轨道晃动中平均漂移与流动的拉格朗日视角与欧拉视角对比
Lagrangian vs Eulerian view on the mean drift and streaming flows in orbital sloshing
AI总结：
本文通过截断渐近近似数值求解接触线区域，对比轨道晃动中拉格朗日平均流与欧拉流动，揭示高粘性流体中Stokes漂移的粘性修正与欧拉流动同等重要，并与实验在非共振频率下取得显著一致。
摘要：
轨道晃动是一种温和混合容器内液体内容物的技术，常用于发酵和细胞培养过程。除了在界面处观察到的丰富波动动力学外，Bouvard等人（2017）[1]揭示了隐藏在流体主体中的拉格朗日平均流结构。该流动表现出与波同向的整体环向（方位角）旋转，以及接触线附近非平凡的极向涡旋。已知旋转晃动波会诱导流体粒子的净运动，因此在欧拉流动（粘性流动）与拉格朗日流动之间存在波平均后的差异，通常称为Stokes漂移。然而，在实验中区分这两个分量具有挑战性，因为它们随强迫振幅和频率的标度相似。因此，它们的相对贡献仍未得到量化，特别是对于高粘性流体，此前基于无粘性假设的分析对此失效。在本工作中，我们构建了该问题的截断渐近近似，其中每一阶的解均通过数值计算，以尽可能准确地描述接触线区域，否则该区域在解析上难以处理。这一弱非线性分析在一阶波和二阶平均流方面的结果随后与Bouvard等人（2017）[1]的实验进行了全面比较，在非共振频率下显示出显著的一致性。当粘性起重要作用时，我们的发现表明，将极向模式仅归因于欧拉流动是不正确的，并且对Stokes漂移的粘性修正对最终的平均拉格朗日流同样重要。
英文摘要：
Orbital sloshing is a technique to gently mix a container's liquid content and it is commonly used in fermentation and cell cultivation processes. Besides the rich wave dynamics observed at the interface, Bouvard et al. (2017) [1] unveiled the structure of the Lagrangian mean flow hiding in the fluid bulk. The latter flow shows a global toroidal (azimuthal) rotation co-directed with the wave and nontrivial poloidal vortices near the contact line. Rotating sloshing waves are known to induce a net motion of fluid particles and hence a wave-averaged difference between the Eulerian flow - viscous streaming - and the Lagrangian flow, that is commonly referred to as Stokes drift. Nevertheless, discerning these two components in an experiment is challenging as they scale similarly with the forcing amplitude and frequency. Their relative contributions remain therefore unquantified, particularly for highly viscous fluids, for which prior analysis, based on inviscid arguments, fails. In this work, we construct a truncated asymptotic approximation of the problem, where the solution at each order is computed numerically to describe as accurately as possible the contact line region, otherwise analytically untractable. The results of this weakly nonlinear analysis in terms of first-order wave and second-order mean flow are then thoroughly compared with the experiments by Bouvard et al. (2017) [1], showing a remarkable agreement for off-resonance frequencies. When viscosity matters, our findings suggest that it is incorrect to attribute the poloidal patterns solely to the Eulerian streaming flow and that viscous corrections to the Stokes drift are equally important in the resulting mean Lagrangian flow.
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