Advertisement

A Note on the Solute Dispersion in a Porous Medium

  • Igor PažaninEmail author
Article
  • 117 Downloads

Abstract

In this paper, we study the solute transport through a semi-infinite channel filled with a fluid saturated sparsely packed porous medium. A small perturbation of magnitude \(\varepsilon \) is applied on the channel’s walls on which the solute particles undergo a first-order chemical reaction. The effective model for solute concentration in the small-Péclet-number regime is derived using asymptotic analysis with respect to the small parameter \(\varepsilon \). The obtained mathematical model clearly indicates the effects of porous medium, chemical reaction and boundary distortion. In particular, the effect of porous medium parameter on the dispersion coefficient is discussed.

Keywords

Solute dispersion Porous medium Chemical reaction Small boundary perturbation Asymptotic analysis 

Notes

Acknowledgements

The author has been supported by the Croatian Science Foundation (Project 3955: Mathematical modeling and numerical simulations of processes in thin or porous domains). The author would like to thank the referees for their helpful comments and suggestions that helped to improve the paper.

References

  1. 1.
    Adler, P.M.: Porous Media: Geometry and Transports. Butterworth-Heinermann Series in Chemical Engineering, Boston (1992)Google Scholar
  2. 2.
    Allaire, G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes I. Abstract framework, a volume distribution of holes. Arch. Rational. Mech. Anal. 113, 209–259 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Darcy, H.: Les fontaines publiques de la ville de Dijon. Victor Darmon, Paris (1856)Google Scholar
  4. 4.
    Bolster, D., Dentz, M., Le Borgne, T.: Solute dispersion in channels with periodically varying apertures. Phys. Fluids 21, 056601 (2009)CrossRefzbMATHGoogle Scholar
  5. 5.
    Brinkman, H.: A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27–34 (1947)zbMATHGoogle Scholar
  6. 6.
    Chandrasekhara, B.C., Rudraiah, N., Nagaraj, S.T.: Velocity and dispersion in porous media. Int. J. Eng. Sci. 18, 921–929 (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cussler, E.L.: Diffusion: Mass Transfer in Fluid Systems, 2nd edn. Cambridge University Press, New York (1997)Google Scholar
  8. 8.
    John Lee, S.-J., Sundararajan, N.: Microfabrication for Microfluidics. Artech House, Boston (2010)Google Scholar
  9. 9.
    Marušić-Paloka, E.: Effects of small boundary perturbation on flow of viscous fluid. ZAMM -J. Appl. Math. Mech. 96, 1103–1118 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Marušić-Paloka, E., Pažanin, I.: On the reactive solute transport through a curved pipe. Appl. Math. Lett. 24, 878–882 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Marušić-Paloka, E., Pažanin, I., Radulović, M.: Flow of a micropolar fluid through a channel with small boundary perturbation. Z. Naturforsch. A 71, 607–619 (2016)Google Scholar
  12. 12.
    Marušić-Paloka, E., Pažanin, I.: On the Darcy–Brinkman flow through a channel with slightly perturbed boundary. Transp. Porous Media 117, 27–44 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Marušić-Paloka, E., Pažanin, I., Marušić, S.: Comparison between Darcy and Brinkman laws in a fracture. Appl. Math. Comput. 218, 7538–7545 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Mikelic, A., Devigne, V., van Duijn, C.J.: Rigorous upscaling of the reactive flow through a pore, under dominant Péclet and Damkohler numbers. SIAM J. Math. Anal. 38, 1262–1287 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ng, C.-O., Wang, C.Y.: Darcy-Brinkman flow through a corrugated channel. Transp. Porous Media 85, 605–618 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Nunge, R.J., Lin, T.-S., Gill, W.N.: Laminar dispersion in curved tubes and channels. J. Fluid Mech. 51, 363–383 (1972)CrossRefzbMATHGoogle Scholar
  17. 17.
    Pal, D.: Effect of chemical reaction on the dispersion of a solute in a porous medium. Appl. Math. Model. 23, 557–566 (1999)CrossRefzbMATHGoogle Scholar
  18. 18.
    Rosencrans, S.: Taylor dispersion in curved channels. SIAM J. Appl. Math. 57, 1216–1241 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Taylor, G.I.: Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. Roy. Soc. Lond. Sect. A 219, 186–203 (1953)CrossRefGoogle Scholar
  20. 20.
    Rubinstein, J., Mauri, R.: Dispersion and convection in periodic porous media. SIAM J. Appl. Math. 46, 1018–1023 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rudraiah, N., Ng, C.-O.: Dispersion in porous media with and without reaction: a review. J. Porous Media 10, 219–248 (2007)CrossRefGoogle Scholar
  22. 22.
    Sanchez-Palencia, E.: On the asymptotics of the fluid flow past an array of fixed obstacles. Int. J. Eng. Sci. 20, 1291–1301 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Valdes-Parada, F.J., Aguiar-Madera, C.G., Alvarez-Ramirez, J.: On diffusion, dispersion and reaction in porous media. Chem. Eng. Sci. 66, 2177–2190 (2011)CrossRefGoogle Scholar
  24. 24.
    Woolard, H.F., Billingham, J., Jensen, O.E., Lian, G.: A multiscale model for solute transport in a wavy-walled channel. J. Eng. Math. 64, 25–48 (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia

Personalised recommendations