Network approximation for effective viscosity of concentrated suspensions with complex geometry

We study suspensions of rigid particles (inclusions) in a viscous incompressible fluid. The particles are close to touching one another, so that the suspension is near the packing limit, and the flow at small Reynolds number is modeled by the Stokes equations. The objective is to determine the dependence of the effective viscosity (μ) on the geometric properties of the particle array. We study spatially irregular arrays, for which the volume fraction alone is not sufficient to estimate the effective viscosity. We use the notion of the interparticle distance parameter δ, based on the Voronoi tessellation, and we obtain a discrete network approximation of (μ), as δ → 0. The asymptotic formulas for (μ), derived in dimensions two and three, take into account translational and rotational motions of the particles. The leading term in the asymptotics is rigorously justified in two dimensions by constructing matching upper and lower variational bounds on (μ). While the upper bound is obtained by patching together local approximate solutions, the construction of the lower bound cannot be obtained by a similar local analysis because the boundary conditions at fluid-solid interfaces must be resolved for all particles simultaneously. We observe that satisfying these boundary conditions, as well as the incompressibility condition, amounts to solving a certain algebraic system. The matrix of this system is determined by the total number of particles and their coordination numbers (number of neighbors of each particle). We show that the solvability of this system is determined by the properties of the network graph (which is uniquely defined by the array of particles) as well as by the conditions imposed at the external boundary.