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Influence of the Reynolds number on non-Newtonian flow in thin porous media

Maria Anguiano, Matthieu Bonnivard, Francisco J. Suarez-Grau

TL;DR

This work analyzes how the Reynolds number affects the flow of a generalized Newtonian fluid through a thin perforated porous medium governed by the Carreau law. It combines homogenization with an unfolding method to derive effective two-dimensional Darcy laws and identifies a critical Reynolds number of order $1/\varepsilon$ (i.e., $\gamma=1$) that separates inertia-free and inertia-influenced regimes. For $\gamma\leq 1$, inertial effects vanish in the limit and the limit equations are linear or nonlinear Darcy laws depending on the flow index $r$ and the regime; for $\gamma>1$ inertia is anticipated to contribute to the homogenized problem. A Newton-type numerical method is developed to solve the nonlinear Darcy laws and validated on several cell problems and geometries, demonstrating practical applicability for predicting flow in thin porous media.

Abstract

We study the effect of the Reynolds number on the flow of a generalized Newtonian fluid through a thin porous medium in $\mathbb{R}^3$. This medium is a domain of thickness $\varepsilon \ll 1$, perforated by periodically distributed solid cylinders of size $\varepsilon$. We consider the nonlinear stationary Navier-Stokes system with viscosity following the Carreau law. Using tools from homogenization theory and assuming that the Reynolds number scales as $\varepsilon^{-γ}$, where $γ$ is a real constant, we prove the existence of a critical Reynolds number of order $1/\varepsilon$, in the sense that the inertial term in the Navier-Stokes system has no influence in the limit if the Reynolds number is of order smaller than or equal to $1/\varepsilon$ (i.e. $γ= 1$). In this case, we derive linear or nonlinear Darcy laws connecting velocity to pressure gradient. Conversely, we expect a contribution from the inertial term in the homogenized problem if the Reynolds number is greater than $1/\varepsilon$. Finally, we propose a numerical method to solve nonlinear Darcy laws describing effective flow in the critical case and demonstrate its practical applicability on several examples.

Influence of the Reynolds number on non-Newtonian flow in thin porous media

TL;DR

This work analyzes how the Reynolds number affects the flow of a generalized Newtonian fluid through a thin perforated porous medium governed by the Carreau law. It combines homogenization with an unfolding method to derive effective two-dimensional Darcy laws and identifies a critical Reynolds number of order (i.e., ) that separates inertia-free and inertia-influenced regimes. For , inertial effects vanish in the limit and the limit equations are linear or nonlinear Darcy laws depending on the flow index and the regime; for inertia is anticipated to contribute to the homogenized problem. A Newton-type numerical method is developed to solve the nonlinear Darcy laws and validated on several cell problems and geometries, demonstrating practical applicability for predicting flow in thin porous media.

Abstract

We study the effect of the Reynolds number on the flow of a generalized Newtonian fluid through a thin porous medium in . This medium is a domain of thickness , perforated by periodically distributed solid cylinders of size . We consider the nonlinear stationary Navier-Stokes system with viscosity following the Carreau law. Using tools from homogenization theory and assuming that the Reynolds number scales as , where is a real constant, we prove the existence of a critical Reynolds number of order , in the sense that the inertial term in the Navier-Stokes system has no influence in the limit if the Reynolds number is of order smaller than or equal to (i.e. ). In this case, we derive linear or nonlinear Darcy laws connecting velocity to pressure gradient. Conversely, we expect a contribution from the inertial term in the homogenized problem if the Reynolds number is greater than . Finally, we propose a numerical method to solve nonlinear Darcy laws describing effective flow in the critical case and demonstrate its practical applicability on several examples.
Paper Structure (19 sections, 12 theorems, 101 equations, 9 figures, 3 tables)

