Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T21:29:26.673Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of a liquid film flowing down an inclined anisotropic and inhomogeneous porous layer: an analytical description

Published online by Cambridge University Press:  18 October 2016

P. Deepu ,
Srinivas Kallurkar ,
Prateek Anand  and
Saptarshi Basu
P. Deepu*
Affiliation:
TIFR Centre for Interdisciplinary Sciences, Narsingi, Hyderabad 500075, India
Srinivas Kallurkar
Affiliation:
Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India
Prateek Anand
Affiliation:
Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India
Saptarshi Basu
Affiliation:
Department of Mechanical Engineering, Indian Institute of Science, Bangalore 560012, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the effect of anisotropy and inhomogeneity in the permeability of the porous layer on the stability of surface waves of an inclined fluid–porous double-layer system. The fluid is assumed to be Newtonian and the porous layer to be Darcian. The porous layer is saturated with the same fluid and the two layers are coupled at the interface via the Beavers–Joseph condition. Linear stability analysis is performed based on a long-wave approximation. The resulting eigenvalue problem is exactly solved up to third order in the wavenumber. The anisotropic behaviour of permeability, cross-stream component of permeability, surface tension and porosity are found to have only higher-order effects on the stability characteristics of the system. On the other hand, the inhomogeneous feature in the streamwise component of permeability play a dominant role in determining the stability of the gravity-driven surface waves; as do other system parameters such as the thickness of the fluid layer relative to that of the porous layer and the Beavers–Joseph coefficient.

JFM classification

Instability: Instability Interfacial Flows (free surface): Interfacial Flows (free surface) Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 807 , 25 November 2016 , pp. 135 - 154
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Auriault, J. L. 2009 On the domain of validity of Brinkman’s equation. Trans. Porous Med. 79, 215–223.CrossRefGoogle Scholar
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30, 197–207.Google Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–573.CrossRefGoogle Scholar
Berryman, J. G. & Greene, R. R. 1980 Discrete inverse methods for elastic waves in layered media. Geophysics 45, 213–233.CrossRefGoogle Scholar
Camporeale, C., Mantelli, E. & Manes, C. 2013 Interplay among unstable modes in films over permeable walls. J. Fluid Mech. 719, 527–550.CrossRefGoogle Scholar
Chang, H. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26, 103–136.CrossRefGoogle Scholar
Chen, F. 1992 Salt-finger instability in an anisotropic and inhomogeneous porous substrate underlying a fluid layer. J. Appl. Phys. 71, 5222–5236.CrossRefGoogle Scholar
De Bruin, G. J. 1974 Stability of a layer of liquid flowing down an inclined plane. J. Engng Maths 8, 259–270.CrossRefGoogle Scholar
Deepu, P., Anand, P. & Basu, S. 2015a Stability of Poiseuille flow in a fluid overlying an anisotropic and inhomogeneous porous layer. Phys. Rev. E 92, 023009.Google Scholar
Deepu, P., Dawande, S. & Basu, S. 2015b Instabilities in a fluid overlying an inclined anisotropic and inhomogeneous porous layer. J. Fluid Mech. 762, R2, 12.CrossRefGoogle Scholar
Durlofsky, L. & Brady, J. F. 1987 Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30, 3329–3341.CrossRefGoogle Scholar
Floryan, J. M., Davis, S. H. & Kelly, R. E. 1987 Instabilities of a liquid film flowing down a slightly inclined plane. Phys. Fluids 30, 983–989.CrossRefGoogle Scholar
Green, T. & Freehill, R. L. 1969 Marginal stability in inhomogeneous porous media. J. Appl. Phys. 40, 1759–1762.CrossRefGoogle Scholar
Hutter, K. 1983 Theoretical Glaciology: Material Science of Ice and the Mechanics of Glaciers and Ice Sheets. Springer.CrossRefGoogle Scholar
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin layers of a viscous fluid. J. Expl Theor. Phys. 19, 105–120.Google Scholar
Liu, R. & Liu, Q. 2009 Instabilities of a liquid film flowing down an inclined porous plane. Phys. Rev. E 80, 036316.Google ScholarPubMed
Manes, C., Pokrajac, D., Mcewan, I. & Nikora, V. 2009 Turbulence structure of open channel flows over permeable and impermeable beds: a comparative study. Phys. Fluids 21, 125109.CrossRefGoogle Scholar
Nguyen, L. T. & Balakotaiah, V. 2000 Modeling and experimental studies of wave evolution on free falling viscous films. Phys. Fluids 12, 2236–2256.CrossRefGoogle Scholar
Ochoa-Tapia, J. A. & Whitaker, S. 1995 Momentum transfer at the boundary between a porous medium and a homogeneous fluid – I. Theoretical development. Intl J. Heat Mass Transfer 38, 2635–2646.CrossRefGoogle Scholar
Pascal, J. P. 1999 Linear stability of fluid flow down a porous inclined plane. J. Phys. D: Appl. Phys. 32, 417–422.CrossRefGoogle Scholar
Pascal, J. P. 2006 Instability of power-law fluid flow down a porous incline. J. Non-Newtonian Fluid Mech. 133, 109–120.CrossRefGoogle Scholar
Pascal, J. P. & D’Alessio, S. J. D. 2010 Instability in gravity-driven flow over uneven permeable surfaces. Intl J. Multiphase Flow 36, 449–459.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 1998 Modeling film flows down inclined planes. Eur. Phys. J. B 6, 277–292.CrossRefGoogle Scholar
Sadiq, I. M. R. & Usha, R. 2008 Thin Newtonian film flow down a porous inclined plane: stability analysis. Phys. Fluids 20, 022105.CrossRefGoogle Scholar
Samanta, A., Goyeau, B. & Ruyer-Quil, C. 2013 A falling film on a porous medium. J. Fluid Mech. 716, 414–444.CrossRefGoogle Scholar
Squire, H. B. 1933 On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proc. R. Soc. Lond. A 142 (847), 621–628.Google Scholar
Straughan, B. & Walker, D. W. 1996 Anisotropic porous penetrative convection. Proc. R. Soc. Lond. A 452, 97–115.Google Scholar
Thiele, U., Goyeau, B. & Velarde, M. G. 2009 Stability analysis of thin film flow along a heated porous wall. Phys. Fluids 21, 014103.CrossRefGoogle Scholar
Tilton, N. & Cortelezzi, L. 2008 Linear stability analysis of pressure-driven flows in channels with porous walls. J. Fluid Mech. 604, 411–445.CrossRefGoogle Scholar
Usha, R. & Millet, S. 2013 Thin film flow down a porous substrate in the presence of an insoluble surfactant: stability analysis. Phys. Fluids 25, 022101, 23.Google Scholar
Yih, S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar
Supplementary material: File

Deepu supplementary material

Deepu supplementary material

Download Deepu supplementary material(File)
File 65.5 KB