Hydrodynamic interactions between two equal spheres in a highly rarefied gas

This paper describes hydrodynamic interactions between two spherical partic les having equal radii, a, and translating with velocities U-1 and U-2 in a highly rarefied gas. The center-to-center distance between the two spheres is a chi. The gas is at rest far from the two particles. The spheres move with speeds that are much smaller than the mean thermal speed of the gas mo lecules so that the Mach number, M equivalent to max(U-1,U-2)/(c) over bar, characterizing the deviation from equilibrium is much less than one. Here (c) over bar is the mean thermal speed of the gas molecules. Gas molecules are assumed to be diffusively reflected from the particle surfaces. Our ana lysis is confined to the case where the particle Knudsen number is very lar ge, i.e., Kn(o)equivalent to lambda(o)/a -->infinity, lambda(o) being the m ean free path of the gas far from the two particles. We first study the fre e-molecular drag on the two sphere configuration for arbitrary translations of the spheres. For small Mach number, the general time-dependent, nonline ar problem may be approximated by a quasisteady, linear problem in which th e spheres are held fixed and molecules reflected from each sphere have a mo dified Maxwell-Boltzmann distribution of velocities. A standard integral eq uation formulation based on flux balances at the particle surfaces is then employed to calculate the drag force acting on the spheres. The results obt ained can be used as leading estimates for the forces acting on the spheres when Kn(o)much greater than 1 and 2 less than or equal to chi much less th an Kn(o). We then consider the case where the flow in the vicinity of each sphere is nearly free-molecular, but the flow in the O(a chi) space between the spheres is nearly continuum in nature. In this limit, the flow in the gap between the spheres is studied using the method of reflections. This ap proach can be used for arbitrary Kn(o) provided Kn(o)<chi < Kn(o) M-1. The leading correction to the drag force due to the hydrodynamic interactions b etween the spheres when Kn(o)much greater than 1 is obtained. In all cases studied, the temperature of the two spheres is assumed to be the same as th at of the surrounding gas. (C) 1999 American Institute of Physics. [S1070-6 631(99)02209-6].