Phase interaction force model for nonuniform particle laden flows

Many models for phase interaction forces, such as drag, are developed in statistically homogeneous flows, while most multiphase flows in nature or in engineering applications are inhomogeneous. This work studies the corrections to the drag model considering effect of statistical inhomogeneity. Let L be the macroscopic length scale. If there is any effect of the inhomogeneity on the phase interaction force, the force should be a function of L, implying that force model cannot be developed by only studying the average forces in homogeneous flows, since such flows contain no information about L.

To address this issue, we note that the gradient operator ∇ on an average quantity is of order 1/L. Accurate to the first order of the length scale ratio between the inter-particle distance d and L, the corrections to phase interaction force density can be expressed as this operator acting on quantities proportional to the length scale d. The dimension of the quantities is the dimension of stress. It turns out there are two such stresses. They are the particle–fluid–particle stress and the diffusion stress. The phase interaction force density is then decomposed into three terms, a force density in a homogeneous flow, and two terms related to the stresses. These terms can be calculated and studied in statistically homogeneous flows.