Pressure and shear-driven flows of a confined film of fluid overlying a periodic one-dimensional topography of arbitrary shape are considered for prediction of the effective hydraulic permeability in the Stokes flow regime. The other surface confining the fluid may be a planar no-slip wall, an identically patterned wall, a free surface or a surface with prescribed shear. Analytical predictions are obtained using spectral analysis and the domain perturbation method under the assumption of small pattern size to pitch ratio. Using a novel decomposition of the channel height effects into exponentially and algebraically decaying components, a simple surface-metrology-dependent relationship which connects the eigenvalues of the effective permeability tensor is obtained. Two representative topographies are assessed numerically: the infinitely differentiable topography of a phase-modulated sinusoid which has multiple local extrema and zero crossings and the non-differentiable triangular-wave topography. Non-differentiability in the form of corners of triangular patterns and the cusps of scalloped patterns are not found to be an impediment to meaningful and numerically accurate asymptotic predictions of effective permeability and effective slip, contradicting an earlier suggestion from the literature. Several distinct applications of the theory to the friction-reduction and shear-stability performance of the recently developed lubricant impregnated patterned surfaces as well as to scalloped and trapezoidal drag-reduction riblets are discussed, with comparison to experimental data from the literature for the last application. Analytical approximations which have an extended domain of numerical accuracy are also proposed.

中文翻译：

---

任意形状壁槽约束流的有效渗透率张量

覆盖在任意形状的周期性一维地形上的受限流体膜的压力和剪切驱动流动被考虑用于预测斯托克斯流态中的有效水力渗透率。限制流体的另一个表面可以是平面的无滑移壁、相同图案的壁、自由表面或具有规定剪切力的表面。在小图案尺寸与间距比的假设下，使用谱分析和域扰动方法获得分析预测。使用通道高度效应到指数和代数衰减分量的新分解，获得连接有效渗透率张量的特征值的简单的表面计量相关关系。两个有代表性的地形进行了数值评估：具有多个局部极值和零交叉的相位调制正弦波的无限微分拓扑和不可微的三角波拓扑。没有发现三角形图案的角和扇形图案的尖端形式的不可微性阻碍对有效渗透率和有效滑移进行有意义且数值准确的渐近预测，这与文献中的早期建议相矛盾。讨论了该理论在最近开发的润滑剂浸渍图案表面以及扇形和梯形减阻肋的摩擦降低和剪切稳定性性能方面的几种不同应用，并与上次应用的文献中的实验数据进行了比较.