The dynamics of a polarizable spheroid in a shear flow subjected to a magnetic field is determined by a balance between the hydrodynamic torque and the magnetic torque due to the induced magnetic moment. The magnetic moment of the particle is an odd function of $\mathit{H}·\widehat{\mathit{z}}$ which saturates to constant for $|\mathit{H}·\widehat{\mathit{z}}|\gg 1$, where $\mathit{H}$ is the magnetic field and $\widehat{\mathit{z}}$ is the orientation vector of the particle. Three different models are used, the realistic Langevin model and the simpler linear and signum approximations. These models contain two dimensionless parameters, $\mathrm{\Sigma }=\left({\mu }_0\chi \right|\mathit{H}{|}^2/\mathrm{\Gamma })$ and ${\mathrm{\Sigma }}_s=({\mu }_0{m}_s\left|\mathit{H}\right|/\mathrm{\Gamma })$, where $\mathrm{\Gamma }$ is the characteristic hydrodynamic torque, ${\mu }_0$ is the magnetic permeability of free space, $\chi$ is the polarizability for low magnetic field, and $m_s$ is the saturation moment. The dynamics of the spheroid is analyzed for the case where the magnetic field is aligned along the flow plane. For the linear model, an analytical solution for the evolution of the particle orientation is obtained; there is a continuous transition between a rotating state and a static state when the parameter $\mathrm{\Sigma }$ exceeds a critical value which depends on the orientation of the magnetic field and the aspect ratio of the particle. The phase portrait for the signum model exhibits a rich variety in dynamical behavior, including continuous and discontinuous transitions between the rotating and static states, and the possibility of multiple steady states. The transition between stationary and rotating states, and the orientation and magnetic torque in both states, are numerically determined for the Langevin model.

中文翻译：

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平行磁场作用下可极化球体在剪切流中的动力学

可极化的椭球体在受到磁场作用的剪切流中的动力学特性是由流体动力转矩与归因于磁矩的磁转矩之间的平衡决定的。粒子的磁矩是$\mathit{H}·\widehat{\mathit{ž}}$ 饱和为常数 $|\mathit{H}·\widehat{\mathit{ž}}|\gg \mathrm{1个}$， 在哪里 $\mathit{H}$ 是磁场， $\widehat{\mathit{ž}}$是粒子的方向向量。使用了三种不同的模型，即逼真的Langevin模型以及更简单的线性和符号近似。这些模型包含两个无量纲参数，$\mathrm{\Sigma }=（{\mu }_0\chi |\mathit{H}{|}^{\mathrm{2个}}/\mathrm{\Gamma }）$ 和 ${\mathrm{\Sigma }}_s=（{\mu }_0{米}_s\left|\mathit{H}\right|/\mathrm{\Gamma }）$， 在哪里 $\mathrm{\Gamma }$ 是典型的流体动力扭矩， ${\mu }_0$ 是自由空间的磁导率， $\chi$ 是低磁场的极化率，并且 ${米}_s$是饱和时刻。对于磁场沿流动平面对齐的情况，分析球体的动力学。对于线性模型，获得了粒子取向演化的解析解；当参数$\mathrm{\Sigma }$超过临界值，该临界值取决于磁场的方向和粒子的纵横比。信号模型的相图在动力学行为方面表现出丰富的变化，包括旋转状态和静态之间的连续和不连续过渡，以及可能存在多个稳态。对于Langevin模型，在数值上确定了稳态和旋转状态之间的过渡以及两种状态下的方向和磁转矩。