# The asymptotic analysis of a Darcy-Stokes system coupled through a curved interface

## TL;DR

This paper performs an asymptotic analysis of a coupled Darcy-Stokes system with a curved interface, deriving a lower-dimensional model as the channel width approaches zero, useful for fluid exchange modeling.

## Contribution

It introduces a novel asymptotic approach for a Darcy-Stokes system with curved interfaces, deriving a simplified coupled model in the narrow channel limit.

## Key findings

- Derived a Darcy-Brinkman lower-dimensional coupled system
- Established the limit as channel width tends to zero
- Used coordinate transformations to handle curved interfaces

## Abstract

The asymptotic analysis of a Darcy-Stokes system modeling the fluid exchange between a narrow channel (Stokes) and a porous medium (Darcy) coupled through a $ C^{2} $ curved interface, is presented. The channel is a cylindrical domain between the interface ($ \Gamma $) and a parallel translation of it ($ \Gamma + \epsilon \, \boldsymbol{\widehat{e}}_{N} $). The introduction of a change variable to fix the domain's geometry and the introduction of two systems of coordinates: the Cartesian and a local one (consistent with the geometry of the surface), permit to find a Darcy-Brinkman lower dimensional coupled system as the limiting form, when the width of the channel tends to zero ($ \epsilon \rightarrow 0 $).