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

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(0)' the mean-free path of the bulk gas molecules, lambda(0), 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, alpha(0) = (h(c)/lambda(0)). 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 alpha(0), the particle Stokes number, St(0), and the initial particle separation, h(0)'. 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, St(1), is determined and is found to have the form St(1) = 2 [1n(h(0)'/l) + C(alpha(0))], where C(alpha(0)) is an O(1) quantity and I is a characteristic length scale defined by l = h(c)(1 + alpha(0))/alpha(0). The total energy dissipated during the approach and rebound of two particles when St(0) much greater than St(1) is also determined in the event of perfectly elastic or inelastic solid-body collisions.