Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-25T00:49:47.608Z Has data issue: false hasContentIssue false
  • English
  • Français

Periodic Steady-state Solutions of a Liquid Film Model via a Classical Method

Published online by Cambridge University Press:  20 November 2018

Ahmad Alhasanat  and
Chunhua Ou
Ahmad Alhasanat
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S6, e-mail: a.alhasanat@mun.ca ou@mun.ca
Chunhua Ou
Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S6, e-mail: a.alhasanat@mun.ca ou@mun.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, periodic steady-state of a liquid film flowing over a periodic uneven wall is investigated via a classical method. Specifically, we analyze a long-wave model that is valid at the near-critical Reynolds number. For the periodic wall surface, we construct an iteration scheme in terms of an integral form of the original steady-state problem. The uniform convergence of the scheme is proved so that we can derive the existence and the uniqueness as well as the asymptotic formula of the periodic solutions.

Keywords

34E0534E1034E15film flowclassical methodsasymptotic analysis
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 1 , 01 March 2018 , pp. 3 - 15
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Alekseenko, S. V., Markovich, D. M., Evseev, A. R., Bobylev, A. V., Tarasov, B. V., and Karsten, V. M., Experimental investigation of liquid distribution over structured packing. AIChE J. 54 (2008), 1424–1430.Google Scholar
[2] Benjamin, T. B., Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2 (1957), 554–574. http://dx.doi.Org/10.1017/S0022112057000373.Google Scholar
[3] Blyth, M. G., Effect ofan electric field on the stability of contaminated film flow down an inclined plane. J. Fluid Mech. 595 (2008), 221–237. http://dx.doi.Org/10.1017/S0022112007009147.Google Scholar
[4] Gonzalez, A. and Castellanos, A., Nonlinear electrohydrodynamic waves on films falling down an inclined plane. Physical Review e 43 (1996), no. 4, 3573–3578.Google Scholar
[5] Hastings, S. P. and McLeod, J. B., Classical methods in ordinary differential equations. With applications to boundary value problems. Graduate Studies in Mathematics, 129, American Mathematical Society, Providence, RI, 2012.Google Scholar
[6] Jacobson, N., Basic algebra. I. W. H. Freeman and Co., San Francisco, CA, 1974.Google Scholar
[7] Kistler, S. F. and Schweizer, P. M., Liquid film coating. Chapman and Hall, New York, 1997.Google Scholar
[8] Stone, H. A. and Kim, S., Microfluidics: Basic issues, applications, and challenges. AIChE Journal 47 (2001), no. 6, 1250–1254.Google Scholar
[9] Tseluiko, D. and Blyth, M. G., Effect ofinertia on electrified film flow over a wavy wall. J. Engrg Math. 65 (2009), 229–242. http://dx.doi.org/10.1007/s10665-009-9283-1.Google Scholar
[10] Tseluiko, D., Blyth, M. G., and Papageoriou, D. T., Stability offilm flow over inclined topography based on a long-wave nonlinear model. J. Fluid Mech. 729 (2013), 638–671. http://dx.doi.org/10.1017/jfm.2013.331.Google Scholar
[11] Weinstein, S. J. and Ruschak, K. J., Coating flows. Annu. Rev. Fluid. Mech. 36 (2004), 29–53.Google Scholar
[12] Whitesides, G. M. and Stroock, A. D., Flexible methods for microfluidics. Physics Today 54 (2001), 42–48.Google Scholar