Effective thermal conductivity of superfluid helium: laminar, turbulent and ballistic regimes

Abstract

In this paper we extend previous results on the effective thermal conductivity of liquid helium II in cylindrical channels to rectangular channels with high aspect ratio. The aim is to compare the results in the laminar regime, the turbulent regime and the ballistic regime, all of them obtained within a single mesoscopic formalism of heat transport, with heat flux as an independent variable.

Thermodynamics of computation and linear stability limits of superfluid refrigeration of a model computing array

“…2), σ is given by the particular system we want to refrigerate, which will be related to the computation rate below, and the main physical challenge is to obtain q, which is the heat transported inside the system by conduction, or by internal convection. Let's briefly recall what we have found in our previous studies [16], [26]. First, let us consider the flow inside the channel with rectangular cross-section (x being the coordinate along the channel, and y and z the transversal coordinates).…”

“…In Refs. [26] and [24] we have evaluated how much the presence of quantized vortices reduces the efficiency of the refrigeration by means of superfluid helium and in Refs. [27,28] how the profile of the heat flux (and the normal component) is modificed by their contribution.…”

“…In phonon hydrodynamics (when the mean free path of the phonons is smaller than the smallest size of the system) it is usually set q |∂σ = 0, as made in Refs. [24,16], whereas in dilute phonon flow (namely, when the mean free path of the phonons is of the same order of the size of the system) the usual constant boundary condition is q |∂σ = C ∂q ∂n |∂σ , with C and being a numerical constant of the order of the unity and the mean free-path of the heat-carriers [29,26]. Here, for the sake of simplicity and lack of suitable information, we choose q |∂σ = 0 as in phonon hydrodynamics.…”

Refrigeration of an Array of Cylindrical Nanosystems by Flowing Superfluid Helium

“…Indeed, heat transport in He II is not described by Fourier's law, but by a more complex equation, including non-local terms, strongly depending on the geometry and the size of the particular system being considered. Heat transport can be investigated in laminar regime, in fully turbulent regime, in the ballistic regime, or in the transition from laminar to turbulent regimes [10,[18][19][20][21]. In the laminar regime, one has for the average heat flux…”

“…has to be smaller than the temperature-dependent critical value Re q,crit (T ) for each cylinder column i, i.e., Re q (T i ) < Re q,crit (T ), in order to avoid quantum turbulence [10,18,20]. Indeed, the critical value of Re q plays a decisive role in the appearance of quantum turbulence [13,18].…”