We study the upscaling of a system of many interacting particles through a heterogenous thin elongated obstacle as modeled via a two-dimensional diffusion problem with a one-directional nonlinear convective drift. Assuming that the obstacle can be described well by a thin composite strip with periodically placed microstructures, we aim at deriving the upscaled model equations as well as the effective transport coefficients for suitable scalings in terms of both the inherent thickness at the strip and the typical length scales of the microscopic heterogeneities. Aiming at computable scenarios, we consider that the heterogeneity of the strip is made of an array of periodically arranged impenetrable solid rectangles and identify two scaling regimes what concerns the small asymptotics parameter for the upscaling procedure: the characteristic size of the microstructure is either significantly smaller than the thickness of the thin obstacle or it is of the same order of magnitude. We scale up the diffusion–polynomial drift model and list computable formulas for the effective diffusion and drift tensorial coefficients for both scaling regimes. Our upscaling procedure combines ideas of two-scale asymptotics homogenization with dimension reduction arguments. Consequences of these results for the construction of more general transmission boundary conditions are discussed. We illustrate numerically the concentration profile of the chemical species passing through the upscaled strip in the finite thickness regime and point out that trapping of concentration inside the strip is likely to occur in at least two conceptually different transport situations: (i) full diffusion/dispersion matrix and nonlinear horizontal drift, and (ii) diagonal diffusion matrix and oblique nonlinear drift.

中文翻译：

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通过具有周期性微观结构的复合薄带放大扩散和多项式漂移之间的相互作用

我们通过具有单向非线性对流漂移的二维扩散问题建模，研究了许多相互作用粒子系统通过异质细长障碍物的放大。假设可以通过具有周期性放置的微结构的薄复合带很好地描述障碍物，我们的目标是根据带的固有厚度和典型长度推导出放大的模型方程以及适当缩放的有效传输系数微观异质性的尺度。针对可计算的场景，我们认为条带的异质性由一系列周期性排列的不可穿透的实心矩形组成，并确定了两个与放大过程的小渐近​​参数有关的缩放机制：微观结构的特征尺寸要么明显小于薄障碍物的厚度，要么处于相同的数量级。我们放大了扩散多项式漂移模型，并列出了两种缩放方式的有效扩散和漂移张量系数的可计算公式。我们的升级程序将两尺度渐近同质化的思想与降维参数相结合。讨论了这些结果对于构建更一般的传输边界条件的后果。我们用数字说明了在有限厚度范围内通过放大带的化学物质的浓度分布，并指出带内浓度的捕获可能发生在至少两种概念上不同的传输情况中：