Hydrodynamics of superfluid turbulence fronts in He II: steady propagation

Abstract

A hydrodynamic theory of superfluid turbulent flow of He II which was developed recently is applied to a specific inhomogeneous flow situation, viz. a superfluid turbulence front propagating into an (unstable) state of zero turbulence. It is shown that in a wide range of experimental flow conditions the two equations governing the evolution of the vortex tangle may be uncoupled from the other equations. In the case where the vortex tangle is in internal equilibrium the two vortex-tangle equations may, in addition, be reduced to one non-linear partial differential equation of the first order. It appears that the waves of permanent form permitted by this equation fall apart in two classes, viz. a class of ‘warm’ fronts propagating in the direction of the heat flow and a class of ‘cold’ fronts moving oppositely. The velocity ranges of the warm and cold fronts are separated by a velocity gap. The initial-value problem for front propagation is solved exactly by means of the method of characteristics. A linear analysis of front stability based on that exact solution yields criteria for the selection of the front velocity by requiring marginal stability of the corresponding warm and cold fronts. The significance of marginal stability as a dynamical mechanism for velocity selection was recently put forward by van Saarloos (1988). It is shown that alternative selection criteria for the velocity of warm and cold fronts are provided by the requirements of minimum rate of line-length production and minimum dissipation rate. The comparison of the theoretical values for the velocities of warm and cold fronts with the experimental front velocities reported by Slegtenhorst et al. (1982) for capillary flow of He II looks promising. Wall effects will be taken into account in a separate paper.