磁性微游泳器的渐近动力学
Asymptotic Dynamics of Magnetic Micro-Swimmers
摘要：
通过旋转磁场使微型游泳者运动，在工程和数学建模方面都提出了有趣的挑战。本文研究了Stokes流中永磁刚体在空间均匀旋转磁场作用下的动力学问题。该系统取决于两个外部参数：与磁场的角速度成正比且与场的大小成反比的马逊数，以及磁场与其旋转轴之间的圆锥角。本文主要研究在低、高梅森数极限下以及在低锥角极限下的渐近动力学。在这三种制度中提供了分析解决方案。在低梅森数的极限下，动力系统存在一个周期解，在该周期解中，游泳者的磁矩趋向于与磁场一致。在大梅森数的极限下，磁矩趋向于与平行于旋转轴的平均磁场一致。在低锥角极限下的渐近动力学允许桥接这两个区域。最后，我们使用数值方法来比较这些解析预测与数值解。
英文摘要：
Micro-swimmers put into motion by a rotating magnetic field have provided interesting challenges both in engineering and in mathematical modelling. We study here the dynamics of a permanent-magnetic rigid body submitted to a spatially-uniform steadily-rotating magnetic field in Stokes flow. This system depends on two external parameters: the Ma- son number, which is proportional to the angular speed of the magnetic field and inversely proportional to the magnitude of the field, and the conical angle between the magnetic field and its axis of rotation. This work focuses on asymptotic dynamics in the limits of low and high Mason number, and in the limit of low conical angle. Analytical solutions are provided in these three regimes. In the limit of low Mason number, the dynamical system admits a periodic solution in which the magnetic moment of the swimmer tends to align with the magnetic field. In the limit of large Mason number, the magnetic moment tends to align with the average magnetic field, which is parallel to the axis of rotation. Asymptotic dynamics in the limit of low conical angle allow to bridge these two regimes. Finally, we use numerical methods to compare these analytical predictions with numerical solutions.
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