Brownian Motion in Multiphase Flow

Abstract

A previous phenomenological theory for Brownian motion of rigid spherical particles in a flowing fluid (Ramshaw, 1979, 1981) is extended to multiphase mixtures and arbitrary flow regimes. It is argued that each phase i possesses its own intrinsic osmotic pressure q i =n i k B T, where n i is the mean number of discrete particles (i.e., inclusions, fragments, blobs, or chunks) of phase i per unit total volume, regardless of their rigidity or their size and shape distributions. The gradient of q i appears as an additional force term in the momentum equation for phase i. The osmotic pressures q i also contribute to the total pressure p of the mixture, so these contributions must be subtracted out before the conventional multiphase pressure forces are computed. The resulting pressure terms in the momentum equation for phase i then become −α i ν(p−q)−νq i , where α i is the volume fraction of phase i and q=∑ i q i . This formulation provides a single unified description of the flow of both multiphase mixtures and multicomponent gases, and exhibits a smooth transition between these two limiting cases as the particle sizes vary from macroscopic to molecular dimensions. The stability properties of the equations are examined in the incompressible limit, and it is found that the Brownian motion stabilizes and regularizes the system only for microscopically small relative velocities.