Isothermal flow of liquid He II through narrow channels

Isothermal flow of liquid He II through narrow slits of width 0.3, 2.3, and 3.4 $\mathrm{\mu }$ has been examined using two different types of experimental arrangements, one in which the liquid flows under its own gravitational head, the other in which the liquid is pushed by a plunger through the slit. In these experiments to a good approximation only the superfluid flows. Values of the superfluid critical velocity, ${\mathbf{v}}_s={\mathbf{v}}_{s,c}$, are found to be 18.0, 5.85, and 2.73 cm/sec for the three slits respectively, independent of temperature for the range $\mathit{1.1}<T<\mathit{2.0}°K$. At the $\mathrm{\lambda }$ point ${\mathbf{v}}_{s,c}$ becomes zero. At low temperatures, the critical velocities as obtained in these flows are the same as found in heat flow measurements using the same slits; but above $\mathit{1.6}°K$ results from the latter show that ${\mathbf{v}}_{s,c}$ increases. Possible reasons for this behavior are discussed in terms of the vortex line model, the principal conclusion being that for both isothermal flow and heat flow in narrow channels the breakdown of superfluidity occurs because of interactions between the superfluid and the wall. At supercritical velocities $({\mathbf{v}}_s>{\mathbf{v}}_{s,c})$ the dissipation effects observed in isothermal flow can be described entirely in terms of a force ${\mathbf{F}}_s$ associated with the superfluid alone, whereas in the case of heat flow the mutual force ${\mathbf{F}}_{sn}$ is dominant. The present experiments show that the pressure gradient is proportional to ${\mathbf{F}}_s$, and ${\mathbf{F}}_s$ is independent of the slit width, has a temperature variation below $\mathit{1.9}°K$ much like that of the superfluid density squared, and depends upon the velocity as ${({\mathbf{v}}_s-{\mathbf{v}}_{s,c})}^n$. For low supercritical velocities $n=\mathit{1}$ and for $({\mathbf{v}}_s-{\mathbf{v}}_{s,c})$ greater than about 1.5 cm/sec $n=\mathit{1.75}$. These results are also considered from the viewpoint of the vortex line model.