Streaming flow due to a quartz tuning fork oscillating in normal and superfluid ${}^4\mathrm{He}$

We visualize the streaming flow due to a rapidly oscillating quartz tuning fork, in both normal He I and superfluid He II, by following the flow-induced motions of relatively small particles suspended in the liquid. Over the investigated temperature range, between 1.2 and 2.3 K, at the experimentally probed length scales, the streaming patterns observed in He II appear identical to those seen in He I and are very similar to those reported to occur in water, outside the Stokes boundary layer. The outcome strongly supports the view that, at scales larger than the quantum length scale of the flow, the mean distance between quantized vortices, mechanically forced turbulent coflows of He II behave classically, due to the dynamical locking of the two components of superfluid ${}^4\mathrm{He}$ by the action of the mutual friction force.

Figure 1

(a) Sketch of the tuning fork immersed in liquid helium and illuminated by the planar laser sheet from the right-hand side; the camera is placed perpendicularly to the laser sheet. (b) Typical (negative) image showing the fork prongs as a pair of rectangles; 13-mm high and 8-mm wide field of view. (c) Dimensions of the fork prongs, length $L=9.0$ mm, width $W=0.9$ mm, and height $H=0.4$ mm.

Figure 2

Trajectories obtained in He I; see the first line of Table 1. All panels (corresponding scales in mm) show the same tracks but they are differently colored by velocity direction (top left), velocity magnitude (top right), horizontal (bottom left), and vertical (bottom right) components of the particle velocity (velocities in mm/s, positive upwards and towards the right-hand side). The gray rectangles display the position of the fork prongs. Relevant color maps are shown under each panel.

Figure 3

Trajectories obtained in He II; see the second line of Table 1. Symbols as in Fig. 2 (note the different velocity magnitudes).

Figure 4

Trajectories obtained in He II; see the penultimate line of Table 1. Symbols as in Fig. 2.

Figure 5

Dependence of the Lagrangian pseudovorticity $\theta \left(\mathbf{r}\right)$ on $R_m$ for a sample case; see Eq. (1) and the fourth line of Table 1 (temperature $T=1.57$ K and driving force $F=41\mu \mathrm{N}$). The position of the prongs is shown as gray rectangles; $\theta \left(\mathbf{r}\right)$ has the same dimensions as vorticity, i.e., $1/\mathrm{s}$. (Top panel) $R_m=0.2$ mm. (Middle panel) $R_m=0.4$ mm. (Bottom panel) $R_m=2$ mm. Note the different color coding for each panel (red denotes positive, anticlockwise rotation, while blue indicates the opposite) and the fact that the entire field of view is shown here.

Figure 6

Lagrangian pseudovorticity $\theta \left(\mathbf{r}\right)$ calculated according to Eq. (1), with $R_m=0.4$ mm, for all data sets; see Table 1. Symbols as in Fig. 5. (Right column) Largest velocity data (their better quality is due to the larger number of measurements). Note that here the field of view in the fork proximity is shown (corresponding scales in mm) and that the pseudovorticity color coding depends on the fork velocity, as indicated.

Figure 7

(a) Schematic view of the streaming flow features in the proximity of the fork top left corner (the fork oscillates in the horizontal direction). (b) Lagrangian pseudovorticity map calculated from all the data obtained in He II, with driving force $F=41\mu \mathrm{N}$; see the penultimate row of Table 1. (c) Corresponding particle trajectories, color coded by velocity magnitude; see the top right panel of Fig. 4. The strong outgoing jets are shown with black arrows, as the weaker incoming jets, orientated along the axes of the prong corners.