  By numerically computing the steady axisymmetric flow of helium II confined
inside a finite aspect ratio Couette annulus, we determine the transition from
Ekman flow to Taylor vortex flow as a function of temperature and aspect
ratio.We find that the low-Reynolds number flow is quite different to that of a
classical fluid, particularly at lower temperatures.At high aspect ratio our
results confirm the existing linear stability theory of the onset of Taylor
vortices, which assumes infinitely long cylinders.
