Effective shear and extensional viscosities of cocentrated disordered suspensions of rigid particles

Abstract

We study effective shear viscosity $μ^\star$ and effective extensional viscosity $λ^\star$ of concentrated non-colloidal suspensions of rigid spherical particles. The focus is on the spatially disordered arrays. We use recently developed discrete network approximation techniques to obtain asymptotic formulas for the viscosities as the typical inter-particle distance $δ$ tends to zero. For disordered arrays, the volume fraction alone does not determine the effective viscosity. Use of the network approximation allows us to study the dependence of the effective viscosities on variable distances between neighboring particles. Our analysis can be characterized as global because it goes beyond the local analysis of the flow between two particles. The principal conclusion in the paper is that, in general, asymptotic formulas obtained by global analysis are different from the formulas obtained from local analysis. In particular, the leading term in the asymptotics of $μ^\star$ is of lower order than suggested by the local analysis (weak blow up), while the order of the leading term in the asymptotics of $λ^\star$ depends on the geometry of the particle array (either weak or strong blow up). We obtain geometric conditions on a random particle array that lead to the strong blow up of $λ^\star$, and show that these conditions are generic. We also provide an example of a closely packed particle array for which the leading term in the asymptotics of $λ^\star$ degenerates (weak blow up).