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Open AccessArticle

A Potential Field Description for Gravity-Driven Film Flow over Piece-Wise Planar Topography

1,*,†, 2,† and 1,2,†
1
Department of Mechatronics and Robotics, Heilbronn University, D-74081 Heilbronn, Germany
2
Department of Engineering, Durham University, Durham DH1 3LE, UK
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Fluids 2019, 4(2), 82; https://doi.org/10.3390/fluids4020082
Received: 7 April 2019 / Accepted: 16 April 2019 / Published: 2 May 2019
(This article belongs to the Special Issue Free surface flows)
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Abstract

Models based on a potential field description and corresponding first integral formulation, embodying a reduction of the associated dynamic boundary condition at a free surface to one of a standard Dirichlet-Neumann type, are used to explore the problem of continuous gravity-driven film flow down an inclined piece-wise planar substrate in the absence of inertia. Numerical solutions of the first integral equations are compared with analytical ones from a linearised form of a reduced equation set resulting from application of the long-wave approximation. The results obtained are shown to: (i) be in very close agreement with existing, comparable experimental data and complementary numerical predictions for isolated step-like topography available in the open literature; (ii) exhibit the same qualitative behaviour for a range of Capillary numbers and step heights/depths, becoming quantitively similar when both are small. A novel outcome of the formulation adopted is identification of an analytic criteria enabling a simple classification procedure for specifying the characteristic nature of the free surface disturbance formed; leading subsequently to the generation of a related, practically relevant, characteristic parameter map in terms of the substrate inclination angle and the Capillary number of the associated flow. View Full-Text
Keywords: film flows; topography; lubrication theory; analytic solutions; FE solutions; first integrals film flows; topography; lubrication theory; analytic solutions; FE solutions; first integrals
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Scholle, M.; Gaskell, P.H.; Marner, F. A Potential Field Description for Gravity-Driven Film Flow over Piece-Wise Planar Topography. Fluids 2019, 4, 82.

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