Elliptical particle suspensions in Couette flow

The rheological properties and microstructure of neutrally buoyant elliptical particle suspensions are studied using the immersed boundary-lattice Boltzmann method. For dilute suspensions containing only one particle, with the increase of aspect ratio Ar, the particle ceases to rotate due to the inertia at a critical aspect ratio ${\text{Ar}}_c$, and the value of ${\text{Ar}}_c$ decreases with increasing Re. The inertia-induced rotation arrest causes a nonmonotonic variation of particle alignment with the flow direction, and thence the relative viscosity, with increasing Ar, for a fixed Re. The relative viscosity first decreases, with increasing Ar due to the increasing alignment (on average) of tumbling particles; however, rotation arrest leads to a subsequent increase in the relative viscosity for $\text{Ar}>{\text{Ar}}_c$. This nonmonotonic variation persists over the entire range of Re and Ar examined here. For dense suspensions containing multiple particles, with increasing Ar, particles align more with the flow, but the orientation becomes almost constant when Ar is greater than a threshold ${\text{Ar}}_t$. Smaller values of ${\text{Ar}}_t$ are observed for higher Re. Meanwhile, ${\eta }_r$ decreases due to particle alignment for $\text{Ar}<{\text{Ar}}_t$ and then increases due to high particle-particle interaction for $\text{Ar}>{\text{Ar}}_t$. Further, the contributions of stresslet ($S$), particle acceleration stress ($P$), and Reynolds stress ($R$) on ${\eta }_r$ and the first normal stress difference $N_1$ are analyzed. In addition to the major contribution of stresslet, Reynolds stress contributes more as Ar increases at high Re. In addition, the microstructure and the probability density functions of lateral velocity $\left(U_y\right)$ and angular velocity (${\omega }_p$) are also analyzed. Our results may be helpful to understand the rheological properties of nonspherical particle suspensions.

Figure 1

Schematic diagram for elliptical particles in Couette flows. In our study, all 2D elliptical particles with different Ar have an identical area, i.e., the effective radii of all particles are identical.

Figure 2

Grid (a) and time-step (b) independence studies for dilute suspension containing only one particle with $\text{Ar}=2, \text{Re}=1$. (c) Grid independence studies for particles with different aspect ratios. For cases in (a), (c), a sufficiently small time step $\delta t=0.00025\frac{1}{\dot{\gamma }}$ is adopted and fixed. For cases in (b), a sufficient spatial resolution $\delta x=\frac{1}{14}{r}_e$ is adopted and fixed. In all cases, we keep a fixed Re by changing $\mu$. (d) Grid independence study for dense cases. The average viscosity as a function of $\varphi$ for $\text{Ar}=2$ with different spatial resolution. Here $\text{Re}=1$ and $\delta t=0.00025\frac{1}{\dot{\gamma }}$ are fixed.

Figure 3

(a) Orientation and angular velocity of an elliptical particle in Couette flow. The solid and dashed lines represent the simulation result at $\text{Re}=0.08$ and Jeffery's analytical solution at $\text{Re}=0$, respectively. (b) Relative viscosity as a function of particle concentration of circular particles. (c) The rate of change of the orientation of an elliptic particle $\omega$ at Re = 15, 28, and 30. (d) The period of the motion of the ellipse increases to infinity as Re goes to the critical value.

Figure 4

The intrinsic viscosity $\left[\eta \right]$ and orientational order parameter $T_{xx}$ as functions of Ar with different $\beta$'s. The solid and hollow symbols correspond to the values of $\left[\eta \right]$ and $T_{xx}$, respectively. The vertical dashed lines are drawn to denote the critical Ar, i.e., ${\text{Ar}}_c$, specifically ${\text{Ar}}_c\approx 8.6$, 8.4, 7.8, and 7.7 for $\beta =7$, 11, 16, and 20, respectively. For these cases, $\text{Re}=0.1$ is fixed.

Figure 5

(a) Time-averaged angular velocity ${\omega }_p$ of a particle as a function of aspect ratio Ar for different Re. (b) Orientational order parameter ($T_{xx}$) as a function of Ar for different Re. (c) Critical Reynolds number ${\text{Re}}_c$ as a function of Ar. $\beta =16$ is fixed.

Figure 6

The relative viscosity as a function of (a) Ar and (b) $T_{xx}$ for different Reynolds number. The black arrows denote the direction of increasing Ar. $\beta =16$ is fixed.

Figure 7

Relative viscosity (a), (c) and first normal stress difference (b), (d) as functions of Ar for $\text{Re}=0.05$ [(a) and (b)] and $\text{Re}=1$ [(c) and (d)]. The corresponding contributions of stresslet ($S$), particle acceleration ($P$), and Reynolds stress ($R$) are shown as well. $\beta =16$ is fixed.

Figure 8

The instantaneous particle distributions and contours of $x$-component velocity ($U_x$) for (a), (d), (g) the cases of $\text{Re}=0.05$, and (b), (e), (h) the cases of $\text{Re}=1$. The upper, middle, and lower rows correspond to cases of $\text{Ar}=1$, 2, and 8, respectively. (c), (f), (i) Time-averaged $U_x$ along the $Y$-axis. The black solid line represents the undisturbed shear flow without particles. In these cases, $\beta =11$ and $\varphi =0.4$ are fixed.

Figure 9

The concentration profiles (particle distributions) in the cases of (a) $\text{Ar}=1$, (b) $\text{Ar}=2$, and (c) $\text{Ar}=8$ with different $\varphi$ at $\text{Re}=0.05$ and 1. The solid lines and dashed lines correspond to $\text{Re}=0.05$ and 1, respectively. (d) $T_{xx}$ as a function of $\varphi$ for different Re and Ar.

Figure 10

The relative viscosity (a) and first normal stress difference (b) as functions of $\varphi$ for different Ar and Re.

Figure 11

(a), (c) Relative viscosity and (b), (d) first normal stress difference as functions of Ar at (a), (b) $\text{Re}=0.05$ and (c), (d) $\text{Re}=1$. The corresponding contributions of stresslet ($S$), particle acceleration ($P$), and Reynolds stress ($R$) are shown as well. The values of $T_{xx}$ (blue solid lines) in (a) and (c) are labeled on the right-hand side of the figure. In these cases, $\varphi =0.3$ is fixed.

Figure 12

The probability density function of $U_y$ at $\text{Re}=0.05$ for different $\varphi$ for (a) $\text{Ar}=1$, (b) $\text{Ar}=2$, and (c) $\text{Ar}=8$. (d) $\text{PDF}\left(U_y\right)$'s for different Ar and Re in the cases of $\varphi =0.4$.

Figure 13

The probability density function of ${\omega }_p$ at $\text{Re}=0.05$ for different $\varphi$ for (a) $\text{Ar}=1$, (b) $\text{Ar}=2$, and (c) $\text{Ar}=8$. (d) PDFs of ${\omega }_p$ for different Ar and Re in the cases of $\varphi =0.4$.