Dynamics of Superflow by Mesoscopic Condensate

The shear viscosity $\eta$ of a quantum liquid in the vicinity of $T_{\lambda}$ is examined. In liquid helium 4 above $T_{\lambda}$ ($T_{\lambda}<T<3.7K$), under a strong effect of Bose statistics, the coherent many-body wave function grows to an intermediate size between a macroscopic level and a microscopic one. These wave functions are qualitatively different from thermal fluctuation, and manifest themselves in the gradual decrease in shear viscosity above $T_{\lambda}$. To formulate this phenomenon, we combine the correlation function with fluid dynamics. Applying the Kramers-Kronig relation to the generalized Poiseuille's formula for capillary flow, we perform a perturbation calculation of the reciprocal $1/\eta$ with respect to the particle interaction, and examine how the growth of coherent wave functions gradually decreases shear viscosity. Comparing with the experimentally determined $\eta (T)$, $\hat {\rho\cdot}_s(T)/\rho\cdot$ of such a mesoscopic condensate is estimated to reach $10^{-5}$ just above $T_{\lambda}$. We examine the effect of condensate size on the stability of such a superflow, and touch upon the superflow in porous media.