When a dipolar spheroid is subjected to a shear flow in the presence of an external field, there is a torque due to the dipole-field interaction which tends to align the spheroid in the direction of the field, and a torque due to the shear flow which tends to rotate the particle in closed `Jeffrey orbits’. When the external field is in the flow plane, depending on the strength and orientation of the field, the phase portrait in orientation space could have 2-6 stationary nodes and/or a limit cycle. When the external field strength is low, there are two stationary points off the flow plane which are both stable/unstable, and an unstable/stable limit cycle on the flow plane. When the external field strength is high, there is one stable node where the particle orientation is parallel to the field, and one unstable node where the particle orientation is anti-parallel to the field. As the external field strength is increased, the manner in which the phase portrait evolves depends on the external field orientation with respect to the flow direction and the particle shape factor. It is shown that complex phase portraits result from the relatively simple trajectories of the stationary nodes in the three-dimensional space spanned by the two orientation angles and the dimensionless parameter $\Sigma$, which is the ratio of the torques due to the shear flow and the external field. Depending on the aspect ratio of the particle, there are up to two saddle-node bifurcations, two sub-critical bifurcations on the flow plane and one reverse saddle-node merger of two stationary points off the flow plane. The limits of an ideal thin rod/disk are shown to be singular limits, where there are no stationary points off the flow plane even for low external field, and the orientation of the particle changes discontinuously as the cross-stream component of the external field changes sign.

中文翻译：

---

受到外场作用的剪切流中偶极子球体动力学的分叉

当偶极子球体在存在外部磁场的情况下受到剪切流作用时，由于偶极子-场相互作用而产生的转矩趋于使球体沿磁场方向对准，而由于剪切流而产生的转矩这倾向于使粒子沿封闭的“杰弗里轨道”旋转。当外部场在流动平面中时，取决于场的强度和方向，定向空间中的相像可以具有2-6个固定节点和/或极限环。当外部场强较低时，在流动平面上会有两个固定点，它们既稳定/不稳定，又在流动平面上具有不稳定/稳定的极限循环。当外部场强较高时，会有一个稳定的节点，其粒子方向与场平行，一个不稳定的节点，其粒子方向与电场反平行。随着外场强度的增加，相像的演化方式取决于外场相对于流动方向和颗粒形状因子的取向。结果表明，复杂的相像是由二维方向和无量纲参数所跨越的三维空间中固定节点的相对简单轨迹产生的$\Sigma$，这是由于剪切流和外场引起的扭矩之比。根据粒子的长宽比，流平面上最多有两个鞍形节点分叉，两个亚临界分叉和两个固定点的反向鞍形节点合并。理想的细杆/圆盘的极限被示为奇异极限，即使对于低外部场，在流动平面上也没有静止点，并且随着外部场的横流分量，粒子的方向会不连续地变化更改标志。