模拟轨道振动圆柱形容器中单波峰到双波峰转变的振幅方程
An amplitude equation modeling the single-to-double crest wave transition in orbital shaken cylindrical containers
摘要：
容器以恒定的角速度沿着平面圆形轨迹的运动，即轨道摇动，在几种工业应用中是令人感兴趣的，例如用于发酵过程或干细胞培养，其中良好的混合和有效的气体交换是主要目标。在这些外部强迫条件下，自由表面通常表现出一个主要的稳态运动，通过一个单峰动态，其波幅，作为外部强迫参数的函数，表现出达芬样的行为。然而，先前在实验室规模的圆柱形容器中的实验已经揭示，由于超谐波的激发，在某些驱动频率范围内可以观察到不同的动力学。在这些超谐波中，双峰动力学是特别相关的，因为它显示出显着的大振幅响应，这是强烈的外部强迫的空间结构的青睐。在无粘极限和关于圆柱形容器，我们正式在这里弱非线性分析通过多时间尺度方法的全流体动力晃荡系统，导致一个振幅方程适合描述这样的超谐动力学和由此产生的单到双波峰波过渡。弱非线性的预测是在相当好的协议与文献中描述的以前的实验。最后，我们讨论了如何通过渐近求解具有小非线性项的受迫Helmholtz-Duffing方程的第一超调和函数来导出类似的振幅方程。
英文摘要：
The container motion along a planar circular trajectory at a constant angular velocity, i.e. orbital shaking, is of interest in several industrial applications, e.g. for fermentation processes or in cultivation of stem cells, where good mixing and efficient gas exchange are the main targets. Under these external forcing conditions, the free surface typically exhibits a primary steady state motion through a single-crest dynamics, whose wave amplitude, as a function of the external forcing parameters, shows a Duffing-like behaviour. However, previous experiments in lab-scale cylindrical containers have unveiled that, owing to the excitation of super-harmonics, diverse dynamics are observable in certain driving-frequency ranges. Among these super-harmonics, the double-crest dynamics is particularly relevant, as it displays a notably large amplitude response, that is strongly favored by the spatial structure of the external forcing. In the inviscid limit and with regards to circular cylindrical containers, we formalize here a weakly nonlinear analysis via multiple timescale method of the full hydrodynamic sloshing system, leading to an amplitude equation suitable to describe such a super-harmonic dynamics and the resulting single-to-double crest wave transition. The weakly nonlinear prediction is shown to be in fairly good agreement with previous experiments described in the literature. Lastly, we discuss how an analogous amplitude equation can be derived by solving asymptotically for the first super-harmonic of the forced Helmholtz-Duffing equation with small nonlinearities.
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