Well-posed Stokes/Brinkman and Stokes/Darcy coupling revisited with new jump interface conditions

Angot, Philippe

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1875-1911.

The global well-posedness in time is proved, with no restriction on the size of the data, for the Stokes/Brinkman and Stokes/Darcy coupled flow problems with new jump interface conditions recently derived by Angot et al. [Phys. Rev. E 95 (2017) 063302-1–063302-16] using asymptotic modelling and shown to be physically relevant. These original conditions include jumps of both stress and tangential velocity vectors at the fluid–porous interface. They can be viewed as generalizations for the multi-dimensional flow of Beavers and Joseph’s jump condition of tangential velocity and Ochoa-Tapia and Whitaker’s jump condition of shear stress. Therefore, they are different from those most commonly used in the literature. The case of Saffman’s approximation is also studied, but with a force balance for the cross-flow including the Darcy drag and inducing a law of pressure jump different from the usual one. The proof of these results follows the general framework briefly introduced by Angot [C. R. Math. Acad. Sci. Paris, Ser. I 348 (2010) 697–702; Appl. Math. Lett. 24 (2011) 803–810.] for the steady flow.

Classification : 35Q30, 35Q35, 65M85, 76D03, 76D05, 76D07, 76S05
Mots clés : Fluid–porous coupled flow, Stokes/Brinkman model, Stokes/Darcy model, jump interface conditions, stress vector jump, tangential velocity jump

[1] V. Adolfsson and D. Jerison, L^p-Integrability of the second order derivatives for the Neumann problem in convex domains. Indiana Univ. Math. J. 43 (1994) 1123–1138. | DOI | MR | Zbl

[2] G. Allaire, Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. I: abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113 (1991) 209–259. | DOI | MR | Zbl

[3] G. Allaire, Homogenization of the Navier–Stokes equations in open sets perforated with tiny holes. II: non-critical sizes of the holes for a volume distribution and a surface distribution of holes. Arch. Ration. Mech. Anal. 113 (1991) 261–298. | DOI | MR | Zbl

[4] Ph. Angot, Analysis of singular perturbations on the Brinkman problem for fictitious domain models of viscous flows. Math. Meth. Appl. Sci. 22 (1999) 1395–1412. | DOI | MR | Zbl

[5] Ph. Angot, A model of fracture for elliptic problems with flux and solution jumps. C. R. Acad. Sci. Paris, Ser. I Math. 337 (2003) 425–430. | DOI | MR | Zbl

[6] Ph. Angot, A unified fictitious domain model for general embedded boundary conditions. C. R. Acad. Sci. Paris, Ser. I Math. 341 (2005) 683–688. | DOI | MR | Zbl

[7] Ph. Angot, A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions. C. R. Math. Acad. Sci. Paris, Ser. I 348 (2010) 697–702. | DOI | MR | Zbl

[8] Ph. Angot, On the well-posed coupling between free fluid and porous viscous flows. Appl. Math. Lett. 24 (2011) 803–810. | DOI | MR | Zbl

[9] Ph. Angot, F. Boyer and F. Hubert, Asymptotic and numerical modelling of flows in fractured porous media. ESAIM:M2AN 43 (2009) 239–275. | DOI | Numdam | MR | Zbl

[10] Ph. Angot, G. Carbou and V. Péron, Asymptotic study for Stokes–Brinkman model with jump embedded transmission conditions. Asymptot. Anal. 96 (2016) 223–249. | MR | Zbl

[11] Ph. Angot, B. Goyeau and A.J. Ochoa-Tapia, Asymptotic modeling of transport phenomena at the interface between a fluid and a porous layer: jump conditions. Phys. Rev. E 95 (2017) 063302-1–063302-16. | DOI | MR

[12] T. Arbogast and D.S. Brunson, A computational method for approximating a Stokes–Darcy system governing a vuggy porous medium. Comput. Geosci. 11 (2007) 207–218. | DOI | MR | Zbl

[13] J.-L. Auriault, About the Beavers and Joseph boundary condition. Transp. Porous Media 83 (2010) 257–266. | DOI

[14] L. Badea,M. Discacciati and A. Quarteroni, Mathematical analysis of the Navier-Stokes/Darcy coupling. Numer. Math. 1152 (2010) 195–227. | DOI | MR | Zbl

[15] J. Barrère, O. Gipouloux and S. Whitaker, On the closure problem for Darcy’s law. Transp. Porous Media 7 (1992) 209–222. | DOI

[16] G.S. Beavers and D.D. Joseph, Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1967) 197–207. | DOI

[17] A.Yu. Beliaev and S.M. Kozlov, Darcy equation for random porous media. Comm. Pure Appl. Math. XLIX (1996) 1–34. | DOI | MR | Zbl

