An introduction to the homogenization modeling of non-Newtonian and electrokinetic flows in porous media
Résumé
The flow of complex fluids through porous media is common to many engineering applications. The upscaling is a powerful tool for modeling nonhomo-geneous media and we consider homogenization of quasi-Newtonian and electroki-netic flows through porous media. For the quasi-Newtonian polymeric fluids, the incompressible Navier-Stokes equations with the invariants dependent viscosity is supposed to hold the pore scale level. The 2-scale asymptotic expansions and the two-scale convergence of the monotone operators are applied to derive the reservoir level filtration law, given as a monotone relation between the filtration velocity and the pressure gradient. The second problem, we consider, is the quasi-static transport of an electrolyte through an electrically charged medium. The physical chemistry modeling is presented and used to get a dimensionless form of the problem. Next the equilibrium solutions are constructed through solving the Poisson-Boltzmann equation. For the solutions being close to the equilibrium, the two-scale convergence is applied to obtain the Onsager relations linking gradients of the pressure and of the chemical potentials to the filtration velocity and the ionic fluxes.
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