Homogenization of an incompressible non-Newtonian flow through a thin porous medium

Homogenization of Bingham flow in thin porous media

“…Now, from the second estimate in (13) and Remark 4.3(i) in [3], we deduce the first estimate in (13). For the case a ε ≫ ε, proceeding similarly with Remark 4.3(ii) in [3], we obtain the desired result.…”

“…Recently, the model of thin porous medium under consideration in this paper was introduced in [15], where the flow of an incompressible viscous fluid described by the stationary Navier-Stokes equations was studied by the multiscale expansion method, which is a formal but powerful tool to analyse homogenization problems. These results were rigorously proved in [4] using an adaptation introduced in [3] of the periodic unfolding method from [12]. This adaptation consists of a combination of the unfolding method with a rescaling in the height variable, in order to work with a domain of fixed height, and to use monotonicity arguments to pass to the limit.…”

“…For the cases a ε ≈ ε or a ε ≪ ε, taking into account Remark 4.3(i) in [3], we obtain the second estimate in (13), and, consequently, from classical Korn's inequality we obtain the last estimate in (13). Now, from the second estimate in (13) and Remark 4.3(i) in [3], we deduce the first estimate in (13). For the case a ε ≫ ε, proceeding similarly with Remark 4.3(ii) in [3], we obtain the desired result.…”

“…This adaptation consists of a combination of the unfolding method with a rescaling in the height variable, in order to work with a domain of fixed height, and to use monotonicity arguments to pass to the limit. In [3], in particular, the flow of an incompressible stationary Stokes system with a nonlinear viscosity, being a power law, was studied. For non-stationary incompressible viscous flow in a thin porous medium see [1], where a non-stationary Stokes system is considered, and [2], where a non-stationary non-newtonian Stokes system, where the viscosity obeyed the power law, is studied.…”

“…(Stokes) There is a unique u ∈ V and a unique (up to an additive real constant) p ∈ L 2 (Π) such that (if < ·, · > is the dual pairing between (H −1 (Π)) 3 and (H 1 0 (Π)) 3 ) A fluid whose stress is not defined by relation (2) is called a non-newtonian fluid. There are several classes of non-newtonian fluids, as the power law, Carreau, Cross, Bingham fluids.…”