On Flexible Tubes Conveying Fluid: Geometric Nonlinear Theory, Stability and Dynamics
On Flexible Tubes Conveying Fluid: Geometric Nonlinear Theory, Stability and Dynamics
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DOI：
10.1007/s00332-015-9246-9
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发表时间：
2015-08-01
影响因子：
3
通讯作者：
Putkaradze, Vakhtang
中科院分区：
文献类型：
--
作者：
Gay-Balmaz, Francois;Putkaradze, Vakhtang
我们推导出一个完全三维的，几何精确的理论输送流体的柔性管。该理论还包括在动态过程中可用于流体运动的横截面的变化。我们的方法是基于简化的，精确的几何描述的弹性杆，再加上流体输运和流体的体积守恒约束。我们首先推导出的运动方程直接，通过使用欧拉-庞加莱，变分原理。然后，我们证明这一推导与一个更一般的理论阐明了有趣的数学概念出现在这个问题上，如部分左（弹性）和右（流体）不变性的系统，增加完整约束（体积）。我们分析了完全非线性行为的模型时，管的轴保持直线。然后，我们进行的线性稳定性分析，并表明我们的理论引入了重要的修正先前得出的结果，无论是在所有波长的一致性和所产生的影响，从动态变化的横截面。最后，我们推导并分析了几个解析的，完全非线性的行波型的二维解。
We derive a fully three-dimensional, geometrically exact theory for flexible tubes conveying fluid. The theory also incorporates the change of the cross section available to the fluid motion during the dynamics. Our approach is based on the symmetry-reduced, exact geometric description for elastic rods, coupled with the fluid transport and subject to the volume conservation constraint for the fluid. We first derive the equations of motion directly, by using an Euler-Poincar, variational principle. We then justify this derivation with a more general theory elucidating the interesting mathematical concepts appearing in this problem, such as partial left (elastic) and right (fluid) invariance of the system, with the added holonomic constraint (volume). We analyze the fully nonlinear behavior of the model when the axis of the tube remains straight. We then proceed to the linear stability analysis and show that our theory introduces important corrections to previously derived results, both in the consistency at all wavelength and in the effects arising from the dynamical change of the cross section. Finally, we derive and analyze several analytical, fully nonlinear solutions of traveling wave type in two dimensions.