Lubrication flows between spherical particles colliding in a compressible non-continuum gas

Abstract

The low-Reynolds-number collision and rebound of two rigid spheres moving in an ideal isothermal gas is studied in the lubrication limit. The spheres are non-Brownian in nature with radii much larger than the mean-free path of the molecules. The nature of the flow in the gap between the particles depends on the relative magnitudes of the minimum gap thickness, h′o, the mean-free path of the bulk gas molecules, λo, and the gap thickness at which compressibility effects become important, hc. Both the compressible nature of the gas and the non-continuum nature of the flow in the gap are included and their effects are studied separately and in combination. The relative importance of these two effects is characterized by a dimensionless number, αo ≡ (hc/λo). Incorporation of these effects in the governing equations leads to a partial differential equation for the pressure in the gap as a function of time and radial position The dynamics of the collision depend on αo, the particle Stokes number, Sto, and the initial particle separation, h′o. While a continuum incompressible lubrication force applied at all separations would prevent particle contact, the inclusion of either non-continuum or compressible effects allows the particles to contact. The critical Stokes number for particles to make contact, St1, is determined and is found to have the form St1 = 2 [ln(h′o/l) + C(αo)], where C(αo) is an O(1) quantity and l is a characteristic length scale defined by l = hc(1 + αo)/αo. The total energy dissipated during the approach and rebound of two particles when Sto ≫ St1 is also determined in the event of perfectly elastic or inelastic solid-body collisions.