Lubrication approximation for micro-particles moving along parallel   walls
  Lubrication expressions for the friction coefficients of a spherical particle moving in a fluid between and along two parallel solid walls are explicitly evaluated in the low-Reynolds-number regime. They are used to determine lubrication expression for the particle free motion under an ambient Poiseuille flow. The range of validity and the accuracy of the lubrication approximation is determined by comparing with the corresponding results of the accurate multipole procedure. The results are applicable for thin, wide and long microchannels, or quasi-two-dimensional systems. 
M. L. Ekiel-Jezewska, E. Wajnryb, J. Blawzdziewicz and F. Feuillebois