With diminishing fraction of their solid portion, compound gas–solid superhydrophobic surfaces exhibit a large amount of slip which allows for appreciable velocity amplification in pressure-driven microchannel flows. We address this small solid-fraction limit in the context of a grating-like configuration, where superhydrophobicity is provided by a periodic array of flat-meniscus bubbles which are trapped in a Cassie state within the grooved channel walls. Asymptotic analysis for both longitudinal and transverse flows reveals a logarithmic scaling of the effective slip length in the solid fraction of the compound boundaries, thus refuting earlier claims of an algebraic singularity. The logarithmic scaling in the longitudinal problem is explained using an analogy between the unidirectional velocity and the velocity potential in two-dimensional irrotational flows. In the transverse problem it has to do with the Stokes paradox. The mechanisms identified herein explain the absence of slip-length singularity in the comparable asymmetric configuration, where only one of the channel walls is superhydrophobic.

中文翻译：

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小固体分数的超疏水光栅之间的压力驱动流中的速度放大

随着固体部分的减少，复合气体-固体超疏水表面表现出大量的滑移，从而允许在压力驱动的微通道流中明显地放大速度。我们在光栅状配置的情况下解决了这个小的固体分数限制，其中超疏水性是由周期性半月板形平面气泡提供的，该平面弯月形气泡以卡西状态被困在开槽的通道壁内。纵向和横向流动的渐近分析揭示了复合边界的实心部分中有效滑动长度的对数换算，因此驳斥了先前关于代数奇异性的主张。使用单向速度和二维非旋流中的速度势之间的类比来解释纵向问题中的对数标度。在横向问题中，它与斯托克斯悖论有关。本文确定的机理解释了在可比的不对称结构中不存在滑移长度奇异性的情况，其中通道壁中只有一个是超疏水的。