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Surface- and interface oscillations in an immiscible spherical visco-elastic system

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Summary

The transcendental frequency equation is presented for various visco-elastic immiscible spherical liquid arrangements exhibiting free surface- and interfacial tension. The system is in a zero-gravity environment. The natural damped frequencies depend on the viscosity, surface tension, density and the Maxwell relaxation time. For a freely floating sphere the numerical results of the frequency equation are presented and exhibit with larger surface tension higher natural frequencies, for small relaxation times stronger and for larger relaxation times weaker decay of the oscillations. The increase of viscosity renders stronger decay of the oscillations. For smaller surface tension the oscillation ceases to exist and yields for small relaxation parameter τν/a 2 an aperiodic motion of the drop, while for higher surface tension the oscillation of the visco-elastic sphere exhibit higher frequencies and larger decay. Increase of visco-elasticity (relaxation time) renders the begin of aperiodic motion of a liquid sphere a larger diameter. For further increased relaxation time, however, the sphere always oscillates.

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Abbreviations

a :

radius of undisturbed outer free liquid surface of sphere

b :

radius of undisturbed interfacial surface

c :

radius of undisturbed inner free liquid surface

I n ,K n :

modified Besselfunctions of ordern and first and second kind resp

p j :

pressure in regionj=1, 2

P n 0 :

Legendre polynomials

P n m :

associated Legendre functions

r, ϑ, ϕ:

spherical coordinates

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

complex, frequency,\(S \equiv \frac{{sa^2 }}{v}\)

t :

time

T 01 :

outer surface tension atr=a

T 12 :

interfacial surface tension atr=b

T 02 :

inner surface tension atr=c

u j :

velocity distribution in radial direction

v j :

velocity distribution in ϑ-direction

x=cos ϑ:

or abbreviation for\(x = a\sqrt {\frac{s}{v}(1 + \tau s)} \)

ϱ j :

density of liquids

η j :

dynamic viscosities of liquids (Newtonian limit viscosity)

v j =η j /ϱ j :

kinematic viscosities of liquids

σ r :

normal stress

τ rϑ :

tangential stress

Ψ j :

streamfunctionsj=1, 2

τ j :

relaxation time

D :

rate-of-strain tensor

T :

stress tensor

W :

antisymmetric part of the rate-of-strain tensor

ω:

circular frequency

ζ:

free liquid surface elevation

σ ϑ :

meridial stress

σ ϕ :

angular stress

e r :

unit vector in radial direction

e ϑ :

unit vector in meridial direction

\( - \bar \sigma \) :

decay magnitude

\(\bar \omega \) :

circular oscillation frequency

References

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Authors and Affiliations

  1. Hochschule der Bundeswehr München, D-8000, München, Federal Republic of Germany

    H. F. Bauer

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  1. H. F. Bauer
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Bauer, H.F. Surface- and interface oscillations in an immiscible spherical visco-elastic system. Acta Mechanica 56, 127–149 (1985). https://doi.org/10.1007/BF01177114

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  • DOI: https://doi.org/10.1007/BF01177114

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