Particle-fluid interaction force in nonuniform particle laden flows*

Particle-fluid interaction force is of the primary importance in modeling particle laden flows. Most models for the force are developed from studies of uniform flows, while in most practical multiphase flows, the particle volumes are not uniform. In this work we study the effect of gradient of the particle volume fraction on the force. We find the particle-fluid interaction force can be decomposed into three terms. The first term is the average fluid force on a pair of nearest particles. Since this force component is the same on the nearest pair, it does not cause relative motion between the pair; and therefore, is not responsible for formation and deformation of particle clusters. In the second term we introduce a diffusion stress. The second term is the product of the diffusion stress and the volume fraction gradient.  This diffusion stress is similar to a tensor diffusion coefficient, but instead of producing a mass flux, it produces a force. The third term is the divergence of particle-fluid-particle (PFP) stress. Both the diffusion stress and PFP stress are results of the fluid mediated particle-particle interactions.  The diffusion stress is only important when there is a gradient in the particle volume fraction. The PFP stress can also be important in uniform particle distributions but with spatially varying velocity fields.  Definitions, properties, numerical results, and physical meanings of these terms will be presented.