Effective boundary conditions, correct to third order in a small parameter $$\epsilon$$ , are derived by homogenization theory for the motion of an incompressible fluid over a rough wall with periodic micro-indentations. The length scale of the indentations is l, and $$\epsilon = l/L \ll 1$$ , with L a characteristic length of the macroscopic problem. A multiple scale expansion of the variables allows to recover, at leading order, the usual Navier slip condition. At next order the slip velocity includes a term arising from the streamwise pressure gradient; furthermore, a transpiration velocity $${\mathcal {O}}(\epsilon ^{2})$$ appears at the fictitious wall where the effective boundary conditions are enforced. Additional terms appear at third order in both wall-tangent and wall-normal components of the velocity. The application of the effective conditions to a macroscopic problem is carried out for the Hiemenz stagnation point flow over a rough wall, highlighting the differences among the exact results and those obtained using conditions of different asymptotic orders.

中文翻译：

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粗糙壁面的有效边界条件：一种高阶均匀化方法

对于不可压缩流体在具有周期性微压痕的粗糙壁上的运动，有效边界条件在小参数 $$\epsilon$$ 中修正为三阶。压痕的长度尺度是 l，并且 $$\epsilon = l/L \ll 1$$ ，其中 L 是宏观问题的特征长度。变量的多尺度扩展允许以领先的顺序恢复通常的 Navier 滑移条件。在下一个顺序中，滑移速度包括一个由流向压力梯度引起的项；此外，蒸腾速度 $${\mathcal {O}}(\epsilon ^{2})$$ 出现在强制执行有效边界条件的虚拟壁上。附加项出现在速度的壁面切线和壁面法向分量中的三阶。