Rheology of particulate suspensions in a Poiseuille flow

Particulate dense suspensions behave as complex fluids. They do not lend themselves easily to analytical solution. We propose an analytical model to mimic this problem. Namely, we consider arrays of long parallel plates which represent a simplification of arrays of chains of spherical particles. This simplified model can be solved analytically. The effect of effective rotation of the spherical particles is taken into account by attributing different velocities on each side of the plate that mimics the fact that particles are subject to shear. This work is an extension of a previous study where particle rotation was disregarded. The flow rate, the dissipation and the apparent viscosity are studied as a function of the underlying structure. For a single plate placed out of the flow center, the viscosity is lower when rotation is taken into account. For two plates, the minimal viscosity corresponds to the situation where the particles are as close as possible to the center and arranged symmetrically with respect to the center. We compute the rheological properties for arbitrary plate positions, and exploit them for a periodic arrangement. For N plates, and in a confined geometry, the viscosity is about twice as small as compared to the situation where rotation is ignored. We have conducted a numerical study of a suspension of spherical particles, and linear chains of spherical particles. The numerical study is in good qualitative and semiquantitative agreement with the analytical theory considering long plates. This agreement highlights the fact that our analytical model captures the essential features of a real suspension. The numerical study is based on a fluid dynamic particle method where the particles are represented by a scalar field having high viscosity inside.