We study the shape and the geometrical properties of sessile drops with translational invariance (namely ‘liquid cylinders’) deposited upon a flat superhydrophobic substrate. We account for the flattening effects of gravity on the shape of the drop using a pendulum rotation motion analogy. In the framework of the inviscid Saint-Venant equations, we show that liquid cylinders are always unstable because of the Plateau–Rayleigh instability. However, a cylindrical drop deposited upon a superhydrophobic non-flat channel (here, wedge-shaped channels) is stabilised beyond a critical cross-sectional area. The critical threshold of the Plateau–Rayleigh instability is analytically computed for various profiles of the channel. The stability analysis is performed in terms of an effective propagation speed of varicose waves. Experiments are performed in order to test these analytical results. We measure the critical drop size at which breakup occurs, together with the decreasing effective propagation speed of varicose waves as the threshold is approached. Our theoretical predictions are in excellent agreement with the experimental measurements.

中文翻译：

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沿液体圆柱体的表面波。第 1 部分 重力对 Plateau-Rayleigh 不稳定性的稳定作用

我们研究了沉积在平坦超疏水基底上的具有平移不变性（即“液体圆柱体”）的静滴的形状和几何特性。我们使用钟摆旋转运动类比来解释重力对液滴形状的扁平化影响。在无粘圣维南方程的框架中，我们表明由于高原-瑞利不稳定性，液体圆柱体总是不稳定的。然而，沉积在超疏水非平坦通道（此处为楔形通道）上的圆柱形液滴在超过临界横截面区域时稳定。高原-瑞利不稳定性的临界阈值是针对通道的各种剖面进行分析计算的。根据曲张波的有效传播速度进行稳定性分析。进行实验以测试这些分析结果。我们测量发生破裂的临界液滴尺寸，以及随着接近阈值而降低的曲张波的有效传播速度。我们的理论预测与实验测量非常吻合。