Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-24T20:34:07.691Z Has data issue: false hasContentIssue false
  • English
  • Français

A NOTE ON APPROXIMATE BENCHMARK SOLUTIONS FOR VISCOUS TWO-LAYER FLOWS

Part of: Two-phase and multiphase flows

Published online by Cambridge University Press:  10 February 2011

M. SELLIER  and
R. D. LENZ
M. SELLIER*
Affiliation:
Department of Mechanical Engineering, The University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand (email: mathieu.sellier@canterbury.ac.nz)
R. D. LENZ
Affiliation:
ExxonMobil Research and Engineering, Fairfax, VA 22037, USA (email: richard.d.lenz@gmail.com)
*
✉For correspondence; e-mail: mathieu.sellier@canterbury.ac.nz
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.

An important test of the quality of numerical methods developed to track the interface between two fluids is their ability to reproduce test cases or benchmarks. However, benchmark solutions are scarce and virtually nonexistent for complex geometries. We propose a simple method to generate benchmark solutions in the context of the two-layer flow problem, a classical multiphase flow problem. The solutions are obtained by considering the inverse problem of finding the required channel geometry to obtain a prescribed interface profile. This viewpoint shift transforms the problem from that of having to solve a complex differential equation to the much easier one of finding the roots of a quartic polynomial.

Keywords

multiphase flowlong wave approximationexact solution

MSC classification

Secondary: 76T99: None of the above, but in this section
Type
Research Article
Information
The ANZIAM Journal , Volume 51 , Issue 4 , April 2010 , pp. 406 - 415
Copyright
Copyright © Australian Mathematical Society 2011

References

[1]Abramowitz, M. and Stegun (eds), I. A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, 9th printing (Dover, New York, 1972).Google Scholar
[2]Decré, M. M. J. and Baret, J. C., “Gravity-driven flows of viscous liquids over two-dimensional topographies”, J. Fluid Mech. 487 (2003) 147–166.CrossRefGoogle Scholar
[3]Gaskell, P. H., Jimack, P. K., Sellier, M., Thompson, H. M. and Wilson, M. C. T., “Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography”, J. Fluid Mech. 509 (2004) 253–280.CrossRefGoogle Scholar
[4]Hocking, G. C., “Cusp-like free surface flows due to a submerged source or sink in the presence of a flat or sloping bottom”, J. Aust. Math. Soc. Ser. B 26 (1985) 470–486.CrossRefGoogle Scholar
[5]Kalliadasis, S., Bielarz, C. and Homsy, G. M., “Steady free-surface thin film flows over topography”, Phys. Fluids 12 (2000) 1889–1898.CrossRefGoogle Scholar
[6]Lenz, R. D., Liquid displacement in lithographic printing: modeling and visualization, Ph. D. Thesis, University of Minnesota, 2007.Google Scholar
[7]Lenz, R. D. and Kumar, S., “Steady two-layer flow in a topographically patterned channel”, Phys. Fluids 19 (2007) 102103.CrossRefGoogle Scholar
[8]Luo, H., Blyth, M. G. and Pozrikidis, C., “Two-layer flow in a corrugated channel”, J. Engrg. Math. 60 (2008) 127–147.CrossRefGoogle Scholar
[9]Luo, H. and Pozrikidis, C., “Shear-driven and channel flow of a liquid film over a corrugated or indented wall”, J. Fluid Mech. 556 (2006) 167–188.CrossRefGoogle Scholar
[10]MacDonald, I., Baines, M. J., Nichols, N. K. and Samuels, P. G., “Analytic benchmark solutions for open-channel flows”, J. Hyd. Eng. 123 (1997) 1041–1045.CrossRefGoogle Scholar
[11]Richardson, A. R., “Stationary waves in water”, Phil. Mag. 40 (1920) 97–110.CrossRefGoogle Scholar
[12]Rudzki, M. P., “Über eine Klasse hydrodynamischer Probleme mit besonderen Grenzbedingungen”, Math. Ann. 50 (1898) 269–281.CrossRefGoogle Scholar
[13]Sautreaux, C., “Sur une question d’hydrodynamique”, Ann. Sci. Éc. Norm. Super. 10 (1893) S.95–S.182.CrossRefGoogle Scholar
[14]Scardovelli, R. and Zaleski, S., “Direct numerical simulation of free-surface and interfacial flow”, Annu. Rev. Fluid Mech. 31 (1999) 567–603.CrossRefGoogle Scholar
[15]Sellier, M., “Substrate design or reconstruction from free surface data for thin film flows”, Phys. Fluids 20 (2008) art:062106.CrossRefGoogle Scholar
[16]Squires, T. M. and Quake, S. R., “Microfluidics: fluid physics at the nano-liter scale”, Rev. Mod. Phys. 77 (2005) 977–1026.CrossRefGoogle Scholar
[17]Tuck, E. O., “On air flow over free surfaces of stationary water”, J. Aust. Math. Soc. Ser. B 19 (1975) 66–80.CrossRefGoogle Scholar