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Asymptotic behavior of a Bingham Flow in thin domains with rough boundary

Giuseppe Cardone, Carmen Perugia, Manuel Villanueva Pesqueira

Abstract

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter $ε$ denotes the thickness of the domain and the roughness periodicity of the boundary. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when $ε$ tends to zero. We obtain the homogenized limit problem for the velocity and the pressure, which preserves the nonlinear character of the flow, and study the effects of the microstructure in the corresponding effective equations. Finally, we give the interpretation of the limit problem in terms of a non linear Darcy law.

Asymptotic behavior of a Bingham Flow in thin domains with rough boundary

Abstract

We consider an incompressible Bingham flow in a thin domain with rough boundary, under the action of given external forces and with no-slip boundary condition on the whole boundary of the domain. In mathematical terms, this problem is described by non linear variational inequalities over domains where a small parameter denotes the thickness of the domain and the roughness periodicity of the boundary. By using an adapted linear unfolding operator we perform a detailed analysis of the asymptotic behavior of the Bingham flow when tends to zero. We obtain the homogenized limit problem for the velocity and the pressure, which preserves the nonlinear character of the flow, and study the effects of the microstructure in the corresponding effective equations. Finally, we give the interpretation of the limit problem in terms of a non linear Darcy law.

Paper Structure

This paper contains 8 sections, 10 theorems, 118 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The setting of the problem
  3. A priori estimates
  4. The unfolding operator
  5. Some convergence results for the velocity
  6. Convergence results for the pressure
  7. The limit problem
  8. Conclusions

Key Result

Lemma 4

For any fixed $\epsilon$, let $({\bf u}_\epsilon, p_\epsilon)$ be the solution of vpfluid. Under the assumption Hf, the following estimates hold with $C$ a positive constant independent of $\epsilon$.

Figures (2)

  • Figure 1: Thin domain with oscillating periodic boundary
  • Figure 2: Representative cell

Theorems & Definitions (16)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 4
  • Definition 5
  • Proposition 6
  • Remark 7
  • Proposition 8
  • Proposition 9
  • Proposition 10
  • ...and 6 more