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We studied the thermal conductivity of superfluid $^3$He in a 2.5 mmeffective diameter and 0.15 m long channel connecting the two volumes of ourexperimental assembly. The main volume contained pure solid $^4$He, pure liquid$^3$He and saturated liquid $^3$He-$^4$He mixture at varying proportions, whilethe separate heat-exchanger volume housed sinter and was filled by liquid$^3$He. The system was cooled externally by a copper nuclear demagnetizationstage, and, as an option, internally by the adiabatic melting of solid $^4$Hein the main volume. The counterflow effect of superfluid just below thetransition temperature $T_c$ resulted in the highest observed conductivityabout five times larger than that of the normal fluid at the $T_c$. Once thehydrodynamic contribution had practically vanished below $0.5T_c$, we firstobserved almost constant conductivity nearly equal to the normal fluid value atthe $T_c$. Finally, below about $0.3T_c$, the conductivity rapidly falls offtowards lower temperatures.

Thermal conductivity of superfluid $^3$He-B in a tubular channel down to 0.1$T_c$ at the $^4$He crystallization pressure