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, λ_o, and the gap thickness at which compressibility effects become important, h_c. 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, α_o≡ (h_c/λ_o). 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 α_o, the particle Stokes number, St_o, 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(α_o)], where C(α_o) is an O(1) quantity and l is a characteristic length scale defined by l≡ h_c(1+α_o)/ α_o. The total energy dissipated during the approach and rebound of two particles when St_o[Gt ]St₁ is also determined in the event of perfectly elastic or inelastic solid-body collisions.