A multi-layer integral model for locally-heated thin film flow

Kay, E.D.; Hibberd, Stephen; Power, H.

Abstract

Based on an approach used to model environmental flows such as rivers and estuaries, we develop a new multi-layered model for thin liquid film flow on a locally-heated inclined plane. The film is segmented into layers of equal thickness with the velocity and temperature of each governed by a momentum and energy equation integrated across each layer individually. Matching conditions applied between the layers ensure the continuity of down-plane velocity, temperature, stress and heat flux. Variation in surface tension of the liquid with temperature is considered so that local heating induces a surface shear stress which leads to variation in the film height profile (the Marangoni effect). Moderate inertia and heat convection effects are also included.

In the absence of Marangoni effects, when the film height is uniform, we test the accuracy of the model by comparing it against a solution of the full heat equation using finite differences. The multi-layer model offers significant improvements over that of a single layer. Notably, with a sufficient number of layers, the solution does not exhibit local regions of negative temperature often predicted using a single-layer model.

With Marangoni effects included the film height varies however we find heat convection can mitigate this variation by reducing the surface temperature gradient and hence the surface shear stress. Numerical results corresponding to the flow of water on a vertical plane show that very thin films are dominated by the Marangoni shear stress which can be sufficiently strong to overcome gravity leading to a recirculation in the velocity field. This effect reduces with increasing film thickness and the recirculation eventually disappears. In this case heating is confined entirely to the interior of the film leading to a uniform height profile.

Citation

Kay, E., Hibberd, S., & Power, H. (2017). A multi-layer integral model for locally-heated thin film flow. Journal of Computational Physics, 336, 51-68. https://doi.org/10.1016/j.jcp.2017.01.066

• Journal Article Type: Article
• Acceptance Date: Jan 29, 2017
• Online Publication Date: Feb 1, 2017
• Publication Date: May 1, 2017
• Deposit Date: Mar 6, 2017
• Publicly Available Date: Mar 6, 2017
• Journal: Journal of Computational Physics
• Print ISSN: 0021-9991
• Electronic ISSN: 1090-2716
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 336
• Pages: 51-68
• DOI: https://doi.org/10.1016/j.jcp.2017.01.066
• Keywords: Thin film flow; Marangoni effect; Layered model; Integral model; Moderate Reynolds number
• Public URL: https://nottingham-repository.worktribe.com/output/858798
• Publisher URL: https://doi.org/10.1016/j.jcp.2017.01.066
• Additional Information: This article is maintained by: Elsevier; Article Title: A multi-layer integral model for locally-heated thin film flow; Journal Title: Journal of Computational Physics; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.jcp.2017.01.066; Content Type: article; Copyright: © 2017 Elsevier Inc. All rights reserved.
• Contract Date: Mar 6, 2017