Non-continuum tangential lubrication gas flow between two spheres

As two particles approach each other, the continuum lubrication force diverges, with decreasing separation preventing contact. However, for separations comparable to the mean free path of the gas, . These functions, in combination with the result by Sundararajakumar & Koch (J. Fluid Mech., vol. 313, 1996, pp. 238–308) and the classical work by Jeffrey & Onishi (J. Fluid Mech., vol. 139, 1984, pp. 261–290), yield a complete formulation for the hydrodynamic interactions of two spheres at all separations, from non-interacting spheres in the extreme far field through all the transitions that occur up to contact. We apply the new formulation to the classical case of a particle settling parallel to a vertical wall. The continuum Stokes equation predicts a settling speed that decreases with decreasing gap separation and vanishes at contact, whereas the non-continuum model developed herein predicts a finite settling speed at contact.