Contribution of the normal component to the thermal resistance of turbulent liquid helium L. Saluto 1 , D. Jou 2 and M.S. Mongiov` ı 1 1 Dipartimento di Energia, ingegneria dell’Informazione e modelli Matematici (DEIM), Universit` a degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy 2 Departament de F´ ısica, Universitat Aut` onoma de Barcelona, 08193 Bellaterra, Catalonia, Spain E-mail addresses: lidia.saluto@unipa.it, David.Jou@uab.cat, m.stella.mongiovi@unipa.it Abstract Previous results for the velocity profile of the normal component of helium II in coun- terflow are used to evaluate the viscous contribution to the effective thermal resistance. It turns out that such contribution becomes considerably higher than the usual Landau estimate, because in the presence of vortices the velocity profile significatively separates from the Poiseuille parabolic profile. Thus, a marked increase of the contribution of the normal component to the thermal resistance with respect to the viscous Landau estimate does not necessarily imply that the normal component is turbulent. Furthermore, we ex- amine the influence of a possible slip flow along the walls when the radius of the tube becomes comparable to the phonon mean-free path; this implies a reduction of the thermal resistance with respect to that obtained for non-slip boundary conditions. Keywords: Thermal resistance – Superfluid helium – Quantum turbulence – Normal component Pacs numbers: 44.15.+a – 47.37.+q – 67.10.Jn – 67.25.bd – 67.25.de 1 Introduction Heat transfer along thin channels filled with superfluid liquid helium is a classical topic in non- Fourier heat transport [1, 2]. For sufficiently low values of the heat flux q, the temperature gradient is proportional to q. For high values of q, the temperature gradient becomes propor- tional to q 3 , because of the appearance of a chaotic tangle of quantized vortices of superfluid, which strongly contributes to the thermal resistance [3–7]. One of the current topics of interest is the flow of the normal component, in particular whether it is laminar or turbulent [8]. In the latter case, both the superfluid and normal components would have a turbulent flow. One way of checking it would be the accurate measurement of the effective thermal resistance (or its reciprocal, the effective thermal conductivity). This would be far easier than measuring the full velocity profile, but it requires a very detailed understanding of the normal viscous contribution to the thermal resistance, which is the aim of this paper. The most explicit and inequivocal way of clarifying the laminar or turbulent character of such flow would be the analysis of the velocity profile of the normal component across the 1