The particle–fluid–particle pressure tensor for ideal-fluid–particle flow

Starting from the coupled Boltzmann–Enskog (BE) kinetic equations for a two-particle system consisting of hard spheres, a hyperbolic two-fluid model for binary, hard-sphere mixtures was derived in Fox (2019, J. Fluid Mech.877, 282). In addition to spatial transport, the BE kinetic equations account for particle–particle collisions, using an elastic hard-sphere collision model, and the Archimedes (buoyancy) force due to spatial gradients of the pressure in each phase, as well as other forces involving spatial gradients. The ideal-fluid–particle limit of this model is found by letting one of the particle diameters go to zero while the other remains finite. The resulting two-fluid model has closed terms for the spatial fluxes and momentum exchange due to the excluded volume occupied by the particles, e.g. a momentum-exchange term F\!\!fp that depends on gradients of the fluid density ρ f , fluid velocity uf and fluid pressure pf . In Zhang et al. (2006, Phy. Rev. Lett.97, 048301), the corresponding unclosed momentum-exchange term depends on the divergence of an unknown particle–fluid–particle (pfp) stress (or pressure) tensor. Here, it is shown that the pfp-pressure tensor x1D64B\!pfp can be found in closed form from the expression for F\!\!fp derived in Fox (2019, J. Fluid Mech.877, 282). Remarkably, using this expression for x1D64B\!pfp ensures that the two-fluid model for ideal-fluid–particle flow is well posed for all fluid-to-particle material-density ratios Z = ρ f / ρ ₚ .