American Institute of Mathematical Sciences

Advanced
March  2010, 5(1): 63-95. doi: 10.3934/nhm.2010.5.63

Flows in porous media with erosion of the solid matrix

Leda Bucciantini 1, , Angiolo Farina 2, and  Antonio Fasano 2,

1. 

Università degli Studi di Firenze, Dipartimento di Fisica, Via Sansone 1, I-50019 Sesto Fiorentino (FI), Italy
2. 

Università degli Studi di Firenze, Dipartimento di Matematica "Ulisse Dini”, Viale Morgagni 67/A, I-50134 Firenze, Italy, Italy

Received  July 2009 Revised  October 2009 Published  February 2010

We consider the flow of an incompressible Newtonian fluid through an idealized porous medium consisting of an array of identical solid symmetric lamellae, whose profile varies in space and time due to a stress induced erosion process. The focus is on the influence of mass exchange between solid and fluid on the macroscopic flow. By means of the upscaling procedure illustrated in [6] we derive the governing system of equations for the macroscopic flow, encompassing various physical situations. We show that Darcy's law no longer applies in the classical sense. The corresponding mathematical problem turns out to be surprisingly complicated. Existence and uniqueness are proved. Numerical simulations are presented.
Keywords: PDE's, free boundary problems., homogenization, Darcy's law.
Mathematics Subject Classification: Primary: 76S05; Secondary: 35R3.
Citation: Leda Bucciantini, Angiolo Farina, Antonio Fasano. Flows in porous media with erosion of the solid matrix. Networks & Heterogeneous Media, 2010, 5 (1) : 63-95. doi: 10.3934/nhm.2010.5.63
[1]

Erik Kropat. Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 51-76. doi: 10.3934/naco.2017003
[2]

Evgeny Galakhov. Some nonexistence results for quasilinear PDE's. Communications on Pure & Applied Analysis, 2007, 6 (1) : 141-161. doi: 10.3934/cpaa.2007.6.141
[3]

Panos K. Palamides, Alex P. Palamides. Singular boundary value problems, via Sperner's lemma. Conference Publications, 2007, 2007 (Special) : 814-823. doi: 10.3934/proc.2007.2007.814
[4]

Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91
[5]

Juhi Jang, Ian Tice. Passive scalars, moving boundaries, and Newton's law of cooling. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1383-1413. doi: 10.3934/dcds.2016.36.1383
[6]

Dmitry Jakobson and Iosif Polterovich. Lower bounds for the spectral function and for the remainder in local Weyl's law on manifolds. Electronic Research Announcements, 2005, 11: 71-77.

[7]

Tohru Nakamura, Shuichi Kawashima. Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law. Kinetic & Related Models, 2018, 11 (4) : 795-819. doi: 10.3934/krm.2018032
[8]

Anatoli Babin, Alexander Figotin. Newton's law for a trajectory of concentration of solutions to nonlinear Schrodinger equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1685-1718. doi: 10.3934/cpaa.2014.13.1685
[9]

Arno Berger. Multi-dimensional dynamical systems and Benford's Law. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 219-237. doi: 10.3934/dcds.2005.13.219
[10]

Sebastián Ferrer, Francisco J. Molero. Andoyer's variables and phases in the free rigid body. Journal of Geometric Mechanics, 2014, 6 (1) : 25-37. doi: 10.3934/jgm.2014.6.25
[11]

Kim Dang Phung. Energy decay for Maxwell's equations with Ohm's law in partially cubic domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2229-2266. doi: 10.3934/cpaa.2013.12.2229
[12]

Ángela Jiménez-Casas, Aníbal Rodríguez-Bernal. PDE problems with concentrating terms near the boundary. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2147-2195. doi: 10.3934/cpaa.2020095
[13]

Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425
[14]

Fatiha Alabau-Boussouira. On the influence of the coupling on the dynamics of single-observed cascade systems of PDE's. Mathematical Control & Related Fields, 2015, 5 (1) : 1-30. doi: 10.3934/mcrf.2015.5.1
[15]

Dario Bambusi, Simone Paleari. Families of periodic orbits for some PDE’s in higher dimensions. Communications on Pure & Applied Analysis, 2002, 1 (2) : 269-279. doi: 10.3934/cpaa.2002.1.269
[16]

Armen Shirikyan, Leonid Volevich. Qualitative properties of solutions for linear and nonlinear hyperbolic PDE's. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 517-542. doi: 10.3934/dcds.2004.10.517
[17]

Ronald Mickens, Kale Oyedeji. Traveling wave solutions to modified Burgers and diffusionless Fisher PDE's. Evolution Equations & Control Theory, 2019, 8 (1) : 139-147. doi: 10.3934/eect.2019008
[18]

Theodore Kolokolnikov, Juncheng Wei. Hexagonal spike clusters for some PDE's in 2D. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020039
[19]

Avner Friedman. Free boundary problems arising in biology. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 193-202. doi: 10.3934/dcdsb.2018013
[20]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences