Nonlocal flow sampling enables vortex trapping of heavy particles

Abstract

Most analyses of inertial particle motion in vortical flows rely on the point-particle approximation, in which the fluid velocity is assumed to be linear at the scale of the particle, and for heavy particles inertia typically leads to centrifugal expulsion from vortex cores. Here, we show that a spatially extended particle, modeled as a rigid symmetric dumbbell of two identical inertial point particles connected by a massless rod that samples the flow at two points, can converge to a vortex-centered spinning state. We study the dynamics of this inertial dumbbell in a steady two-dimensional Lamb-Oseen vortex and identify three qualitatively distinct long-time behaviors controlled by the Stokes number. In the weak-inertia limit, the motion remains bounded and traces spirographic-like trajectories around the vortex center, while at sufficiently large inertia centrifugal effects dominate and trajectories spiral outward, approaching inertial point-particle behavior. Between these limits, the dumbbell can reach a trapped spinning state in which the center-of-mass converges to the vortex center and spins steadily, with accessibility determined by the initial conditions. Basin-of-attraction maps and ensemble statistics reveal a non-monotonic dependence of the accessibility of the spinning state on inertia, with basins of finite measure occurring only over an intermediate range of Stokes numbers. Linear stability is governed by the logarithmic slope of the vortex angular-velocity profile, and for the Lamb-Oseen vortex the spinning state is stable for all Stokes numbers. These results highlight how nonlocal flow sampling by spatially extended inertial particles can fundamentally alter transport and long-time behavior in vortical flows.

Figures (6)

• Figure 1: Schematic of a rigid dumbbell of length $\ell$ in a Lamb--Oseen vortex. The color map indicates the fluid angular velocity $\Omega(r)$. The inset shows the coordinate system and notation: $\bm{r}_1$ and $\bm{r}_2$ denote the bead positions, and $\bm{r}_c$ denotes the center-of-mass position. The dumbbell orientation is given by the angle $\theta$ measured from the $x$-axis, and $\alpha$ is the angle between the rod and $\bm{r}_c$.
• Figure 2: Trajectories of the center-of-mass of the dumbbell in the $(x_c,y_c)$ plane for different Stokes numbers: (a) $\mathrm{St}=10^{-5}$, (b) $\mathrm{St}=10^{-2}$, (c) $\mathrm{St}=5\times10^{-2}$, and (d) $\mathrm{St}=10^{-1}$. The green marker indicates the initial position of the center-of-mass at $(x_c(0),y_c(0))=(1,0)$, and the red marker denotes the position at time $t=50$. At $t=0$, the orientation is $\theta(0)=0$, and the initial translational and angular velocities are zero, $v_{cx}(0)=v_{cy}(0)=0$ and $\omega(0)=0$.
• Figure 3: Long-time evolution of a dumbbell in a Lamb--Oseen vortex for $\mathrm{St}=10^{-2}$ (blue), $\mathrm{St}=5 \times 10^{-2}$ (green), and $\mathrm{St}=10^{-1}$ (red), with the same initial conditions as in Fig. \ref{['fig:trajectories']}. Panels show (a) the radial distance $r_c(t)$, (b) the radial velocity $v_r(t) = \dot{r}_c$ and (c) the azimuthal velocity $v_\phi(t) = r_c \dot{\phi}_c$ of the dumbbell center-of-mass, and (d) the angular velocity $\omega(t) = \dot{\theta}$ of the dumbbell. The inset in (b) magnifies the late-time interval corresponding to the dashed box.
• Figure 4: Basins of attraction of the spinning state at the vortex center in the plane of initial conditions $(r_c(0),\alpha(0))$ for different Stokes numbers: (a) $\mathrm{St}=10^{-2}$, (b) $\mathrm{St}=5\times 10^{-2}$, and (c) $\mathrm{St}=1$. Purple and yellow regions indicate, respectively, initial conditions that lead to the trapped spinning state at the origin and the outward-spiraling state. (d) Fraction of initial conditions leading to the spinning state, $f_{\mathrm{spin}}$ (blue, left axis), and the corresponding critical initial radius $r_c^{\mathrm{crit}}$ (red, right axis, see text for definition) as functions of the Stokes number.
• Figure 5: Linear stability analysis of the spinning state. (a) Real parts of the four eigenvalues $\lambda_{1,\dots,4}$ of $\mathrm{J}_{\mathrm{rot}}$ governing perturbations about the spinning state. (b) Imaginary parts of the same eigenvalues, showing the conjugate pairing $\pm\,\Im(\lambda)$.