Derivation of the Reynolds type equation with memory effects, governing transient flow of lubric

In this paper we derive the Reynolds type lubrication equation with memory effects for lubrication of a rotating shaft. Starting from the nonstationary Stokes equations and using the asymptotic expansion we obtain a law that is nonlocal in time and gives a linear connection between velocity and pressure gradient. Our law tends to the classical Reynolds equation when time tends to infinity. We prove a convergence theorem for velocity and pressure in appropriate functional space