Hydrodynamic interactions between two equal spheres in a highly rarefied gas

Abstract

This paper describes hydrodynamic interactions between two spherical particles having equal radii, a, and translating with velocities U1 and U2 in a highly rarefied gas. The center-to-center distance between the two spheres is aχ. 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 molecules so that the Mach number, M≡max(U1,U2)/c̄, characterizing the deviation from equilibrium is much less than one. Here c̄ is the mean thermal speed of the gas molecules. Gas molecules are assumed to be diffusively reflected from the particle surfaces. Our analysis is confined to the case where the particle Knudsen number is very large, i.e., Kno≡λo/a→∞, λo being the mean free path of the gas far from the two particles. We first study the free-molecular drag on the two sphere configuration for arbitrary translations of the spheres. For small Mach number, the general time-dependent, nonlinear problem may be approximated by a quasisteady, linear problem in which the spheres are held fixed and molecules reflected from each sphere have a modified Maxwell–Boltzmann distribution of velocities. A standard integral equation formulation based on flux balances at the particle surfaces is then employed to calculate the drag force acting on the spheres. The results obtained can be used as leading estimates for the forces acting on the spheres when Kno≫1 and 2⩽χ≪Kno. We then consider the case where the flow in the vicinity of each sphere is nearly free-molecular, but the flow in the O(aχ) 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 approach can be used for arbitrary Kno provided Kno≪χ≪Kno M−1. The leading correction to the drag force due to the hydrodynamic interactions between the spheres when Kno≫1 is obtained. In all cases studied, the temperature of the two spheres is assumed to be the same as that of the surrounding gas.