Roughness-induced effects on the convection–diffusion–reaction problem in a thin domain

Abstract

In this paper, we investigate a convection–diffusion–reaction problem in a thin domain endowed with the Robin-type boundary condition describing the reaction catalyzed by the upper wall. Motivated by the microfluidic applications, we allow the oscillating behavior of the upper boundary and analyze the resonant case where the amplitude and period of the oscillation have the same small order as the domain's thickness. Depending on the magnitude of the reaction mechanism, we rigorously derive three different asymptotic models via the unfolding operator method. In particular, we identify the critical case in which the effects of the domain's geometry and all physically relevant processes become balanced.

Keywords:

• Convection–diffusion–reaction equation
• Robin boundary condition
• thin domain
• rough boundary
• unfolding method

AMS Subject classifications:

• 35B25
• 35B40
• 35J25

Acknowledgements

We also thank an anonymous referee for comments and observations that contributed to greatly improve this article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author (JCN) is partially supported by CNPq 141675/2015-2 (Brazil). The second one (IP) is supported by the Croatian Science Foundation (scientific project: Asymptotic analysis of the boundary value problems in continuum mechanics - AsAn) and the third author (MCP) by CNPq 303253/2017-7 and FAPESP 2017/02630-2 (Brazil).