The Coupling Navier-Stokes/Darcy Systems by Vorticity-Velocity-Pressure Formulation

  • Authors

    • Anis Younes Faculty of sciences of Tunisia
    • Yousra Ouali
    2014-06-21
    https://doi.org/10.14419/gjma.v2i3.2850

  • Abstract

    In this paper We consider the Coupling Navier-Stokes/Darcy equations  in a two or three dimensional domain provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity.  We establish a coupled variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. We discuss coupling conditions and we analyze the global coupled model in order to prove its well-posedness and to characterize effective algorithms to compute the solution of its numerical approximation.

    Keywords: coupled problem, Darcy's equation, Navier-Stokes equation, Finite element.

  • References

    1. Amrouche,C., Bernardi,C., Dauge,M., Giraut,V.: Vector potentials in three-dimensional nonsmooth domains, [Math.Meth.in the applied Sciences] No.21(1998), 823-864.
    2. Azaiez,M., Bernardi,C., Chor,N.:Spectral discretization of the vorticity, velocity and pressure formulation of the Navier-Stokes equations, Laboratoire Jacques-Louis Lions, 2005.
    3. Bernardi,C., Hecht,F., Pironneau,O.:Coupling Darcy And Stokes Equations for Porous Media With Cracks, M2AN ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 39, No 1, 2005, pp.7 to 35
    4. Badea,L., Discacciati,M. , Quarteroni,A.: Mathematical analysis of the Navier-Stokes/Darcy coupling, Modeling and scientic computing, MOX Report, No. 97, dicembre 2006.
    5. Dubois,F.: Vorticity-velocity-pressure formulation for the Stokes problem, Math. Meth. in the Applied Sciences, 25 (2002), 1091-1119.
    6. Dubois,F.: Vorticity-velocity-pressure formulation for the Stokes problem, Math. Meth. Appl. Sci, 25 (2002) 1091-1119.
    7. Salmon,S.:Dveloppement numrique de la formulation tourbillon-vitesse-pression pour le problme de Stokes, Thesis, Universit Pierre et Marie Curie, Paris (1999).
    8. Girault,V., Raviart,P.A.:Finite Element Methods for the Navier Stokes Equations, Theory and Algorithms, Springer- Verlag (1986).
    9. Nedelec,J.C.: Mixed nite element in R3, Numer. Math, 35, pp. 315-341 (1980).
    10. Nedelec,J.C.: Elments nis mixtes incompressibles pour l'quation de Stokes dans R3, Numer. Math, 39, pp. 97-112 (1982).
    11. El Chami,F., Mansour,G., Sayah,T.:Error studies of the coupling DARCY-STOKES system with velocity-pressure formulation, hal-00511792, version, 3-27, Apr 2011.
    12. Girault,V., Rivire,B.:DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saman in-terface condition, SIAM Journal on Numerical Analysis, 2009.
    13. Discacciati,M.: Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows, PhD thesis, Ecole Polytechnique Fdrale de Lausanne, Switzerland, 2004.
    14. F. Hecht and O. Pironneau, FreeFem++, see www.freefem.org.

  • Downloads

  • How to Cite

    Younes, A., & Ouali, Y. (2014). The Coupling Navier-Stokes/Darcy Systems by Vorticity-Velocity-Pressure Formulation. Global Journal of Mathematical Analysis, 2(3), 80-90. https://doi.org/10.14419/gjma.v2i3.2850

    Received date: 2014-05-20

    Accepted date: 2014-06-14

    Published date: 2014-06-21