# Coupling of linear and angular momentum in concentrated suspensions of spheres

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## Abstract

In Newtonian fluids, the proportionality constant that relates the flux of linear momentum to gradients in the velocity is the shear viscosity. Similarly, the flux of angular momentum is related to gradients in the spin field by the spin viscosity for simple fluids. In fluids in which external torque per unit volume is applied, the equations of linear and angular momentum are coupled. In simple fluids, the equations are coupled through the vortex viscosity, which relates the transfer of angular to linear momentum to the difference in the local spin rate and one-half of the vorticity of the fluid. Apparent viscosity is normally measured, for example, in a pressure driven flow experiment, as a proportionality constant that relates the force acted on system and the volume flow rate of the system. The purpose of this study is to evaluate shear viscosity, vortex viscosity, and spin viscosity in suspensions by solving the coupled momentum equations and study their relationships to the apparent viscosity and energy dissipation.

In this study, we investigated the influence of particle spin imposed by external torque on the apparent viscosity of non-colloidal suspension by using a simulation based on the boundary element method (BEM). The numerical results reveal that particles spin has a pronounced effect on apparent viscosity of the suspension. For example, in suspension subjected to a pressure gradient in a tube, vary the spin gradient from positive to negative by exerting external torque on the particles, the apparent viscosity changes from close to zero to infinite with asymptotic change to apparent negative viscosity. Apparent "negative" viscosity is due to torque driven flow through vortex viscosity that is the proportionality constant between the transfer of the linear to angular momentum flux and the difference between the spin rate and the one-half local vorticity. Measurements of vortex viscosities from the numerical experiments agreed the theoretical prediction of vortex viscosity of dilute suspensions and extended the results to concentrated suspensions. The energy efficiency of torque driven flow is compared to the energy efficiency of force driven flow.

External torques are applied to neutrally buoyant particles in boundary element simulations to generate a cubic velocity profile in a suspension confined in a rectangular channel. The quantitative values of spin viscosity are determined as a function of solids fraction by calculating the volume averaged stress and kinematics of these flows and combining these results with a theoretical analysis of the coupled equations describing the conservation of linear and angular momentum. Many configurations of particles at each concentration are generated and analyzed until reproducible averages of the spin viscosity at each solids fraction are determined. The scaling of the spin viscosity with the square of the particle size predicted in earlier theoretical studies is verified in this investigation.

The conservation equation describing the rate of energy dissipation in suspensions is derived, and the contributions due to spin viscosity are included. The energy dissipation rate is shown to consist of the linear combination of terms that include the shear viscosity, vortex viscosity and spin viscosity. Predictions from these equations using the coefficients determined in the this work are compared with the energy dissipation rates obtained from a macroscopic balance using the force and velocity on the boundaries of the suspension in numerical simulations of a number of different flow fields. Over the range of our data, these independent predictions are shown to be in excellent agreement for both dilute and concentrated suspensions.