Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-28T11:43:20.614Z Has data issue: false hasContentIssue false
  • English
  • Français

Upscaled model for unsteady slip flow in porous media

Published online by Cambridge University Press:  02 August 2021

Didier Lasseux Open the ORCID record for Didier Lasseux [Opens in a new window] ,
Francisco J. Valdés-Parada Open the ORCID record for Francisco J. Valdés-Parada [Opens in a new window]  and
Alessandro Bottaro Open the ORCID record for Alessandro Bottaro [Opens in a new window]
Didier Lasseux
Affiliation:
I2M, UMR 5295, CNRS, Univ. Bordeaux, 351, Cours de la Libération, 33405Talence CEDEX, France
Francisco J. Valdés-Parada*
Affiliation:
División de Ciencias Básicas e Ingeniería, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, col. Vicentina, 09340, Mexico
Alessandro Bottaro
Affiliation:
Dipartimento di Ingegneria Civile, Chimica e Ambientale, Scuola Politecnica, Università di Genova, Via Montallegro 1, Genova16145, Italy
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This work reports on modelling unsteady gas flow in porous media at the macroscopic scale in the slip regime, a topic of interest in a wide range of applications. The slip effect is modelled by means of a Navier-type boundary condition. A macroscopic model is derived from the initial-boundary-value problem governing unsteady, single-phase flow of a Newtonian fluid through homogeneous porous media in the creeping, isothermal and slightly compressible slip regime. For momentum transport, the macroscopic model involves two terms. The first consists of a convolution product between the macroscopic pressure gradient and the temporal derivative of an apparent dynamic permeability tensor; the second accounts for the memory of the initial condition. Both contributions are predicted from the solution of a unique closure problem that is independent of the initial flow configuration and of the macroscopic pressure gradient. The accuracy of the model is assessed by comparisons with direct numerical simulations performed at the pore-scale, which find excellent agreement. The simulations also show that a classical heuristic model, which is the consequence of assuming a separation of time scales between the pore-scale and the macroscale, is inadequate, in general, to correctly predict the macroscopic velocity. Results from this work provide a formal clear insight about unsteady flow in porous media in the slip regime, motivating further theoretical and experimental work.

