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

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

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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