The Coupling Navier-Stokes/Darcy Systems by Vorticity-Velocity-Pressure Formulation
DOI:
https://doi.org/10.14419/gjma.v2i3.2850Abstract
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
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.
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.
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
Badea,L., Discacciati,M. , Quarteroni,A.: Mathematical analysis of the Navier-Stokes/Darcy coupling, Modeling and scientic computing, MOX Report, No. 97, dicembre 2006.
Dubois,F.: Vorticity-velocity-pressure formulation for the Stokes problem, Math. Meth. in the Applied Sciences, 25 (2002), 1091-1119.
Dubois,F.: Vorticity-velocity-pressure formulation for the Stokes problem, Math. Meth. Appl. Sci, 25 (2002) 1091-1119.
Salmon,S.:Dveloppement numrique de la formulation tourbillon-vitesse-pression pour le problme de Stokes, Thesis, Universit Pierre et Marie Curie, Paris (1999).
Girault,V., Raviart,P.A.:Finite Element Methods for the Navier Stokes Equations, Theory and Algorithms, Springer- Verlag (1986).
Nedelec,J.C.: Mixed nite element in R3, Numer. Math, 35, pp. 315-341 (1980).
Nedelec,J.C.: Elments nis mixtes incompressibles pour l'quation de Stokes dans R3, Numer. Math, 39, pp. 97-112 (1982).
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.
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.
Discacciati,M.: Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows, PhD thesis, Ecole Polytechnique Fdrale de Lausanne, Switzerland, 2004.
F. Hecht and O. Pironneau, FreeFem++, see www.freefem.org.
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal''s published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).