Attraction of neutrally buoyant deformable particles towards a vortex

We conduct direct numerical simulations (DNS) by the full Eulerian finite difference method of a neutrally buoyant hyperelastic particle in Taylor–Green vortical flow. We investigate the case that the initial shape of the particle is spherical with a diameter comparable with the vortex radius. The particle orbit depends on the degree of its deformation due to the flow. When the particle is stiffer (i.e., the capillary number Ca is smaller than about 0.1), the particle is hardly deformed and slowly swept out from the vortex. In contrast, softer particles with $\text{Ca}\gtrsim 0.1$ are significantly deformed, in an initial period, and they are attracted towards the vortex; then, afterward, the deformation is relaxed so that they can stay around the vortex center. Such an attraction of anisotropic neutrally buoyant particles towards a vortex occurs, even if particles are rigid. We demonstrate this phenomenon by additional DNS of a rigid prolate spheroidal particle in the vortical flow. We also develop analytical arguments to show that the angle between the major axis of the particle and the pathline determines whether the vortex attracts or repulses the particle. This feature well explains the radial motion of neutrally buoyant elastic particles in vortical flow.

Figure 1

(a) Location and deformation of an elastic particle with $\text{Ca}=10$ at (i) $t^{\prime }=0$, (ii) 2, (iii) 4, and (iv) 6. The blue curve denotes the orbit during (ii) $0\le t^{\prime }\le 2$, (iii) $2\le t^{\prime }\le 4$, and (iv) $4\le t^{\prime }\le 6$. The solid black line is the axis of the Taylor–Green vortex. (b) Similar to panel (a) but for a particle with $\text{Ca}=0.01$. Supplemental movie for panel (a) is available online [43].

Figure 2

(a) Temporal evolution of the distance $r\left(t\right)$ between the elastic particle's mass center and the axis of the Taylor–Green vortex. The initial condition is $r\left(0\right)=5\mathcal{L}/16$. Different lines correspond to results with different values of the capillary number: $\text{Ca}=10$, 1, 0.1, and 0.01 from dark to light colors. (b) The Ca-dependence of the distance $r\left(t^{\prime }\right)$ at time $t^{\prime }=2$ ($•$) and $t^{\prime }=4$ ($\circ$).

Figure 3

Initial-condition dependence of the location and deformation of the elastic particle with $\text{Ca}=10$. We show (a) the distance $r\left(t^{\prime }\right)$ from the vortex axis and (b) Taylor parameter $T\left(t^{\prime }\right)$, which is defined by Eq. (17), as functions of the normalized time $t^{\prime }$. Four different curves denote results with different initial locations: $r\left(0\right)=i\mathcal{L}/16$ with $i=3$, 4, 5 and 6 from lighter (thicker) to darker (thinner) lines.

Figure 4

(a) Temporal evolution of the Taylor parameter $T\left(t\right)$ defined by Eq. (17). Similarly to Fig. 2, different lines correspond to different Ca. (b) The Ca-dependence of the temporal average ${\langle T\rangle }_{0\le t^{\prime }\le 2}$ of $T$ in the period $0\le t^{\prime }\le 2$. The initial condition is $r\left(0\right)=5\mathcal{L}/16$.

Figure 5

Location and orientation of the rigid prolate spheroidal particle with $T=0.5$ at (a) $t^{\prime }=0$, (b) 1.5, and (c) 3. The black line is the vortex axis, and the blue line is the trajectory of the particle in the periods (b) $0\le t^{\prime }\le 1.5$ and (c) $1.5\le t^{\prime }\le 3$.

Figure 6

(a) Temporal evolution of the distance $r$ between the vortex axis and the particle center. The light gray, gray and black lines show the results for $T=0.2$, 0.33, and 0.5, respectively. (b) Temporal evolution of the angle $\psi$ [see Fig. 7] for $T=0.5$. The dashed line indicates $\psi =0$. Shaded periods in both panels correspond to the internals with $\psi <0$ for $T=0.5$.

Figure 7

The definition of $\psi$, which is the angle between the major axis of the prolate spheroidal particle and the radial direction. Here, $O$ denotes the center of the vortex. (b) The rod model examined in Sec. 4b, which is composed of two neurally buoyant particles connected by a massless rod with length $L$. The angle $\psi$ is defined similarly to panel (a).

Figure 8

(a) Temporal evolution of $\psi$ of the elastic particles. Different lines correspond to results with different values of Ca: 10, 1, 0.1 and 0.01 from dark to light colors. (b) The Ca-dependence of the temporal average of $\langle Tsin\psi \rangle$ over the period $0\le t^{\prime }\le 2$.

Figure 9

Temporal evolution of the rotational speed for a rigid spheroidal particle in the Couette flow. Solid lines denote the analytical solutions by Jeffery [49], and circles are the result of the present DNS. Different colors correspond to different values of the Taylor parameter: red, $T=1.5$; black, 2; blue, 3.