Pré-Publication, Document De Travail Année : 2025

Asymptotic expansions of Stokes flows in finite periodic channels

Résumé

It is widely admitted in the heat transfer community that a steady laminar flow in a periodic channel becomes periodically developed after a small distance from the channel inlet. In this article, we provide a mathematical proof of this result for steady incompressible Stokes flows. Assuming the strip is a juxtaposition of $N$ periodic unit cells with a prescribed velocity at the inlet and a zero normal-stress boundary condition at the outlet, we prove that the velocity and pressure fields can be approximated up to exponentially small errors as $N\rightarrow +\infty$ by the sum of two terms: (i) a periodic velocity field driven by a constant pressure gradient propagating the input mass flow through the channel, and (ii) two boundary layer terms carrying zero mass and decaying exponentially away from the inlet and outlet boundaries. The periodically developed part of the flow can be computed by solving a periodic cell problem very similar to the one commonly encountered in periodic homogenization. The two boundary layer terms are solution to two Stokes problems posed in semi-infinite strips; they capture the transitory behavior between the inlet and outlet boundaries and the periodically developed region of the flow. We demonstrate the consistency of our mathematical result by numerical verifications on a number of two-dimensional flow geometries which showcase an excellent accuracy of the obtained asymptotic expansions. We complete our work by stating a homogenization result for a fluid flowing in a thin periodic strip of width $\epsilon$ with prescribed inlet and outlet boundary conditions. We obtain that the effective velocity and pressure fields are governed by Darcy's law with a permeability constant predicted by classical periodic homogenization. Moreover, we identify the effective boundary conditions satisfied by the homogenized flow field, which are (i) a prescribed normal velocity at the inlet and (ii) a zero pressure boundary condition at the outlet.

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Dates et versions

hal-04443652 , version 1 (07-02-2024)
hal-04443652 , version 2 (18-09-2024)
hal-04443652 , version 3 (25-02-2025)

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  • HAL Id : hal-04443652 , version 3

Citer

F Feppon. Asymptotic expansions of Stokes flows in finite periodic channels. 2025. ⟨hal-04443652v3⟩

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