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On travelling wave solutions of a model of a liquid film flowing down a fibre

Part of: Equations of mathematical physics and other areas of application Parabolic equations and systems Incompressible viscous fluids Foundations, constitutive equations, rheology

Published online by Cambridge University Press:  12 August 2021

HANGJIE JI Open the ORCID record for HANGJIE JI [Opens in a new window] ,
ROMAN TARANETS  and
MARINA CHUGUNOVA
HANGJIE JI
Affiliation:
Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095, USA email: hangjie@math.ucla.edu
ROMAN TARANETS
Affiliation:
Institute of Applied Mathematics and Mechanics of the NASU, Dobrovol’s’koho Str. 1, Sloviansk 84100, Ukraine email: taranets_r@yahoo.com
MARINA CHUGUNOVA
Affiliation:
Claremont Graduate University, 150 E. 10th Street, Claremont, CA 91711, USA email: marina.chugunova@cgu.edu
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Abstract

Existence of non-negative weak solutions is shown for a full curvature thin-film model of a liquid thin film flowing down a vertical fibre. The proof is based on the application of a priori estimates derived for energy-entropy functionals. Long-time behaviour of these weak solutions is analysed and, under some additional constraints for the model parameters and initial values, convergence towards a travelling wave solution is obtained. Numerical studies of energy minimisers and travelling waves are presented to illustrate analytical results.

Keywords

Travelling wavesthin film equationfourth-order parabolic partial differential equationsexistence of weak solutions

