A study on the small-amplitude oscillation of a viscoelastic droplet suspended in a viscoelastic host liquid is carried out. The inner and outer liquids are assumed to be immiscible and their viscoelasticity is described by Jeffreys equation. The analytical characteristic equation is derived and the complex frequency is solved numerically. The effect of the outer host liquid on the oscillation of the droplet is examined for the fundamental mode $n=2$. It is found that the damping rate and the frequency of oscillation of the droplet are decreased as the host liquid gets denser in the inviscid case. The boundary between periodic oscillation and aperiodic decay in the parametric plane of Ohnesorge number and relative stress relaxation time is captured. The viscosity of the outer liquid may result in the disappearance of bifurcations and periodic oscillations of the droplet, while its elasticity plays a destabilization role in the oscillation behavior of the droplet.

中文翻译：

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悬浮在粘弹性基质液体中的粘弹性液滴的形状振荡

对悬浮在粘弹性基质液体中的粘弹性液滴的小振幅振荡进行了研究。假定内部和外部液体是不溶混的，它们的粘弹性由Jeffreys方程描述。推导了解析特性方程，并通过数值求解了复数频率。对于基本模式，检查了外部基质液体对液滴振荡的影响$ñ=2$。已经发现，在无粘性的情况下，随着主体液体变得更稠密，液滴的阻尼速率和振荡频率降低。捕获了Ohnesorge数和相对应力松弛时间的参数平面中的周期性振动和非周期性衰减之间的边界。外部液体的粘度可能导致分叉的消失和液滴的周期性振荡，而其弹性在液滴的振荡行为中起不稳定作用。