Effective shear and extensional viscosities of concentrated disordered suspensions of rigid particles.
Leonid V. Berlyand, Alexander Panchenko · arXiv (Cornell University) · 2004
We study effective shear viscosity � ⋆ and effective extensional viscosity λ ⋆ of concentrated non-colloidal suspensions of rigid spherical particles. The focus is on the spatially disordered arrays, but periodic arrays are considered as well. We use recently developed discrete network approximation techniques to obtain asymptotic formulas for � ⋆ and λ ⋆ as the typical interparticle distance δ tends to zero, assuming that the fluid flow is governed by Stokes equations. 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 � ⋆ and λ ⋆ on variable distances between neighboring particles in such arrays. Our analysis, carried out for a two-dimensional model, can be characterized as global because it goes beyond the local analysis of flow between two particles and takes into account hydrodynamical interactions in the entire particle array. Previously, asymptotic formulas for � ⋆ and λ ⋆ were obtained via asymptotic analysis of lubrication effects in a single thin gap between two closely spaced particles. The principal conclusion in the paper is that, in general, asymptotic formulas for � ⋆ and λ ⋆ obtained by global analysis are different from the formulas obtained from local analysis. In particular, we show that the leading term in the asymptotics of � ⋆ 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 ⋆ depends on the geometry of the particle array (either weak or strong blow up). We obtain geometric conditions on a random particle array under which the asymptotic order of λ ⋆ coincides with the order of the local dissipation in a gap between two neighboring particles, and show that these conditions are generic. We also provide an example of a uniformly closely packed particle array for which the leading term in the asymptotics of λ ⋆ degenerates (weak blow up).