Toy model for vortex-ring-assisted particle drag in superfluid counterflow

The interpretation of data obtained from particle image and tracking velocimetry in the study of superfluid flows has been so far a challenging task. Tracking particles (as solid hydrogen or deuterium) are attracted to the cores of quantized vortices, so that their dynamics can be strongly affected by the surrounding vortex tangle. Previous phenomenological arguments indicate that tracking particles and microsized vortex rings could form bound states (denoted here as VRP states). While a comprehensive description of the vortex ring-particle bonding mechanism has to deal with somewhat involved flow configurations, we introduce a simplified two-dimensional model of VRP states, which captures essential qualitative features of their three-dimensional counterparts. Besides an account of known experimental and numerical observations, the model proves to be of great heuristic interest. In particular, it sheds light on the important role played by viscous dissipation (due to the normal component of the fluid), the Magnus force, and topologically excited vortex rings in the stability and dynamics of VRP states.

Figure 1

A particle can get trapped to the core of a quantized vortex filament, due to the strong pressure depletion around it. When this happens, the particle's trajectory will be essentially ruled by the evolution of the host vortex filament.

Figure 2

(a) Sketch of the formation of VRP states as a three-stage process $\mathrm{A}\to \mathrm{B}\to \mathrm{C}$. The vortex ring evolution is represented on a symmetry plane cut, as seen from the particle's reference frame (see the text, Sec. 2, for details); (b) a vortex dipole hydrodynamically interacts with a solid disk in the presence of a two-dimensional superfluid stream.

Figure 3

Phase diagrams of flow regimes are depicted for some values of $b$: (a) $b=0.5$, (b) $b=1.0$, (c) $b=2.0$, and (d) $b=4.0$. The numerical simulations, which have explored the regions $0\le a\le 0.2$ and $0\le c\le 1$, have all been carried out for $\alpha =0.2$ and ${\alpha }^{\prime }=0$. Red, blue, and green points correspond to values of parameters associated, respectively, to particle-vortex decoupled flow regimes, backward-moving VRP states, and forward-moving VRP states. It turns out that backward-moving VRP states are approximately bounded from above and from below by physical lines, as particularly evidenced in the phase diagram for the case $b=4.0$ (straight dashed lines).

Figure 4

Mean particle velocities $\overline{u}$ along the physical lines $c=2.5a$ (black line) and $c=10a$ (red line) as $a$ varies. They are associated to simulation parameters $\alpha =0.2$, ${\alpha }^{\prime }=0$, and $b=4.0$.

Figure 5

Returning points of vertical (A and C) and horizontal (B and D) vortex motions, as seen from the particle's reference frame. The positive (negative) vortex orbit, represented by the red (blue) line, is counterclockwise (clockwise) oriented. Simulation parameters are $\alpha =0.2$, ${\alpha }^{\prime }=0$, $a=0.01$, $b=1.0$, and $c=0.5$.

Figure 6

Dynamical balance of a stable limit orbit, as observed from the particle's reference frame, for the (a) vertical and (b) horizontal velocity contributions. The crossing points A, B, C, and D refer to the points indicated in Fig. 5. Simulation parameters are $\alpha =0.2$, ${\alpha }^{\prime }=0$, $a=0.01$, $b=1.0$, and $c=0.5$.

Figure 7

Stable cycles for $\alpha =0.0$ (blue), 0.1 (green), and 0.2 (red). In all the simulations, ${\alpha }^{\prime }=0.0$, $a=0.01$, $b=1.0$, and $c=0.5$. The tiny orbit for the case $\alpha =0.0$ is detailed in the inset.