Microscopic foundation for the hydrodynamic description of flow near a macroparticle

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Abstract
The two-fluid mean-field theory of the preceding paper is applied to the special case of a macroscopic test particle immersed in a quasicontinuous background fluid. Here, as in the preceding investigation, the particle interactions are taken to be pair-additive sums of long-ranged continuous and short-ranged, rigid-core contributions. The kinetic equation for the conditional singlet distribution function of untagged solvent particles is recast in terms of a related smoothly varying function which satisfies a boundary condition at the interface where this fluid comes into contact with the test particle. This formulation of the theory is then used to examine the case when the test particle is much larger and more massive than a solvent particle and when the free path length of the background medium is significantly smaller than the test particle diameter. In this limit the moments of the background fluid kinetic equation reduce to the linearized Navier–Stokes equations for pure solvent and the velocity moment of the boundary condition becomes identical with the macroscopic slip-flow condition of hydrodynamics. From these, one can obtain the usual Stokes’ formula for the frictional force on the test particle. The theory also can be described by a kinetic equation for the background fluid which incorporates singular source terms arising from the rigid-core interactions between the solvent and the test particle. In the case of a large, very massive test particle the moments of this equation are found to be generalizations to high density of the ‘‘source hydrodynamic equations’’ of Cukier, Kapral, Lebenhaft, and Mehaffey.