On the Flow of a Viscoplastic Fluid in a Thin Periodic Domain

Abstract

We study in this paper the steady incompressible nonlinear flow of a Bingham fluid in a thin periodic domain, which is a model of porous media. The model of thin porous media of thickness much smaller than the parameter of periodicity was introduced in [Zh08], where a stationary incompressible Navier-Stokes flow was studied. Recently, the model of the thin porous medium under consideration in this paper was introduced in [FaEtAl16], where the flow of an incompressible viscous fluid described by the stationary Navier-Stokes equations was studied by the multiscale asymptotic expansion method, which is a formal but powerful tool to analyse homogenization problems. These results were rigorously proved in [AS18] using an adaptation (introduced in [AS17]), of the unfolding method from [CiEtAl08]. 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 [AS17], in particular, the flow of an incompressible stationary Stokes system with a nonlinear viscosity, being a power law, was studied. For nonstationary incompressible viscous fluid flow in a thin porous medium we refer to [An17], where a nonstationary Stokes system is considered, and [An217], where a nonstationary non-Newtonian Stokes system, where the viscosity obeys the power law, is studied. For the unfolding method applied to the study of problems stated in other type of thin periodic domains we refer for instance to [Gr04] for crane type structures and to [GrEtAl17] for thin layers with thin beams structures, where elasticity problems are studied.

On the Bingham Flow in a Thin Y-Like Shaped Structure

“…In the recent paper [5], the authors present a model of Bingham flow in a thin periodic domain which contains an array of obstacles modelized as vertical cylinders. Up to our knowledge, the first convergence result on Bingham flow in thin periodic domains is due to [1], followed by [2]. We refer to [18] for Bingham flows in periodic domains of infinite length.…”

Sharp Pressure Estimates for the Navier–Stokes System in Thin Porous Media

“…Theorem 4.3 (Darcy's law for VTPM). Consider δ ∈ (0, 1), γ ≤ 1 and C(δ) defined by (8). Then, there exists ṽ ∈ H 1 (0, 1; L 2 (ω) 3 ) with ṽ3 = 0 and ṽ = 0 on z 3 = {0, 1}, and P ∈ L C(1) 0…”

“…Studies related to Newtonian fluids through TPM can be found in Anguiano and Suárez-Grau [11,14], Armiti-Juber [16], Bayada and et al [17], Larsson et al [26], Suárez-Grau [31], Valizadeh and Rudman [32], Wagner et al [33] and Zhengan and Hongxing [36]. Concerning generalized Newtonian fluids see Anguiano and Suárez-Grau [7,12,15], for Bingham fluids see Anguiano and Bunoiu [8,9], for compressible and piezo-viscous flow see Pérez-Ràfols et al [27], and for micropolar fluids see Suárez-Grau [30] and for diffusion problems see Anguiano [5,6] and Bunoiu and Timofte [21].…”

Carreau law for non-newtonian fluid flow through a thin porous media

“…In particular, the generalized Newtonian fluids obeying the power law in the thin porous media Ω ε have been studied rigorously in Anguiano and Suárez-Grau [10] where we have obtained a 2D Darcy's law when the domain thickness tends to zero (see also [15] for the extension to the case of a thin porous media with an array of cylinders with small diameter). Also, the Bingham plastic behavior in the thin porous media Ω ε has been studied in [8,9]. For other studies concerning thin porous media, we refer to Anguiano [3,4,5,6,7], Anguiano and Suárez-Grau [11,12,14], Jouybari and T. S. Lundström [29], Prat and Agaësse [33], Suárez-Grau [35], Yeghiazarian et al [36] and Zhengan and Hongxing [37].…”