for the distribution functions. Two versions of the model are developed: one valid for arbitrary particle concentration and the other, carried to more details, for a special case of dilute suspensions. The governing equations for dilute suspensions are then solved for the linear response to imposed oscillatory flows. The result is a sum of three terms: the first two express the linear response in pure particle and pure polymer suspensions, the third expresses the coupling effect. In particular, the effective viscosity for suspension in a polymeric solution is found to be ηeff/η0=1+ζ+(52)φ+(257)ζφ+O(φ2), where φ is a volume fraction of suspended particles and ζ is dimensionless parameter proportional to the concentration and the characteristic relaxation time of the polymer molecules.