MSC classification

Primary: 35K35: Initial-boundary value problems for higher-order parabolic equations
Secondary: 35K55: Nonlinear parabolic equations 35K65: Degenerate parabolic equations 76A20: Thin fluid films 35Q35: PDEs in connection with fluid mechanics 76D08: Lubrication theory
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 5 , October 2022 , pp. 864 - 893
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Sadeghpour, A., Oroumiyeh, F., Zhu, Y., Ko, D. D., Ji, H., Bertozzi, A. L. & Ju, Y. S. (2021) Experimental study of a string-based counterflow wet electrostatic precipitator for collection of fine and ultrafine particles. J. Air Waste Manag. Assoc. 71(7), 851–865 doi: 10.1080/10962247.2020.1869627.CrossRefGoogle Scholar
Beretta, E., Bertsch, M. & Dal Passo, R. (1995) Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. Arch. Rational Mech. Anal. 129(2), 175–200.CrossRefGoogle Scholar
Bernis, F. & Friedman, A. (1990) Higher order nonlinear degenerate parabolic equations. J. Differ. Equations 83(1), 179–206.CrossRefGoogle Scholar
Chang, H.-C. & Demekhin, E. A. (1999) Mechanism for drop formation on a coated vertical fibre. J. Fluid Mech. 380, 233–255.CrossRefGoogle Scholar
Chou, K.-S. & Du, S.-Z. (2008) Estimates on the hausdorff dimension of the rupture set of a thin film. SIAM J. Math. Anal. 40(2), 790–823.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. (2006) On viscous beads flowing down a vertical fibre. J. Fluid Mech. 553, 85–105.CrossRefGoogle Scholar
de Gennes, P. G. (1985) Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.CrossRefGoogle Scholar
Eidel’man, S. D. (1969) Parabolic Systems, North Holland Publishing Company, Amsterdam, Netherlands.Google Scholar
Frenkel, A. (1992) Nonlinear theory of strongly undulating s flowing down vertical cylinders. EPL (Europhys. Lett.) 18(7), 583.CrossRefGoogle Scholar
Ji, H., Falcon, C., Sadeghpour, A., Zeng, Z., Ju, Y. & Bertozzi, A. (2019) Dynamics of thin liquid films on vertical cylindrical fibres. J. Fluid Mech. 865, 303–327.CrossRefGoogle Scholar
Ji, H. & Li, L. (2019) Numerical methods for thermally stressed shallow shell equations. J. Comput. Appl. Math. 362, 626–652.CrossRefGoogle Scholar
Ji, H., Sadeghpour, A., Ju, Y. & Bertozzi, A. (2020) Modelling film flows down a fibre influenced by nozzle geometry. J. Fluid Mech. 901, R6. doi: 10.1017/jfm.2020.605.CrossRefGoogle Scholar
Ji, H. & Witelski, T. P. (2018) Instability and dynamics of volatile thin films. Phys. Rev. Fluids 3(2), 024001.CrossRefGoogle Scholar
Ji, H. & Witelski, T. P. (2020) Steady states and dynamics of a thin-film-type equation with non-conserved mass. Eur. J. Appl. Math. 31(6), 968–1001.CrossRefGoogle Scholar
Kalliadasis, S. & Chang, H.-C. (1994) Drop formation during coating of vertical fibres. J. Fluid Mech. 261, 135–168.CrossRefGoogle Scholar
Keller, H. B. (1987) Lectures on numerical methods in bifurcation problems. Appl. Math. 217, 50.Google Scholar
Kliakhandler, I., Davis, S. & Bankoff, S. (2001) Viscous beads on vertical fibre. J. Fluid Mech. 429, 381–390.CrossRefGoogle Scholar
Marzuola, J. L., Swygert, S. R. & Taranets, R. (2020) Nonnegative weak solutions of thin-film equations related to viscous flows in cylindrical geometries. J. Evol. Equations 20, 1227–1249.CrossRefGoogle Scholar
Rosenau, P. & Oron, A. (1989) Evolution and breaking of liquid film flowing on a vertical cylinder. Phys. Fluids A Fluid Dyn. 1(11), 1763–1766.CrossRefGoogle Scholar
Ruyer-Quil, C., Treveleyan, P., Giorgiutti-Dauphiné, F., Duprat, C. & Kalliadasis, S. (2008) Modelling film flows down a fibre. J. Fluid Mech. 603, 431–462.CrossRefGoogle Scholar
Ruyer-Quil, C., Trevelyan, S. P. M. J., Giorgiutti-Dauphiné, F., Duprat, C. & Kalliadasis, S. (2009) Film flows down a fiber: modeling and influence of streamwise viscous diffusion. Eur. Phys. J. Spec. Top. 166(1), 89–92.CrossRefGoogle Scholar
Sadeghpour, A., Zeng, Z., Ji, H., Ebrahimi, N. D., Bertozzi, A. & Ju, Y. (2019) Water vapor capturing using an array of traveling liquid beads for desalination and water treatment. Sci. Adv. 5(4), eaav7662.CrossRefGoogle Scholar
Sadeghpour, A., Zeng, Z. & Ju, Y. S. (2017) Effects of nozzle geometry on the fluid dynamics of thin liquid films flowing down vertical strings in the Rayleigh-Plateau regime. Langmuir 33, 6292–6299.CrossRefGoogle ScholarPubMed
Shlang, T. & Sivashinsky, G. (1982) Irregular flow of a liquid film down a vertical column. J. de Physique 43(3), 459–466.CrossRefGoogle Scholar
Solonnikov, V. A. (1965) On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Matematicheskogo Instituta Imeni VA Steklova 83, 3–163.Google Scholar
Thiele, U. (2011) On the depinning of a drop of partially wetting liquid on a rotating cylinder. J. Fluid Mech. 671, 121–136.CrossRefGoogle Scholar
Trifonov, Y. Y. (1992) Steady-state traveling waves on the surface of a viscous liquid film falling down on vertical wires and tubes. AIChE J. 38(6), 821–834.CrossRefGoogle Scholar
Zeng, Z., Sadeghpour, A. & Ju, Y. S. (2018) Thermohydraulic characteristics of a multi-string direct-contact heat exchanger. Int. J. Heat Mass Transfer 126, 536–544.CrossRefGoogle Scholar
Zeng, Z., Sadeghpour, A. & Ju, Y. S. (2019) A highly effective multi-string humidifier with a low gas stream pressure drop for desalination. Desalination 449, 92–100.CrossRefGoogle Scholar