Particle-Fluid-Particle Stress in Multiphase Flows

Multiphase flows with particles modeled using an Euler-Euler formulation often neglect the stress coming from the particle-fluid-particle (PFP) interactions. Consider an array of particles moving in otherwise quiescent inviscid fluid. For a potential uniform flow over a single sphere, the forces are zero due to symmetry, implying that the presence of the fluid has no effect on the particles. In reality, the PFP interactions will break this symmetry which leads to a non-zero force. This force is represented by a stress gradient term in the momentum equation which to our best knowledge has never been quantified and modeled. In this presentation, we discuss the definition and calculation of this stress.

We first present the calculation of the stress in the potential flow limit. For finite Reynolds numbers, the stress can be approximately calculated using the Pairwise Interaction Point Particle (PIEP) model (Akiki et al. JFM 2017) valid up to a moderate volume fractions of 0.2. To calculate the stress, we need to compute the average forces acting on particles conditional on the nearest particle location and then integrate over the relative pair locations over the space. The use of quantities conditional on the nearest particles ensures the convergence of the integral.