Steady and rotating states of a polarizable spheroid subjected to a magnetic field and a shear flow

The orientation vector of a spheroid subjected to a shear flow undergoes periodic Jeffrey orbits on a unit sphere. If the spheroid is polarizable and is subjected to a magnetic field, there is a transition between a rotating state at low magnetic field strength and a steady orientation along the field direction at high magnetic field strength. For the special case where the magnetic field is parallel to the flow plane, it has been shown [V. Kumaran, Phys. Rev. Fluids 6, 043702 (2021)] that the spheroid exhibits complex dynamics, including a transition from rotating to static states and the possibility of multiple steady states. Here the more general case is analyzed where the magnetic field is not parallel to the flow plane, for Langevin, linear, and signum magnetization models. At low field strength, there are two possible states, a steady orientation almost perpendicular to the flow plane and a rotating state with orientation close to the flow plane. The magnetic field orientation for transition between these two states is derived for a general magnetic model. For high magnetic field strength, the particle aligns close to the magnetic field direction. The special cases of a spherical particle and a thin rod are analyzed for arbitrary magnetic field strength. Analytical results are obtained for the critical magnetic field strength for transition between rotating and steady states, and the scaling of the frequency of rotation with magnetic field strength in the rotating state for the linear and signum models. These are compared with numerical results for the Langevin model. It is found that a thin rod has rotating states only when the cross-stream component of the magnetic field is antiparallel to the velocity gradient, and the orientation is always steady when the cross-stream component of the magnetic field is parallel to the velocity gradient. For steady orientation, the torque in the streamwise direction is zero, and the ratio of the torques in the cross-stream and spanwise directions is only a function of the magnetic field orientation.

Figure 1

The magnitude of the magnetic moment as a function of the component of the magnetic field along the particle axis for the Langevin model (black, solid line), linear model (red, dashed line), and signum model (blue, dotted line).

Figure 2

The orientation angles of the spheroid (a) and the magnetic field (b) referenced to a fixed coordinate system in which the flow is in the $X$ direction and the velocity gradient is in the $Y$ direction.

Figure 3

The transition value ${\mathrm{\Sigma }}_t^{sp}$ as a function of $1-sin\left(\xi \right)$ for a spherical particle (blue solid lines) for the linear model $(\circ )$ and for ${\mathrm{\Sigma }}_s=1\left(▵\right)$, and the ratio $({\mathrm{\Sigma }}_t^{sp}/{\mathrm{\Sigma }}_s)$ (red dashed lines) as a function of $1-sin\left(\xi \right)$ for ${\mathrm{\Sigma }}_s=0.7$ ($\nabla$), ${\mathrm{\Sigma }}_s=0.3$ ($◃$), ${\mathrm{\Sigma }}_s=0.1$ ($▹$), and ${\mathrm{\Sigma }}_s=0.01$ ($\diamond$).

Figure 4

The frequency of the stable limit cycle for a spherical particle for (a) $\xi =(\pi /3)$ and $\mathrm{\Sigma }\le {\mathrm{\Sigma }}_t^{sp}$ and (b) $sin\left(\xi \right)=\sqrt{0.95}$ and $\mathrm{\Sigma }\le {\mathrm{\Sigma }}_t^{sp}$ for the linear model ($\circ$), Langevin model with $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=0.3$ ($▵$), and 1.0 ($\diamond$). The filled symbols in panel (a) show the frequency at transition from a rotating to a stationary state.

Figure 5

The scaled torques $(T_X^M/\mathrm{\Gamma })$ (red, dashed line) and $(T_Z/\mathrm{\Gamma })$ (blue, solid line) as a function of $\mathrm{\Sigma }$ (a) and ${\mathrm{\Sigma }}_s$ (b) for a spherical particle $\xi =(\pi /6)$ and $\eta =0$. The parameter regime $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)\le 1$ is shown in panel (a) for $\circ$ linear model, $▵(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=0.3, \nabla (\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=1$; and the parameter regime $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)\ge 1$ is shown in panel (b) for the signum model ($\circ$), $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=3$ ($▵$) and $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=1\left(\nabla \right)$.

Figure 6

The scaled torques $T_X^m/\mathrm{\Gamma }$ (a), (c) and $T_Z^m/\mathrm{\Gamma }$ (b), (d) for the steady solutions shown by the solid lines, and the time-averaged torques ${\overline{T}}_X^m/\mathrm{\Gamma }$ (a), (c) and ${\overline{T}}_Z^m/\mathrm{\Gamma }$ (b), (d) for the stable limit cycle shown by the dashed lines, as a function of $\mathrm{\Sigma }$ (a), (b) and ${\mathrm{\Sigma }}_s$ (c), (d) for a spherical particle for $\xi =(\pi /3)$ and $\eta =0$. The parameter regime $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)\le 1$ is shown in panels (a) and (b) for $\circ$ linear model, $▵(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=0.3, \nabla (\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=1$; and the parameter regime $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)\ge 1$ is shown in panels (c) and (d) for the signum model ($\circ$), $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=3$ ($▵$) and $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=1\left(\nabla \right)$. The average torque for the rotating state is shown by the dashed line. The dotted line in panel (a) is the $O\left(\mathrm{\Sigma }\right)$ contribution to the average torque $({\overline{T}}_X/\mathrm{\Gamma })$ for the linear model determined from Eq. (33).

