Page 131 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 119

z z

Ω Ω

Flu id
Fl u i d
(a)

r r
α α

v =Ω= r ⋅ Ω r ⋅
v
α α
(b)
Figure 3-20. (a) Normal fluid in rotating vessel acquires meniscus with shape depending only
on angular velocity Ω . Top view of fluid-containing vessel rotating with angular velocity
Ω . The normal fluid acquires an eddy current with velocity v .
α
then the fluid velocity may be calculated as,

v α () r = 1 ∂ α φ = 2 Γ r π , (118)
r ∂

and, since the velocity decays with distance, this is the profile of a vortex.
Now, calculation of the circulation of this vortex gives,

³ v α ⋅dl = ∆ φ = Γ . (119)

Examination of Eq. (119) reveals that if the circulation (potential change) is
zero, one still has the conflict between the mathematical violation of
vorticity and the experimental observation of vortices. However, if the angle
2
α is not uniquely defined, except up to modulus π, then it would be
possible to reconcile the two if the potential φ were not single-valued. This,
in turn, would be the case if the phase of the wavefunction was not unique,
∆
but also defined modulo π2 , so that χ = 2π N . In this case, the circulation
(119) would be expressed as,