[18] C. Bernardi, T.-C. Rebollo, F. Hecht and Z. Mghazli, Mortar finite element discretization of a model coupling Darcy and Stokes equations. ESAIM:M2AN 42 (2008) 375–410. | DOI | Numdam | MR | Zbl

[19] F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier–Stokes equations and related models. Vol. 83 of Appl. Math. Sci. Springer, New York (2013). | DOI | MR | Zbl

[20] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer-Verlag, New York (2011). | DOI | MR | Zbl

[21] A. Brillard, Asymptotic analysis of incompressible and viscous fluid flow through porous media; Brinkman’s law via epi-convergence methods. Annales Faculté des Sciences de Toulouse 8 (1986) 225–252. | DOI | Numdam | MR | Zbl

[22] A. Brillard, J.El. Amrani and M. Jarroudi, Derivation of a contact law between a free fluid and thin porous layers via asymptotic analysis methods. Appl. Anal. 92 (2013) 665–689. | DOI | MR | Zbl

[23] H.C. Brinkman, A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particules. Appl. Sci. Res. A1 (1947) 27–34. | Zbl

[24] H.C. Brinkman, On the permeability of media consisting of closely packed porous particules. Appl. Sci. Res. A1 (1947) 81–86.

[25] J.-P. Caltagirone, Physique des Ecoulements Continus. Vol. 74 of Mathématiques et Applications. SMAI & Springer-Verlag, Berlin (2013). | DOI

[26] Y. Cao, M. Gunzburger, F. Hua and X. Wang, Coupled Stokes–Darcy model with Beavers–Joseph interface boundary condition. Comm. Math. Sci. 8 (2010) 1–25. | DOI | MR | Zbl

[27] Y. Cao, M. Gunzburger, X. Hu, F. Hua, X. Wang and W. Zhao, Finite element approximations for Stokes–Darcy flow with Beavers-Joseph interface conditions. SIAM J. Numer. Anal. 47 (2010) 4239–4256. | DOI | MR | Zbl

[28] P.C. Carman, Fluid flow through granular beds. Trans. Inst. Chem. Eng. 15 (1937) 150–166.

[29] T. Carraro, C. Goll, A. Marciniak-Czochra and A. Mikelić, Pressure jump interface law for the Stokes–Darcy coupling: confirmation by direct numerical simulations. J. Fluid. Mech. 732 (2013) 510–536. | DOI | MR | Zbl

[30] M. Cieszko and J. Kubik, Derivation of matching conditions at the contact surface between fluid–saturated porous solid and bulk fluid. Transp. Porous Media 34 (1999) 319–336. | DOI

[31] R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology. II. Evolution Problems. Springer-Verlag, Berlin (1993), Vol. 6. | MR | Zbl

[32] M. Discacciati and A. Quarteroni, Navier-Stokes/Darcy coupling: modeling, analysis and numerical approximation. Rev. Math. Complut. 22 (2009) 315–426. | MR | Zbl

[33] M. Discacciati, A. Quarteroni and A. Valli, Robin–Robin domain decomposition methods for the Stokes–Darcy coupling. SIAM J. Numer. Anal. 45 (2007) 1246–1268. | DOI | MR | Zbl

[34] E. Fabes, O. Mendez and M. Mitrea, Boundary Layers on Sobolev-Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159 (1998) 323–368. | DOI | MR | Zbl

[35] V. Girault and P.A. Raviart, Finite element methods for the Navier–Stokes equations, Vol. 5 of Springer Series in Comput. Math. Springer-Verlag, Berlin (1986). | DOI | MR | Zbl

[36] V. Girault and B. Rivière, DG approximation of coupled Navier-Stokes and Darcy equations by Beavers-Joseph-Saffman interface condition. SIAM J. Numer. Anal. 47 (2009) 2052–2089. | DOI | MR | Zbl

[37] V. Girault, D. Vassilev and I. Yotov, Mortar multiscale finite element methods for Stokes–Darcy flows. Numer. Math. 127 (2014) 93–165. | DOI | MR | Zbl

[38] B. Goyeau, D. Lhuillier, D. Gobin and M.G. Velarde, Momentum transport at a fluid–porous interface. Int. J. Heat Mass Transf . 46 (2003) 4071–4081. | DOI | Zbl

[39] W.G. Gray and K. O’Neill, On the general equations for flow in porous media and their reduction to Darcy’s law. Water Resour. Res. 12 (1976) 148–154. | DOI

[40] P. Grisvard, Elliptic problems in nonsmooth domains, Vol. 24 of Monographs and Studies in Mathematics. Adv. Publish. Prog., Pitman, Boston (1985). | MR | Zbl

[41] J. Happel, Viscous flow relative to arrays of cylinders. AIChE J. 5 (1959) 174–177. | DOI

[42] G.W. Jackson and D.F. James, The permeability of fibrous porous media. Can. J. Chem. Eng. 64 (1986) 364–374. | DOI

