Experimental investigation of the dynamics of a vibrating grid in superfluid ${}^4\mathrm{He}$ over a range of temperatures and pressures

In an earlier paper [Nichol et al., Phys. Rev. E, 70, 056307 (2004)] some of the present authors presented the results of an experimental study of the dynamics of a stretched grid driven into vibration at or near its resonant frequency in isotopically pure superfluid ${}^4\mathrm{He}$ over a range of pressures at a very low temperature, where the density of normal fluid is negligible. In this paper we present the results of a similar study, based on a different grid, but now including the temperature range where the normal fluid density is no longer insignificant. The new grid is very similar to the old one except for a small difference in the character of its surface roughness. In many respects the results at low temperature are similar to those for the old grid. At low amplitudes the results are somewhat history dependent, but in essence there is no damping greater than that in vacuo. At a critical amplitude corresponding to a velocity of about $50\mathrm{mm}{\mathrm{s}}^{-1}$ there is a sudden and large increase in damping, which can be attributed to the generation of new vortex lines. Strange shifts in the resonant frequency at intermediate amplitudes observed with the old grid are no longer seen, however they must therefore have been associated with the different surface roughness, or perhaps were due simply to some artifact of the old grid, the details of which we are currently unable to determine. With the new grid we have studied both the damping at low amplitudes due to excitations of the normal fluid, and the dependence of the supercritical damping on temperature. We present evidence that in helium at low amplitudes there may be some enhancement in the effective mass of the grid in addition to that associated with potential flow of the helium. In some circumstances small satellite resonances are seen near the main fundamental grid resonance, which are attributed to coupling to some other oscillatory system within the experimental cell.

Figure 1

Electron micrographs of the gold-evaporated grid used in Ref. 15 (top) and in the present measurements (bottom).

Figure 2

Schematic circuit diagram indicating the various electrical connections to the experimental cell.

Figure 3

Resonance characteristics of the grid obtained in vacuum during cooling from $77\mathrm{K}$ down to about $8\mathrm{mK}$.

Figure 4

Initial resonance characteristics as a function of applied driving voltage (a) obtained at a pressure of $5\text{bar}$, at base temperature. Shown in (b) are the corresponding resonance curves after the grid was “cleaned” (see text). Note that the resonance frequency of the grid in He II is shifted relative to that in vacuo by $\Delta f_0\sim 40\mathrm{Hz}$.

Figure 5

(a) “Regular” resonance characteristics obtained at a pressure of $10\text{bar}$ and base temperature, as a function of the applied drive voltage. Shown in (b) are higher-drive characteristics, plotted over a wider frequency range.

Figure 6

Shown in (a) is the resonant response of the grid for various temperatures ($200\mathrm{mK}\text{to}1.25\mathrm{K}$, top-to-bottom), for a drive voltage of $10\mathrm{mVpp}$ and a pressure of $15\text{bar}$. Graph (b) shows the variation of the linewidth (FWHM) of the resonance curves as a function of temperature. The values plotted with circle (엯) markers were measured from the curves shown in the top graph, while those plotted with cross (×) markers represent estimates (see text).

Figure 7

Shown in (a) is the variation of the response at resonance as a function of the driving voltage, for a range of temperatures at a pressure of $15\text{bar}$. The lower inset shows part of the $500\mathrm{mK}$ plot, exhibiting hysteresis. Hysteresis also appears on the corresponding resonance characteristic, as shown in (b). The upper inset of (a) shows a closer view of the large drive region of the graph for selected temperatures.

Figure 8

The resonant response at the onset of dissipation (extracted from the data plotted in Fig. 7) as a function of temperature, at a pressure of $15\text{bar}$.

Figure 9

The measured resonant frequency of the grid as a function of the density of He II at a drive of $1\mathrm{mVpp}$); the value of density was extracted from the applied pressure using the data provided by Donnelly and Barenghi [26]. The dashed line is a straight-line fit to the points excluding the one at zero density; the continuous line is a similar fit taking into account all data points. The error bars are too small to show on this scale.

Figure 10

A simple model comprising two, linearly coupled, harmonic oscillators, used to reproduce the experimental data containing satellite resonance peaks—see text for more details.

Figure 11

The continuous line is the resonance characteristic shown in Fig. 5b for a drive voltage of $100\mathrm{mVpp}$. The dashed line is the steady-state response of mass $m_1$, of the coupled oscillator model shown in Fig. 10.