On the mathematical paradoxes for the flow of a viscoplastic film down an inclined surface

In this paper we consider the motion of thin visco-plastic Bingham layer over an inclined surface whose profile is not flat. We assume that the ratio between the thickness and the length of the layer is small, so that the lubrication approach is suitable. Under specific hypotheses (e.g. creeping flow) we analyze two cases: finite tilt angle and small tilt angle. In both cases we prove that the physical model generates two mathematical problems which do not admit non-trivial solutions. We show that, though the relevant physical quantities (e.g. stress, velocity, shear rate, etc.) are well defined and bounded, the mathematical problem is inherently ill posed. In particular, exploiting a limit procedure in which the Bingham model is retrieved from a linear bi-viscous model we eventually prove that the underlying reason of the inconsistency has to be sought in the hypothesis of perfect stiffness of the unyielded part. We therefore conclude that: either the Bingham model is inappropriate to describe the lubrication motion over a non-flat surface, or the lubrication technique fails in approximating thin Bingham films.