Compression effects in heterogeneous media
Didier Bresch; Šárka Nečasová; Charlotte Perrin
Journal de l'École polytechnique — Mathématiques, Volume 6  (2019), p. 433-467

We study in this paper compression effects in heterogeneous media with maximal packing constraint. Starting from compressible Brinkman equations, where maximal packing is encoded in a singular pressure and a singular bulk viscosity, we show that the global weak solutions converge (up to a subsequence) to global weak solutions of the two-phase compressible/incompressible Brinkman equations with respect to a parameter ε which measures effects close to the maximal packing value. Depending on the importance of the bulk viscosity with respect to the pressure in the dense regimes, memory effects are activated or not at the limit in the congested (incompressible) domain.

Nous étudions dans cet article des effets de compression dans des milieux hétérogènes soumis à une contrainte d’entassement maximal. Partant des équations de Brinkman compressibles où la contrainte maximale est prise en compte au sein d’une pression et d’une viscosité de volume toutes deux singulières, nous montrons que les solutions faibles globales convergent (à une sous-suite près) vers des solutions faibles globales d’un modèle biphasique de type compressible/incompressible quand le paramètre ε, mesurant l’intensité de la résistance à la compression au voisinage de l’entassement maximal, tend vers 0. En fonction de la prédominance relative de la viscosité de volume par rapport à la pression dans les régimes denses, nous mettons en évidence l’activation d’effets de mémoire à la limite dans le domaine congestionné (incompressible).

Received : 2018-07-12
Accepted : 2019-06-06
Published online : 2019-06-19
DOI : https://doi.org/10.5802/jep.98
Classification:  35Q35,  35B25,  76T20
Keywords: Compressible Brinkman equations, maximal packing, singular limit, free boundary problem, memory effect
@article{JEP_2019__6__433_0,
     author = {Didier Bresch and \v S\'arka Ne\v casov\'a and Charlotte Perrin},
     title = {Compression effects in heterogeneous media},
     journal = {Journal de l'\'Ecole polytechnique --- Math\'ematiques},
     publisher = {\'Ecole polytechnique},
     volume = {6},
     year = {2019},
     pages = {433-467},
     doi = {10.5802/jep.98},
     zbl = {07070266},
     mrnumber = {3974475},
     language = {en},
     url = {https://jep.centre-mersenne.org/item/JEP_2019__6__433_0}
}
Bresch, Didier; Nečasová, Šárka; Perrin, Charlotte. Compression effects in heterogeneous media. Journal de l'École polytechnique — Mathématiques, Volume 6 (2019) , pp. 433-467. doi : 10.5802/jep.98. https://jep.centre-mersenne.org/item/JEP_2019__6__433_0/

[1] G. Allaire - “Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes”, Arch. Rational Mech. Anal. 113 (1990) no. 3, p. 209-259 | Article | MR 1079189 | Zbl 0724.76020

[2] B. Andreotti, Y. Forterre & O. Pouliquen - Granular media. Between fluid and solid, Cambridge University Press, Cambridge, 2013 | Zbl 1388.76001

[3] F. Berthelin - “Existence and weak stability for a pressureless model with unilateral constraint”, Math. Models Methods Appl. Sci. 12 (2002) no. 2, p. 249-272 | Article | MR 1892581 | Zbl 1027.35079

[4] F. Berthelin - “Theoretical study of a multi-dimensional pressureless model with unilateral constraint”, SIAM J. Math. Anal. 49 (2017) no. 3, p. 2287-2320 | Article | Zbl 1370.35216

[5] F. Bouchut, Y. Brenier, J. Cortes & J.-F. Ripoll - “A hierarchy of models for two-phase flows”, J. Nonlinear Sci. 10 (2000) no. 6, p. 639-660 | Article | MR 1799394 | Zbl 0998.76088

[6] D. Bresch & P.-E. Jabin - “Global existence of weak solutions for compresssible Navier–Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor”, Ann. of Math. (2) 188 (2018) no. 2, p. 577-684 | Article | Zbl 1405.35133

