Page 130 - Principles and Applications of NanoMEMS Physics

118 Chapter 3

G =
v = ∇ χ , (113)
m

that is, the velocity is related to the phase, χ , of the wave function, so one
can rewrite (113) as,
G
v = ∇ φ , (114)

which clearly expresses that the flow is a potential flow, since the curl of any

gradient is zero, and the potential is given by,

φ = = χ . (115)
m

A further phenomenon accomplanying superfluidity, and elucidated by
first-principles considerations, pertains to the dynamics of superfluids when
placed in a rotating container. In particular, it is experimentally found, Fig.
3-19, in a vessel containing a mixture of normal and superfluid components,
G
and rotating at an angular velocity Ω , that the dynamic behavior of the two
components is quite different. On the one hand, as is expected from classical
hydrodynamics, the normal component rotates with the vessel (i.e., it is
carried along with the vessel due to friction), so that it acquires an eddy
G
G
G
current v = Ω × r , and this velocity, in turn, gives rise to an accompanying
n
G G G
vortex, since ×∇ v = 2 Ω , see Fig. 3-19. The superfluid component, on the
n
other hand, becomes populated by a distribution of vortices. This appearance
of vortices in the superfluid component would appear to contradict the
fundamental condition for superfluidity of zero vorticity, see Eq.(95). The
clue to this behavior was to be found in the recognition that potential flow,
characterized by (95), may also be obtained whenever the equivalent form,
based on Stokes’ theorem,
G G
v s ³ r d = 0, (116)

is satisfied. In particular, if the potential of the rotating fluid is proportional
to the angle, see Fig. 3-20, so that one has,
Γ
φ = α , (117)
2 π