A study of axisymmetric shape oscillations of viscoelastic drops in a vacuum is conducted, using the method of weakly nonlinear analysis. The motivation is the relevance of the shape oscillations for transport processes across the drop surface, as well as fundamental interest. The study is performed for, but not limited to, the two-lobed mode of initial drop deformation. The Oldroyd-B model is used for characterizing the liquid rheological behaviour. The method applied yields a set of governing equations, as well as boundary and initial conditions, for different orders of approximation. In the present paper, the equations and solutions up to second order are presented, together with the characteristic equation for the viscoelastic drop. The characteristic equation has an infinite number of roots, which determine the time dependency of the oscillations. Solutions of the characteristic equation are validated against experiments on acoustically levitated individual viscoelastic aqueous polymer solution drops. Experimental data consist in decay rate and oscillation frequency of free damped drop shape oscillations. With these data, solutions of the characteristic equation dominating the oscillations are identified. The theoretical analysis reveals nonlinear effects, such as the excess time in the prolate shape and frequency change for varying initial deformation amplitude. The influences of elasticity, measured by the stress relaxation and deformation retardation time scales, are quantified, and the effects are compared to the Newtonian case in the moderate-amplitude regime.

中文翻译：

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粘弹性液滴形状振荡的非线性效应

采用弱非线性分析方法对真空中粘弹性液滴的轴对称形状振动进行了研究。动机是形状振荡与液滴表面传输过程的相关性以及基本利益。该研究针对但不限于初始跌落变形的两叶模式进行。Oldroyd-B 模型用于表征液体流变行为。所应用的方法针对不同阶的近似产生一组控制方程以及边界和初始条件。在本文中，给出了二阶方程和解，以及粘弹性液滴的特征方程。特征方程有无数个根，它们决定了振荡的时间依赖性。特征方程的解通过声悬浮单个粘弹性聚合物水溶液滴的实验进行验证。实验数据包括自由阻尼液滴形状振荡的衰减率和振荡频率。利用这些数据，可以确定主导振荡的特征方程的解。理论分析揭示了非线性效应，例如扁长形状的多余时间和变化的初始变形幅度的频率变化。通过应力松弛和变形延迟时间尺度测量的弹性的影响被量化，并将其影响与中等振幅状态下的牛顿情况进行比较。