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A Darcy law with memory by homogenisation for evolving microstructure

David Wiedemann, Malte A. Peter

Abstract

We consider the homogenisation of the instationary Stokes equations in a porous medium with an a-priori given evolving microstructure. In order to pass to the homogenisation limit, we transform the Stokes equations to a domain with a fixed periodic microstructure. The homogenisation result is a Darcy-type equation with memory term and has the form of an integro-differential equation. The evolving microstructure leads to a time and space dependent permeability coefficient and the local change of the porosity causes an additional source term for the pressure.

A Darcy law with memory by homogenisation for evolving microstructure

Abstract

We consider the homogenisation of the instationary Stokes equations in a porous medium with an a-priori given evolving microstructure. In order to pass to the homogenisation limit, we transform the Stokes equations to a domain with a fixed periodic microstructure. The homogenisation result is a Darcy-type equation with memory term and has the form of an integro-differential equation. The evolving microstructure leads to a time and space dependent permeability coefficient and the local change of the porosity causes an additional source term for the pressure.
Paper Structure (16 sections, 19 theorems, 97 equations, 2 figures)

This paper contains 16 sections, 19 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The $\varepsilon$-scaled problem
  3. Reference structure
  4. Macroscopic domain:
  5. Reference pore geometry:
  6. The $\varepsilon$-scaled reference domains:
  7. Evolving microdomain
  8. Assumptions on the transformations:
  9. Two-scale limit domain:
  10. Transformation of the micromodel
  11. Transformation of the data:
  12. Existence and a-priori estimates for the microscopic problem
  13. Homogenisation of the substitute problem
  14. A Darcy law with memory for evolving microstructure
  15. Back-transformation of the two-pressure Stokes system
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $\hat{f}_\varepsilon$, $\hat{v}_{\Gamma_\varepsilon}$, $\hat{p}_b$ and $w_\varepsilon^\mathrm{in}$ by given as above. Then there exists a constant $C>0$ such that Moreover, $\hat{w}_\varepsilon^\mathrm{in}$ is compatible i.e. $\hat{w}_\varepsilon^\mathrm{in} \in H^1_{{\Gamma_\varepsilon}}(\Omega_\varepsilon;\mathbb{R}^d)$ and $\operatorname{div}(\hat{w}_\varepsilon^\mathrm{in}) = -\operatorna

Figures (2)

  • Figure 1: Illustration of amicroscopically evolving geometry at two different points in time
  • Figure 2: Two-scale transformation method

Theorems & Definitions (37)

  • Remark 2.2
  • Lemma 3.1: Uniform bounds of the transformed data
  • proof
  • Theorem 4.1: Existence, uniqueness and a-priori estimates of the solution of the Stokes equations
  • Proposition 4.2: Existence result for operator differential--algebraic equations
  • proof
  • Lemma 4.3: $\varepsilon$-scaled Poincaré inequality
  • proof
  • Lemma 4.4: $\varepsilon$-scaled right-inverse of the divergence operator
  • Lemma 4.5
  • ...and 27 more