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 $\boldsymbol {F}_{\!\!fp}$ that depends on gradients of the fluid density $\rho _f$ , fluid velocity $\boldsymbol{u}_{f}$ and fluid pressure $p_f$ . 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 ${\unicode{x1D64B}}_{\!pfp}$ can be found in closed form from the expression for $\boldsymbol {F}_{\!\!fp}$ derived in Fox (2019, J. Fluid Mech. 877 , 282). Remarkably, using this expression for ${\unicode{x1D64B}}_{\!pfp}$ ensures that the two-fluid model for ideal-fluid–particle flow is well posed for all fluid-to-particle material-density ratios $Z = \rho _f / \rho _p$ .