Effect of fluid and particle inertia on the rotation of an oblate spheroidal particle suspended in linear shear flow

This work describes the inertial effects on the rotational behavior of an oblate spheroidal particle confined between two parallel opposite moving walls, which generate a linear shear flow. Numerical results are obtained using the lattice Boltzmann method with an external boundary force. The rotation of the particle depends on the particle Reynolds number, ${\mathrm{Re}}_p=G{d}^2{\nu }^{-1}$ ($G$ is the shear rate, $d$ is the particle diameter, $\nu$ is the kinematic viscosity), and the Stokes number, $\mathrm{St}=\alpha {\mathrm{Re}}_p$ ($\alpha$ is the solid-to-fluid density ratio), which are dimensionless quantities connected to fluid and particle inertia, respectively. The results show that two inertial effects give rise to different stable rotational states. For a neutrally buoyant particle $\left(\mathrm{St}={\mathrm{Re}}_p\right)$ at low ${\mathrm{Re}}_p$, particle inertia was found to dominate, eventually leading to a rotation about the particle's symmetry axis. The symmetry axis is in this case parallel to the vorticity direction; a rotational state called log-rolling. At high ${\mathrm{Re}}_p$, fluid inertia will dominate and the particle will remain in a steady state, where the particle symmetry axis is perpendicular to the vorticity direction and has a constant angle ${\varphi }_c$ to the flow direction. The sequence of transitions between these dynamical states were found to be dependent on density ratio $\alpha$, particle aspect ratio $r_p$, and domain size. More specifically, the present study reveals that an inclined rolling state (particle rotates around its symmetry axis, which is not aligned in the vorticity direction) appears through a pitchfork bifurcation due to the influence of periodic boundary conditions when simulated in a small domain. Furthermore, it is also found that a tumbling motion, where the particle symmetry axis rotates in the flow-gradient plane, can be a stable motion for particles with high $r_p$ and low $\alpha$.

Figure 1

Rotational states of the oblate spheroid in a linear shear flow; (a) tumbling; (b) log-rolling; (c) kayaking; (d) inclined rolling; (e) steady state (the particle symmetry axis is located in the $xy$ plane with an angle ${\varphi }_c$ to the $x$ direction).

Figure 2

Flow problem studied: An oblate spheroidal particle is suspended between two parallel moving walls, creating a linear shear flow in the $x$ direction; the particle orientation is determined by the angles $\theta$ and $\varphi$. Due to rotational symmetry the angle around the symmetry axis, $\psi$ is of less interest.

Figure 3

Qualitative flow field around a steady particle at high Reynolds number, ${\mathrm{Re}}_p\ge {\mathrm{Re}}_c$; the stresses on the particle surface from the primary flow (1) give a torque in vorticity direction, whereas the secondary flow (2) gives a torque opposite to the vorticity. The dashed streamlines separates the two flow regions and lead to stagnation points on the particle surface.

Figure 4

Trajectories of the particle $\left(r_p=4,\alpha =1\right)$ rotation projected through $(s_x,s_y)=(sin\theta cos\varphi ,sin\theta sin\varphi )$ with increased ${\mathrm{Re}}_p$; the upper figures show schematically how the trajectories behave, while lower figures show the actual simulated trajectories; solid circles denote the initial orientation in the lower figures; for clarification of the symmetry in the problem, the equivalent trajectories of $(-s_x,-s_y)$ are also drawn; (a) ${\mathrm{Re}}_p=10$; (b) ${\mathrm{Re}}_p=26$; (c) ${\mathrm{Re}}_p=31$; (d) ${\mathrm{Re}}_p=32$; (e) ${\mathrm{Re}}_p=36$; (f) ${\mathrm{Re}}_p=52$.

Figure 5

Schematical illustration of the bifurcation occurring at ${\mathrm{Re}}_p={\mathrm{Re}}_c$; (a) ${\mathrm{Re}}_p\lesssim {\mathrm{Re}}_c$; (b) ${\mathrm{Re}}_p={\mathrm{Re}}_c$; (c) ${\mathrm{Re}}_p\gtrsim {\mathrm{Re}}_c$.

Figure 6

Intrinsic viscosity $\eta$ as function of ${\mathrm{Re}}_p$ for a neutrally buoyant oblate spheroidal particle of $r_p=4$; LR = log-rolling, T = tumbling, S = steady state.

