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: multiphysics, weak solution, interface conditions, Beavers-Joseph-Saffman
@article{M2AN_2013__47_2_539_0, author = {Cesmelioglu, Aycil and Girault, Vivette and Rivi\`ere, B\'eatrice}, title = {Time-dependent coupling of {Navier-Stokes} and {Darcy} flows}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {539--554}, publisher = {EDP-Sciences}, volume = {47}, number = {2}, year = {2013}, doi = {10.1051/m2an/2012034}, mrnumber = {3021697}, zbl = {1267.76096}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012034/} }
TY - JOUR AU - Cesmelioglu, Aycil AU - Girault, Vivette AU - Rivière, Béatrice TI - Time-dependent coupling of Navier-Stokes and Darcy flows JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 539 EP - 554 VL - 47 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012034/ DO - 10.1051/m2an/2012034 LA - en ID - M2AN_2013__47_2_539_0 ER -
%0 Journal Article %A Cesmelioglu, Aycil %A Girault, Vivette %A Rivière, Béatrice %T Time-dependent coupling of Navier-Stokes and Darcy flows %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 539-554 %V 47 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012034/ %R 10.1051/m2an/2012034 %G en %F M2AN_2013__47_2_539_0
Cesmelioglu, Aycil; Girault, Vivette; Rivière, Béatrice. Time-dependent coupling of Navier-Stokes and Darcy flows. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 2, pp. 539-554. doi : 10.1051/m2an/2012034. http://www.numdam.org/articles/10.1051/m2an/2012034/
[1] Sobolev Spaces. Academic Press, New-York (1975). | MR | Zbl
,[2] A computational method for approximating a Darcy-Stokes system governing a vuggy porous medium. Comput. Geosci. 11 (2007) 207-218. | MR | Zbl
and ,[3] Homogenization of a Darcy-Stokes system modeling vuggy porous media. Comput. Geosci. 10 (2006) 291-302. | MR | Zbl
and ,[4] Un théorème de compacité. CRAS Paris Sér. I 256 (1963) 5042-5044. | Zbl
,[5] Mathematical analysis of the Navier-Stokes/Darcy coupling. Numer. Math. 1152 (2010) 195-227. | MR
, and ,[6] Boundary conditions at a naturally impermeable wall. J. Fluid. Mech. 30 (1967) 197-207.
and ,[7] A unified stabilized method for Stokes and Darcy's equations. J. Computat. Appl. Math. 198 (2007) 35-51. | MR | Zbl
and ,[8] Coupled Stokes-Darcy model with Beavers-Joseph interface boundary condition. Commun. Math. Sci. 8 (2010) 1-25. | MR | Zbl
, , and ,[9] Analysis of time-dependent Navier-Stokes flow coupled with Darcy flow. J. Numer. Math. 16 (2008) 249-280. | MR | Zbl
and ,[10] Primal discontinuous Galerkin methods for time-dependent coupled surface and subsurface flow. J. Sci. Comput. 40 (2009) 115-140. | Zbl
and ,[11] On the solution of the coupled Navier-Stokes and Darcy equations. Comput. Methods Appl. Mech. Eng. 198 (2009) 3806-3820. | MR | Zbl
and ,[12] Numerical modelling of coupled surface and subsurface flow systems. Adv. Water Resour. 33 (2010) 92-105.
and ,[13] Theory of differential equations. McGraw-Hill, New York (1955). | Zbl
and ,[14] Domain Decomposition Methods for the Coupling of Surface and Groundwater Flows. Ph.D. thesis, Ecole Polytechnique Fédérale de Lausanne, Switzerland (2004).
,[15] 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. | MR | Zbl
and ,[16] Navier-Stokes/Darcy coupling : Modeling, analysis, and numerical approximation. Rev. Mat. Comput. 22 (2009) 315-426. | MR | Zbl
and ,[17] Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM J. Numer. Anal. 45 (2007) 1246-1268. | MR | Zbl
, and ,[18] DG approximation of coupled Navier-Stokes and Darcy equations by Beaver-Joseph-Saffman interface condition. SIAM J. Numer. Anal. 47 (2009) 2052-2089. | MR
and ,[19] Elliptic problems in nonsmooth domains. Pitman, Boston, MA. Monogr. Stud. Math. 24 (1985). | MR | Zbl
,[20] Numerical analysis of coupled Stokes/Darcy flows in industrial filtrations. Transp. Porous Media 64 (2006) 1573-1634.
, , and ,[21] 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. | MR | Zbl
and ,[22] On the interface boundary condition of Beavers, Joseph and Saffman. SIAM J. Appl. Math. 60 (2000) 1111-1127. | MR | Zbl
and ,[23] A strongly conservative finite element method for the coupling of Stokes and Darcy flow. J. Computat. Phys. 229 (2010) 5933-5943. | MR
and ,[24] Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2003) 2195-2218. | MR | Zbl
, and ,[25] Equations différentielles opérationnelles et problèmes aux limites. Springer-Verlag, Berlin, Heidelberg, New York (1961). | Zbl
,[26] Non-homogeneous boundary value problems and applications. I. Springer-Verlag, New York (1972). | MR | Zbl
and ,[27] A robust finite element method for Darcy-Stokes flow. SIAM J. Numer. Anal. 40 (2002) 1605-1631 (electronic). | MR | Zbl
, and ,[28] 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. | MR | Zbl
and ,[29] Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | Zbl
,[30] Analysis of a discontinuous finite element method for the coupled Stokes and Darcy problems. J. Sci. Comput. 22 (2005) 479-500. | MR | Zbl
,[31] Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal. 42 (2005) 1959-1977. | MR | Zbl
and ,[32] On the boundary condition at the surface of a porous media. Stud. Appl. Math. 50 (1971) 292-315. | Zbl
,[33] Compact sets in the space Lp(0,T;B). Ann. Math. Pures Appl. 146 (1990) 1093-1117. | MR | Zbl
,[34] Coupling Stokes-Darcy flow with transport. SIAM J. Sci. Comput. 31 (2009) 3661-3684. | MR
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