We study the dynamics of fragmentation for Newtonian and viscoelastic liquids in rotary atomization. In this common industrial process centripetal acceleration destabilizes the liquid torus that forms at the rim of a spinning cup or disk due to the Rayleigh-Taylor instability. The resulting ligaments leave the liquid torus with a remarkably repeatable spacing that scales inversely with the rotation rate. The fluid filaments then follow a well-defined geometrical path-line that is described by the involute of a circle. Knowing the geometry of this phenomenon we derive the detailed kinematics of this process and compare it with the experimental observations. We show that the ligaments elongate tangentially along the involute of the circle and thin radially as they separate from the cup. We use these kinematic conditions to develop an expression for the spatial variation of the filament deformation rate and show that it decays away from the spinning cup. Once the ligaments are sufficiently far from the cup, they are not stretched sufficiently fast to overcome the critical rate of capillary thinning and consequently undergo capillary-driven breakup forming droplets. We couple these kinematic considerations with the known properties of several Newtonian and viscoelastic test liquids to develop a quantitative understanding of this commercially important fragmentation process that can be compared in detail with experimental observations. We also investigate the resulting droplet size distributions and observe that the appearance of satellite droplets during the pinch-off process lead to the emergence of bidisperse droplet size distributions. These binary distributions are well described by the superposition of two separate $\mathrm{\Gamma }$ distributions that capture the physics of the disintegration process for the main and satellite droplets, respectively. Furthermore, as we consider more viscous Newtonian liquids or weakly viscoelastic test fluids, we show that changes in the liquid viscosity or viscoelasticity have a negligible effect on the average droplet size. However, incorporation of viscous/viscoelastic effects delays the thinning dynamics in the ligaments and thus results in broader droplet size distributions. The ratio of the primary to satellite droplet size increases monotonically with both viscosity and viscoelasticity. We develop a simple physical model that rationalizes the observed experimental trends and provides us a better understanding of the principal dynamical features of rotary fragmentation for both Newtonian and weakly viscoelastic liquids.

中文翻译：

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牛顿液体和粘弹性液体的旋转雾化

我们研究牛顿和粘弹性液体在旋转雾化中的破碎动力学。在这种常见的工业过程中，由于瑞利-泰勒（Rayleigh-Taylor）的不稳定性，向心加速度使在旋转杯或圆盘边缘形成的液体圆环不稳定。产生的韧带以明显可重复的间距离开液体环面，该间距与转速成反比。然后，流体细丝遵循定义良好的几何路径，该路径由圆的渐开线描述。了解了这种现象的几何形状后，我们得出了该过程的详细运动学并将其与实验观察结果进行比较。我们显示韧带沿圆的渐开线沿切线方向延伸，并在与杯分离时沿径向变细。我们使用这些运动学条件来表达长丝变形率的空间变化表达式，并表明它从纺杯中衰减。一旦韧带距离杯足够远，它们就不会足够快地拉伸以克服毛细血管变薄的临界速率，从而导致毛细血管驱动的破裂形成液滴。我们将这些运动学方面的考虑因素与几种牛顿和粘弹性测试液体的已知特性相结合，以建立对该商业上重要的破碎过程的定量理解，可以将其与实验观察结果进行详细比较。我们还研究了所得的液滴尺寸分布，并观察到夹捏过程中卫星液滴的出现会导致双分散液滴尺寸分布的出现。这些二元分布很好地描述为两个独立的叠加$\mathrm{\Gamma }$分别捕获主要液滴和卫星液滴崩解过程物理学的分布。此外，当我们考虑使用更粘的牛顿液体或弱粘弹性的测试流体时，我们发现液体粘度或粘弹性的变化对平均液滴尺寸的影响可忽略不计。然而，粘性/粘弹性效应的引入延迟了韧带的变薄动力学，因此导致更宽的液滴尺寸分布。初级液滴与卫星液滴尺寸的比率随粘度和粘弹性二者单调增加。我们开发了一个简单的物理模型，可以合理化观察到的实验趋势，并为我们提供了对牛顿型和弱粘弹性液体旋转破碎的主要动力学特征的更好的理解。