[43] W. Jäger and A. Mikelić, On the interface boundary condition of Beavers & Joseph and Saffman. SIAM J. Appl. Math. 60 (2000) 1111–1127. | DOI | MR | Zbl

[44] W. Jäger and A. Mikelić, On the boundary conditions at the interface between a porous medium and a free fluid. Transp. Porous Media 78 (2009) 489–508. | MR

[45] J. Kubik and M. Cieszko, Analysis of matching conditions at the boundary surface of a fluid-saturated porous solid and a bulk fluid: the use of Lagrange multipliers. Continuum Mech. Thermodyn. 17 (2005) 351–359. | DOI | MR | Zbl

[46] W.L. Layton, F. Schieweck and I. Yotov, Coupling fluid flow with porous media flow. SIAM J. Numer. Anal. 40 (2003) 2195–2218. | DOI | MR | Zbl

[47] T. Lévy, Fluid flow through an array of fixed particules. Int. J. Eng. Sci. 21 (1983) 11–23. | DOI | MR | Zbl

[48] T. Lévy and E. Sanchez-Palencia, Suspension of solid particules in a newtonian fluid. J. Non-Newtonian Fluid Mech. 13 (1983) 63–78. | DOI | Zbl

[49] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod & Gauthier-Villars, Paris (1969). | MR | Zbl

[50] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Springer-Verlag, New York (1972), Vol. I. | MR | Zbl

[51] M.J. Macdonald, C.C. Chu, P.P. Guilloit and K.M. Ng, A generalized Blake-Kozeny equation for multi-sized spherical particles. AIChE J. 37 (1991) 1583–1588. | DOI

[52] A. Marciniak-Czochra and A. Mikelić, Effective pressure interface law for transport phenomena between an unconfined fluid and a porous medium using homogenization. SIAM Multiscale Model. Simul. 10 (2012) 285–305. | DOI | MR | Zbl

[53] J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles de type elliptique, voisine de la variationnelle. Ann. Scuola Norm. Sup. Pisa 16 (1962) 305–326. | Numdam | MR | Zbl

[54] J. Nečas, Les méthodes directes en théorie des équations elliptiques. Masson, Paris (1967). | MR | Zbl

[55] D.A. Nield, The Beavers-Joseph boundary condition and related matters: a historical and critical review. Transp. in Porous Media 78 (2009) 537–540. | DOI

[56] J.A. Ochoa-Tapia and S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid I: theoretical development. Int. J. Heat Mass Transf. 38 (1995) 2635–2646 | DOI | Zbl

[57] J.A. Ochoa-Tapia and S. Whitaker, Momentum transfer at the boundary between a porous medium and a homogeneous fluid II: comparison with experiment. Int. J. Heat Mass Transf . 38 (1995) 2647–2655. | DOI | Zbl

[58] L.E. Payne and B. Straughan, Analysis of the boundary condition at the interface between a viscous fluid and a porous medium and related modelling questions. J. Math. Pures Appl. 77 (1998) 317–354. | DOI | MR | Zbl

[59] B. Rivière and I. Yotov, Locally conservative coupling of Stokes and Darcy flow. SIAM J. Numer. Anal. 42 (2005) 1959–1977. | DOI | MR | Zbl

[60] J. Rubinstein, Effective equations for flow in random porous media with large number of scales. J. Fluid Mech. 170 (1986) 379–383. | DOI | Zbl

[61] J. Rubinstein, On the macroscopic description of slow viscous flow past a random array of spheres. J. Stat. Phys. 44 (1986) 849–863. | DOI | MR | Zbl

[62] J. Rubinstein and S. Torquato, Flow in random porous media: mathematical formulation, variational principles and rigourous bounds. J. Fluid Mech. 206 (1989) 25–46. | DOI | MR | Zbl

[63] P.G. Saffman, On the boundary condition at the surface of a porous medium. Stud. Appl. Math. L50 (1971) 93–101. | DOI | Zbl

[64] R. Temam, Navier–Stokes Equations: Theory and Numerical Analysis. North-Holland, Amsterdam (1986). | MR | Zbl

[65] F.J. Valdés-Parada, C.G. Aguilar-Madera, J.A. Ochoa-Tapia and B. Goyeau, Velocity and stress jump conditions between a porous medium and a fluid. Adv. Water Res. 62 (2013) 327–339. | DOI

[66] S. Whitaker, Flow in porous media I: a theoretical derivation of Darcy’s law. Transp. Porous Media 1 (1986) 3–25. | DOI

[67] S. Whitaker, The Forchheimer equation: a theoretical development. Transp. Porous Media 25 (1996) 27–61. | DOI

[68] S. Whitaker, The method of volume averaging. Vol. 13 of Theory and Applications of Transport in Porous Media. Kluwer Acad. Publ., Dordrecht (1999). | DOI