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. In particular we derive the well known “lubrication paradox” and we show that such a paradox is not due to the divergence of the stress at the yield surface, but actually to the constitutive structure of the Bingham model, since we show that the relevant physical quantities (e.g. stress, velocity, shear rate, etc.) are well defined and bounded. Exploiting a limit procedure in which the Bingham model is retrieved from a linear bi-viscous model we eventually prove that the inconsistency is due to the hypothesis of perfect stiffness of the unyielded part, showing that the Bingham model may not be adequate to describe the lubrication motion over a non-flat surface.