Summary
For a finite solidly rotating cylindrical liquid column the damped natural axisymmetric frequencies have been determined. The liquid was considered incompressible and viscous. The cases of freely slipping edges and that of anchored edges have been treated. It was found that instability appears in a purely aperiodic root for the spinning liquid bridge. This is in contrast to the instability appearing in the damped oscillatory natural frequency of a nonspinning liquid column at\(\frac{h}{a} \geqq 2\pi\). The spinning viscous liquid column exhibits the same instability as the frictionless liquid. It appears at\(\frac{h}{a} \geqq 2\pi /\sqrt {1 + We}\) for axisymmetric oscillations.
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Abbreviations
- a :
-
radius of liquid column
- I m :
-
modified Bessel function of first kind and orderm
- s :
-
complex frequency (\(\left( {S \equiv \frac{{sa^2 }}{v}} \right)\))
- r, ϕ,z :
-
polar cylindrical coordinates
- p :
-
pressure
- t :
-
time
- u, v, w :
-
radial-, azimuthal- and axial velocities of liquid, respectively
- \(We = \frac{{\rho a^3 \Omega _0 ^2 }}{\sigma }\) :
-
Weber number
- h :
-
height of liquid column
- η:
-
dynamic viscosity of liquid
- v :
-
kinematic viscosity of liquid (v=η/ρ)
- ρ:
-
density of liquid
- σ:
-
surface tension of liquid
- τ rϕ , τ rz :
-
shear stress
- Φ(r, z, t):
-
circulation
- Ψ(r, z, t):
-
streamfunction
- Ω 0 :
-
angular velocity of liquid column about the axis of symmetry
- ζ(ϕ,t):
-
free surface displacement
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Bauer, H.F. Natural damped frequencies and axial response of a rotating finite viscous liquid column. Acta Mechanica 93, 29–52 (1992). https://doi.org/10.1007/BF01182571
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DOI: https://doi.org/10.1007/BF01182571