A non-Reynolds lubrication model and application to droplet levitation

An attempt for extending the applicability of the Reynolds lubrication theory is presented. By considering a larger surface-to-surface distance than that for the Reynolds lubrication theory, the effect of the non-negligible pressure gradient in the surface-normal direction is incorporated into the lubrication model. The analysis shows that the local pressure is separated into (i) a base component satisfying the Reynolds lubrication theory and (ii) an adjusting component varying in the surface-normal direction, and the second component is found to be related with the velocity of the local Couette-Poiseuille flow. The lubrication model is verified in a non-Reynolds regime of a flow between a moving curved object and stationary object, and good agreement in the pressure distributions by analytical and numerical methods is observed in both wall-normal and longitudinal directions. The lubrication model is also applied to a droplet levitation problem over a moving wall (Sawaguchi 2019) involving a $\mu$m-thin air film, and the levitation lift reproduced from the full drop profile is found to provide a more accurate and complete view of the levitation problem.