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Travelling wave solutions for a thin-film equation related to the spin-coating process

Published online by Cambridge University Press:  17 July 2017

M. V. GNANN ,
H. J. KIM  and
H. KNÜPFER
M. V. GNANN
Affiliation:
Center for Mathematics, Technical University of Munich, Boltzmannstr. 3, 85747 Garching near Munich, Germany email: gnann@ma.tum.de
H. J. KIM
Affiliation:
Institute of Applied Mathematics and IWR, University of Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany emails: kim.hyeonjeong@math.uni-heidelberg.de, knuepfer@uni-heidelberg.de
H. KNÜPFER
Affiliation:
Institute of Applied Mathematics and IWR, University of Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany emails: kim.hyeonjeong@math.uni-heidelberg.de, knuepfer@uni-heidelberg.de
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Abstract

We study a problem related to the spin-coating process in which a fluid coats a rotating surface. Our interest lies in the contact-line region for which we propose a simplified travelling wave approximation. We construct solutions to this problem by a shooting method that matches solution branches in the contact-line region and in the interior of the droplet. Furthermore, we prove uniqueness and qualitative properties of the solution connected to the fourth-order nature of the equation, such as a global maximum in the film height close to the contact line, elevated from the average height of the film.

Keywords

PDEs for fluid mechanicstravelling wave solutionsnon-linear fourth-order equationsexistence and uniquenessthin fluid films
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 3 , June 2018 , pp. 369 - 392
Copyright
Copyright © Cambridge University Press 2017 

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