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

Abstract

Summary We consider the flow of generalized Newtonian fluid through a thin porous media. The media under consideration is a bounded perforated three dimensional domain confined between two parallel plates, where the distance between the plates is described by a small parameter $\varepsilon$. The perforation consists in an array of solid cylinders, which connect the plates in perpendicular direction, with diameter of size $\varepsilon$ and distributed periodically with period $\varepsilon$. The flow is described by the three dimensional incompressible stationary Stokes system with a nonlinear viscosity following the Carreau law. We study the limit when the thickness tends to zero and prove that the averaged velocity satisfies a nonlinear two-dimensional homogenized law of Carreau type. We illustrate our homogenization result by numerical simulations showing the influence of the Carreau law on the behavior of the limit system, in the case where the flow is driven by a constant pressure gradient and for different geometries of perforations.

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

“…Lemma 5.2 (Velocity estimates). Let ṽ a weak solution of (7). Then: (i) In the case HTPM (δ > 1), there exists a positive constant C, independent of , such that we have…”

“…We consider ϕ(x , z) ∈ D(ω; C ∞ # (Z) 3 ) with ϕ(x , z) = 0 in ω × S. Considering ϕ (x , z 3 ) = ϕ(x , x / δ , z 3 ) as test function in (7), integrating by parts and taking into account the extension of ṽ and P , we have…”

“…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].…”

Hydrodynamics of Incompressible Creeping Couple Stress Fluid Through a Uniformly Porous Channel in Presence of Darcy Resistance: Exact Solution and Physical Insights Using the Inverse Method Approach

“…Anguiano et al [5] investigated the flow of Carreau's law refers to the nonfluid flow through a thin porous medium.…”

On p-Laplacian Reaction–Diffusion Problems with Dynamical Boundary Conditions in Perforated Media

“…Another possible way is to study this parabolic model in a thin porous media (see, for instance, [2,8,10,12,42,43] for more details on the importance of this type of domains). Finally, another problem could be to consider a porous media containing a thin fissure.…”