Figure 7

The value of ${\mathrm{\Sigma }}_t^{tr}$ (a) and ${\mathrm{\Sigma }}_{st}^{tr}$ (b) as a function of $\eta$ for the transition between rotating solutions (below) and steady solutions (above) for a thin rod for $\xi =\pi /6$ ($\circ$), $\xi =\pi /4$ ($▵$), $\xi =\pi /3$ ($\nabla$), and $\xi =\pi /2$ ($\diamond$). In panel (a), the solid line is the result for the linear model [Eq. (66)], the dotted and dashed lines are the results for $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=1$ and $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=3$, respectively. In panel (b), the solid line is the result for the signum model [Eq. (72)], the dotted and dashed lines are the results for $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=100$ and $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=30$, respectively. The thick solid blue and thick dashed red lines in panel (b) are ${\mathrm{\Sigma }}_{s+}^{tr}$ and ${\mathrm{\Sigma }}_{s-}^{tr}$ [Eq. (C5)], respectively. The phase portraits at the red filled and blue open circles in (a) are shown in Fig. 8, and at the red filled and blue open circles in (b) are shown in Fig. 11 for the signum model.

Figure 8

Phase portraits in the $\theta$ vs $\varphi -\eta$ plane and sample trajectories for a thin rod and the linear model for the magnetic moment. The external field orientation $\xi =\pi /4$ and $\eta =\pi /4$, and external field strength $\mathrm{\Sigma }=0.6$ (a) and 1.4 (b) at the red filled and blue open circles in Fig. 7, respectively. The thick blue line is the stable limit cycle, the blue open circles and red filled circles are the stable and unstable fixed points, the brown plus is a saddle point, and the dashed brown lines are separatrices between different basins of attraction/repulsion.

Figure 9

The scaled frequency of rotation for a thin rod for the linear model (solid blue lines) and signum model (dashed red lines) as a function of $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_t^{tr})$ and $({\mathrm{\Sigma }}_s/{\mathrm{\Sigma }}_{st}^{tr})$ (a) and $(1-\mathrm{\Sigma }/{\mathrm{\Sigma }}_t^{tr})$ and $(1-{\mathrm{\Sigma }}_s/{\mathrm{\Sigma }}_{st}^{tr})$ (b) when the orientation of the magnetic field is $\eta =\pi /4$ and $\xi =\pi /6$ ($\circ$), $\pi /4$ ($▵$), $\pi /3$ ($\nabla$) and $\pi /2$ ($\diamond$). Here ${\mathrm{\Sigma }}_t^{tr}$ and ${\mathrm{\Sigma }}_{st}^{tr}$ are given in Eqs. (66) and (72), respectively. The blue line is Eq. (69) for the linear model.

Figure 10

The scaled torque $(T_Z^M/\mathrm{\Gamma })$ for the steady solution (solid lines) and the average torque ${\overline{T}}_Z^m/\mathrm{\Gamma }$ for the rotating limit cycle (dashed line) for a thin rod as a function of $\mathrm{\Sigma }$ for the linear model (a) and as a function of ${\mathrm{\Sigma }}_s$ for the signum model (b) for $\xi =\pi /4$ and $\eta =0$ ($\circ$), $\pi /4$ ($▵$), and $3\pi /4$ ($\diamond$).

Figure 11

Phase portraits in the $\theta$ vs $\varphi \eta$ plane and sample trajectories for a thin rod and the signum model for the magnetic moment. The external field orientation is $\xi =\pi /4$ and $\eta =\pi /4$, and external field strength is ${\mathrm{\Sigma }}_s=0.15$ (a) and 0.22 (b), at the red filled and blue open circles in Fig. 7, respectively. The blue circle is the stable fixed point, the blue line is the stable limit cycle, and $\widehat{\mathit{z}}·\widehat{\mathit{H}}=0$ along the black dashed line.

Figure 12

The scaled frequency of rotation, $\omega /{\mathrm{\Sigma }}_t^{tr}$ (blue solid lines) and $(\omega /{\left({\mathrm{\Sigma }}_{st}^{tr}\right)}^{2/3})$ (red dashed lines) for a thin rod for the Langevin model as a function of (a) $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_t^{tr})$ (blue) and $({\mathrm{\Sigma }}_s/{\mathrm{\Sigma }}_{st}^{tr})$ (red) and (b) $(1-\mathrm{\Sigma }/{\mathrm{\Sigma }}_t^{tr})$ (blue) and $(1-{\mathrm{\Sigma }}_s/{\mathrm{\Sigma }}_{st}^{tr})$ (red) when the orientation of the magnetic field is $\xi =\pi /4$ and $\eta =\pi /4$ for $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=0.3$ ($\square$) 1.0 ($\circ$), 3.0 ($▵$), 10.0 ($\nabla$), and 30.0 ($\diamond$). The thick blue and red lines are the results for the linear model and signum model, respectively.

