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Homogenized limits of Stokes flow and advective transport in thin perforated domains

Markus Gahn, Vlad Revnic

TL;DR

This work delivers a rigorous framework for simultaneous homogenization and dimension reduction of Stokes flow and advection-diffusion transport in thin, ε-periodic perforated layers with thickness ε^α (0<α<1). By developing a two-scale convergence theory adapted to thin layers and constructing a pressure Bogovskii operator in perforated geometries, the authors derive macroscopic Darcy-type equations for the fluid and diffusion-advection equations for transport, with effective coefficients obtained from cell problems. Two distinct macroscopic transport models emerge depending on the diffusion scaling: a fully anisotropic diffusion with vertical advection in the high-diffusion case, and a predominantly vertical diffusion with vertical advection in the slow-horizontal-diffusion case. Special geometries, such as cylindrical inclusions, are treated to illustrate reduced-dimensional coupling in the Darcy velocity. Overall, the results provide systematic procedures and explicit effective coefficients for modeling flow and transport in thin porous layers across complex micro-geometries.

Abstract

We deal with the rigorous homogenization and dimension reduction of flow and transport problems posed in thin $\varepsilon$-periodic perforated layers with thickness of order $\varepsilon^α$ with $α\in (0,1)$ and therefore the thickness of the layer is large compared its porosity. The aim is the derivation of effective models for $\varepsilon\to 0 $, when the thickness of the layer tends to zero. For the flow problem we consider incompressible Stokes equations with a pressure boundary condition on the top/bottom of the layer, and the transport problem is given by reaction-diffusion-advection problem with advective flow governed from the fluid velocity from the Stokes model and different scalings for the diffusion coefficient modelling low and fast diffusion in the horizontal direction. In the limit, a Darcy-type law is obtained for the Stokes flow with Darcy-velocity depending only on the derivative of the Darcy-pressure in the vertical direction. The effective equation for the transport problem is again of diffusion-advection-type including homogenized coefficients, and with advective flow given by the Darcy-velocity and only taking place in the vertical direction. In the case of slow diffusion in the vertical direction, effective diffusion only takes place in the vertical direction, where in the case of high diffusion in horizontal direction, we obtain effective diffusion in all space directions. To pass to the limit we use the method of two-scale convergence adapted to our microscopic geometry, which is based on uniform a priori estimates. Critical parts in the derivation of the macro-models are the control of the fluid pressure, for which we construct a Bogovskii-operator for thin perforated domains, as well as the strong two-scale convergence for the microscopic solution of the transport equation, necessary to pass to the limit in the advective term.

Homogenized limits of Stokes flow and advective transport in thin perforated domains

TL;DR

This work delivers a rigorous framework for simultaneous homogenization and dimension reduction of Stokes flow and advection-diffusion transport in thin, ε-periodic perforated layers with thickness ε^α (0<α<1). By developing a two-scale convergence theory adapted to thin layers and constructing a pressure Bogovskii operator in perforated geometries, the authors derive macroscopic Darcy-type equations for the fluid and diffusion-advection equations for transport, with effective coefficients obtained from cell problems. Two distinct macroscopic transport models emerge depending on the diffusion scaling: a fully anisotropic diffusion with vertical advection in the high-diffusion case, and a predominantly vertical diffusion with vertical advection in the slow-horizontal-diffusion case. Special geometries, such as cylindrical inclusions, are treated to illustrate reduced-dimensional coupling in the Darcy velocity. Overall, the results provide systematic procedures and explicit effective coefficients for modeling flow and transport in thin porous layers across complex micro-geometries.

Abstract

We deal with the rigorous homogenization and dimension reduction of flow and transport problems posed in thin -periodic perforated layers with thickness of order with and therefore the thickness of the layer is large compared its porosity. The aim is the derivation of effective models for , when the thickness of the layer tends to zero. For the flow problem we consider incompressible Stokes equations with a pressure boundary condition on the top/bottom of the layer, and the transport problem is given by reaction-diffusion-advection problem with advective flow governed from the fluid velocity from the Stokes model and different scalings for the diffusion coefficient modelling low and fast diffusion in the horizontal direction. In the limit, a Darcy-type law is obtained for the Stokes flow with Darcy-velocity depending only on the derivative of the Darcy-pressure in the vertical direction. The effective equation for the transport problem is again of diffusion-advection-type including homogenized coefficients, and with advective flow given by the Darcy-velocity and only taking place in the vertical direction. In the case of slow diffusion in the vertical direction, effective diffusion only takes place in the vertical direction, where in the case of high diffusion in horizontal direction, we obtain effective diffusion in all space directions. To pass to the limit we use the method of two-scale convergence adapted to our microscopic geometry, which is based on uniform a priori estimates. Critical parts in the derivation of the macro-models are the control of the fluid pressure, for which we construct a Bogovskii-operator for thin perforated domains, as well as the strong two-scale convergence for the microscopic solution of the transport equation, necessary to pass to the limit in the advective term.

Paper Structure

This paper contains 18 sections, 24 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. The microscopic models and main results
  4. The fluid problem
  5. The transport problem
  6. The microscopic domain
  7. The two-scale convergence for thin heterogeneous layers
  8. The fluid problem
  9. A priori estimates for the microscopic solutions $v_{\varepsilon,\alpha}$ and $p_{\varepsilon,\alpha}$
  10. Two-scale compactness for the microscopic solutions $v_{\varepsilon,\alpha}$ and $p_{\varepsilon,\alpha}$
  11. Derivation of the macroscopic model
  12. The case of cylindrical inclusions $Y_s$
  13. The transport problem
  14. A priori estimates for the microscopic solution $c_{\varepsilon,\alpha}$
  15. Two-scale compactness for the microscopic solutions $c_{\varepsilon,\alpha}$
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $(u_{\varepsilon,\alpha},p_{\varepsilon,\alpha})$ be the weak solution of the microscopic problem eq:Stokes_micro_strong. Then, there exists $u_0 \in L^2(\Omega,H_{\mathrm{per}}^1(Y))^n$ with $u_0 = 0$ in $Y\setminus Y_f$ and $\nabla_y \cdot u_0=0$ in $\Omega\times Y$, and $p_0 \in L^2(\Omega)$

Theorems & Definitions (49)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Definition 1
  • Remark 3.1
  • Lemma 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 39 more