Dynamics of paramagnetic and ferromagnetic ellipsoidal particles in shear flow under a uniform magnetic field

We investigate the two-dimensional dynamic motion of magnetic particles of ellipsoidal shapes in shear flow under the influence of a uniform magnetic field. In the first part, we present a theoretical analysis of the rotational dynamics of the particles in simple shear flow. By considering paramagnetic and ferromagnetic particles, we study the effects of the direction and strength of the magnetic field on the particle rotation. The critical magnetic-field strength, at which particle rotation is impeded, is determined. In a weak-field regime (i.e., below the critical strength) where the particles execute complete rotations, the symmetry property of the rotational velocity is shown to depend on the direction of the magnetic field. In a strong-field regime (i.e., above the critical strength), the particles are impeded at steady angles and the stability of these angles is examined. Under a uniform field, paramagnetic and ferromagnetic particles behave differently, in terms of the critical strength, symmetry property of the rotational velocity, and steady angles. In the second part, we use two-dimensional numerical simulations to study the implications of rotational dynamics for lateral migration of the particles in wall-bound shear flows. In the weak-field regime, the paramagnetic prolate particles migrate away when the field is applied perpendicular to the flow and towards the bounded wall when the field is applied parallel to the flow. Ferromagnetic particles exhibit negligible migration under fields that are parallel or perpendicular to the flow. The different lateral migration behaviors are due to the difference in the symmetry property of particle rotational velocity. In the strong-field regime, the particles are impeded at different stable steady angles, which result in different lateral migration behaviors as well. The fundamental insights from our work demonstrate various feasible strategies for manipulating paramagnetic and ferromagnetic particles.

Figure 1

Schematic of a magnetic prolate ellipsoid (major axis $2a$ and minor axis $2b=2c)$ in a simple shear flow. A uniform magnetic field is imposed at an arbitrary direction, characterized by $\beta$. The particle orientation is denoted by $\varphi$, and later ${\varphi }_p$ and ${\varphi }_f$ will be used to distinguish paramagnetic and ferromagnetic particles.

Figure 2

(a) Critical-field strength $S_p^{\mathrm{cr}}$ and (b) critical steady angle ${\varphi }_p^{\mathrm{cr}}$ as a function of field direction $\beta$ for a paramagnetic particle with $r=2$, 3, and 4.

Figure 3

Steady angles ${\varphi }_p^s$ of the paramagnetic particle and the derivatives $\frac{d\dot{\varphi }}{d{\varphi }_p}{|}_{{\varphi }_p^s}$ for field strength $S_p>S_p^{\mathrm{cr}},r=4$, and (a) $\beta =0$ and (b) $\beta =\pi /2$.

Figure 4

Influence of magnetic-field strength $S_p$ and direction $\beta$ on the rotational period when $S_p<S_p^{\mathrm{cr}}$ and $r=4$.

Figure 5

Dependence of ${\tau }_p$ on $\beta$ for a paramagnetic particle with field strength $S_p<1$ and $r=4$.

Figure 6

(a) Critical-field strength $S_f^{\mathrm{cr}}$ and (b) critical steady angle ${\varphi }_f^{\mathrm{cr}}$ as a function of field direction $\beta$ for a ferromagnetic particle with $r=1.4$, 2, 3, and 4.

Figure 7

Phase diagram of the number of steady angles for ferromagnetic particles for (a) $r=1.4$, (b) $r=\sqrt{2}$, (c) $r=2$, and (d) $r=4$.

Figure 8

Impeded angles ${\varphi }_f^s$ of the ferromagnetic particle and derivatives $\frac{d\dot{{\varphi }_f}}{d{\varphi }_f}{|}_{{\varphi }_f^s}$ for field strength $S_f>S_f^{\mathrm{cr}}$ and (a) $\beta =0$ and (b) $\beta =\pi /2$. The particle aspect ratio is $r=4$.

Figure 9

Dimensionless period of particle rotational as a function of $S_f$ for various magnetic field directions $\beta$.

Figure 10

Dependence of (a) ${\tau }_{f,1}$ and (b) ${\tau }_{f,2}$ on $\beta$ for different field strengths $S_p<1$ and $r=4$.

Figure 11

Lateral migration of (a) paramagnetic and (b) ferromagnetic particles in the weak-field regime $(S_p\approx 0.664$ and $S_f\approx 0.664)$ over a rotation of $2\pi$ with $r=4$.

Figure 12

Lateral migration of paramagnetic and ferromagnetic particles in a strong field with $r=4$. (a) Theoretical values of $sin\left(2{\varphi }_{p,\beta =0}^{ss}\right)$ (solid line) and $sin\left(2{\varphi }_{f,\beta =0}^{ss}\right)$ (dashed line) for $\beta =0$. (b) Lateral positions of the particles as a function of the dimensionless time $\dot{\gamma }t$ from the numerical simulations when $\beta =0$. (c) Theoretical values of $sin\left(2{\varphi }_{p,\beta =\pi /2}^{ss}\right)$ (solid line), $sin\left(2{\varphi }_{f,\beta =\pi /2,1}^{ss}\right)$ (dashed line), and $sin\left(2{\varphi }_{f,\beta =\pi /2,2}^{ss}\right)$ (dot-dashed line) for $\beta =\pi /2$. (d) Lateral positions of the particles as a function of the dimensionless time $\dot{\gamma }t$ from the numerical simulations when $\beta =\pi /2$. Depending on its initial orientation, the ferromagnetic particle shows two different lateral migration behaviors for $S_f=15$.

Figure 13

Steady angles ${\varphi }_p^s$ of the paramagnetic particle and the derivatives $\frac{d\dot{\varphi }}{d{\varphi }_p}{|}_{{\varphi }_p^s}$ for field strength $S_p>S_p^{\mathrm{cr}},r=4$, and (a) $\beta =\pi /4$ and (b) $\beta =3\pi /4$.

Figure 14

Steady angles ${\varphi }_f^s$ of the ferromagnetic particle and derivatives $\frac{d\dot{{\varphi }_f}}{d{\varphi }_f}{|}_{{\varphi }_f^s}$ for field strength $S_f>S_f^{\mathrm{cr}}$ and (a) $\beta =\pi$ and (b) $\beta =3\pi /2$. The particle aspect ratio is $r=4$.

Figure 15

Steady angles ${\varphi }_f^s$ of a ferromagnetic particle with $r=1.4$ and derivatives $\frac{d\dot{{\varphi }_f}}{d{\varphi }_f}{|}_{{\varphi }_f^s}$ for field strength $S_f>S_f^{\mathrm{cr}}$ and (a) $\beta =0$, (b) $\beta =\pi /2$, (c) $\beta =\pi$, and (d) $\beta =3\pi /2$.

Figure 16

Schematic of numerical model of an ellipsoidal particle suspended in a simple shear flow under the influence of a uniform magnetic field.

Figure 17

Comparison between numerical simulations and theories: (a) particle rotation in a simple shear flow, (b) rotational period of paramagnetic particles in a magnetic field of $\beta =0$, and (c) rotational period of ferromagnetic particles in a magnetic field of $\beta =0$. A particle aspect ratio $r=4$ is used in these comparisons.