The effects of harmonically co-oscillating the inner and outer cylinders about zero mean rotation in a Taylor–Couette flow are examined numerically using Floquet theory, for the case where the fluid confined between the cylinders obeys the upper convected Maxwell model. Although stability diagrams and mode competition involved in the system were clearly elucidated recently by Hayani Choujaa et al. (2021) in weakly elastic fluids, attention is focused, in this paper, on the dynamic of the system at higher elasticity with emphasis on the nature of the primary bifurcation. In this framework, we are dealing with pure inertio-elastic parametric resonant instabilities where the elastic and inertial mechanisms are considered of the same order of magnitude. It turns out, on the one hand, that the fluid elasticity gives rise, at the onset of instability, to the appearance of a family of new harmonic modes having different axial wavelengths and breaking the spatio-temporal symmetry of the base flow: invariance in the axial direction generating the symmetry group and a half-period-reflection symmetry in the azimuthal direction generating a spatio-temporal symmetry group. On the other hand, new quasi-periodic flow emerging in the high frequency limit and other interesting bifurcation phenomena including bi and tricritical states are also among the features induced by the fluid elasticity. Lastly, and in comparison with the Newtonian configuration of this system, the fluid elasticity leads to a total suppression of the non-reversing flow besides emergence of instabilities with lower wavelengths. Such a comparison provides insights into the dynamics of elastic hoop stresses in altering the flow reversal in modulated Taylor–Couette flow.

中文翻译：

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由限制在两个零均值左右的同振荡圆柱体之间的上对流麦克斯韦流体的非线性弹性触发的三临界态和准周期性

对于限制在圆柱体之间的流体遵循上部对流麦克斯韦模型的情况，使用 Floquet 理论对泰勒-库埃特流中内圆柱体和外圆柱体关于零平均旋转的谐波共振荡的影响进行了数值检验。尽管 Hayani Choujaa 等人最近清楚地阐明了系统中涉及的稳定性图和模式竞争。(2021) 在弱弹性流体中，本文将注意力集中在较高弹性下系统的动态，重点是初级分岔的性质。在这个框架中，我们正在处理纯惯性弹性参数共振不稳定性，其中弹性和惯性机制被认为具有相同的数量级。事实证明，一方面，在不稳定开始时，流体弹性导致出现一系列具有不同轴向波长并打破基流时空对称性的新谐波模式：轴向方向产生对称群，方位方向上的半周期反射对称产生时空对称群。另一方面，在高频极限中出现的新的准周期流动以及包括双临界态和三临界态在内的其他有趣的分叉现象也是由流体弹性引起的特征。最后，与该系统的牛顿结构相比，除了出现较低波长的不稳定性之外，流体弹性还导致不可逆流动的完全抑制。这种比较可以深入了解弹性环向应力在改变调制泰勒-库埃特流中的流动反转时的动力学。