Boundary conditions at a liquid/air interface in lubrication flows

Abstract

A difficulty in applying the lubrication approximation to flows where a liquid/air interface forms lies in supplying boundary conditions at the point of formation of the interface that are consistent with the lubrication approximation. The method of matched asymptotic expansions is applied to the flow between partially submerged, counter-rotating rollers, a representative problem from this class, and the lubrication approximation is found to generate the first term of an outer expansion of the problem solution. The first term of an inner expansion describes the two-dimensional flow in the vicinity of the interface, and approximate results are found by the finite-element method. Matching between the inner and outer solutions determines boundary conditions on the pressure and the pressure gradient at the point of formation of the interface which allow the solution to the outer, lubrication flow to be completed.