Ill posedness of Bingham-type models for the downhill flow of a thin film on an inclined plane

Abstract

In this paper we consider the flow of a thin layer of a Bingham-type material over an inclined plane with “small” tilt angle. A Bingham-type continuum is a material which behaves as a viscous fluid above a certain threshold (tied to the shear stress) and as a solid below such a threshold. We consider creeping flow and that the ratio between the thickness and the length of the layer is small, so that the lubrication approach is suitable. The unknowns of the model are the layer thickness, the position of the yield surface and the position of the advancing front. We first show that, though diverging in a neighborhood of the wetting front, the shear stress is integrable so that total dissipation is bounded. We then prove that the mathematical problem is inherently ill posed independently on the constitutive model selected for the solid domain. We therefore conclude that either the Bingham-type models are inappropriate to describe the thin film motion on an inclined surface or the lubrication technique fails in approximating such flows.

The dam-break problem for eroding viscoplastic fluids

“…The experimental results were compared with the predictions from lubrication theory. Within this theory, the momentum balance equations can be simplified by neglecting inertia terms and making use of the flow's shallowness [23][24][25][26][27][28][29][30].…”

“…Making use of the constitutive equation then leads to the velocity profile and, finally, the depth-averaged mass conservation provides the evolution equation for the flow depth h(x, t). There is a large body of work applying this theory to viscoplastic flows [23,26,27,30,51]; it is succinctly summarised in the next section.…”

“…The viscoplastic dam-break problem is a typical example of time-dependent flow, in which the flow passes through different phases from release to runout. This problem has been studied within the frameworks of the Saint-Venant equations [19][20][21][22] and lubrication theory [23][24][25][26][27][28][29][30]. Based on the assumptions that the flow is shallow (i.e.…”

Creeping flow of a Herschel–Bulkley fluid with pressure-dependent material moduli

“…The momentum equation in the unyielded part cannot be derived from the classical local differential formulation (2.3), since in a visco-plastic fluid, the stress is not defined below the yield limit. This is a tricky issue which we have addressed recently in a series of paper, [6][7][8][9] regarding the modelling of a Bingham fluid in lubrication approximation. Indeed, following [7], the integral formulation of the linear momentum balance in the plug is given by *…”