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

• Published in: Fluids (Basel) 2019-06, Vol.4 (2), p.82
• Main Authors: Scholle, Markus, Gaskell, Philip H., Marner, Florian
• Format: Article
• Language: English
• Subjects: analytic solutions; Approximation; Boundary conditions; Dirichlet problem; FE solutions; film flows; first integrals; Free surfaces; Inclination angle; Integral equations; Kinematics; lubrication theory; Numerical prediction; Potential fields; Reynolds number; Substrates; Topography; Velocity
• ISSN: 2311-5521; 2311-5521
• DOI: 10.3390/fluids4020082

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.