Particle tracking velocimetry applied to thermal counterflow in superfluid ${}^4\mathrm{He}$: Motion of the normal fluid at small heat fluxes

This paper is concerned with thermal counterflow in superfluid ${}^4\mathrm{He}$, particularly with the motion of the normal component at small heat fluxes when this motion is believed to be laminar. Recent experiments in which this motion is traced with micron-sized particles of solid deuterium show that the motion is not spatially uniform on a scale of order the spacing of the quantized vortex lines in the superfluid component. It is argued that this lack of uniformity has its origin in the fact that the force of mutual friction, which limits the thermal conductivity, is concentrated near the cores of the vortex lines. Possible effects of this concentration of force are discussed, and it is concluded that the experimental observations can be explained only if the vortex lines are arranged randomly in space, so that the observed lack of uniformity in the normal-fluid velocity can be regarded as being due partly to spatial variations in the vortex line density. However, problems remain, in that the form of the observed velocity correlation function has still to be understood.

Figure 1

The calculated streamwise particle velocity PDFs at 1.85 K for heat fluxes of 38 and 320 mW/${\mathrm{cm}}^2$. The solid lines are Gaussian fits to the data.

Figure 2

Calculated streamwise and transverse velocity standard deviations for the G2 particles at three temperatures.

Figure 3

Calculated streamwise velocity correlation functions for the G2 particles at different temperatures and heat fluxes.

Figure 4

The observed broadening of the excimer lines at 1.85 K with increasing normal-fluid velocity, when the flow of normal fluid is observed to be laminar. The red triangles show the broadening expected from the PTV results reported in this paper

Figure 5

The wake generated in the normal fluid by a single rectilinear vortex line.

Figure 6

Illustrating the effect of the small radial flows.

Figure 7

Rectilinear vortices in random positions.

Figure 8

(a, b) The wakes generated in the normal fluid by a randomly spaced array of rectilinear vortices; the dimensionless velocity in the wakes as a function of $\tilde{x}$ and $\tilde{z}$. Figure 8 relates to the distribution of vortices in Fig. 7.

Figure 9

The calculated correlation function, $r\left(\tilde{z}\right)$, plotted against $\tilde{z}$.