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Falling film on an anisotropic porous medium

Published online by Cambridge University Press:  25 August 2022

Sanghasri Mukhopadhyay Open the ORCID record for Sanghasri Mukhopadhyay [Opens in a new window] ,
Nicolas Cellier Open the ORCID record for Nicolas Cellier [Opens in a new window] ,
Usha R Open the ORCID record for Usha R [Opens in a new window] ,
Marx Chhay Open the ORCID record for Marx Chhay [Opens in a new window]  and
Christian Ruyer-Quil Open the ORCID record for Christian Ruyer-Quil [Opens in a new window]
Sanghasri Mukhopadhyay
Affiliation:
Université Savoie Mont Blanc, CNRS, LOCIE, 73000 Chambéry, France
Nicolas Cellier
Affiliation:
Université Savoie Mont Blanc, CNRS, LOCIE, 73000 Chambéry, France
Usha R
Affiliation:
Department of Mathematics, IIT Madras, Chennai, 600036 Tamil Nadu, India
Marx Chhay
Affiliation:
Université Savoie Mont Blanc, CNRS, LOCIE, 73000 Chambéry, France
Christian Ruyer-Quil*
Affiliation:
Université Savoie Mont Blanc, CNRS, LOCIE, 73000 Chambéry, France
*
†Email address for correspondence: christian.ruyer-quil@univ-smb.fr
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Abstract

The stability and dynamics of a falling liquid film over an anisotropic porous medium are studied using a one-domain approach. Our stability analysis shows a significant departure from the effective no-slip boundary condition in the isotropic case. Anisotropy does not affect the threshold of linear instability. However, a non-trivial dual effect of anisotropy on the film stability is observed depending on the permeability of the porous medium. This dual effect results from the net balance of the enhancement of viscous diffusion at the top Brinkman sublayer and the mitigation of viscous damping in the core Darcy sublayer. Three-equation models have been derived from the lubrication theory approximation in terms of the exact mass balance and averaged momentum balances in the porous and liquid layers. In the nonlinear regime, anisotropy has a dual effect by damping capillary waves at large permeabilities and enhancing them at low permeabilities. Anisotropy also affects wave speeds and shapes, modifies travelling-wave branches of solutions, affects the development of a time-periodic wavetrain by inlet forcing and alters the noise-driven dynamics of the flow. These effects result from the mitigation of mass exchange at the liquid–porous interface and the contribution of the cross-stream permeability in the Brinkman top sublayer to the viscous diffusion.

