Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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


Authors: L. Fusi, A. Farina and F. Rosso
Journal: Quart. Appl. Math. 73 (2015), 615-627
MSC (2010): Primary 76A05, 74D10, 76D08
DOI: https://doi.org/10.1090/qam/1391
Published electronically: September 15, 2015
MathSciNet review: 3432275
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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.


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Additional Information

L. Fusi
Affiliation: Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy
Email: fusi@math.unifi.it

A. Farina
Affiliation: Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy
Email: farina@math.unifi.it

F. Rosso
Affiliation: Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy
Email: rosso@math.unifi.it

DOI: https://doi.org/10.1090/qam/1391
Received by editor(s): December 13, 2013
Published electronically: September 15, 2015
Article copyright: © Copyright 2015 Brown University

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