Averaged Reynolds equation for flows between rough surfaces in sliding motion

The flow between rough surfaces in sliding motion with contacts between these surfaces, is analyzed through the volume averaging method. Assuming a Reynolds (lubrication) approximation at the roughness scale, an average flow model is obtained combining spatial and time average. Time average, which is often omitted in previous works, is specially discussed. It is shown that the effective transport coefficients, traditionally termed 'flow factors' in the lubrication literature, that appear in the average equations can be obtained from the solution to two closure problems. This allows for the numerical determination of flow factors on firmer bases and sheds light on some arguments to the literature. Moreover, fluid flows through fractures form an important subset of problems embodied in the present analysis, for which macroscopisation is given.