Finite-size effects on the thermal conductivity of $^4$He confined in rectangular channels.

We report results for the thermal conductivity $\lambda(t)$ of $^4$He confined in glass capillary arrays with rectangular channels of size $1\times10\times1000~\mu{\rm m}^3$ near the bulk super-fluid transition temperature $T_\lambda$ as a function of the reduced temperature $t \equiv T/T_{\lambda} - 1$. Even close to $T_\lambda$ we found that $\lambda(t)$ differs very little from similar measurements for cylindrical channels of radius $1~\mu$m, \footnote{D. Murphy, E. Genio, G. Ahlers, F.C. Liu, and Y. Liu, Phys. Rev. Lett. {\bf 90}, 025301 (2003).} indicating similar scaling functions for the two geometries. This differs from the finite-size effects on thermodynamic properties, which have very different scaling functions near the transition for parallel-plate and cylindrical geometries.\footnote{M. O. Kimball, K. P. Mooney, and F. M. Gasparini, Phys. Rev. Lett. {\bf 92}, 115301 (2004).}