标题： 在小直径圆柱体穿孔的薄域中建模非牛顿流体 显示英文标题

标题： Modeling non-Newtonian fluids in a thin domain perforated with cylinders of small diameter

摘要： 我们考虑一种广义牛顿流体通过高度为$h_\varepsilon$的薄多孔介质的流动，该介质中以$\varepsilon$周期性分布的固体圆柱体穿孔，圆柱体的直径非常小，为$\varepsilon\delta_\varepsilon$，其中小参数$\varepsilon, \delta_\varepsilon$和$h_\varepsilon$被用于趋于零。我们假设流体由非线性幂律粘度的三维不可压缩斯托克斯系统描述，流动指数为$1<r<2$（剪切变稀）。 特定情况$h_\varepsilon=\sigma_\varepsilon$，其中$\sigma_\varepsilon:=\varepsilon/\delta_\varepsilon^{{2-r\over r}}\to 0$，最近发表于 (Anguiano and Suárez-Grau,\emph{地中海期刊数学}(2021) 18:175)。在本文中，我们对任何$h_\varepsilon$进行了先前研究的推广，并提供了对由小直径圆柱体组成的薄多孔介质中非牛顿流体渐近行为更完整的描述。 我们证明了根据$\lambda:=\lim_{\varepsilon\to 0}\sigma_\varepsilon/h_\varepsilon\in [0,+\infty]$的值，存在三种低维渐近模型：在$\lambda=0$的情况下为非线性达西定律，在$\lambda\in (0,+\infty)$的情况下为非线性布林克曼型定律，在$\lambda=+\infty$的情况下为非线性雷诺兹定律。 显示英文摘要

摘要： We consider the flow of a generalized Newtonian fluid through a thin porous medium of height $h_\varepsilon$ perforated with $\varepsilon$-periodically distributed solid cylinders of very small diameter $\varepsilon\delta_\varepsilon$, where the small parameters $\varepsilon, \delta_\varepsilon$ and $h_\varepsilon$ are devoted to tend to zero. We assume that the fluid is described by the 3D incompressible Stokes system with a non-linear power law viscosity of flow index $1<r<2$ (shear thinning). The particular case $h_\varepsilon=\sigma_\varepsilon$, where $\sigma_\varepsilon:=\varepsilon/\delta_\varepsilon^{{2-r\over r}}\to 0$, was recently published in (Anguiano and Su\'arez-Grau, \emph{Mediterr. J. Math.} (2021) 18:175). In this paper, we generalize previous study for any $h_\varepsilon$ and we provide a more complete description on the asymptotic behavior of non-Newtonian fluids in a thin porous medium composed by cylinders of small diameter. We prove that depending on the value of $\lambda:=\lim_{\varepsilon\to 0}\sigma_\varepsilon/h_\varepsilon\in [0,+\infty]$, there exist three types of lower-dimensional asymptotic models: a non-linear Darcy law in the case $\lambda=0$, a non-linear Brinkman-type law in the case $\lambda\in (0,+\infty)$, and a non-linear Reynolds law in the case $\lambda=+\infty$.