Heat conduction by liquid helium II in capillary tubes III. mutual friction

Abstract

Measurements are described of the stable supercritical thermal resistance of liquid helium II between 1.2°k and 2.0°K in capillary tubes of diameter 52.0 microns, 107.6 microns and 366 microns. The results are presented in terms of the quantized vortex line theory, and values of L ₀, the effective length of vortex line per unit volume, are calculated. Graphs of L ₀ 1/2 against relative velocity v are straight lines at high velocities, corresponding to a cube law of mutual friction of the form F _sn = A _ρsρn v(v−v ₀)², as expected from the theory of Vinen. Values of A are independent of tube diameter and length, and increase with temperature in the same way as found by Vinen, but are somewhat larger. At low velocities, where the average spacing of vortex lines becomes comparable with the channel width, the effective value of A drops rather abruptly by about one-half. In this region, Vinen's theory can no longer be expected to apply, but a plot of the dimensionless variables (DL ₀ ^1/2) against (mvD/h) (where D is the ratio of area to perimeter of the channel) gives approximately a universal curve for all tube sizes at a given temperature, as would be expected from a dimensional analysis.