A homogenization model for the rheology and local field statistics of suspensions of particles in yield stress fluids

Abstract

We investigate the rheological behavior of athermal particle suspensions using experiments and theory. A generalized version of the homogenization estimates of Ponte Castañeda and Willis [J. Mech. Phys. Solids, 43(12), 1919–1951 (1995)] is presented for the effective viscosity of athermal suspensions accounting for additional microstructural features (e.g., polydispersity) via an empirical parameter, λ. For the case of identically sized spheres dispersed with statistical isotropy in a Newtonian fluid, the parameter λ is estimated from the results of Batchelor and Green [J. Fluid Mech. 56(2), 375–400 (1972)] for the Huggins coefficient. Predictions for the macroscopic viscosity are found to be in good agreement with measurements for monodisperse polymethyl methacrylate (PMMA) spheres in glycerol, as well as for the empirical Krieger–Dougherty equation for the shear viscosity. The proposed estimates have the added benefit that they can also be used to get information on the statistics of the stress and strain-rate fields in the fluid and particle phases. In addition, results for the effective shear viscosity are used in combination with the linear comparison method of Ponte Castañeda [J. Mech. Phys. Solids 39(1), 45–71 (1991)] to generate the corresponding estimates for the effective macroscopic behavior and field statistics of particle suspensions in (viscoplastic) yield stress fluids. Good agreement is also found between the theoretical estimates and experimental results for the effective yield and flow stress of suspensions with monodisperse PMMA spheres in Carbopol. Finally, it is argued that the results for the phase averages and fluctuations of the stress and strain-rate fields can be used to provide a physical interpretation for the parameter λ in terms of the polydispersity of the suspension and its implications for the percolation threshold.