Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit

The hydrodynamic reciprocal theorem for Stokes flows is generalized to incorporate the Navier slip boundary condition, which can be derived from Onsager's variational principle of least energy dissipation. The hydrodynamic reciprocal relations and the Jeffery orbit, both of which arise from the motion of a slippery anisotropic particle in a simple viscous shear flow, are investigated theoretically and numerically using the fluid particle dynamics method [Phys. Rev. Lett. 85, 1338 (2000)]. For a slippery elliptical particle in a linear shear flow, the hydrodynamic reciprocal relations between the rotational torque and the shear stress are studied and related to the Jeffery orbit, showing that the boundary slip can effectively enhance the anisotropy of the particle. Physically, by replacing the no-slip boundary condition with the Navier slip condition at the particle surface, the cross coupling between the rotational torque and the shear stress is enhanced, as manifested through a dimensionless parameter in both of the hydrodynamic reciprocal relations and the Jeffery orbit. In addition, simulations for a circular particle patterned with portions of no-slip and Navier slip are carried out, showing that the particle possesses an effective anisotropy and follows the Jeffery orbit as well. This effective anisotropy can be tuned by changing the ratio of no-slip portion to slip potion. The connection of the present work to nematic liquid crystals' constitutive relations is discussed.

Figure 1

$l_{\text{s}}/\xi$ plotted as a function of ${\eta }_l/{\eta }_i$. The symbols represent the results from the FPD simulations with slippery fluid-solid interface; the solid line is a linear fitting expressed by Eq. (30) for $l_s/\xi \ge 3$. The inset shows ${log}_{10}(\eta /{\eta }_l)$ plotted as a function of $\varphi$ for $\eta \left(\varphi \right)$ in Eq. (29), with ${\eta }_s/{\eta }_l=50$ and ${\eta }_i/{\eta }_l=0.1$.

Figure 2

A schematic illustration for a two-dimensional fluid-particle system.

Figure 3

Snapshots of the velocity field (arrows) and the particle (level curve of $\varphi =0.5$), obtained for the numerical verification of the hydrodynamic reciprocal relations. (a) Case 1 where the two walls are fixed (with $W=0$) and the elliptical particle rotates (with angular velocity $\omega$). (b) Case 2 where the particle is fixed (with $\omega =0$) and the two walls move with the velocities $\pm W\widehat{x}$. The slip length is $l_s=5\xi$. The fluid-particle system measures $2\times 2$, the major axis of the particle is $a=0.25$, and the aspect ratio is $e=0.5$. The fluid-solid interfacial thickness is $\xi =0.01$, the viscosity ratio is $r_{\eta }\equiv {\eta }_s/{\eta }_l=50$, the angular velocity in case 1 is $\omega =-1$, and the wall speed in case 2 is $W=2$.

Figure 4

Orientational dependence of the generalized forces, obtained for the numerical verification of the hydrodynamic reciprocal relations. (a) The integrated stress $S$ (by the upper wall on the fluid) plotted as a function of the particle orientation $\theta$ in case 1. (b) The torque $T$ (by the particle on the fluid) plotted as a function of the particle orientation $\theta$ in case 2. The symbols represent the results from the FPD simulations; the solid lines are the fitting curves according to Eq. (36). The insets in (a) and (b) illustrate case 1 and case 2, respectively. All the parameter values listed in the caption to Fig. 3 are used here.

Figure 5

The anisotropy parameter $p$ plotted as a function of the slip length $l_s$ at the particle surface. The major axis of the elliptical particle is $a=0.25$, and the aspect ratio is $e=0.5$. The fluid-solid interfacial thickness is $\xi =0.01$, and the slip length $l_s$ varies from $3\xi$ to $10\xi$. Here $p_S \left(▴\right)$ and $p_T \left(▾\right)$ are obtained through the numerical verification of the hydrodynamic reciprocal relations: $p_S$ is determined by fitting $S\left(\theta \right)$ and $p_T$ is determined by fitting $T\left(\theta \right)$ according to Eq. (36). In addition, $p_J (•)$ is obtained from the FPD simulation of the Jeffery orbit: $p_J$ is determined by fitting the angular velocity $\omega \left(\theta \right)$ (shown in Fig. 6) according to Eq. (37). Note that in the limit of no slip $\left(l_s=0\right)$, the values of $p$ agree with Jeffery's result $p=0.6$ from Eq. (35) (within $\sim 3\%$).

Figure 6

The FPD simulation results for the Jeffery orbit of a slippery elliptical particle in a simple shear flow. The fluid-particle system measures $2\times 2$, the shear rate is $\dot{\gamma }=2W/H=2$, the major axis of the particle is $a=0.25$, and the aspect ratio is $e=0.5$. The fluid-solid interfacial thickness is $\xi =0.01$, and the slip length is $l_s=5\xi$. (a) A snapshot of the velocity field (arrows) and the particle (level curve of $\varphi =0.5$). (b) The angular velocity $\omega$ plotted as a function of the particle orientation $\theta$. The symbols represent the simulation results; the solid line is the fitting curve according to Eq. (37).

Figure 7

A schematic illustration of a two-dimensional circular patchy particle. The particle surface is patterned with four sections, two with no-slip boundary condition and the other two with Navier boundary condition. The four sections are arranged in such a way that the patchy particle possesses the same symmetry as an ellipse. The diameter connecting the centers of two no-slip sections is used to denote the particle orientation. The particle radius is $a$, and $2\phi$ is the angular width of a no-slip section. The fraction covered by the no-slip sections is $c\equiv 2\phi /\pi$.

Figure 8

The FPD simulation results for the Jeffery orbit of a circular patchy particle in a simple shear flow. The fluid-particle system measures $2\times 2$, the shear rate is $\dot{\gamma }=2W/H=2$, the particle radius is $a=0.2$, the fluid-solid interfacial thickness is $\xi =0.01$, the no-slip fraction is $c=0.5$, and the slip length in the slip sections is $l_s=0.15$. (a) A snapshot of the velocity field (arrows) and the particle (level curve of $\varphi =0.5$). (b) The angular velocity $\omega$ plotted as a function of the particle orientation $\theta$. The symbols represent the simulation results; the solid line is the fitting curve according to Eq. (37).

Figure 9

The anisotropy parameter $p$ of a circular patchy particle from the FPD simulations, plotted as a function of the no-slip fraction $c$, for different values of the slip length in the slip sections $l_s$. The particle radius is $a=0.2$, and the fluid-solid interfacial thickness is $\xi =0.01$. Simulations are carried out for the no-slip fraction $c=1/6,1/3,1/2,2/3$, and $5/6$, and the slip length in the slip sections $l_s=0.10 \left(\blacksquare \right),0.15 (\square ),0.20 (•),0.25 (\circ )$, and $0.30 \left(▾\right)$.