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Axisymmetric viscous liquid oscillations in a cylindrical container

Axialsymmetrische Schwingungen einer viskosen Flüssigkeit in einem zylindrischen Behälter

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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

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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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