Convection in a Porous Medium and Mimetic Scheme in Polar Coordinates

  • Bülent Karasözen
  • Anastasia Trofimova
  • Vyacheslav Tsybulin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6885)


Analytical investigation of natural convection of the incompressible fluid in the porous media based on the Darcy hypothesis (Lapwood convection) gives intriguing branching off of one-parameter family of convective patterns. This scenario may be suppressed in computations when governing equations are approximated by schemes which do not preserve the cosymmetry property. We consider the problem in polar coordinates and construct a mimetic finite-difference scheme using computer algebra tools. The family of steady states is computed and it is demonstrated that this family disappears under non-mimetic approximation.


Porous Medium Nusselt Number Rayleigh Number Convective Cell Discrete Symmetry 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Nield, D.A., Bejan, A.: Convection in porous media. Springer, New York (2006)zbMATHGoogle Scholar
  2. 2.
    Margolin, L., Shashkov, M.: Finite volume methods and the equations of finite scale: A mimetic approach. Int. J. Numer. Meth. Fluids 56, 991–1002 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Yudovich, V.I.: Cosymmetry, degeneracy of the solutions of operator equations, and the onset of filtrational convection. Math. Notes 49, 540–545 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Lyubimov, D.V.: On the convective flows in the porous medium heated from below. J. Appl. Mech. Techn. Phys. 16, 257–261 (1975)CrossRefGoogle Scholar
  5. 5.
    Tsybulin, V.G., Nemtsev, A.D., Karasözen, B.: A mimetic finite-difference scheme for convection of multicomponent fluid in a porous medium. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 322–333. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Karasözen, B., Tsybulin, V.G.: Finite-difference approximation and cosymmetry conservation in filtration convection problem. Phys. Letters A 262, 321–329 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Karasözen, B., Tsybulin, V.G.: Conservative finite difference schemes for cosymmetric systems. In: Proc. 4th Conf. on Computer Algebra in Scientific Computing, pp. 363–375. Springer, Heidelberg (2001)Google Scholar
  8. 8.
    Karasözen, B., Tsybulin, V.G.: Cosymmetry preserving finite-difference methods for equations of convection in a porous medium. Appl. Num. Math. 55, 69–82 (2005)CrossRefzbMATHGoogle Scholar
  9. 9.
    Arakawa, A.: Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. J. Comp. Phys. 1, 119–143 (1966)CrossRefzbMATHGoogle Scholar
  10. 10.
    Govorukhin, V.N.: Numerical simulation of the loss of stability for secondary steady regimes in the Darcy plane-convection problem. Doklady Akademii Nauk. 363, 806–808 (1998)Google Scholar
  11. 11.
    Ganzha, V.G., Vorozhtsov, E.V.: Numerical Solutions for Partial Differential Equations. Problem Solving Using Mathematica. CRC Press, Boca Raton (1996)zbMATHGoogle Scholar
  12. 12.
    Gerdt, V.P., Blinkov, Y.A.: Involution and Difference Schemes for the Navier–Stokes Equations. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 94–105. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Bülent Karasözen
    • 1
  • Anastasia Trofimova
    • 2
  • Vyacheslav Tsybulin
    • 2
  1. 1.Department of Mathematics & Institute of Applied MathematicsMiddle East Technical UniversityAnkaraTurkey
  2. 2.Southern Federal UniversityRostov-on-DonRussia

Personalised recommendations