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Axisymmetric natural damped frequencies of a viscous liquid in a circular cylindrical Container — An alternative semi-analytical solution

Axialsymmetrische schwingungen einer viskosen Flüssigkeit im zylindrischen Tank. — Eine alternative semi-analytische Lösung

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Abstract

For axisymmetric liquid motion (m=0) in a circular cylindrical Container nlled with an incompressible and viscous liquid the natural damped frequencies are determined. The analysis satisfies the side wall conditions exactly while only the normal Container bottom condition could be satisfied. This restricts the Solution results to liquid height flllings h/a to larger values (h/a>1). It was found that with increase of the liquid height ratio h/a the oscillation frequency and decay magnitude are both increasing.

Zusammenfassung

Es werden die axialsymmetrischen gedämpften Eigenfrequenzen einer viskosen Flüssigkeit im kreiszylindrischen Behälter bestimmt. Dabei werden die Zylinderwandbedingungen exakt erfüllt und für den Tankboden nur die Normalbedingung berücksichtigt. Da das Schwappen der Flüssigkeit im oberen Teil des Behälters stattfindet gilt die hier behandelte Lösung für größere Flüssigkeitshöhenverhältnisse h/a>1. Die numerischen Ergebnisse zeigen, daß mit wachsender Flüssigkeitshöhe die Schwappfrequenzen und die Abklingraten zunehmen.

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Abbreviations

a :

radius of container

g :

gravity constant (g*=ga 3/ν 2)

h :

liquid height

J m :

Bessel function of the first kind

p :

pressure

r, ϕ, z :

cylindricalpolar coordinates

\(s = \bar \sigma + i\bar \omega \) :

complex frequency (S=sa 2/ν)

t :

time

u, w :

radial and axial liquid velocity

α 2=ϱga 2/σ :

parameter (α 2=g*/σ *=Bo) Bond number

σ :

liquid surface tension (σ*=σa/(ϱν 2))

Λ n :

roots of Eq. (18) (Λ n =λ n a)

ϱ :

mass density

η :

dynamic viscosity

ν=η/ϱ :

kinematic viscosity

Ψ :

stream function

Σ:

free liquid surface displacement

τ rz :

shear stress

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Authors and Affiliations

  1. Institut für Raumfahrttechnik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, D-85577, Neubiberg, Germany

    H. F. Bauer & W. Eidel

Authors
  1. H. F. Bauer
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  2. W. Eidel
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Bauer, H.F., Eidel, W. Axisymmetric natural damped frequencies of a viscous liquid in a circular cylindrical Container — An alternative semi-analytical solution. Forsch Ing-Wes 65, 191–199 (1999). https://doi.org/10.1007/PL00010871

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  • DOI: https://doi.org/10.1007/PL00010871

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