JFM classification

Low-Reynolds-number flows: Porous media Low-Reynolds-number flows: Stokesian dynamics Mathematical Foundations: General fluid mechanics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 923 , 25 September 2021 , A37
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Allaire, G. 1991 Homogenization of the Navier–Stokes equations with a slip boundary condition. Commun. Pure Appl. Maths XLIV, 605–641.CrossRefGoogle Scholar
Allaire, G. 1992 Progress in partial differential equations: calculus of variations, application. In Homogenization of the Unsteady Stokes Equations in a Porous Medium (ed. C. Bandle, J. Bemelmans & M. Chipot), pp. 109–123. Longman Scientific and Technical.Google Scholar
Auriault, J.-L. 1980 Dynamic behaviour of a porous medium saturated by a Newtonian fluid. Intl J. Engng Sci. 18 (6), 775–785.CrossRefGoogle Scholar
Auriault, J.-L., Boutin, C. & Geindreau, C. 2009 Homogenization of Coupled Phenomena in Heterogenous Media. ISTE LTD.CrossRefGoogle Scholar
Bear, J. 2018 Modeling Phenomena of Flow and Transport in Porous Media. Springer.CrossRefGoogle Scholar
Bottaro, A. 2019 Flow over natural or engineered surfaces: an adjoint homogenization perspective. J. Fluid Mech. 877, P1.CrossRefGoogle Scholar
Breugem, W.P., Boersma, B.J. & Uittenbogaard, R.E. 2006 The influence of wall permeability on turbulent channel flow. J. Fluid Mech. 562, 35–72.CrossRefGoogle Scholar
Chastanet, J., Royer, P. & Auriault, J.-L. 2004 Does Klinkenberg's law survive upscaling? Transp. Porous Media 56 (2), 171–198.CrossRefGoogle Scholar
Choi, J. & Dong, H. 2021 Green functions for the pressure of Stokes systems. Intl Math. Res. Notices 2021, 1699–1759.CrossRefGoogle Scholar
Cioranescu, D., Donato, P. & Ene, H. 1996 Homogenization of the Stokes problem with non-homogeneous slip boundary conditions. Math. Meth. Appl. Sci. 19, 857–881.3.0.CO;2-D>CrossRefGoogle Scholar
Cowling, T.G. 1950 Molecules in Motion, chap IV. Hutchinson.Google Scholar
Das, M.K., Mukherjee, P.P. & Muralidhar, K. 2018 Modeling Transport Phenomena in Porous Media with Applications. Springer.CrossRefGoogle Scholar
Gray, W.G. 1975 A derivation of the equations for multi-phase transport. Chem. Engng Sci. 30 (2), 229–233.CrossRefGoogle Scholar
Haberman, R. 2012 Applied Partial Differential Equations with Fourier Series and Boundary Value Problems (Featured Titles for Partial Differential Equations), 5th edn. Pearson.Google Scholar
Haddad, O.M., Al-Nimr, M.A. & Sari, M.S. 2007 Forced convection gaseous slip flow in circular porous micro-channels. Transp. Porous Media 70, 167–179.CrossRefGoogle Scholar
Hayek, M. 2015 Exact solutions for one-dimensional transient gas flow in porous media with gravity and Klinkenberg effects. Transp. Porous Media 107, 403–417.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
Howes, F.A. & Whitaker, S. 1985 The spatial averaging theorem revisited. Chem. Engng Sci. 40 (8), 1387–1392.CrossRefGoogle Scholar
Jin, Y. & Kuznetsov, A.V. 2017 Turbulence modeling for flows in wall bounded porous media: an analysis based on direct numerical simulations. Phys. Fluids 29 (4), 045102.CrossRefGoogle Scholar
Khuzhayorov, B., Auriault, J.-L. & Royer, P. 2000 Derivation of macroscopic filtration law for transient linear viscoelastic fluid flow in porous media. Intl J. Engng Sci. 38 (5), 487–504.CrossRefGoogle Scholar
Klinkenberg, L.J. 1941 The permeability of porous media to liquids and gases. Am. Petrol. Inst. 2 (2), 200–213.Google Scholar
Lasseux, D. & Valdés-Parada, F.J. 2017 Symmetry properties of macroscopic transport coefficients in porous media. Phys. Fluids 29 (4), 043303.CrossRefGoogle Scholar
Lasseux, D., Valdés-Parada, F.J. & Bellet, F. 2019 Macroscopic model for unsteady flow in porous media. J. Fluid Mech. 862, 283–311.CrossRefGoogle Scholar
Lasseux, D., Valdés-Parada, F.J., Ochoa-Tapia, J.A. & Goyeau, B. 2014 A macroscopic model for slightly compressible gas slip-flow in homogeneous porous media. Phys. Fluids 26 (5), 053102.CrossRefGoogle Scholar
Lasseux, D., Valdés-Parada, F.J. & Porter, M.L. 2016 An improved macroscale model for gas slip flow in porous media. J. Fluid Mech. 805, 118–146.CrossRefGoogle Scholar
Lauga, E., Brenner, M. & Stone, H. 2007 Microfluidics: The No-Slip Boundary Condition, pp. 1219–1240. Springer.Google Scholar
Lockerby, D.A., Reese, J.M., Emerson, D.R. & Barber, R.W. 2004 Velocity boundary condition at solid walls in rarefied gas calculations. Phys. Rev. E 70, 017303.CrossRefGoogle ScholarPubMed
Mikelić, A. 1994 Mathematical derivation of the Darcy-type law with memory effects, governing transient flow through porous medium. Glasnik Matematicki 29 (49), 57–77.Google Scholar
Nandal, J., Kumari, S. & Rathee, R. 2019 The effect of slip velocity on unsteady peristalsis MHD blood flow through a constricted artery experiencing body acceleration. Intl J. Appl. Mech. Engng 24 (3), 645–659.CrossRefGoogle Scholar
Navier, M. 1822 Mémoire sur les lois du mouvement des fluides, vol. 6. l'Académie Royale des Sciences.Google Scholar
Nield, D.A. & Bejan, A. 2013 Convection in Porous Media. Springer.CrossRefGoogle Scholar
Pasquier, S., Quintard, M. & Davit, Y. 2017 Modeling flow in porous media with rough surfaces: effective slip boundary conditions and application to structured packings. Chem. Engng Sci. 165, 131–146.CrossRefGoogle Scholar
Pavan, V. & Oxarango, L. 2007 A new momentum equation for gas flow in porous media: the Klinkenberg effect seen through the kinetic theory. J. Stat. Phys. 126 (2), 355–389.CrossRefGoogle Scholar
Perrier, P., Graur, I.A., Ewart, T. & Méolans, J.G. 2011 Mass flow rate measurements in microtubes: from hydrodynamic to near free molecular regime. Phys. Fluids 23 (4), 042004.CrossRefGoogle Scholar
Polubarinova-Kochina, P.Ya. 1962 Theory of Ground Water Movement (Translated from the Russian edition by J. M. Roger de Wiest). Princeton University Press.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Qayyum, M., Khan, H., Rahim, M.T. & Ullah, I. 2015 Modeling and analysis of unsteady axisymmetric squeezing fluid flow through porous medium channel with slip boundary. PLoS ONE 1, e0117368.CrossRefGoogle Scholar
Samanta, A. 2020 Linear stability of a plane Couette–Poiseuille flow overlying a porous layer. Intl J. Multiphase Flow 123, 103160.CrossRefGoogle Scholar
Ullah, I., Khan, I. & Shafie, S. 2017 Heat and mass transfer in unsteady MHD slip flow of Casson fluid over a moving wedge embedded in a porous medium in the presence of chemical reaction: numerical solutions using Keller–Box method. Numer. Meth. Partial Differ. Equ. 34 (5), 1867–1891.CrossRefGoogle Scholar
Whitaker, S. 1996 The Forchheimer equation: a theoretical development. Transp. Porous Media 25 (1), 27–61.CrossRefGoogle Scholar
Whitaker, S. 1999 The Method of Volume Averaging. Springer.CrossRefGoogle Scholar
Wu, Y.-S., Pruess, K. & Persoff, P. 1998 Gas flow in porous media with Klinkenberg effects. Transp. Porous Media 32, 117–137.CrossRefGoogle Scholar