[7] D. Bresch, C. Perrin & E. Zatorska - “Singular limit of a Navier-Stokes system leading to a free/congested zones two-phase model”, Comptes Rendus Mathématique 352 (2014) no. 9, p. 685-690 | Article | MR 3258257 | Zbl 1302.35293

[8] D. Bresch & M. Renardy - “Development of congestion in compressible flow with singular pressure”, Asymptot. Anal. 103 (2017) no. 1-2, p. 95-101 | Article | MR 3657424 | Zbl 1378.35231

[9] P. Coussot - Rheometry of pastes, suspensions, and granular materials: applications in industry and environment, John Wiley & Sons, 2005 | Article

[10] R. Danchin & P. B. Mucha - “Compressible Navier-Stokes system: large solutions and incompressible limit”, Adv. Math. 320 (2017), p. 904-925 | Article | MR 3709125 | Zbl 1384.35058

[11] P. Degond, J. Hua & L. Navoret - “Numerical simulations of the Euler system with congestion constraint”, J. Comput. Phys. 230 (2011) no. 22, p. 8057-8088 | Article | MR 2835410 | Zbl 1408.76379

[12] P. Degond, P. Minakowski, L. Navoret & E. Zatorska - “Finite volume approximations of the Euler system with variable congestion”, Comput. & Fluids 169 (2018), p. 23-39 | Article | MR 3811030 | Zbl 1410.76216

[13] B. Desjardins, E. Grenier, P.-L. Lions & N. Masmoudi - “Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions”, J. Math. Pures Appl. (9) 78 (1999) no. 5, p. 461-471 | Article | MR 1697038 | Zbl 0992.35067

[14] L. Desvillettes, F. Golse & V. Ricci - “The mean-field limit for solid particles in a Navier-Stokes flow”, J. Statist. Phys. 131 (2008) no. 5, p. 941-967 | Article | MR 2398959 | Zbl 1154.76018

[15] B. Ducomet & Š. Nečasová - “On the 2D compressible Navier-Stokes system with density-dependent viscosities”, Nonlinearity 26 (2013) no. 6, p. 1783-1797 | Article | MR 3065932 | Zbl 1291.35174

[16] S. Énault - Modélisation de la propagation d’une tumeur en milieu faiblement compressible, ENS Lyon (2010), Ph. D. Thesis

[17] E. Feireisl - “Compressible Navier-Stokes equations with a non-monotone pressure law”, J. Differential Equations 184 (2002) no. 1, p. 97-108 | Article | MR 1929148 | Zbl 1012.76079

[18] E. Feireisl, Y. Lu & J. Málek - “On PDE analysis of flows of quasi-incompressible fluids”, Z. Angew. Math. Mech. 96 (2016) no. 4, p. 491-508 | Article | MR 3489305

[19] E. Feireisl & A. Novotný - Singular limits in thermodynamics of viscous fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, Cham, 2017 | Zbl 06755653

[20] X. Huang & J. Li - “Existence and blowup behavior of global strong solutions to the two-dimensional barotrpic compressible Navier-Stokes system with vacuum and large initial data”, J. Math. Pures Appl. (9) 106 (2016) no. 1, p. 123-154 | Article | MR 3505779 | Zbl 1342.35252

[21] A. Lefebvre - “Numerical simulation of gluey particles”, ESAIM Math. Model. Numer. Anal. 43 (2009) no. 1, p. 53-80 | Article | MR 2494794 | Zbl 1163.76056

[22] A. Lefebvre-Lepot & B. Maury - “Micro-macro modelling of an array of spheres interacting through lubrication forces”, Adv. Math. Sci. Appl. 21 (2011) no. 2, p. 535-557 | MR 2953131 | Zbl 1329.35136

[23] P.-L. Lions - Mathematical topics in fluid mechanics. Vol. 2: Compressible models, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998 | Zbl 0908.76004

