Abstract
The response of a viscous liquid layer around a center-core to axial excitation has been determined analytically. For a simplified analytical treatment the no-slip condition at the top and the bottom of the liquid column had to be abandoned while the normal condition could be satisfied. The free liquid surface elevation, the radial- and axial velocity distribution have been presented for various surface tension parameters σa/ρv3, aspect ratios h/a and at various locations inside and along the liquid bridge as functions of the forcing frequency parameter Ω a2/ν. It could be noticed that due to viscosity the resonance amplitudes are finite and that the lowest resonance is sharply tuned.
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Abbreviations
- a :
-
radius free liquid surface
- b :
-
radius of center-core
- a−b :
-
thickness of liquid layer
- \(\vec e_r ,\vec e_\varphi ,\vec k\) :
-
unit vectors in radial-, angular- and axial direction resp
- I 0,I 1 :
-
modified Besselfunctions of first kind and zeroth and first order
- k=b/a :
-
diameter ratio
- K 0,K 1 :
-
modified Besselfunctions of second kind and zeroth and first order
- h :
-
length of liquid bridge
- p :
-
pressure
- r, φ,z :
-
cylindrical coordinates
- \(s = \bar \sigma + i\bar \omega \) :
-
complex frequency
- S≡sa 2/ν:
-
for damped motion
- S 0 :
-
for undamped motion
- t :
-
time
- u, w :
-
velocity components in radial- and axial direction resp
- \(\bar z_0 \) :
-
axial excitation amplitude
- \(\bar Z \equiv \bar z_0 v/h/a\) :
-
“reduced” amplitude
- σ *≡a/ρv 2 :
-
surface tension parameter
- σ:
-
surface tension
- η:
-
dynamic viscosity
- ν=η/ρ:
-
kinematic viscosity
- ρ:
-
density of liquid
- ζ:
-
free liquid surface elevation
- τ=νt/a 2 :
-
dimensionless time
- τ rz :
-
shear stress
- Ω:
-
forcing frequency
- \(\bar \Omega = \Omega a^2 /v\) :
-
“reduced” (dimensionless) forcing frequency
- ω2n-1 :
-
natural axisymmetric frequencies means differentiation to the argument
Literatur
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Bauer, H.F.: Response of a viscous liquid layer around a centercore to axial excitation in zero-gravity. Forschungsbericht der Universität der Bundeswehr München. LRT-WE-9-FB-9 (1990).
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Bauer, H.F. Response of a viscous liquid layer around a center-core to axial excitation in zero-gravity. Forsch Ing-Wes 57, 14–21 (1991). https://doi.org/10.1007/BF02574942
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DOI: https://doi.org/10.1007/BF02574942