Figure 7

Equilibrium angles of the oblate spheroid of $r_p=4$ as function of ${\mathrm{Re}}_p$; by fitting the data points to a polynomial function, the critical Reynolds number is determined to be ${\mathrm{Re}}_c=31$.

Figure 8

The critical density ratio ${\alpha }_c$ as function of ${\mathrm{Re}}_p$ (thin blue line) for an oblate spheroid of $r_p=4$; the dashed black line shows the fitted rational function to ${\alpha }_c$ and the thick red line is the simulated value of ${\alpha }_T$. The error bars correspond to the size of the minimum interval $\mathrm{\Delta }\alpha =10/{\mathrm{Re}}_p$; the black crosses indicate $\alpha$ and ${\mathrm{Re}}_p$ for which the graphs in Fig. 9 are taken.

Figure 9

Angular velocity and angle $\theta$ as function of time for an oblate spheroid at ${\mathrm{Re}}_p>{\mathrm{Re}}_c$ and $\alpha ={\alpha }_c+10/{\mathrm{Re}}_p$; although initialized in a tumbling motion, all cases lead to a steady state [$\theta =\pi /2$ and $(\dot{\varphi },\dot{\theta },\dot{\psi })=(0,0,0)$]. (a) ${\mathrm{Re}}_p=40,\alpha =4.309$; (b) ${\mathrm{Re}}_p=50,\alpha =4.524$; (c) ${\mathrm{Re}}_p=75,\alpha =4.122$; (d) ${\mathrm{Re}}_p=100,\alpha =3.807$.

Figure 10

Stable rotational states depending on St and ${\mathrm{Re}}_p$ for an oblate spheroid of $r_p=4$; LR = log-rolling, T = tumbling, S = steady state. At $\mathrm{St}>{\mathrm{St}}_c={\alpha }_c{\mathrm{Re}}_p$ and at $\mathrm{St}>{\mathrm{St}}_T={\alpha }_T{\mathrm{Re}}_p$ a tumbling solution exists, but it is unstable.

Figure 11

Critical Reynolds numbers ${\mathrm{Re}}_c$ and ${\mathrm{Re}}_{\mathrm{PF}}$ as functions of particle aspect ratio $r_p$.

Figure 12

Critical density ratios ${\alpha }_{\mathrm{cd}2}$ and ${\alpha }_{c,\max }$ as functions of $r_p$.

Figure 13

Trajectories of a neutrally buoyant oblate spheroid of $r_p=2$ in a highly confined domain $\left(\kappa =0.5\right)$ plotted the same way as in Fig. 4. (a) ${\mathrm{Re}}_p=100$ (log-rolling); (b) ${\mathrm{Re}}_p=130$ (inclined rolling); (c) ${\mathrm{Re}}_p=160$ (steady state).

Figure 14

Schematical illustration of the bifurcation occurring at ${\mathrm{Re}}_p={\mathrm{Re}}_c$ at high confinement for an oblate spheroid of $r_p=2$. (a) ${\mathrm{Re}}_p\lesssim {\mathrm{Re}}_c$; (b) ${\mathrm{Re}}_p={\mathrm{Re}}_c$; (c) ${\mathrm{Re}}_p\gtrsim {\mathrm{Re}}_c$.

Figure 15

The critical Reynolds numbers ${\mathrm{Re}}_c$ and ${\mathrm{Re}}_{\mathrm{PF}}$ as function of channel confinement $\kappa$ for an oblate spheroid of $r_p=2$; the critical confinement above which ${\mathrm{Re}}_c>{\mathrm{Re}}_{\mathrm{PF}}$ is found at ${\kappa }_c\approx 0.24$. $\mathrm{LR}=$ log-rolling; $\mathrm{T}=$ tumbling; $\mathrm{S}=$ steady state; $\mathrm{IR}=$ inclined rolling.

Figure 16

The final angle ${\theta }_{\mathrm{final}}$ as function of ${\mathrm{Re}}_p$ for an oblate spheroid of $r_p=2$ at two different confinements $\kappa =0.2<{\kappa }_c$ and $\kappa =0.5>{\kappa }_c$; the inclined rolling branch connecting the log-rolling and steady state branches is stable at $\kappa >{\kappa }_c$ and unstable at $\kappa <{\kappa }_c$.