Dynamics and hydrodynamics of suspensions of translational–rotational Brownian particles at finite concentrations

Abstract

We consider the hydrodynamics of a suspension of rigid particles in a continuum fluid. The motion of the fluid and particles is coupled by the introduction of hydrodynamic boundary conditions. The dynamics of the particles is governed by rigid body translational and rotational classical equations which are derived for the particles in the suspension at finite concentrations. The appropriate random forces are included, making these equations the Langevin equations for translational and rotational Brownian motion for particles in suspensions at finite concentration. For no‐slip boundary conditions, the suspension velocity field is first derived to lowest order in the concentration, and the results are applied to spheres to evaluate the suspension zero frequency shear viscosity as η =η₀(1+hv_sc) where η₀ is the pure fluid viscosity and v_sc is the volume fraction of the spheres. The constant h equals 5/2 when both translational and rotational Brownian motion is allowed; h=4 if rotations are suppressed; while previous work on stationary suspensions yields h=5/2. A general effective medium theory is introduced to handle suspensions at finite concentrations. A series expansion provides a systematic method for evaluating the corrections to this effective medium theory which arise from correlated disturbances of the fluid velocity field by two or more suspended particles. The zero frequency shear viscosity and friction coefficients for spheres are obtained from the effective medium theory as η =η₀(1−hcv_s)⁻¹ and ζ=ζ₀(1−hcv_s)⁻¹, respectively, where ζ₀ is the single sphere translational (or rotational) friction coefficient. In the static limit hydrodynamic screening vanishes to all orders in concentration for rigid Brownian spheres.