In two-fluid models of multiphase flow the dispersed phase is treated as a continuum and here we explore the consequences of the dispersed phase being in equilibrium with the surrounding flow. Under equilibrium the particle velocity field can be expressed as an explicit expansion in the surrounding fluid velocity. To the leading order the particle velocity is the same as the local fluid velocity and the first order correction will be shown to capture important physics such as preferential accumulation of particles and turbuphoretic particle migration. To verify the equilibrium expansion, we have performed direct numerical simulations of particles and bubbles in the canonical problems of isotropic turbulence and a turbulent channel flow, where “true” particles moved according to their Lagrangian equations of motion are compared with “test” particles moved according to the equilibrium particle velocity. For small particles the equilibrium expansion is shown to converge rapidly. The equilibrium concept will be extended to the rotational motion and thermal field of the particle. The equilibrium approximation for the particle provides a clean mechanism for two-way coupling and a rigorous set of equations for the description of multi-phase flow turbulence, in the limit of a dilute dispersion of small particles.

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