Dispersion and particle pressure in sedimenting suspensions with hydrodynamic interactions and particle inertia

Particle inertia is a feature of suspension dynamics that is often neglected. However, its neglect changes the order of the governing equations of particle motion. In this work, we examine the impact of the inertia of particles over their velocity fluctuations and the resulting stresses. In order to observe effects of particle inertia, we consider dusty-gas suspensions of spherical weakly Brownian microparticles sedimenting in a Newtonian fluid at low Reynolds numbers. We first investigate the motion of a single spherical particle. The simulations for this simple case allow us to define the appropriate physical parameters of the flow and to validate the numerical calculation of velocity statistics over a range of Peclet and Stokes numbers. Next, we perform Langevin dynamics simulations with periodic boundary conditions to integrate the equations governing the translational and rotational motions of N hydrodynamically interacting particles suspended in the viscous fluid. The first aim of this paper is to examine the behavior of the velocity variance of inertial particles, with small fluctuations produced by Brownian motion at the moderate Peclet numbers. The interplay between mild Brownian forces and hydrodynamic interactions decreases the anisotropy of velocity fluctuations observed in non-Brownian suspensions (Pe,1) of sedimenting particles. The simulation results suggest that particle inertia attenuates the magnitude of velocity fluctuations produced by both Brownian motion and viscous hydrodynamic interactions. Additionally, we study the long-time behavior of the velocity fluctuations by calculating their autocorrelation functions and the particle diffusivities. The numerical simulations show clear evidence of particle pressure arising from the flow disturbance produced by hydrodynamic interactions in a suspension. From the particle velocity variance obtained in the present numerical simulations, we propose a simple model for particle-phase pressure as a linear function of the particle volume fraction phi <= 5%, which is usually a closure quantity required in models of more complex particulate systems such as fluidized beds.