Heavy ellipsoids in creeping shear flow: Transitions of the particle rotation rate and orbit shape

The motion of an inertial ellipsoid in a creeping linear shear flow of a Newtonian fluid is studied numerically. This constitutes a fundamental system that is used as a basis for simulations and analysis of flows with heavy nonspherical particles. The torque on the ellipsoid is given analytically by Jeffery [Proc. R. Soc. London, Ser. A 102, 161 (1922)]. This torque is coupled with the angular-momentum equation for the particle. The motion is then governed by the Stokes number $\text{St}={\rho }_e\dot{\gamma }{l}^2/\mu$, where ${\rho }_e$ is the density of the ellipsoid, $\dot{\gamma }$ is the rate of shear, $l$ is the length of the major axis of the ellipsoid, and $\mu$ is the dynamic viscosity of the fluid. For low St (the numerical value depends on the aspect ratio of the particle), the particle motion is similar to the Jeffery orbits obtained for inertia-free particles with the addition of an orbit drift so that the particle eventually lies in the flow-gradient plane. At higher St, more drastic effects are seen. For particles oriented in the flow-gradient plane, the rotation rate increases rather abruptly to half the shear rate in a narrow range of St. For particles with other orientations, the motion goes from a kayaking motion to rotation around an oblique axis. It is suggested that, depending on aspect and density ratios, particle inertia might be sufficient to explain and model orbit drift observed previously at low Reynolds numbers. It is discussed how and when the assumption of negligible fluid inertia and strong particle inertia can be justified from a fundamental perspective for particles of different aspect ratios.

Figure 1

(Color online) Physical situation under study in (a) and definition of the angles $\theta$ and $\phi$ for an oriented spheroid in (b).

Figure 2

(Color online) (a) Rotation angle $\phi$ as a function of time for $k_b=k_c=0.1$ and $\text{St}=0.1$ (blue, thin), and $\text{St}=1000$ (red, thick). (b) Rotation period $T$ as a function of St for $k_b=0.01$ (green, thick) and $k_b=0.1$ (purple, thin) and $k_c=0.01$ and $k_c=0.1$ (solid and dashed, respectively).

Figure 3

(Color online) Contours of the logarithm of ${\text{St}}_{0.5}$ as a function of $k_b$ and $k_c$. The solid contour is 100 and the dashed contours are ${10}^{1.5}$, ${10}^{2.5}$, ${10}^3$, and ${10}^{3.5}$. The thick solid line shows the line $k_b=k_c$, which indicates prolate spheroids.

Figure 4

(Color online) Trajectory of the spheroid end point (thick black) and its projection on the $x^{\prime }{y}^{\prime }$ and $x^{\prime }{z}^{\prime }$ planes (red, thin). $k_b=k_c=0.1$, ${\theta }_0=\pi /8$ and $\text{St}=100$, 316, 562, 1000, 3162, and $10000$ from (a) to (f). Time goes from 0 to 225, except for (b) where the thinner gray extension of the trajectory shows the motion up to $t=800$.

Figure 5

(Color online) (a) Orbit parameter $C_B$ and (b) rotation period $T$ as a function of time. $\text{St}=10$ (blue, dotted), 100 (red, dashed), 316 (green, dash-dotted), 1000 (black, solid), and $10000$ (gray, solid).

Figure 6

(Color online) Trajectory of the spheroid end point (black, thick) and its projection on the $x^{\prime }{y}^{\prime }$- and $x^{\prime }{z}^{\prime }$-planes (red, thin). $k_b=k_c=0.1$. (a) and (b) $\text{St}=1000$, ${\theta }_0=3\pi /8$, and $\pi /4$. Time goes from 0 to 225.

Figure 7

(Color online) (a) Orbit parameter $C_B$ and (b) rotation period $T$ as a function of time for $\text{St}=1000$, ${\theta }_0=\pi /8$ (green, solid), $\pi /4$ (red, dashed), and $3\pi /8$ (black, dash-dotted). (c) $t_{99}$ for $k_b=k_c=0.3$ (blue, thin) and 0.1 (red, thick), ${\theta }_0=\pi /8$ (solid), $\pi /4$ (dashed), and $3\pi /8$ (dash-dotted). The vertical dashed lines indicate the value of ${\text{St}}_{0.5}$ for the respective aspect ratio.

Figure 8

(Color online) Orbit drift parameter $c^{\prime }$ as a function of St for the present case without fluid inertia (blue, ${}^{\ast }$) and from simulations including fluid and particle inertia with $\kappa =1.001$ by Yu et al. [24] (red, ○). $k_b=k_c=0.5$.

Figure 9

(Color online) (a) The function $f_{\alpha }\left(k_b,k_c\right)$. Contours are 2 (solid), 3, 4, 10, 20, and 25. (b) $l$ (solid) and $\dot{\gamma }$ (dash-dotted) for different levels magnitudes of $a_g$, from $a_g=g$ (blue, thick) to $a_g={10}^{-6}g$ (purple, thin) where $g$ is the gravitational acceleration on earth. The intermediate lines are ${10}^{-2}g$ and ${10}^{-4}g$.