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Time-dependent coupling of Navier–Stokes and Darcy flows∗

Published online by Cambridge University Press:  11 January 2013

Aycil Cesmelioglu ,
Vivette Girault  and
Béatrice Rivière
Aycil Cesmelioglu
Affiliation:
Institute for Mathematics and its Applications, University of Minnesota, 207 Church Street, Minneapolis, 55455 MN, USA.
Vivette Girault
Affiliation:
Université Pierre et Marie Curie, Paris VI, Laboratoire Jacques–Louis Lions, 4 place Jussieu, 75252 Paris Cedex 05, France
Béatrice Rivière
Affiliation:
Rice University, Department of Computational and Applied Mathematics, 6100 Main St. MS-134, Houston, 77005 TX, USA.. riviere@caam.rice.edu
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Abstract

A weak solution of the coupling of time-dependent incompressible Navier–Stokes equations with Darcy equations is defined. The interface conditions include the Beavers–Joseph–Saffman condition. Existence and uniqueness of the weak solution are obtained by a constructive approach. The analysis is valid for weak regularity interfaces.

Keywords

Multiphysicsweak solutioninterface conditionsBeavers–Joseph–Saffman
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 539 - 554
Copyright
© EDP Sciences, SMAI, 2013

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References

Références

R. Adams, Sobolev Spaces. Academic Press, New-York (1975).
Arbogast, T. and Brunson, D., A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Comput. Geosci. 11 (2007) 207–218. Google Scholar
Arbogast, T. and Lehr, H., Homogenization of a Darcy–Stokes system modeling vuggy porous media. Comput. Geosci. 10 (2006) 291–302. Google Scholar
Aubin, J., Un théorème de compacité. CRAS Paris Sér. I 256 (1963) 5042–5044. Google Scholar
Badea, L., Discacciati, M. and Quarteroni, A., Mathematical analysis of the Navier–Stokes/Darcy coupling. Numer. Math. 1152 (2010) 195–227. Google Scholar
Beavers, G. and Joseph, D., Boundary conditions at a naturally impermeable wall. J. Fluid. Mech. 30 (1967) 197–207. Google Scholar
Burman, E. and Hansbo, P., A unified stabilized method for Stokes and Darcy’s equations. J. Computat. Appl. Math. 198 (2007) 35–51. Google Scholar
Cao, Y., Gunzburger, M., Hua, F. and Wang, X., Coupled Stokes-Darcy model with Beavers–Joseph interface boundary condition. Commun. Math. Sci. 8 (2010) 1–25. Google Scholar
Çeşmelioğlu, A. and Rivière, B., Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. J. Numer. Math. 16 (2008) 249–280. Google Scholar
Çeşmelioğlu, A. and Rivière, B., Primal discontinuous Galerkin methods for time-dependent coupled surface and subsurface flow. J. Sci. Comput. 40 (2009) 115–140. Google Scholar
Chidyagwai, P. and Rivière, B., On the solution of the coupled Navier-Stokes and Darcy equations. Comput. Methods Appl. Mech. Eng. 198 (2009) 3806–3820. Google Scholar
Chidyagwai, P. and Rivière, B., Numerical modelling of coupled surface and subsurface flow systems. Adv. Water Resour. 33 (2010) 92–105. Google Scholar
E.A. Coddington and N. Levinson, Theory of differential equations. McGraw–Hill, New York (1955).
M. Discacciati, Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland (2004).
M. Discacciati and A. Quarteroni, Analysis of a domain decomposition method for the coupling of Stokes and Darcy equations. in Numerical Analysis and Advanced Applications ENUMATH 2001. Springer, Milan (2003) 3–20.
Discacciati, M. and Quarteroni, A., Navier-Stokes/Darcy coupling : Modeling, analysis, and numerical approximation. Rev. Mat. Comput. 22 (2009) 315–426. Google Scholar
Discacciati, M., Quarteroni, A. and Valli, A., Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal. 45 (2007) 1246–1268. Google Scholar
Girault, V. and Rivière, B., DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Numer. Anal. 47 (2009) 2052–2089. Google Scholar
Grisvard, P., Elliptic problems in nonsmooth domains. Pitman, Boston, MA. Monogr. Stud. Math. 24 (1985). Google Scholar
Hanspal, N., Waghode, A., Nassehi, V. and Wakeman, R., Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations. Transp. Porous Media 64 (2006) 1573–1634. Google Scholar
Heywood, J. and Rannacher, R., Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982) 275–311. Google Scholar
Jäger, W. and Mikelić, A., On the interface boundary condition of Beavers, Joseph and Saffman. SIAM J. Appl. Math. 60 (2000) 1111–1127. Google Scholar
Kanschat, G. and Rivière, B., A strongly conservative finite element method for the coupling of Stokes and Darcy flow. J. Computat. Phys. 229 (2010) 5933–5943. Google Scholar
Layton, W., Schieweck, F. and Yotov, I., Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2003) 2195–2218. Google Scholar
J.-L. Lions, Equations différentielles opérationnelles et problèmes aux limites. Springer-Verlag, Berlin, Heidelberg, New York (1961).
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. I. Springer-Verlag, New York (1972).
Mardal, K.A., Tai, X.-C. and Winther, R., A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal. 40 (2002) 1605–1631 (electronic). Google Scholar
Mu, M. and Xu, J., A two-grid method of a mixed Stokes-Darcy model for coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 45 (2007) 1801–1813. Google Scholar
J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967).
Rivière, B., Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22 (2005) 479–500. Google Scholar
Rivière, B. and Yotov, I., Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal. 42 (2005) 1959–1977. Google Scholar
Saffman, P., On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50 (1971) 292–315. Google Scholar
Simon, J., Compact sets in the space Lp(0,T;B). Ann. Math. Pures Appl. 146 (1990) 1093–1117. Google Scholar
Vassilev, D. and Yotov, I., Coupling Stokes-Darcy flow with transport. SIAM J. Sci. Comput. 31 (2009) 3661–3684. Google Scholar