Representing the Drafting-kissing-tumbling Interaction by the Particle-fluid-particle Stress

Drafting-kissing-tumbling is a well observed and numerically studied interaction among particles in multiphase flows. However, it has not been represented in related models. This interaction is not only a particle-fluid interaction but a particle-fluid-particle (PFP) interaction, hence, cannot be simply represented by a particle-fluid interaction force vector in a model. In a particle cloud, this PFP interaction changes relative particle positions and the shape of the cloud, which should be described by a tensor. In terms of dynamics, the tensor is a stress.  To define a stress for the PFP interaction, conventional methods encounter divergent integrals, because of the long-range hydrodynamic interactions among particles. 

 

To avoid this mathematical difficulty, in this work we use the nearest particle statistics to perform ensemble averages. We first show how the nearest particle statistics correctly considers the interaction among all the particles, not only the nearest pair. Using this approach, a PFP stress is defined, and numerically calculated.  We then show this stress represents the drafting-kissing-tumbling interactions among the particles.