Homogenization and simulation of heat transfer through a thin grain layer

Tom Freudenberg; Michael Eden; Tom Freudenberg; Michael Eden

doi:10.3934/nhm.2024025

Networks and Heterogeneous Media

2024, Volume 19, Issue 2: 569-596. doi: 10.3934/nhm.2024025

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Research article

Homogenization and simulation of heat transfer through a thin grain layer

Tom Freudenberg ^{1
,
,},
Michael Eden ²

1.
Center for Industrial Mathematics, University of Bremen, Bibliothekstraße 5, 28359 Bremen, Germany
2.
Department of Mathematics and Computer Science, Karlstad University, Sweden

Received: 05 December 2023 Revised: 15 March 2024 Accepted: 04 April 2024 Published: 05 June 2024

We investigated the effective influence of grain structures on the heat transfer between a fluid and solid domain using mathematical homogenization. The presented model consists of heat equations inside the different domains, coupled through either perfect or imperfect thermal contact. The size and the period of the grains are of order $\varepsilon$, therefore forming a thin layer. The equation parameters inside the grains also depend on $\varepsilon$. We considered two distinct scenarios: Case (a), where the grains are disconnected, and Case (b), where the grains form a connected geometry but in a way such that the fluid and solid are still in contact. In both cases, we determined the effective differential equations for the limit $\varepsilon \to 0$ via the concept of two-scale convergence for thin layers. We also presented and studied a numerical algorithm to solve the homogenized problem.
- Homogenization,
- mathematical modeling,
- effective interface conditions,
- numerical simulations
Citation: Tom Freudenberg, Michael Eden. Homogenization and simulation of heat transfer through a thin grain layer[J]. Networks and Heterogeneous Media, 2024, 19(2): 569-596. doi: 10.3934/nhm.2024025

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Abstract

We investigated the effective influence of grain structures on the heat transfer between a fluid and solid domain using mathematical homogenization. The presented model consists of heat equations inside the different domains, coupled through either perfect or imperfect thermal contact. The size and the period of the grains are of order $\varepsilon$, therefore forming a thin layer. The equation parameters inside the grains also depend on $\varepsilon$. We considered two distinct scenarios: Case (a), where the grains are disconnected, and Case (b), where the grains form a connected geometry but in a way such that the fluid and solid are still in contact. In both cases, we determined the effective differential equations for the limit $\varepsilon \to 0$ via the concept of two-scale convergence for thin layers. We also presented and studied a numerical algorithm to solve the homogenized problem.

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