Flow interaction between a plain-fluid region in contact with a porous layer attracted significant attention from modelling and analysis sides due to numerous applications in biology, environment and industry. In the most widely used coupled model, fluid flow is described by the Stokes equations in the free-flow domain and Darcy's law in the porous medium, and complemented by the appropriate interface conditions. However, traditional coupling concepts are restricted, with a few exceptions, to one-dimensional flows parallel to the fluid-porous interface. In this work, we use an alternative approach to model interaction between the plain-fluid domain and porous medium by considering a transition zone, and propose the full- and hybrid-dimensional Stokes-Brinkman-Darcy models. In the first case, the equi-dimensional Brinkman equations are considered in the transition region, and the appropriate interface conditions are set on the top and bottom of the transition zone. In the latter case, we perform a dimensional model reduction by averaging the Brinkman equations in the normal direction and using the proposed transmission conditions. The well-posedness of both coupled problems is proved, and some numerical simulations are carried out in order to validate the concepts.

翻译：流体区域与多孔层接触时的流动相互作用，因在生物、环境和工业中的广泛应用，吸引了建模与分析领域的大量关注。在最广泛使用的耦合模型中，流体流动在自由流区域由 Stokes 方程描述，在多孔介质中由 Darcy 定律描述，并辅以合适的界面条件。然而，除少数例外，传统耦合概念局限于平行于流体-多孔界面的一维流动。本研究采用替代方法，通过引入过渡区来模拟流体区域与多孔介质之间的相互作用，提出全维与混合维 Stokes-Brinkman-Darcy 模型。在前一种情况下，在过渡区内考虑等维 Brinkman 方程，并在过渡区顶部和底部设定适当的界面条件；在后一种情况下，通过沿法向对 Brinkman 方程进行平均并采用所提出的传输条件，实施维度模型降阶。证明了两个耦合问题的适定性，并通过数值模拟验证了相关概念。