JFM classification

Interfacial Flows (free surface): Thin films Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 947 , 25 September 2022 , A26
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Aguilar-Madera, C.G., Valdés-Parada, F.J., Goyeau, B. & Ochoa-Tapia, J.A. 2011 Convective heat transfer in a channel partially filled with a porous medium. Intl J. Therm. Sci. 50 (8), 1355–1368.CrossRefGoogle Scholar
Amini, Y., Karimi-Sabet, J., Nasr Esfahany, M., Haghshenasfard, M. & Dastbaz, A. 2019 Experimental and numerical study of mass transfer efficiency in new wire gauze with high capacity structured packing. Sep. Sci. Technol. 54 (16), 2706–2717.CrossRefGoogle Scholar
Angot, P., Goyeau, B. & Ochoa-Tapia, J.A. 2017 Asymptotic modeling of transport phenomena at the interface between a fluid and a porous layer: jump conditions. Phys. Rev. E 95 (6), 063302.CrossRefGoogle Scholar
Arquis, E. & Caltagirone, J.P. 1984 Sur les conditions hydrodynamiques au voisinage d'une interface milieu fluide-milieux poreux : applicationàapplicationà la convection naturelle. C. R. Acad. Sci. Paris (II) 299, 1–4.Google Scholar
Beavers, G.S. & Joseph, D.D. 1967 Boundary conditions at a naturally permeable wall. J.Fluid Mech. 30 (1), 197–207.CrossRefGoogle Scholar
Beavers, G.S., Sparrow, E.M. & Magnuson, R.A. 1970 Experiments on coupled parallel flows in a channel and a bounding porous medium. Trans. ASME J. Basic Engng 92 (4), 843–848.CrossRefGoogle Scholar
Beckermann, C., Viskanta, R. & Ramadhyani, S. 1988 Natural convection in vertical enclosures containing simultaneously fluid and porous layers. J.Fluid Mech. 186, 257–284.CrossRefGoogle Scholar
Brinkman, H.C. 1949 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. 1 (1), 27.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
Castinel, G. & Combarnous, M. 1975 Convection naturelle dans une couche poreuse anisotrope. Rev. Therm. 168, 937–947.Google Scholar
Cellier, N. & Ruyer-Quil, C. 2019 Scikit-finite-diff, a new tool for PDE solving. J.Open Source Softw. 4 (38), 1356.CrossRefGoogle Scholar
Chandesris, M. & Jamet, D. 2006 Boundary conditions at a planar fluid–porous interface for a Poiseuille flow. Intl J. Heat Mass Transfer 49 (13), 2137–2150.CrossRefGoogle Scholar
Chang, H.-C. & Demekhin, E.A. 2002 Complex Wave Dynamics on Thin Films. Studies in Interface Science, vol. 14. Elsevier.Google Scholar
Chang, H.-C., Demekhin, E.A. & Kopelevitch, D.I. 1993 Nonlinear evolution of waves on a vertically falling film. J.Fluid Mech. 250, 433–480.CrossRefGoogle Scholar
Chang, M.-H., Chen, F. & Straughan, B. 2006 Instability of Poiseuille flow in a fluid overlying a porous layer. J.Fluid Mech. 564, 287–303.CrossRefGoogle Scholar
Chen, H. & Wang, X.-P. 2014 A one-domain approach for modeling and simulation of free fluid over a porous medium. J.Comput. Phys. 259, 650–671.CrossRefGoogle Scholar
Craster, R.V. & Matar, O.K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.CrossRefGoogle Scholar
Deepu, P., Kallurkar, S., Anand, P. & Basu, S. 2016 Stability of a liquid film flowing down an inclined anisotropic and inhomogeneous porous layer: an analytical description. J.Fluid Mech. 807, 135–154.CrossRefGoogle Scholar
Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Yu.A., Sandstede, B. & Wang, X. 2008 AUTO 97: continuation and bifurcation software for ordinary differential equations (with HomCont).Google Scholar
Duarte, S.I.P., Ferretti, O.A. & Lemcoff, N.O. 1984 A heterogeneous one-dimensional model for non-adiabatic fixed bed catalytic reactors. Chem. Engng Sci. 39 (6), 1025–1031.CrossRefGoogle Scholar
Epherre, J.F. 1975 Critère d'apparition de la convection naturelle dans une couche poreuse anisotrope. Rev. Therm. 168, 949–950.Google Scholar
Evans, J.P., Forster, C.B. & Goddard, J.V. 1997 Permeability of fault-related rocks, and implications for hydraulic structure of fault zones. J.Struct. Geol. 19 (11), 1393–1404.CrossRefGoogle Scholar
Farrell, N.J.C., Healy, D. & Taylor, C.W. 2014 Anisotropy of permeability in faulted porous sandstones. J.Struct. Geol. 63, 50–67.CrossRefGoogle Scholar
Frisk, D.P. & Davis, E.J. 1972 The enhancement of heat transfer by waves in stratified gas-liquid flow. Intl J. Heat Mass Transfer 15 (8), 1537–1552.CrossRefGoogle Scholar
