Flow Instability in Liquid Helium II

The stability of steady, low-velocity laminar flow of the normal component of liquid helium II is shown to follow from the principle of minimum entropy production. It is pointed out that for two-fluid equations containing a nonlinear mutual friction term, this justification of stability is no longer valid when the energy dissipation by mutual friction is comparable to that by the viscosity of the normal-fluid component. An instability condition which is characterized by a dimensionless parameter G (Gorter number) formed from the ratio of these two dissipative terms, is shown to predict the magnitude and temperature dependence of the critical velocities observed in heat conduction, boundary movement, and isothermal flow experiments in channels wider than 10^-3 cm. Instabilities observed in heat conduction at higher velocities are shown to correlate with a Reynolds number of the usual form. The onset of the mutual friction force is discussed in the light of this phenomenological theory.