Compression effects in heterogeneous media

Bresch, Didier; Nečasová, Šárka; Perrin, Charlotte

Journal de l’École polytechnique — Mathématiques, Tome 6 (2019), pp. 433-467.

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 $\epsilon$, 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).

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 $\epsilon$ 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.

Classification : 35Q35, 35B25, 76T20
Keywords: Compressible Brinkman equations, maximal packing, singular limit, free boundary problem, memory effect
Mot clés : Équations de Brinkman compressibles, contrainte d’entassement maximal, limite singulière, problème à frontière libre, effet mémoire

[1] Allaire, Grégoire 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., Volume 113 (1990) no. 3, pp. 209-259 | DOI | MR | Zbl

[2] Andreotti, Bruno; Forterre, Yoël; Pouliquen, Olivier Granular media. Between fluid and solid, Cambridge University Press, Cambridge, 2013 | Zbl

[3] Berthelin, Florent Existence and weak stability for a pressureless model with unilateral constraint, Math. Models Methods Appl. Sci., Volume 12 (2002) no. 2, pp. 249-272 | DOI | MR | Zbl

[4] Berthelin, Florent Theoretical study of a multi-dimensional pressureless model with unilateral constraint, SIAM J. Math. Anal., Volume 49 (2017) no. 3, pp. 2287-2320 | DOI | Zbl

[5] Bouchut, F.; Brenier, Y.; Cortes, J.; Ripoll, J.-F. A hierarchy of models for two-phase flows, J. Nonlinear Sci., Volume 10 (2000) no. 6, pp. 639-660 | DOI | MR | Zbl

[6] Bresch, Didier; Jabin, P.-E. Global existence of weak solutions for compresssible Navier–Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor, Ann. of Math. (2), Volume 188 (2018) no. 2, pp. 577-684 | DOI | Zbl

[7] Bresch, Didier; Perrin, Charlotte; Zatorska, Ewelina Singular limit of a Navier-Stokes system leading to a free/congested zones two-phase model, Comptes Rendus Mathématique, Volume 352 (2014) no. 9, pp. 685-690 | DOI | MR | Zbl

[8] Bresch, Didier; Renardy, Michael Development of congestion in compressible flow with singular pressure, Asymptot. Anal., Volume 103 (2017) no. 1-2, pp. 95-101 | DOI | MR | Zbl

[9] Coussot, Philippe Rheometry of pastes, suspensions, and granular materials: applications in industry and environment, John Wiley & Sons, 2005 | DOI

[10] Danchin, Raphaël; Mucha, Piotr Bogusław Compressible Navier-Stokes system: large solutions and incompressible limit, Adv. Math., Volume 320 (2017), pp. 904-925 | DOI | MR | Zbl

[11] Degond, Pierre; Hua, Jiale; Navoret, Laurent Numerical simulations of the Euler system with congestion constraint, J. Comput. Phys., Volume 230 (2011) no. 22, pp. 8057-8088 | DOI | MR | Zbl

[12] Degond, Pierre; Minakowski, Piotr; Navoret, Laurent; Zatorska, Ewelina Finite volume approximations of the Euler system with variable congestion, Comput. & Fluids, Volume 169 (2018), pp. 23-39 | DOI | MR | Zbl

[13] Desjardins, B.; Grenier, E.; Lions, Pierre-Louis; Masmoudi, Nader Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl. (9), Volume 78 (1999) no. 5, pp. 461-471 | DOI | MR | Zbl

[14] Desvillettes, Laurent; Golse, François; Ricci, Valeria The mean-field limit for solid particles in a Navier-Stokes flow, J. Statist. Phys., Volume 131 (2008) no. 5, pp. 941-967 | DOI | MR | Zbl

[15] Ducomet, Bernard; Nečasová, Šárka On the 2D compressible Navier-Stokes system with density-dependent viscosities, Nonlinearity, Volume 26 (2013) no. 6, pp. 1783-1797 | DOI | MR | Zbl

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

[17] Feireisl, Eduard Compressible Navier-Stokes equations with a non-monotone pressure law, J. Differential Equations, Volume 184 (2002) no. 1, pp. 97-108 | DOI | MR | Zbl

