Homogenized limits of Stokes flow and advective transport in thin perforated domains

Markus Gahn; Vlad Revnic; Markus Gahn; Vlad Revnic

doi:10.3934/nhm.2026034

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 801-847. doi: 10.3934/nhm.2026034

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Homogenized limits of Stokes flow and advective transport in thin perforated domains

Markus Gahn ¹,
Vlad Revnic ^{2
,
,}

1.
University of Augsburg, Institute of Mathematics, Universitätsstraße 12a, 86159 Augsburg, Germany
2.
Heidelberg University, Institute for Mathematics, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

Received: 03 December 2025 Revised: 14 April 2026 Accepted: 16 April 2026 Published: 29 April 2026

We deal with the rigorous homogenization and dimension reduction of flow and transport problems posed in thin $ \varepsilon $-periodic perforated layers wit a thickness of order $ \varepsilon^{\alpha} $ with $ \alpha \in (0, 1) $. Therefore the thickness of the layer is large compared with its porosity. The aim is the derivation of effective models for $ \varepsilon\to 0 $, when the thickness of the layer tends to zero. For the flow problem, we consider incompressible Stokes equations with a pressure boundary condition on the top/bottom of the layer. The transport problem is given by reaction–diffusion–advection problem with advective flow governed by the fluid velocity from the Stokes model. Furthermore, we treat different scalings for the diffusion coefficient modelling low and fast diffusion in the horizontal direction. In the limit, a Darcy-type law is obtained for the Stokes flow with the Darcy velocity depending only on the derivative of the Darcy pressure in the vertical direction. The effective equation for the transport problem is again one of the diffusion advection-type including homogenized coefficients, and with advective flow given by the Darcy velocity and only taking place in the vertical direction. In the case of slow diffusion in the vertical direction, effective diffusion only takes place in the vertical direction, where, in the case of high diffusion in the horizontal direction, we obtain effective diffusion in all space directions. To pass to the limit, we use the method of two-scale convergence adapted to our microscopic geometry, which is based on uniform a priori estimates. Critical parts in the derivation of the macro-models are the control of the fluid pressure, for which we construct a Bogovskii operator for thin perforated domains with arbitrary boundary conditions on the top/bottom, and the strong two-scale convergence for the microscopic solution of the transport equation, which is necessary to pass to the limit in the advective term. This strong convergence is established by using a Kolmogorov–Simon compactness argument.
- homogenization,
- dimension reduction,
- Stokes equation,
- two-scale convergence,
- reaction–diffusion–advection equation
Citation: Markus Gahn, Vlad Revnic. Homogenized limits of Stokes flow and advective transport in thin perforated domains[J]. Networks and Heterogeneous Media, 2026, 21(3): 801-847. doi: 10.3934/nhm.2026034

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Abstract

We deal with the rigorous homogenization and dimension reduction of flow and transport problems posed in thin $ \varepsilon $-periodic perforated layers wit a thickness of order $ \varepsilon^{\alpha} $ with $ \alpha \in (0, 1) $. Therefore the thickness of the layer is large compared with its porosity. The aim is the derivation of effective models for $ \varepsilon\to 0 $, when the thickness of the layer tends to zero. For the flow problem, we consider incompressible Stokes equations with a pressure boundary condition on the top/bottom of the layer. The transport problem is given by reaction–diffusion–advection problem with advective flow governed by the fluid velocity from the Stokes model. Furthermore, we treat different scalings for the diffusion coefficient modelling low and fast diffusion in the horizontal direction. In the limit, a Darcy-type law is obtained for the Stokes flow with the Darcy velocity depending only on the derivative of the Darcy pressure in the vertical direction. The effective equation for the transport problem is again one of the diffusion advection-type including homogenized coefficients, and with advective flow given by the Darcy velocity and only taking place in the vertical direction. In the case of slow diffusion in the vertical direction, effective diffusion only takes place in the vertical direction, where, in the case of high diffusion in the horizontal direction, we obtain effective diffusion in all space directions. To pass to the limit, we use the method of two-scale convergence adapted to our microscopic geometry, which is based on uniform a priori estimates. Critical parts in the derivation of the macro-models are the control of the fluid pressure, for which we construct a Bogovskii operator for thin perforated domains with arbitrary boundary conditions on the top/bottom, and the strong two-scale convergence for the microscopic solution of the transport equation, which is necessary to pass to the limit in the advective term. This strong convergence is established by using a Kolmogorov–Simon compactness argument.

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