Abstract
The axisymmetric natural damped frequencies (m=0) of a viscous liquid in a cylindrical container are obtained for slipping and anchored contact line at the container wall. The results may also be applied to viscous liquid in a micro gravity environment, if the contact angle of the contact line to the container wall is in the vicinity of π/2, indicating that the free liquid surface remains a plane surface in equilibrium. It was found that there exists, in contrast to frictionless liquid, a range where only aperiodic motion of the liquid is possible. This appears for small liquid height ratios h/a.
Zusammenfassung
Es werden die axialsymmetrischen gedämpften Eigenfrequenzen einer viskosen Flüssigkeit im zylindrischen Behälter bestimmt. Dabei werden die Fälle der gleitenden und verankerten Kontaktlinie an der Zylinderwand untersucht. Die Ergebnisse können auch für den Fall der Schwerelosigkeit benutzt werden, wenn der Kontaktwinkel der Kontaktlinie an der Behälterwand in der Nähe von neunzig Grad liegt und somit die Gleichgewichtslage der freien Flüssigkeitsoberfläche eine ebene Oberfläche bildet. Zum Unterschied zu reibungsfreien Flüssigkeitsschwingungen gibt es Gebiete, in denen die viskose Flüssigkeit nur aperiodische Bewegungen ausführen kann. Dies gilt für kleine Flüssigkeitshöhenverhältnisse h/a.
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Abbreviations
- a :
-
radius of container (outer radius)
- b :
-
inner radius
- g :
-
gravity constant \(\left( {\bar g = {{ga^3 } \mathord{\left/ {\vphantom {{ga^3 } {v^2 }}} \right. \kern-\nulldelimiterspace} {v^2 }}} \right)\)
- h :
-
liquid height
- J n , Y n :
-
Bessel functions of the order n and first and second kind
- I n , K n :
-
modified Bessel functions of the order n and first and second kind
- \(\bar k = {b \mathord{\left/ {\vphantom {b a}} \right. \kern-\nulldelimiterspace} a}\) :
-
diameter ratio
- k :
-
wave number (K=kh)
- n :
-
axisymmetric mode number
- p :
-
liquid pressure
- r, ϕ, z :
-
cylindrical polar coordinates
- \(s = \bar \sigma + i\bar \omega \) :
-
complex frequency (S=sa 2/v)
- t :
-
time
- u, w :
-
radial and axial liquid velocity
- ɛ n :
-
zeros of J1(ɛ)=0
- σ :
-
liquid surface tension
- \(\sigma * = \frac{{\sigma a}}{{\rho v^2 }}\) :
-
surface tension parameter
- ϱ :
-
mass density
- η :
-
dynamic viscosity
- ν=η/ϱ :
-
kinematic viscosity
- ξ n :
-
zeros of \(J_1 \left( \xi \right)Y_1 \left( {\bar k\xi } \right) - J_1 \left( {\bar k\xi } \right)Y_1 \left( \xi \right) = 0\)
- ζ :
-
free liquid surface displacement
- τ rz \(\mu _n^2 = \frac{{\varepsilon _n^2 }}{{a^2 }} + \frac{s}{v}\) :
-
shear stress
References
Abramson HN (ed.) (1966) The dynamic behavior of liquids in moving containers. NASA-SP-106
Graham EW, Rodriguez AM (1952) The characteristics of fuel motion which affect airplane dynamics. J Appl Mech 19: 381–388
Bauer HF, Villanueva J (1966) Theory of liquid sloshing in a rectangular container with numerical examples for C-5A wing. Lockheed-Gerorgia Comp. Rep. No ER-8390
Bauer HF (1964) Treibstoffschwingungen in Raketenbehältern und ihr Einfluß auf die Gesamtstabilität, Teil I. ZFW 12: 85–101
Bauer HF (1963) Theory of liquid sloshing in compartmented cylindrical tanks due to bending exitation. AIAA 1: 1590–1595
Bauer HF (1963) Liquid sloshing in a cylindrical quartertank. AIAA J 1: 2601–2606
Bauer HF (1966) Response of liquid in a rectangular container. J Engng Mech Div. Proc. American Soc. Civil Engrs. 92: 1–23
Bauer HF (1981) Flüssigkeitsschwingungen mit freier Oberfläche in keilförmigen Behältern. Acta Mechanica 38: 31–34
Bauer HF (1982) Flüssigkeitsschwingungen in Kegelbehälterformen. Acta Mechanica 43: 185–200
Bauer HF (1981) Freie Flüssigkeitsschwingungen in parabolischen Behälterformen. ZFW 5: 249–253
Bauer HF (1984) Forced liquid oscillations in paraboloidcontainers. ZFW 8: 49–55
Bauer HF, Eidel W (1989) Liquid oscillations in a prolate speroid. Ing Arch 5: 371–381
Budiansky B (1960) Sloshing of liquids in circular canals and spherical tanks. J Aerospace Sci 27: 161–173
Chun WH (1964) Fuel sloshing in a spherical tank filled to an arbitrary depth. AIAA J 2: 1972–1979
Chun WH (1960) Sloshing of liquids in cylindrical tanks of elliptical cross section. ARS-J 30: 360–363
Rattayya J (1965) Sloshing of liquids in axisymmetric ellipsoidal tanks. AIAA-paper 65-114. AIAA-2, Aerospace Sci. Meeting, New York
Levin E (1973) Oscillations of a fluid in a rectilinear conical container. AIAA-J 1: 1447
Dokuchaev LV (1964) On the solution of a boundary value problem on the sloshing of a liquid in conical cavities. PMM 28: 151–154
Lamb H (1945) Hydrodynamics, 6th ed. Dover Publ., New York
Bauer HF (1992) Liquid oscillations in a circular cylindrical container with “sliding” contact line. Forsch Inges 58: 240–251
Watson GN (1922) A treatise on the theory of Bessel functions. Univ. Press, Cambridge
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Bauer, H.F., Eidel, W. Axisymmetric viscous liquid oscillations in a cylindrical container. Forsch Ing-Wes 63, 189–201 (1997). https://doi.org/10.1007/PL00010831
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DOI: https://doi.org/10.1007/PL00010831