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Calculation of effective viscosity of concentrated solid particle suspensions using a crystal model

V.E. Zgaevskii and Yu. G. Yanovskii

#### Abstract:

The concentrated suspensions are modeled by identical particles that are located at the modes of a lattice of a given symmetry. As the particles move relative to one another there arise dissipative forces that are described in the approximation of lubrication theory. We examine pairwise interaction of the particles and describe the energy dissipation power density for given strain rate gradients. We use the mathematical apparatus of microscopic crystal theory and write the tensor of the viscous moduli of the lattice of a given symmetry. Macroscopic isotropy of the suspensions is achieved by converting to a polycrystalline body with equiprobable orientation of the monocrystals. We obtain the analytic dependence of the effective viscosity on three structural parameters of the model: the relative distance between the centers of the nearest particles, the relative distance between the surfaces of the nearest particles, and the number of nearest neighbors of the particle. The dependence of the effective viscosity on the relationship between the loading geometry and the characteristic dimensional parameter of the structure is noted. It is shown that: 1) the effective viscosity for the same volumetric filling increases exponentially with increase of the number of nearest neighbors, 2) the technique that is used in the literature to evaluate the effective viscosity on the basis of the element of the first row and the first column of the tensor of the viscous moduli of the model of cubic symmetry yields a value that is high by at least half an order of magnitude in comparison with the viscous modulus of the isotropic body, 3) the order of the singularity as the volumetric filling approaches the maximal value is described by a value that is significantly larger than the theoretical value known from the literature and is close to the relation proposed by Simha.

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