Homogenized non-Newtonian viscoelastic rheology of a suspension of interacting particles in a viscous Newtonian fluid

Our study is motivated by an attempt to develop a rigorous mathematical model of a suspension highly filled with a large number of small solid particles, which interact due to surface forces. We use asymptotic analysis in the small parameter e and consider irregular (nonperiodic) geometries for which the sizes of particles and the distances between them are of order 6. We present conditions under which the homogenization of a Newtonian fluid with interacting particles leads to a single medium which is an anisotropic, non-Newtonian viscoelastic fluid with memory described by a relaxation term. We derive formulas for the calculation of the effective viscosity tensor and the relaxation integral kernel. For periodic arrays of particles we show how this tensor can be explicitly computed and compute the distribution of the relaxation times, which is the main quantity of interest in the rheological studies. We also show how the particles' shapes affect this distribution.