Gobin, D., Goyeau, B. & Songbe, J.-P. 1998 Double diffusive natural convection in a composite fluid–porous layer. Trans. ASME J. Heat Transfer 120 (1), 234–242.CrossRefGoogle Scholar
Hill, A.A. & Straughan, B. 2008 Poiseuille flow in a fluid overlying a porous medium. J.Fluid Mech. 603, 137–149.CrossRefGoogle Scholar
Hirata, S.C., Goyeau, B. & Gobin, D. 2009 Stability of thermosolutal natural convection in superposed fluid and porous layers. Transp. Porous Med. 78 (3), 525–536.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M.G. 2012 Falling Liquid Films. Applied Mathematical Sciences, vol. 176. Springer.CrossRefGoogle Scholar
Liu, J. & Gollub, J.P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712.CrossRefGoogle Scholar
Liu, J., Schneider, J.B. & Gollub, J.P. 1995 Three-dimensional instabilities of film flows. Phys. Fluids 7 (1), 55–67.CrossRefGoogle Scholar
Liu, R. & Liu, Q. 2009 Instabilities of a liquid film flowing down an inclined porous plane. Phys. Rev. E 80 (3), 036316.CrossRefGoogle ScholarPubMed
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 (14), 2635–2646.CrossRefGoogle Scholar
Pascal, J.P. 1999 Linear stability of fluid flow down a porous inclined plane. J.Phys. D: Appl. Phys. 32 (4), 417–422.CrossRefGoogle Scholar
Pradas, M., Tseluiko, D. & Kalliadasis, S. 2011 Rigorous coherent-structure theory for falling liquid films: viscous dispersion effects on bound-state formation and self-organization. Phys. Fluids 23 (4), 044104.CrossRefGoogle Scholar
Rackauckas, C. & Nie, Q. 2017 Differentialequations.jl–a performant and feature-rich ecosystem for solving differential equations in julia. J.Open Res. Softw. 5 (1), 15.CrossRefGoogle Scholar
Revels, J., Lubin, M. & Papamarkou, T. 2016 Forward-mode automatic differentiation in Julia. arXiv:1607.07892 [cs.MS].Google Scholar
Ruyer-Quil, C. & Manneville, P. 2002 Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14 (1), 170–183.CrossRefGoogle Scholar
Sadiq, I.M.R. & Usha, R. 2008 Thin Newtonian film flow down a porous inclined plane: stability analysis. Phys. Fluids 20 (2), 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
Samanta, A., Ruyer-Quil, C. & Goyeau, B. 2011 A falling film down a slippery inclined plane. J.Fluid Mech. 684, 353–383.CrossRefGoogle Scholar
Scheid, B., Ruyer-Quil, C., Kalliadasis, S., Velarde, M.G. & Zeytounian, R.Kh. 2005 Thermocapillary long waves in a liquid film flow. Part 2. Linear stability and nonlinear waves. J.Fluid Mech. 538, 223–244.CrossRefGoogle Scholar
Shkadov, V.Ya. 1967 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 2, 43–51, en. trans. in Fluid Dyn. 2 (Faraday Press, N.Y., 1970), 29–34.Google Scholar
Shkadov, V.Ya. & Sisoev, G.M. 2004 Waves induced by instability in falling films of finite thickness. Fluid Dyn. Res. 35, 357–389.CrossRefGoogle Scholar
Silveira, A.C.P., Tanguy, G., Perrone, Í.T., Jeantet, R., Ducept, F., de Carvalho, A.F. & Schuck, P. 2015 Flow regime assessment in falling film evaporators using residence time distribution functions. J.Food Engng 160, 65–76.CrossRefGoogle Scholar
Sisoev, G.M. & Shkadov, V.Ya. 1999 A two-parameter manifold of wave solutions to an equation for a falling film of viscous fluid. Dokl. Phys. 44, 454–459.Google Scholar
Sparrow, E.M., Beavers, G.S., Chen, T.S. & Lloyd, J.R. 1973 Breakdown of the laminar flow regime in permeable-walled ducts. Trans. ASME J. Appl. Mech. 40 (2), 337–342.CrossRefGoogle Scholar
Tilton, N. & Cortelezzi, L. 2006 The destabilizing effects of wall permeability in channel flows: a linear stability analysis. Phys. Fluids 18 (5), 051702.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
Valdés-Parada, F.J. & Lasseux, D. 2021 A novel one-domain approach for modeling flow in a fluid–porous system including inertia and slip effects. Phys. Fluids 33 (2), 022106.CrossRefGoogle Scholar
de Wasch, A.P. & Froment, G.F. 1972 Heat transfer in packed beds. Chem. Engng Sci. 27 (3), 567–576.CrossRefGoogle Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6 (3), 321.CrossRefGoogle Scholar
Yoshimura, P.N., Nosoko, T. & Nagata, T. 1996 Enhancement of mass transfer into a falling laminar liquid film by two-dimensional surface waves—some experimental observations and modeling. Chem. Engng Sci. 51 (8), 1231–1240.CrossRefGoogle Scholar