Lubrication approximation of flows of a special class of non-Newtonian fluids defined by rate type constitutive equations

We consider the flow of a rate-type fluid defined by an implicit constitutive equation in a channel with non flat walls. We also assume that the channel characteristic width is small in comparison to the channel length, so that the lubrication approximation can be applied. The model developed is mainly motivated by the evidence that many lubricants seem to be well described by implicit rate type fluid models. The mathematical problem is reduced to an integro-differential equation for the pressure that is solved numerically for several channel profiles.