Homogenization of MHD flows in porous media

The paper is concerned with the homogenization of a nonlinear differential system describing the flow of an electrically conducting, incompressible and viscous Newtonian fluid through a periodic porous medium, in the presence of a magnetic field. We introduce a variational formulation of the differe...

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Bibliographic Details
Published in:Journal of Differential Equations 2022-12, Vol.339, p.90-133
Main Authors: Amirat, Youcef, Hamdache, Kamel, Shelukhin, Vladimir V.
Format: Article
Language:English
Subjects:
Analysis of PDEs
Averaging of a nonlinear differential system
Electromagnetism
Engineering Sciences
Fluids mechanics
Homogenization
Magnetohydrodynamic (MHD) flow in porous media
Mathematics
Maxwell equations
Mechanics
Stokes equations
Two-scale equations
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Summary:The paper is concerned with the homogenization of a nonlinear differential system describing the flow of an electrically conducting, incompressible and viscous Newtonian fluid through a periodic porous medium, in the presence of a magnetic field. We introduce a variational formulation of the differential system equipped with boundary conditions. We show the existence of a solution of the variational problem, and derive uniform estimates of the solutions depending on the characteristic parameters of the flow. Using the two-scale convergence method, we rigorously derive a two-scale equation for the two-scale current density, and a two-pressure Stokes system. We derive, in the case of constant magnetic permeability, an explicit relation expressing the macroscopic velocity as a function of the macroscopic Lorentz force, the pressure gradient, the external body force, and the macroscopic current density, via two permeability filtration tensors. When the magnetic field is absent, this relation reduces to the Darcy law.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2022.08.014