Abstract:In this paper we study a two-phase one-dimensional free boundary problem for parabolic equation, arising from a mathematical model for Bingham-like fluids with visco-elastic core presented in [L. Fusi, A. Farina, A mathematical model for Bingham-like fluids with visco-elastic core, Z. Angew. Math. Phys. 55 (2004) 826-847]. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Local existence is prov… Show more
“…; see e.g. [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]) 616 L. FUSI, A. FARINA, AND F. ROSSO otherwise. We also refer the reader to [4], where interesting experimental investigations on these kinds of materials are reported.…”
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.
“…; see e.g. [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]) 616 L. FUSI, A. FARINA, AND F. ROSSO otherwise. We also refer the reader to [4], where interesting experimental investigations on these kinds of materials are reported.…”
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.