[18] Feireisl, Eduard; Lu, Yong; Málek, Josef On PDE analysis of flows of quasi-incompressible fluids, Z. Angew. Math. Mech., Volume 96 (2016) no. 4, pp. 491-508 | DOI | MR

[19] Feireisl, Eduard; Novotný, Antonín Singular limits in thermodynamics of viscous fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, Cham, 2017 | Zbl

[20] Huang, Xiangdi; Li, Jing 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), Volume 106 (2016) no. 1, pp. 123-154 | DOI | MR | Zbl

[21] Lefebvre, Aline Numerical simulation of gluey particles, ESAIM Math. Model. Numer. Anal., Volume 43 (2009) no. 1, pp. 53-80 | DOI | MR | Zbl

[22] Lefebvre-Lepot, Aline; Maury, Bertrand Micro-macro modelling of an array of spheres interacting through lubrication forces, Adv. Math. Sci. Appl., Volume 21 (2011) no. 2, pp. 535-557 | MR | Zbl

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

[24] Lions, Pierre-Louis; Masmoudi, Nader Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl. (9), Volume 77 (1998) no. 6, pp. 585-627 | DOI | MR | Zbl

[25] Lions, Pierre-Louis; Masmoudi, Nader On a free boundary barotropic model, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 16 (1999) no. 3, pp. 373-410 | DOI | MR

[26] Maury, Bertrand Prise en compte de la congestion dans les modeles de mouvements de foules, 2012 (Actes des colloques Caen, docplayer.fr/32954222)

[27] Maury, Bertrand; Preux, A. Pressureless Euler equations with maximal density constraint: a time-splitting scheme, Topological optimization and optimal transport (Radon Ser. Comput. Appl. Math.), Volume 17, De Gruyter, Berlin, 2017, pp. 333-355 | MR | Zbl

[28] Mecherbet, Amina; Hillairet, Matthieu $L^p$ estimates for the homogenization of Stokes problem in a perforated domain, J. Inst. Math. Jussieu (2018), p. 1–28 | DOI

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

[30] Nasser El Dine, Houssein; Saad, Mazen; Talhouk, Raafat Existence results for a monophasic compressible Darcy–Brinkman’s flow in porous media, J. Elliptic Parabol. Equ., Volume 5 (2019) no. 1, pp. 125-147 | DOI | MR | Zbl

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

[32] Perepelitsa, Mikhail On the global existence of weak solutions for the Navier-Stokes equations of compressible fluid flows, SIAM J. Math. Anal., Volume 38 (2006) no. 4, pp. 1126-1153 | DOI | MR | Zbl

[33] Perrin, Charlotte Pressure-dependent viscosity model for granular media obtained from compressible Navier-Stokes equations, Appl. Math. Res. Express. AMRX (2016) no. 2, pp. 289-333 | DOI | MR | Zbl

[34] Perrin, Charlotte Modelling of phase transitions in granular flows, LMLFN 2015—low velocity flows—application to low Mach and low Froude regimes (ESAIM Proc. Surveys), Volume 58, EDP Sciences, Les Ulis, 2017, pp. 78-97 | MR | Zbl

[35] Perrin, Charlotte; Zatorska, Ewelina Free/congested two-phase model from weak solutions to multi-dimensional compressible Navier-Stokes equations, Comm. Partial Differential Equations, Volume 40 (2015) no. 8, pp. 1558-1589 | DOI | MR | Zbl

[36] Perthame, Benoît; Quirós, Fernando; Vázquez, Juan Luis The Hele-Shaw asymptotics for mechanical models of tumor growth, Arch. Rational Mech. Anal., Volume 212 (2014) no. 1, pp. 93-127 | DOI | MR | Zbl

[37] Perthame, Benoît; Vauchelet, Nicolas Incompressible limit of a mechanical model of tumour growth with viscosity, Philos. Trans. Roy. Soc. A, Volume 373 (2015), 20140283, 16 pages | DOI | MR | Zbl

[38] Vaĭgant, V. A.; Kazhikhov, A. V. On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid, Sibirsk. Mat. Zh., Volume 36 (1995) no. 6, pp. 1283-1316 | DOI | MR

[39] Vauchelet, Nicolas; Zatorska, Ewelina Incompressible limit of the Navier-Stokes model with a growth term, Nonlinear Anal., Volume 163 (2017), pp. 34-59 | DOI | MR | Zbl