Figure 13

The scaled magnetic torque $(T_Z^m/\mathrm{\Gamma })$ for the stationary states (solid lines), and the average scaled torque $({\overline{T}}_Z^m/\mathrm{\Gamma })$ for the rotating states (dashed lines) for the Langevin model as a function of $\mathrm{\Sigma }$ (a) for $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=0.1$ (blue), 0.3 (red), and 1 (black) and as a function of ${\mathrm{\Sigma }}_s$ (b) for $(\mathrm{\Sigma }/{\mathrm{\Sigma }}_s)=10$ (blue), 3 (red), and 1 (black). The orientation of the magnetic field is $\xi =\pi /4$ and $\eta =0$ ($\circ$), $\pi /4$ ($▵$), and $3\pi /4$ ($\diamond$).

Figure 14

The phase portrait in the $\theta \text{−}\varphi$ plane and sample trajectories for shape factor $B=0.8$ and the linear model for the magnetic field, for external field strength $\mathrm{\Sigma }=5$ and external field orientation $(\xi =\pi /5,\eta =\pi /4)$ (a) and $(\xi =2\pi /5,\eta =\pi /4)$ (b). The open blue circles are stable stationary points, the filled red circles are unstable stationary points, the red dashed line in (a) is an unstable limit cycle, and the brown dashed lines in (b) are separatrices between different basins of attraction/repulsion.

Figure 15

Lines demarcating regions with one stable stationary node and one unstable limit cycle (left of the line) and one stable, one unstable, and one saddle node (right of the line) in the limit $\mathrm{\Sigma },{\mathrm{\Sigma }}_s\gg 1$ in the $\xi \text{−}\eta$ orientation space of the external field, for $B=0.1$ ($\circ$), 0.3 ($▵$), 0.5 ($\nabla$), 0.7 ($◃$), 0.9 ($▹$), 0.95 ($\diamond$), 0.99 ($\square$).

Figure 16

For a spherical particle, as a function of $1-sin\left(\xi \right)$, the values of ${\mathrm{\Sigma }}_t^{sp}$ [blue lines (a)] for the transition between one unstable fixed point and three fixed points, ${\mathrm{\Sigma }}_{+}^{sp}$ [red lines (a)] for the transition between three fixed points and one stable fixed point, and ${\mathrm{\Sigma }}_t^{sp}$ [black lines (a)] and (${\mathrm{\Sigma }}_t^{sp}/{\mathrm{\Sigma }}_s$) (b) for the transition between one unstable and one stable fixed point. In panel (a), the symbols represent $\circ$ linear model [Eqs. (B4) and (B10)], $▵{\mathrm{\Sigma }}_s=1, \nabla {\mathrm{\Sigma }}_s=0.7, \diamond {\mathrm{\Sigma }}_s=0.542$. In panel (b), the symbols represent $\circ {\mathrm{\Sigma }}_s=0.01, ▵{\mathrm{\Sigma }}_s=0.1, \nabla {\mathrm{\Sigma }}_s=0.3, \diamond {\mathrm{\Sigma }}_s=0.541$. The dashed line in panel (a) shows a slope of $-\frac{1}{2}$, and in panel (b) shows a slope of $-\frac{3}{2}$.

Figure 17

The phase portrait in the $\theta$ vs $\varphi \text{,}\eta$ plane and sample trajectories for a spherical particle and the linear model for the magnetic moment. The external field orientation $\xi =\pi /3$ and external field strength $\mathrm{\Sigma }=0.25$ (a) and 0.75 (b) at the points P1 and P2 in Fig. 16 respectively. The open blue and filled red circles are stable and unstable fixed points; the blue dashed line in (a) is a stable limit cycle, and the red dashed line in (b) is an unstable limit cycle.

Figure 18

The phase portrait in the $\theta$ vs $\varphi \text{−}\eta$ plane and sample trajectories for a spherical particle and the linear model for the magnetic moment. The external field orientation $sin{\left(\xi \right)}^2=0.95$ and external field strength $\mathrm{\Sigma }=0.5$ (a), 1.0 (b), and 2.0 (c) at the points P3, P4, and P5 in Fig. 16, respectively. The open blue and filled red circles are stable and unstable fixed points, the brown plus is a saddle node, the blue dashed line in (a) is a stable limit cycle, the brown dashed lines in (b) are separatrices between different basins of attraction, and the red dashed line in (c) is an unstable limit cycle.