[24] P.-L. Lions & N. Masmoudi - “Incompressible limit for a viscous compressible fluid”, J. Math. Pures Appl. (9) 77 (1998) no. 6, p. 585-627 | Article | MR 1628173 | Zbl 0937.35132

[25] P.-L. Lions & N. Masmoudi - “On a free boundary barotropic model”, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) no. 3, p. 373-410 | Article | MR 1687274

[26] B. Maury - “Prise en compte de la congestion dans les modeles de mouvements de foules”, 2012, Actes des colloques Caen, docplayer.fr/32954222

[27] B. Maury & A. Preux - “Pressureless Euler equations with maximal density constraint: a time-splitting scheme”, in Topological optimization and optimal transport, Radon Ser. Comput. Appl. Math., vol. 17, De Gruyter, Berlin, 2017, p. 333-355 | MR 3729382 | Zbl 1380.49040

[28] A. Mecherbet & M. Hillairet - “L p estimates for the homogenization of Stokes problem in a perforated domain”, J. Inst. Math. Jussieu (2018), 1–28 pages | Article

[29] H. Nasser El Dine - Étude mathématique et numérique pour le modèle Darcy-Brinkman pour les écoulements diphasiques en milieu poreux, Lebanese University-EDST; Ecole Centrale de Nantes (ECN) (2017), Ph. D. Thesis

[30] H. Nasser El Dine, M. Saad & R. Talhouk - “Existence results for a monophasic compressible Darcy–Brinkman’s flow in porous media”, J. Elliptic Parabol. Equ. 5 (2019) no. 1, p. 125-147 | Article | MR 3960055 | Zbl 07086145

[31] A. Novotný & I. Straškraba - Introduction to the mathematical theory of compressible flow, Oxford Lecture Series in Mathematics and its Applications, vol. 27, Oxford University Press, Oxford, 2004 | MR 2084891 | Zbl 1088.35051

[32] M. Perepelitsa - “On the global existence of weak solutions for the Navier-Stokes equations of compressible fluid flows”, SIAM J. Math. Anal. 38 (2006) no. 4, p. 1126-1153 | Article | MR 2274477 | Zbl 1116.76070

[33] C. Perrin - “Pressure-dependent viscosity model for granular media obtained from compressible Navier-Stokes equations”, Appl. Math. Res. Express. AMRX (2016) no. 2, p. 289-333 | Article | MR 3551778 | Zbl 06943990

[34] C. Perrin - “Modelling of phase transitions in granular flows”, in LMLFN 2015—low velocity flows—application to low Mach and low Froude regimes, ESAIM Proc. Surveys, vol. 58, EDP Sciences, Les Ulis, 2017, p. 78-97 | MR 3734210 | Zbl 1387.35461

[35] C. Perrin & E. Zatorska - “Free/congested two-phase model from weak solutions to multi-dimensional compressible Navier-Stokes equations”, Comm. Partial Differential Equations 40 (2015) no. 8, p. 1558-1589 | Article | MR 3355504 | Zbl 1331.35265

[36] B. Perthame, F. Quirós & J. L. Vázquez - “The Hele-Shaw asymptotics for mechanical models of tumor growth”, Arch. Rational Mech. Anal. 212 (2014) no. 1, p. 93-127 | Article | MR 3162474 | Zbl 1293.35347

[37] B. Perthame & N. Vauchelet - “Incompressible limit of a mechanical model of tumour growth with viscosity”, Philos. Trans. Roy. Soc. A 373 (2015), article ID 20140283, 16 pages | Article | MR 3393324 | Zbl 1353.35294

[38] V. A. Vaĭgant & A. V. Kazhikhov - “On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid”, Sibirsk. Mat. Zh. 36 (1995) no. 6, p. 1283-1316 | Article | MR 1375428

[39] N. Vauchelet & E. Zatorska - “Incompressible limit of the Navier-Stokes model with a growth term”, Nonlinear Anal. 163 (2017), p. 34-59 | Article | MR 3695967 | Zbl 1370.35234