A macroscopic condition to simulate the interaction between an incompressible fluid flow and a permeable micro-structured rigid surface (i.e. a thin membrane) has been developed using multiscale homogenization and matching asymptotic expansions between the near membrane and the far region. The condition allows us to write the fluid velocity across the membrane, seen macroscopically as a smooth equivalent surface, as an effective jump between the stresses computed on the two faces of this surface. The coefficients appearing in the jump condition are the entries of tensors which solve Stokes problems within the pores, enforced by boundary conditions depending on the flow outside the membrane. These problems, found via homogenization, definitely characterize the microscopic geometrical properties of thin permeable micro-structured sheets. The new macroscopic model is validated by comparisons with direct numerical simulations of the fluid flow across membranes in different configurations, proving that the formalism adopted to write the jump conditions is valid. As a result, a rational tool able to join a microscopic and a macroscopic analysis of fluid flows across membranes is delivered, showing some potentialities to provide advancement in membrane design. It suggests that the concept of permeability has to be substituted by a more general tensor, called here the Navier tensor, which plays the role of permeability only in some particular situations.

中文翻译：

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跨膜的有效应力跳跃

使用多尺度均质化和匹配近膜和远区域之间的渐近膨胀，已经开发了模拟不可压缩流体流和可渗透微结构刚性表面（即薄膜）之间相互作用的宏观条件。该条件允许我们写出穿过膜的流体速度，宏观上被视为平滑的等效表面，作为在该表面两个面上计算的应力之间的有效跳跃。出现在跳跃条件中的系数是解决孔内斯托克斯问题的张量的条目，由取决于膜外流动的边界条件强制执行。通过均质化发现的这些问题明确表征了薄可渗透微结构片材的微观几何特性。通过与不同配置的跨膜流体流动的直接数值模拟进行比较，验证了新的宏观模型，证明了用于编写跳跃条件的形式主义是有效的。因此，提供了一种能够将跨膜流体流动的微观和宏观分析结合起来的合理工具，显示出在膜设计方面提供进步的一些潜力。这表明渗透率的概念必须被更一般的张量取代，这里称为纳维张量，它仅在某些特定情况下才发挥渗透率的作用。提供了一种能够将跨膜流体流动的微观和宏观分析结合起来的合理工具，显示了在膜设计方面提供进步的一些潜力。这表明渗透率的概念必须被更一般的张量取代，这里称为纳维张量，它仅在某些特定情况下才发挥渗透率的作用。提供了一种能够将跨膜流体流动的微观和宏观分析结合起来的合理工具，显示了在膜设计方面提供进步的一些潜力。这表明渗透率的概念必须被更一般的张量取代，这里称为纳维张量，它仅在某些特定情况下才发挥渗透率的作用。