This paper contains 19 sections, 12 theorems, 101 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setting of the problem and main result
  3. Geometrical setting.
  4. Extra notation
  5. Statement of the problem.
  6. Proof of the main results
  7. A priori estimates on the velocity
  8. A priori estimates for the pressure
  9. Pseudoplastic or Newtonian fluids: case $1< r\leq 2$
  10. Dilatant fluids: case $r> 2$
  11. Adaptation of the unfolding method
  12. Variational formulation of the unfolded functions
  13. Proof of Theorem \ref{['mainthmPseudo']}. Pseudoplastic or Newtonian fluids: case $1< r\leq 2$
  14. Proof of Theorem \ref{['mainthmDilatant']}. Dilatant fluids: case $r>2$
  15. Numerical simulations
  16. ...and 4 more sections

Key Result

Theorem 2.2

Let $1< r\leq 2$, $\gamma\leq1$ and $C(r)$ be defined as follows: Then, there exist $\tilde{u}\in H^1_0(0,1;L^2(\omega)^3)$, with $\tilde{u}=0$ on $\omega\times \{0,1\}$ and $\tilde{u}_3\equiv 0$, and $\tilde{P}\in L^{C(r)}_0(\omega)$, such that the extension $(\tilde{u}_{\varepsilon},\tilde{P}_{\varepsilon})$ of a solution of (2) satisfies the following convergen Besides, define $V:\omega\righta

Figures (9)

  • Figure 1: Left: mesh of the 2D domain $Y_f'$ in the case of a circular-based cylindrical inclusion. Right: mesh of the lower half $Y_f'\times (0,1/2)$ of the 3D cell $Y_f$, obtained by extending the 2D mesh of $Y_f'$ in the $z$ direction using the FreeFem++ command buildlayers. The cell problem \ref{['LocalProblemNonNewtonian']} is solved in this half cell, using symmetric boundary condition on $y_3=1/2$. In this example, the 3D mesh contains $8736$ tetrahedra.
  • Figure 2: Magnitude of the solution $w_{\xi'}$ to the cell problem \ref{['LocalProblemNonNewtonian']}, with $\mu=0.1$, in the horizontal cross section $Y_f'\times\{1/2\}$ of the cell $Y_f$, in the case of circular-based cylindrical inclusions. The parameters of the Carreau law \ref{['Carreau']} are $r=1.3$, $\eta_0=1$, $\eta_\infty=0$. First line : $\lambda=1$, second line : $\lambda=1000$. Left column : $\xi'=(1,0)$, right column : $\xi'=(\sqrt{2}/2,\sqrt{2}/2)$.
  • Figure 3: Left: mesh of the 2D domain $Y_f'$ in the case of an elliptic-based cylindrical inclusion. Right: mesh of the lower half $Y_f'\times (0,1/2)$ of the 3D cell $Y_f$, obtained by extending the 2D mesh of $Y_f'$ in the $z$ direction using the FreeFem++ command buildlayers. The cell problem \ref{['LocalProblemNonNewtonian']} is solved in this half cell, using symmetric boundary condition on $y_3=1/2$. In this example, the 3D mesh contains $8106$ tetrahedra.
  • Figure 4: Magnitude of the solution $w_{\xi'}$ to the cell problem \ref{['LocalProblemNonNewtonian']}, with $\mu=0.1$, in the horizontal cross section $Y_f'\times\{1/2\}$ of the cell $Y_f$, in the case of elliptic-based cylindrical inclusions. The parameters of the Carreau law \ref{['Carreau']} are $r=1.3$, $\eta_0=1$, $\eta_\infty=0$. First line : $\lambda=1$, second line : $\lambda=1000$. Left column : $\xi'=(1,0)$, right column : $\xi'=(\sqrt{2}/2,\sqrt{2}/2)$.
  • Figure 5: Domain $\Omega_\varepsilon$ corresponding to $\omega=(0,1)\times (0,1/2)$ and $\varepsilon=0.1$, in the case of circular-based cylindrical inclusions.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Remark 1.1
  • Remark 2.1
  • Theorem 2.2: Pseudoplastic or Newtonian fluids
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5: Dilatant fluids
  • Remark 2.6
  • Lemma 3.1: Remark 4.3-(i) in Anguiano_SuarezGrau
  • Lemma 3.2
  • proof
  • ...and 20 more