A common realization of superhydrophobic surfaces comprises of a periodic array of cylindrical bubbles which are trapped in a periodically grooved solid substrate. We consider the thermocapillary animation of liquid motion by a macroscopic temperature gradient which is longitudinally applied over such a bubble mattress. Assuming a linear variation of the interfacial tension with the temperature, at slope $\sigma _T$, we seek the effective velocity slip attained by the liquid at large distances away from the mattress. We focus upon the dilute limit, where the groove width $2c$ is small compared with the array period $2l$. The requisite velocity slip in the applied-gradient direction, determined by a local analysis about a single bubble, is provided by the approximation $$\begin{align*}& \pi \frac{G\sigma_T c^2}{\mu l} I(\alpha), \end{align*}$$wherein $G$ is the applied-gradient magnitude, $\mu $ is the liquid viscosity and $I(\alpha )$, a non-monotonic function of the protrusion angle $\alpha $, is provided by the quadrature, $$\begin{align*}& I(\alpha) = \frac{2}{\sin\alpha} \int_0^\infty\frac{\sinh s\alpha}{ \cosh s(\pi-\alpha) \sinh s \pi} \, \textrm{d} s. \end{align*}$$

中文翻译：

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纵向热毛细血管绕着一个由突出气泡组成的稀周期性床垫滑动

超疏水表面的常见实现包括周期性排列的圆柱形气泡，这些气泡被捕获在周期性开槽的固体基质中。我们通过纵向施加在这种气泡床垫上的宏观温度梯度来考虑液体运动的热毛细动画。假设界面张力随温度线性变化，在斜率 $\sigma _T$ 处，我们寻求液体在远离床垫很远的地方获得的有效速度滑移。我们关注稀释极限，其中凹槽宽度 $2c$ 与阵列周期 $2l$ 相比较小。通过对单个气泡的局部分析确定的应用梯度方向上必要的速度滑移由近似值 $$\begin{align*}& \pi \frac{G\sigma_T c^2}{\mu l} I(\alpha), \end{align*}$$其中 $G$ 是应用梯度幅度，$\mu $ 是液体粘度，$I(\alpha )$ 是突出角 $\alpha $ 的非单调函数，由正交提供，$$\begin{align*}& I(\alpha) = \frac{2}{\sin\alpha} \int_0^\infty\frac{\sinh s\alpha}{ \cosh s (\pi-\alpha) \sinh s \pi} \, \textrm{d} s。\end{对齐*}$$