Modelling gravity-driven film flow on inclined corrugated substrate using a high fidelity weighted residual integral boundary-layer method

Veremieiev, Sergii; Wacks, Daniel H.

doi:10.1063/1.5063013

Cited by 19 publications

(9 citation statements)

References 46 publications

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“…Although available information on finite Reynolds number behavior of films over substrates with topographical features is well‐studied, 1,53,78 in the present study, an attempt has been made to revisit the problem and address it using a consistent formulation and obtaining the solution using a meshless RBF numerical scheme.…”

Section: Discussionmentioning

confidence: 99%

A consistent energy integral model for a film over a substrate featuring topographies

Pal

Sanyasiraju

Usha

2021

Numerical Methods in Fluids

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A depth‐averaged model, based on energy integral method (EIM) has been employed to capture the steady‐state free surface profiles in a gravity‐driven thin film flow of a viscous, incompressible fluid over inclined substrates with topographical features. The model incorporates inertial effects, energy balance and is consistent up to O(ε), where ε≪1, is the film thickness parameter. The governing equations are solved using a meshless numerical scheme based on radial basis functions. The predictions of the effects of inertia, inclination of the substrate, and aspect ratio of the topographical features are compared with the available finite‐element solutions and experimental observations. The steady‐state profiles generated using the EIM equations are in close agreement with the experimental observations. Being a consistent model, EIM model predictions are expected to be more reliable and accurate than the results based on the depth averaged form of the momentum equations, which are known to be not consistent. This feature is the motivating factor for the present work which is also a first attempt in implementing the meshless numerical scheme to free surface problems. The analysis paves a way for similar investigations on three‐dimensional flows related to film flow over chemically patterned substrates.

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Section: Discussionmentioning

confidence: 99%

A consistent energy integral model for a film over a substrate featuring topographies

Pal

Sanyasiraju

Usha

2021

Numerical Methods in Fluids

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“…. As in the isothermal flow case (Veremieiev & Wacks 2019), but without formally establishing the necessity of a parabolic temperature profile, (2.39) can be arrived at using weighted residuals. The procedure for this being: (i) higher-order terms involving are approximated by the leading expansions (2.32); (ii) the double derivatives with respect to are evaluated using the complete expansions (2.17) with given by (2.30); (iii) weighting of the momentum equation (2.13 b ) by the Nusselt parabolic velocity profile, , and of the energy equation (2.13 c ) by the Nusselt linear temperature profile, ; and finally (iv) integrating the equations between and .…”

Section: Problem Formulation and Modelling Approachmentioning

confidence: 99%

“…Moderate substrate corrugations have been discovered to delay the onset of surface instabilities when compared with their planar counterpart (Wierschem & Aksel 2003;Wierschem, Lepski & Aksel 2005), with subsequent investigations (Oron & Henining 2008;D'Alessio, Pascal & Jasmine 2009;Cao, Vlachogiannis & Bontozoglou 2013;Pollak & Aksel 2013) revealing a rich variety of stability phenomena including: short-wave transitions and isles of stability (Trifonov 2014); the combined effect of corrugation amplitude and wavelength on flow stabilisation (Schörner, Reck & Aksel 2015, 2016; and culminating in the identification of six characteristic stability regimes for film flow over topography . Recently, Veremieiev & Wacks (2019) have shown the qualitative stability behaviour caused by substrate topography can be captured by a WIBL model.…”

Section: Introductionmentioning

confidence: 99%

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Gravity-driven film flow down a uniformly heated smoothly corrugated rigid substrate

2021

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Gravity induced film flow over a rigid smoothly corrugated substrate heated uniformly from below, is explored. This is achieved by reducing the governing equations of motion and energy conservation to a manageable form within the mathematical framework of the well-known long-wave approximation; leading to an asymptotic model of reduced dimensionality. A key feature of the approach and to solving the problem of interest, is proof that the leading approximation of the temperature field inside the film must be nonlinear to accurately resolve the thermodynamics beyond the trivial case of ‘a flat film flowing down a planar uniformly heated incline.’ Superior predictions are obtained compared with earlier work and reinforced via a series of corresponding solutions to the full governing equations using a purpose written finite element analogue, enabling comparisons to be made between free-surface disturbance and temperature predictions, as well as the streamline pattern and temperature contours inside the film. In particular, the free-surface temperature is captured extremely well at moderate Prandtl numbers. The stability characteristics of the problem are examined using Floquet theory, with the interaction between the substrate topography and thermo-capillary instability modes investigated as a set of neutral stability curves. Although there are no relevant experimental data currently available for the heated film problem, recent existing predictions and experimental data concerning the behaviour of corresponding isothermal flow cases are taken as a reference point from which to explore the effect of both heating and cooling.

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“…Complementary stability analyses have become increasingly available [28][29][30], revealing interesting features concerning the effect of periodically repeating topography on the stability of films, showing the critical Reynolds number signalling the onset of instability, compared to the corresponding value predicted for film flow on totally planar substrate, is increased. Resulting even in the re-establishment of islands of stability as the Reynolds number is increased further still-in this respect the reader is directed to the following recent contributions [31][32][33][34][35] In connection with the issue of minimising/controlling the free surface disturbance associated with film flow over topography, others have considered the inverse problem of substrate design based on information concerning the free surface, [36][37][38], non-Newtonian effects [39] and the the influence of electric fields [40,41].…”

Section: Introductionmentioning

confidence: 99%

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

Scholle

Gaskell

Marner

2019

Fluids

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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.

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Modelling gravity-driven film flow on inclined corrugated substrate using a high fidelity weighted residual integral boundary-layer method

Cited by 19 publications

References 46 publications

A consistent energy integral model for a film over a substrate featuring topographies

A consistent energy integral model for a film over a substrate featuring topographies

Gravity-driven film flow down a uniformly heated smoothly corrugated rigid substrate

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

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