The minimization of the Ginzburg-Landau (GL) free energy with respect to two independent variables, the vector potential and complex order parameter leads to two independent equations, the GL equations. The equation given by the variation of the vector potential yields an equation similar to the London equation, whereas variation of the order parameter yields anequation describing the spatial dependence of the order parameter, which indicates that the number of superelectrons is not constant but depends on position, which is a feature not included in the London theory. The two GL equations yield two characteristic lengths, the penetration depth of the magnetic field similar in form to the London penetration depth and temperature-dependent GL coherence length, which characterizes the spatial variation of the order parameter.
The heat transfer of superfluid helium (He II) in a rectangular channel has been investigated both experimentally and numerically. The channel end is kept open to a He II bath under an atmospheric pressure condition. A one-dimensional heat transfer model, which can predict the temperature rise at the end of the He II channel and λ-transition heat flux, is proposed in the present study. This model is developed on the basis of heat conduction by taking into account heat transfer not only in the channel but also in the He II bath. The supply of heat to the channel results in a temperature rise at the channel end. Larger temperature rise at the channel end leads to a smaller value of λ-transition heat flux in the channel. In the case of λ-transition heat flux being supplied to the channel, an increase in aspect ratio, which is defined as the ratio of the gap to the length of the channel, results in an increase in the value of temperature rise at the channel end. A satisfactory agreement between the measured data and the results of calculation on the basis of the present model is observed for the range of aspect ratio tested in this study.