Title: Modeling Carreau fluid flows through a very thin porous medium Show Chinese title

Title: 通过非常薄的多孔介质的Carreau流体流动建模

Abstract: This study investigates three-dimensional, steady-state, and non-Newtonian flows within a very thin porous medium (VTPM). The medium is modeled as a domain confined between two parallel plates and perforated by solid cylinders that connect the plates and are distributed periodically in perpendicular directions. We denote the order of magnitude of the thickness of the domain by $\epsilon$ and define the period and order of magnitude of the cylinders' diameter by $\epsilon$^l, where 0 < l < 1 is fixed. In other words, we consider the regime $\epsilon$ $\ll$ $\epsilon$^l. We assume that the viscosity of the non-Newtonian fluid follows Carreau's law and is scaled by a factor of $\epsilon$^$\gamma$, where $\gamma$ is a real number. Using asymptotic techniques with respect to the thickness of the domain, we perform a new, complete study of the asymptotic behaviour of the fluid as $\epsilon$ tends to zero. Our mathematical analysis is based on deriving sharp a priori estimates through pressure decomposition, and on compactness results for the rescaled velocity and pressure, obtained using the unfolding method. Depending on $\gamma$ and the flow index r, we rigorously derive different linear and nonlinear reduced limit systems. These systems allow us to obtain explicit expressions for the filtration velocity and simpler Darcy's laws for limit pressure. Show Chinese abstract

Abstract: 本研究探讨了非常薄多孔介质（VTPM）中的三维、稳态和非牛顿流。 该介质被建模为两个平行板之间的区域，并由连接板的固体圆柱体穿孔，这些圆柱体在垂直方向上周期性分布。 我们将该区域厚度的数量级表示为$\epsilon$，并定义圆柱体直径的周期和数量级为$\epsilon$^l，其中 0 < l < 1 是固定值。 换句话说，我们考虑的是情形$\epsilon$ $\ll$ $\epsilon$ ^l。 我们假设非牛顿流体的粘度遵循Carreau定律，并按因子$\epsilon$^$\gamma$进行缩放，其中$\gamma$是一个实数。 使用关于区域厚度的渐近技术，我们对流体在$\epsilon$趋于零时的渐近行为进行了新的、完整的分析。 我们的数学分析基于通过压力分解导出精确的先验估计，以及利用展开方法得到的缩放速度和压力的紧性结果。 根据$\gamma$和流动指数 r，我们严格推导出不同的线性和非线性降阶极限系统。 这些系统使我们能够获得过滤速度的显式表达式以及极限压力的简化Darcy定律。