Non-continuum lubrication flows between particles colliding in a gas

<jats:p>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 λ<jats:sub>0</jats:sub> of the gas, the discrete molecular nature of the gas becomes important. For particles of radii <jats:italic>a</jats:italic> smaller than about 50 μm 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.</jats:p><jats:p>The nature of the flow in the gap depends on the relative magnitudes of the minimum gap thickness <jats:italic>h</jats:italic><jats:sub>0</jats:sub> ≡ <jats:italic>a</jats:italic>ε, the mean-free path λ<jats:sub>0</jats:sub>, and the distance <jats:italic>a</jats:italic>ε<jats:sup>1/2</jats:sup> over which the effects of curvature become important. The slip-flow regime, <jats:italic>a</jats:italic>[Gt ]λ<jats:sub>0</jats:sub>, was analysed by Hocking (1973) using the Maxwell slip boundary condition at the particle surface. To find the lubrication force in the transition regime (<jats:italic>a</jats:italic>ε ∼ O(λ<jats:sub>0</jats:sub>)), we use the results of Cercignani & Daneri (1963) for the flux as a function of the pressure gradient in a Poiseuille channel flow. When <jats:italic>a</jats:italic>ε[Lt ]λ<jats:sub>0</jats:sub>[Lt ]<jats:italic>a</jats:italic>ε<jats:sup>1/2</jats:sup>, 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 <jats:italic>h</jats:italic><jats:sub>0</jats:sub>c log(λ<jats:sub>0</jats:sub>/<jats:italic>h</jats:italic><jats:sub>0</jats:sub>), where c 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 6πμ<jats:italic>U</jats:italic><jats:sub>0</jats:sub><jats:italic>a</jats:italic><jats:sup>2</jats:sup>(log <jats:italic>h</jats:italic><jats:sub>0</jats:sub>/λ<jats:sub>0</jats:sub> – 1.28), where 2<jats:italic>U</jats:italic><jats:sub>0</jats:sub> is the relative velocity of the particles.</jats:p><jats:p>When λ<jats:sub>0</jats:sub>[Gt ]<jats:italic>a</jats:italic>ε<jats:sup>1/2</jats:sup>, 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.</jats:p>