Thermal Conduction in Liquid Helium II. I. Temperature Dependence

The thermal conductivity of liquid helium contained in a cylindrical stainless-steel capillary 0.080 cm in diam by 5.16 cm long has been studied between 0.9°K and the $\lambda$ point. The relation between temperature gradient and heat current density $W$ for heat currents greater than the critical heat current density $W_c$ is best expressed in the form $\mathrm{grad}T=D{W}^n$, where $D$ is a temperature-dependent constant and $n$ varies from about 3.0 at low temperatures to about 3.5 above 1.7°K. Below $W_c$ the temperature gradient is much smaller, and is determined entirely by the viscosity of the normal component. $W_c$ was measured over the entire temperature range by a combination of two methods, which are in complete agreement in the region of overlap. The results suggest that, below about 1.7°K, $W_c$ is the result of some sort of normal turbulence describable by a Reynolds number involving the normal fluid velocity but the total density. At higher temperatures such an explanation is no longer adequate, and some other type of critical velocity must be invoked.