稠密气泡床纵向流的奇异有效滑移长度
Singular effective slip length for longitudinal flow over a dense bubble mattress
摘要：
我们认为有效的疏水性的周期性槽表面浸入液体中，与被困的无剪切气泡之间的无滑动脊突出的$π/2$接触角。具体而言，我们进行了奇异摄动分析的限制$ε\ll1$气泡是紧密间隔的，发现有效的滑移长度（归一化的气泡半径）为纵向流动沿脊$π/{\sqrt{2ε}}-(12/π)\ln 2 + (13π/24)\sqrt{2ε}+ o(\sqrtε)$，小参数$ε$是平面形状的固体分数。平方根发散突出了这种配置的强疏水性;这个领先的奇异项（以及第三项）来自气泡之间的间隙区域的局部润滑分析，以及一般的匹配考虑和全局守恒关系。$O(1)$常数项通过与“外”区域中的领先阶解匹配来找到，在该区域中气泡似乎是接触的。我们发现我们的滑移长度公式和最近用“统一变换”方法导出的数值方案（D。Crowdy，IMA J.应用数学，80 1902，2015）。比较表明，我们的渐近公式，连同直径的“稀限”近似（D。Crowdy，J. Fluid Mech.，791 R7，2016）提供了基本上任意无滑移分数的基本分析描述。
英文摘要：
We consider the effective hydrophobicity of a periodically grooved surface immersed in liquid, with trapped shear-free bubbles protruding between the no-slip ridges at a $π/2$ contact angle. Specifically, we carry out a singular-perturbation analysis in the limit $ε\ll1$ where the bubbles are closely spaced, finding the effective slip length (normalised by the bubble radius) for longitudinal flow along the the ridges as $π/{\sqrt{2ε}}-(12/π)\ln 2 + (13π/24)\sqrt{2ε}+ o(\sqrtε)$, the small parameter $ε$ being the planform solid fraction. The square-root divergence highlights the strong hydrophobic character of this configuration; this leading singular term (along with the third term) follows from a local lubrication-like analysis of the gap regions between the bubbles, together with general matching considerations and a global conservation relation. The $O(1)$ constant term is found by matching with a leading-order solution in the "outer'" region, where the bubbles appear to be touching. We find excellent agreement between our slip-length formula and a numerical scheme recently derived using a "unified-transform" method (D. Crowdy, IMA J. Appl. Math., 80 1902, 2015). The comparison demonstrates that our asymptotic formula, together with the diametric "dilute-limit" approximation (D. Crowdy, J. Fluid Mech., 791 R7, 2016), provides an elementary analytical description for essentially arbitrary no-slip fractions.
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