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Viscous response of an anchored spinning liquid column to various axial excitations

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Summary

For a solidly rotating viscous cylindrical liquid column of finite length the response to axial synchronous, counter- and one-sided excitation is determined for anchored contact lines at the disc-rim. For a rotating column additional responses of inertial waves (hyperbolic range) appear forΩ < 2Ω 0, while in the elliptic rangeΩ < 2Ω 0 the sloshing response occurs. The various responses for the free surface displacement have been numerically evaluated. Only in the one-sided exitation case all resonance peaks appear, while for synchronous excitation only the odd resonances and for counter-excitation only the even resonance peaks occur.

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Abbreviations

a :

radius of column

h :

length of liquid bridge

I n :

modified Bessel function

p :

liquid pressure

r, φ,z :

cylindrical polar coordinates

t :

time

u, v, w :

velocity distribution

\(We = \frac{{\varrho a^3 \Omega o^2 }}{\sigma }\) :

Weber number

z 0 :

excitation amplitude

ϱ:

liquid density

σ:

surface tension

\(\sigma ^ * = \frac{{\sigma a}}{{\varrho v^2 }}\) :

surface tension parameter

\(Oh = \frac{1}{{\sqrt {\sigma ^ * } }}\) :

Ohnesorge number

ζ:

liquid surface displacement

\(v = \frac{\eta }{\varrho }\) :

kinematic viscosity

Ω 0 :

rotational speed

\(\omega _0 = \frac{{\Omega _0 a^2 }}{v}\) :

dimensionless rotational speed

Ω:

forcing frequency

\(\Omega ^ * = \frac{{\Omega a^2 }}{v}\) :

dimensionless forcing frequency

\(\bar \Omega = \frac{\Omega }{{\sqrt {\frac{\sigma }{{\varrho a^3 }}} }}\) :

dimensionless forcing frequency for non-viscous liquid

λa=Λ:

root of bi-cubic Eq.(33)

\(\chi _n = \frac{{n\pi a}}{h}\) :

root of bi-cubic Eq.(33)

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Author notes

  1. H. F. Bauer &  W. Eidel

    Present address: Institut für Raumfahrttechnik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, D-85577, Neubiberg, Germany

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    1. H. F. Bauer
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    2. W. Eidel
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    Bauer, H.F., Eidel, W. Viscous response of an anchored spinning liquid column to various axial excitations. Acta Mechanica 116, 153–170 (1996). https://doi.org/10.1007/BF01171427

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