Open AccessIll posedness of Bingham-type models for the downhill flow of a thin film on an inclined plane
Author(s) -
Lorenzo Fusi,
Angiolo Farína,
Fábio Rosso
Publication year - 2015
Publication title -
quarterly of applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.603
H-Index - 41
eISSN - 1552-4485
pISSN - 0033-569X
DOI - 10.1090/qam/1391
Subject(s) - lubrication , bingham plastic , inclined plane , mechanics , plane (geometry) , shear (geology) , dissipation , type (biology) , lubrication theory , shear stress , bounded function , mathematics , materials science , geometry , mathematical analysis , classical mechanics , physics , rheology , composite material , thermodynamics , geology , paleontology , quantum mechanics
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.