Nonlinear higher-order boundary value problems describing thin viscous flows near edges
type: Item
instance of: scholarly article
title: Nonlinear higher-order boundary value problems describing thin viscous flows near edges
zbMATH Open document ID: 1184.76618
author: Lorenzo Giacomelli
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-07-10T00:00:00Z
review text: The author analyzes two boundary value problems recently proposed in [\textit{C. I. Barenblatt, M. Bertsch} and \textit{L. Giacomelli}, Steady and quasi-steady thin viscous flows near the edge of a solid surface. Preprint No. 6/2007, Dipartimento Mo. Mo. Mat., submitted for publication]. Of particular interest is the flow behaviour near the edges of the surface. The first problem concerns steady states and consists of a third-order ordinary differential equation for the liquid height; the equation contains an unknown parameter, the flux, and the boundary conditions relate, near the edges of the substrate, the height and its second derivative to the flux itself. For this problem, non-existence and non-uniqueness results are proved depending on the behavior of the height and flux near the edge, as the flux approaches zero. The second problem examines quasi-steady states and consists of a fourth-order ordinary differential equation for the liquid height; nonlinear boundary conditions also relate the height to the flux near the edges. For this problem, the existence of a solution is proved for suitable height functions.
reviewed by: Elena Gavrilova
Mathematics Subject Classification ID: 76A20
Mathematics Subject Classification ID: 76D08
Mathematics Subject Classification ID: 35Q35
zbMATH DE Number: 5299467
zbMATH Keywords: lubrication theory
zbMATH Keywords: height function
zbMATH Keywords: non-existence
zbMATH Keywords: non-uniqueness
zbMATH Keywords: existence
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.04.038
OpenAlex ID: W2025526622
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DOI: 10.1016/J.JMAA.2008.04.038
description: scientific article; zbMATH DE number 5299467