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Non-linear oscillations of a non-viscous cylindrical liquid system in weightlessness condition

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Abstract

For an immiscible cylindrical liquid system in a circular cylindrical container and a liquid layer around a rigid center core consisting of incompressible and non-viscous liquid the natural non-linear frequencies are determined, if the system is in a zero-gravity environment. The frequencies are determined as a function of the surface elevation and are presented as backbone curves.

It was found, that the immiscible liquid system shows hardening effect depending on the length-to-diameter ratio of the column, the density ratio and the diameter ratio. This means that with increasing interface amplitude the natural frequency increases above the linearized frequency. Decreasing density ratio (inner to outer density), increasing diameter ratio and decreasing aspect ratio yield stronger hardening effects. For a liquid layer around a rigid center core the system shows stronger effects, but exhibits softening behavior, i.e. lower natural frequencies for increasing liquid surface amplitude. Decreasing aspect ratio and decreasing layer thickness show stronger softening effects, as do increasing mode numbers. Close to the Rayleigh instability the liquid system shows extreme non-linear effects.

Zusammenfassung

Es werden die nichtlinearen Frequenzen eines nichtmischbaren Flüssigkeitssystems in einem Kreiszylinderbehälter und einer Flüssigkeitsschicht um einen Zylinder im schwerelosen Zustand bestimmt. Die Frequenzen werden als Funktion der Oberflächenauslenkung dargestellt.

Es wurde festgestellt, da\ die Trennflächenfrequenzen des nichtmischbaren Systems überlinear sind, wobei die Stärke der Nichtlinearität vom Streckungs-, Dichte- und Durchmesserverhältnis abhängt. Ein kleiner werdendes Dichteverhältnis (innerer zu äu\erer Dichte), grö\er werdendes Durchmesserverhältnis und kleineres Streckungsverhältnis ergeben stärkere Überlinearität. Für eine Flüssigkeitsschicht um einen Zylinder konnte unterlineares Verhalten festgestellt werden. Kleiner werdendes Streckungs- und Dickenverhältnis sowie höhere Schwingungsformen ergeben stärkere Unterlinearität. In der Nähe der Rayleigh-Instabilität zeigt das Flüssigkeilssystem extreme Nichtlinearitäten.

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Abbreviations

a :

radius of container wall or rigid center core

b :

radius of interfacial or free liquid surface (see Fig. 1)

L :

length of liquid column

m,n :

angular mode number of liquid, axial mode number

I m,K m :

modified Besselfunctions of first and second kind andmth order

r,ϕ,z :

polar coordinates

R 1,R 2 :

radii of curvature

t :

time

φ 1,φ 2 :

velocity potential of liquid outside and inside

ζ(ϕ, z, t):

free liquid- or interfacial elevation

ρ 1;2 :

liquid densities

σ :

liquid surface tension (or interface tension)

ω (0)mn :

linear natural frequency of liquid

ω :

frequency of liquid (non-linear)

References

  1. L. Rayleigh,On the capillary phenomena of jets. Proc. Roy. Soc. London29, 71–97 (1879).

    Google Scholar 

  2. L. Rayleigh,The theory of sound. New York 1945.

  3. H. Lamb,Hydrodynamics. Cambridge University Press, 1932.

  4. H. F. Bauer,Natural damped frequencies of an infinitely long column of immiscible viscous liquids. Forsch. Ing. Wes.49, 117–126 (1983).

    Google Scholar 

  5. H. F. Bauer,Natural damped frequencies of an infinitely long column of immiscible viscous liquids. ZAMM64, 475–490 (1984).

    Google Scholar 

  6. H. F. Bauer,Free surface- and interface oscillations of an infinitely long visco-elastic liquid column. Acta Astronautica13, 9–22 (1986).

    Google Scholar 

  7. W. Eidel,Freie und erzwungene nichtlineare Schwingungen eines reibungsfreien Flüssigkeitstropfens unter Schwerelosigkeit. Ing. Arch.56, 343–350 (1986).

    Google Scholar 

  8. H. F. Bauer and W. Eidel,Non-linear liquid oscillations in spherical systems under zero-gravity. Acta Mech.65, 107–126 (1986).

    Google Scholar 

  9. H. F. Bauer,Non-linear oscillations of axially independent liquid column under zero-gravity environment. Forsch. Ingwes.54, 82–93 (1988).

    Google Scholar 

  10. W. Eidel and H. F. Bauer,Non-linear oscillations of an inviscid liquid column under zero-gravity, Ing. Arch.58, 276–284 (1988).

    Google Scholar 

  11. H. F. Bauer and W. Eidel,Non-linear oscillations of a non-viscous cylindrical liquid system in weightlessness condition, Forschungsbericht der Universität der Bundeswehr München, LRT-WE-9-FB-14, 1987.

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Authors and Affiliations

  1. Institut für Raumfahrttechnik, Universität der Bundeswehr, Werner-Heisenberg-Weg 39, D-8014, Neubiberg, Federal Republic of Germany

    Helmut F. Bauer & Werner Eidel

Authors
  1. Helmut F. Bauer
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  2. Werner Eidel
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Bauer, H.F., Eidel, W. Non-linear oscillations of a non-viscous cylindrical liquid system in weightlessness condition. Z. angew. Math. Phys. 40, 510–525 (1989). https://doi.org/10.1007/BF00944804

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  • DOI: https://doi.org/10.1007/BF00944804

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