Thermal transport properties of helium near the superfluid transition. II. Dilute³He-⁴He mixtures in the superfluid phase

Abstract

Measurements of the average thermal conductivity κ_exp ≡hQ/ΔT and of the thermal relaxation time τ to reach steady-state equilibrium conditions are reported in the superfluid phase for dilute mixtures of³He in⁴He. Hereh is the cell height,Q is the heat flux, andΔT is the temperature difference across the fluid layer. The measurements were made over the impurity range 2×10⁻⁹<X(³He)<3×10⁻² and with heat fluxes 0.3<Q<160 µW/cm². Assuming the boundary resistanceR _b , measured forX<10⁻⁵, to be independent ofX over the whole range ofX, a calculation is given for κ_exp. ForQ smaller than a well-defined critical heat fluxQ _c (X) ∝X ^0.9, κ_exp is independent of Q and can be identified with the local conductivity κ_eff, which is found to be independent of the reduced temperature ɛ = (T−T_λ)/T_λ for −ɛ≤10⁻². Its extrapolated value at T_λ is found to depart forX≲10⁻³ from the prediction κ_λ ∝X ⁻¹, tending instead to a weaker divergence κ_λ ∝X ^−a witha≈0.08. A finite conductivity asX tends to zero is not excluded by the data, however. ForQ >Q _c (X), a nonlinear regime is entered. ForX≲10⁻⁶, the measurements with the available temperature resolution are limited to the nonlinear conditions, but can be extrapolated into the linear regime forX≳2×10⁻⁷. The results for κ_exp(Q),Q _c (X), and κ_eff(XX) are found to be internally consistent, as shown by comparison with a theory by Behringer based on Khalatnikov's transport equations. Furthermore, the observed relaxation times τ(X) in the linear regime are found to be consistent forX>10⁻⁵ with the hydrodynamic calculations using the measured κ_eff(X). ForX<10⁻⁵, a faster relaxation mechanism than predicted seems to dominate. The transport properties in the nonlinear regimes are presented and unexplained observations are discussed.