Cusp singularities in fluids have been experimentally demonstrated in the past only at a low Reynolds number, Re ≪ 1, and large capillary number, Ca ≫ 1, in Newtonian or non-Newtonian fluids. Here, we show that the collapse of a free surface wave depression cavity can lead to inertial-viscous cusp formation at local Re > 1 and Ca > 1, which gives rise to extreme events, i.e., very high-velocity surface jets. The cavities are generated in a cylindrical container (2R = 10 cm), partially filled with glycerine–water solution, by parametrically forcing the axi-symmetric wave mode beyond the breaking limit. By varying the forcing amplitude and the fluid viscosity, parabolic or cusp singularities manifest, depending on the last stable wave amplitude b that determines the cavity shape. Cusp formation in collapse without bubble pinch-off, leading to very high-velocity surface jets, is obtained when b is close to the singular wave amplitude bs and Ca > 1. The free surface shape is self-similar, changing from an inertial to a viscous regime when the singularity is approached. At cusp singularity, the cavity shape takes the form of (z − Z0)/R ∼ −(r/R)2/3, where Z0 is the final cavity depth. Cavity collapse with bubble pinch-off, which occurs when b > bs, also exhibits a cusp singularity when bs 1, but surface jet velocities are much less because about half of the wave energy is lost.

中文翻译：

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与流体中惯性粘性尖端形成相关的极端奇异事件

流体中的尖点奇点在过去仅在牛顿流体或非牛顿流体中的低雷诺数 Re ≪ 1 和大毛细管数 Ca ≫ 1 下被实验证明。在这里，我们表明自由表面波凹陷腔的坍塌会导致在局部 Re > 1 和 Ca > 1 处形成惯性粘性尖峰，这会引起极端事件，即非常高速的表面喷射。通过参数迫使轴对称波模式超出断裂极限，在圆柱形容器（2R = 10 cm）中产生腔，部分填充有甘油 - 水溶液。通过改变强迫振幅和流体粘度，抛物线或尖点奇点显现出来，这取决于决定空腔形状的最后一个稳定波振幅 b。在没有气泡夹断的情况下在塌陷中形成尖头，当 b 接近奇异波振幅 bs 并且 Ca > 1 时，会获得导致非常高速的表面射流。自由表面形状是自相似的，当接近奇异点时，从惯性状态变为粘性状态。在尖点奇点处，空腔形状采用 (z − Z0)/R ∼ −(r/R)2/3 的形式，其中 Z0 是最终的空腔深度。当 b > bs 时发生气泡夹断的空腔坍塌，当 bs 1 时也表现出尖点奇异性，但表面射流速度要小得多，因为大约一半的波能损失了。空腔形状采用 (z − Z0)/R ∼ −(r/R)2/3 的形式，其中 Z0 是最终的空腔深度。当 b > bs 时发生气泡夹断的空腔坍塌，当 bs 1 时也表现出尖点奇异性，但表面射流速度要小得多，因为大约一半的波能损失了。空腔形状采用 (z − Z0)/R ∼ −(r/R)2/3 的形式，其中 Z0 是最终的空腔深度。当 b > bs 时发生气泡夹断的空腔坍塌，当 bs 1 时也表现出尖点奇异性，但表面射流速度要小得多，因为大约一半的波能损失了。