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On the effects of surface roughness in non-isothermal porous medium flow

María Anguiano, Igor Pažanin, Francisco J. Suárez-Grau

TL;DR

This paper develops a rigorous homogenization framework for non-isothermal Darcy-Brinkman flow in a thin domain with a periodically rough boundary. By combining asymptotic analysis with a tailored periodic unfolding method, it identifies how roughness scale ℓ influences the coupling between flow and heat production via viscous dissipation. The authors derive limit models across three regimes: 0<ℓ<1, ℓ=1, and ℓ>1, each presenting a two-pressure reduced Stokes/Reynolds system coupled to a cell problem, and they show strong regime-dependent interactions absent in smooth domains. The results yield explicit effective descriptions involving the tensor A_M built from local cell problems, and they connect macroscopic Reynolds equations to microscopic geometry and heat transfer, providing insights for engineering designs involving rough porous surfaces.

Abstract

We analyze a non-isothermal Darcy-Brinkman thin-film flow with a periodically oscillating boundary and viscous dissipation acting as a heat source. Using asymptotic analysis and the periodic unfolding method, we establish the convergence of velocity, pressure, and temperature fields as the small parameter (related to the film thickness and the period of the roughness) tends to zero. The limit problems depend on the relative scaling of the roughness wavelength and consist of coupled elliptic systems combining Reynolds-type equations with Darcy-Brinkman cell problems and reduced energy equation. In the critical roughness regime, the effective model exhibits a strong coupling induced by the oscillatory geometry, which does not occur in a smooth-boundary case.

On the effects of surface roughness in non-isothermal porous medium flow

TL;DR

This paper develops a rigorous homogenization framework for non-isothermal Darcy-Brinkman flow in a thin domain with a periodically rough boundary. By combining asymptotic analysis with a tailored periodic unfolding method, it identifies how roughness scale ℓ influences the coupling between flow and heat production via viscous dissipation. The authors derive limit models across three regimes: 0<ℓ<1, ℓ=1, and ℓ>1, each presenting a two-pressure reduced Stokes/Reynolds system coupled to a cell problem, and they show strong regime-dependent interactions absent in smooth domains. The results yield explicit effective descriptions involving the tensor A_M built from local cell problems, and they connect macroscopic Reynolds equations to microscopic geometry and heat transfer, providing insights for engineering designs involving rough porous surfaces.

Abstract

We analyze a non-isothermal Darcy-Brinkman thin-film flow with a periodically oscillating boundary and viscous dissipation acting as a heat source. Using asymptotic analysis and the periodic unfolding method, we establish the convergence of velocity, pressure, and temperature fields as the small parameter (related to the film thickness and the period of the roughness) tends to zero. The limit problems depend on the relative scaling of the roughness wavelength and consist of coupled elliptic systems combining Reynolds-type equations with Darcy-Brinkman cell problems and reduced energy equation. In the critical roughness regime, the effective model exhibits a strong coupling induced by the oscillatory geometry, which does not occur in a smooth-boundary case.

Paper Structure

This paper contains 10 sections, 10 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and problem description
  3. Adaptation of the unfolding method
  4. Convergence results
  5. Limit problems
  6. The case $0<\ell<1$
  7. The case $\ell=1$
  8. The case $\ell>1$
  9. Estimates for velocity and temperature.
  10. Convergences for velocity and temperature.

Key Result

Proposition 2.2

Assume $q\in(1,3/2)$ and let $(\widetilde{\bf u}^\varepsilon, \widetilde{T}_\varepsilon)$ be a solution of the dilated problem (system_1_dil)-(BC_dil). Then, there hold the following estimates:

Theorems & Definitions (22)

  • Remark 2.1
  • Proposition 2.2: Proposition 1 in Pazanin_SG_Darcy
  • Remark 2.3: Extension of $\widetilde{\bf u}_\varepsilon$ and $\widetilde{T}_\varepsilon$ to $\Omega=\omega\times (0, h_{\rm max})$
  • Proposition 2.4: Proposition 2 and Corollary 1 in Pazanin_SG_Darcy
  • proof
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 4.1: Convergence of the velocity
  • proof
  • ...and 12 more