Nonlinear higher-order boundary value problems describing thin viscous flows near edges
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.
scientific article; zbMATH DE number 5299467
scientific article