Non-continuum lubrication flows between particles colliding in a gas

Solid-body collisions between smooth particles in a gas would not occur if the lubrication force for a continuum incompressible fluid were to hold at all particle separations. When the gap between the particles is of the order of the mean free path lambda(0) of the gas, the discrete molecular nature of the gas becomes important. For particles of radii a smaller than about 50 mu m colliding in air at a relative velocity comparable to their terminal velocity, the effects of compressibility of the gas in the gap are not important. The nature of the flow in the gap depends on the relative magnitudes of the minimum gap thickness h(0) = a epsilon, the mean-free path lambda(0), and the distance a epsilon(1/2) over which the effects of curvature become important. The slip-flow regime, a epsilon much greater than lambda(0), was analysed by Hocking (1973) using the Maxwell slip boundary condition at the particle surface. To find the lubrication force in the transition regime (a epsilon similar to O(lambda(0))), we use the results of Cercignani & Daneri (1963) for the flux as a function of the pressure gradient in a Poiseuille channel flow. When a epsilon much less than lambda(0) much less than a epsilon(1/2), one might expect the local flow in the gap to be governed by Knudsen diffusion. However, an attempt to calculate the Knudsen diffusivity between parallel plates leads to a logarithmic divergence, which is cut off by intermolecular collisions, and the flux is therefore proportional to <h(0)(c)over bar> log(lambda(0)/h(0)), where (c) over bar is the mean molecular speed. The non-continuum lubrication force is shown to have a weak, log-log divergence as the particle separation goes to zero. As a result, the energy dissipated in the collision is finite. In the limit of large particle inertia, the energy dissipated is 6 pi mu U(0)a(2) (log h(0)/lambda(0) - 1.28), where 2U(0) is the relative velocity of the particles. When lambda(0) much greater than a epsilon(1/2), we have a free molecular flow in the gap. In this case, owing to the curvature of the particles, the flux versus pressure gradient relation is non-local. We analyse the free molecular flow between two cylinders and obtain scalings for the lubrication force.