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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
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