Orientational dynamics of a triaxial ellipsoid in simple shear flow: Influence of inertia

The motion of a single ellipsoidal particle in simple shear flow can provide valuable insights toward understanding suspension flows with nonspherical particles. Previously, extensive studies have been performed on the ellipsoidal particle with rotational symmetry, a so-called spheroid. The nearly prolate ellipsoid (one major and two minor axes of almost equal size) is known to perform quasiperiodic or even chaotic orbits in the absence of inertia. With small particle inertia, the particle is also known to drift toward this irregular motion. However, it is not previously understood what effects from fluid inertia could be, which is of highest importance for particles close to neutral buoyancy. Here, we find that fluid inertia is acting strongly to suppress the chaotic motion and only very weak fluid inertia is sufficient to stabilize a rotation around the middle axis. The mechanism responsible for this transition is believed to be centrifugal forces acting on fluid, which is dragged along with the rotational motion of the particle. With moderate fluid inertia, it is found that nearly prolate triaxial particles behave similarly to the perfectly spheroidal particles. Finally, we also are able to provide predictions about the stable rotational states for the general triaxial ellipsoid in simple shear with weak inertia.

Figure 1

(a) Illustration of the flow problem of a triaxial ellipsoid suspended in simple shear flow; (b) definitions of angles in the rotated system; (c) typical setup in the numerical simulations, where the shear flow is generated by two parallel walls moving in opposite directions with velocity $U=GL/2$ in a cubical domain with side $L$.

Figure 2

Poincaré surface-of-section at $\varphi =2\pi n$ of an ellipsoid with $r_p=4$; (a) perfect prolate $r_m=1$; (b) triaxial particle of $r_m=1.1$; (c) triaxial particle of $r_m=4/3$.

Figure 3

Stable rotational states of a perfectly spheroidal particle of $r_p=4$ obtained through LB-EBF simulations by Rosén et al. [18]; SA = (“viscous”) short axis tumbling; SA* = “inertial” short axis tumbling; LA = long axis rotation; I = inclined rotation; SS = steady state; dashed lines indicate the accurate values at $\kappa \to 0$ of ${\text{Re}}_{\text{LR}}$ and ${\text{Re}}_{\text{PF}}$ obtained with global stability analysis and ${\text{Re}}_c$ obtained with stationary fluid simulations.

Figure 4

Time series of $m_z={\mathit{e}}_z·\mathit{m}$ for a particle starting close to MA at ${\text{Re}}_p=1$ and $\text{St}=75$; the lower envelope (solid black line) is fitted to an exponential function of form $1-C_{1}exp\lambda t$ (black dashed line) with $C_1=2.2\times {10}^{-3}$ and $\lambda =-8.2\times 10-4$; the negative value of $\lambda$ indicates that MA is stable.

Figure 5

Stability exponent $\lambda$ for the principal rotational states LA, MA, and SA from DNS simulations at low ${\text{Re}}_p$ (left) and moderate ${\text{Re}}_p$ (right) with arrows showing the effect of increasing ${\text{Re}}_p$; results are compared with the Stokes flow model by Lundell [20].

Figure 6

Stability of principal rotations in the ${\text{Re}}_p,\text{St}$-parameter space for a triaxial particle with $r_p=4$ and $r_m=4/3$ at (a) low ${\text{Re}}_p$ (log scale) and (b) moderate ${\text{Re}}_p$ (linear scale); the shaded regions indicate regions with stable rotations of SA, MA, and LA; SS denotes region with steady state; the ${\text{Re}}_p$ at which bifurcations occur (${\text{Re}}_{\text{LR}},{\text{Re}}_{\text{PF}}$, and ${\text{Re}}_c$) are shown with the solid, near vertical lines and the corresponding lines for a perfectly prolate spheroid ($r_m=1$) obtained by Rosén et al. [18, 38] are dashed; the small black dots indicate the simulated parameters that are used to sketch the transitional curves; the white region denoted with C is a region where none of the principal rotational states are stable and where the chaotic rotation is probable; the white region C* also indicates a region where none of the principal rotations are stable, but there might be other stable complex states; the single asterisk (*) indicates a region where MA might be stable, but can only be determined by investigating more initial conditions.

Figure 7

Streamlines in the flow-gradient plane around a perfectly prolate spheroid of $r_p=4$ and $r_m=1$ (a) during the acceleration phase of the tumbling state (SA/MA) at ${\text{Re}}_p=10$; (b) during the deceleration phase of the tumbling state (SA/MA) at ${\text{Re}}_p=10$; (c) during steady state (SS) at ${\text{Re}}_p=110$; the insets show the approximate appearance of the local flow field; the saddle points in (a) and (b) move closer to the particle surface as ${\text{Re}}_p$ increases; the thick dashed lines in (c) show the approximate directions of the uniaxial strain flow surrounding the particle.

Figure 8

Stability exponent $\lambda$ from Stokes flow model by Lundell [20] with flow modifications using parameters ${\epsilon }_1$ (causing inflow along vorticity axis and outflow in flow-gradient plane if ${\epsilon }_1>0$) and ${\epsilon }_2$ (causing more strain in the flow-gradient plane if ${\epsilon }_2>0)$.

Figure 9

Stability exponent $\lambda$ for different ellipsoid shapes ranging from a prolate spheroid $r_m=1$ to an oblate spheroid $r_m=r_p$; the base line case (thick line in all three subfigures) is the exponent at ${\text{Re}}_p=\text{St}={\epsilon }_1={\epsilon }_2=0$ and $r_p=4$; (a) effect of increasing $\text{St}$; (b) effect of increasing ${\epsilon }_1$ (connected to weak ${\text{Re}}_p$ effects); the cross symbols show results from DNS simulations at ${\text{Re}}_p=\text{St}=10$; (c) effect of increasing aspect ratio $r_p$.