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Experimental and theoretical studies of the motion generated by a two-frequency magnetic field at the free surface of a gallium pool

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Abstract

The behavior of a free surface of liquid metal in the presence of a periodic magnetic field pulsating at two frequencies (a midrange frequency f 1 and a low frequency f 2, referred to as the modulation frequency) has been investigated. An expression for the forces induced in an electroconductive material by such a field is established for f 2 ≪ f 1, and a heuristic model is developed to describe the free surface axisymmetric deformations of an inviscid liquid metal submitted to modulated electromagnetic forces. The electromagnetic forces generate standing waves with an oscillation frequency of f 2 or 2f 2. Experiments performed on a cylindrical gallium pool show that the observed surface waves are concentric. The maximum amplitude of the surface oscillations and their frequency are measured by means of a contact probe and analyzed with respect to the modulation frequency f 2. The oscillation amplitude exhibits resonance peaks for values of f 2 corresponding to the eigenfrequencies of the free surface modes. The various free surface oscillation modes are identified. Depending on the value of f 2, the free surface oscillates at the frequency f 2 or at the frequency 2f 2. The experimental features of the free surface oscillations agree qualitatively with those predicted by the proposed heuristic model.

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Abbreviations

A 0 :

electromagnetic vector potential

a :

pool radius

B 0 :

magnetic field strength

F 0 :

electromagnetic forces

f :

frequency of the magnetic field

f 1 :

basic frequency

f 2 :

modulation frequency

f 0,n :

eigenfrequencies of the modes (0, n)

f FS :

frequency of the free surface deformation

f(r/a):

shape function

g(r/a):

shape function

H 0 :

pool height

h m :

dome height

I 0 :

coil current modulus

J 0(x):

zero-order Bessel function

l c :

capillary length

m(t):

modulation function

m 20 :

direct component of the modulation function

m(t):

oscillating component of the modulation function

P m :

magnetic pressure

δ p :

pressure jump at the free surface

R ω :

shield number

U A :

Alfven speed

U b :

order of magnitude of the bulk velocities

U s :

free surface velocity

V e :

excitation voltage

α :

actual modulation coefficient

α 0 :

applied modulation coefficient

δ :

electromagnetic skin depth

Φ:

velocity potential

γ :

surface tension

η :

free surface deformation

η 0 :

free surface static deformation

λ na:

nth zero of J’ o(x)

μ :

magnetic permeability

ρ :

density

σ :

electrical conductivity

ω :

angular frequency

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Authors and Affiliations

  1. Enshmg BP 95, 38402, St Martin d’Hères Cedex, France

    D. Perrier CNRS-EPM (Postdoctor), Y. Fautrelle INPG (Professor) & J. Etay CNRS (Senior Scientist)

Authors
  1. D. Perrier CNRS-EPM
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  2. Y. Fautrelle INPG
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  3. J. Etay CNRS
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Perrier, D., Fautrelle, Y. & Etay, J. Experimental and theoretical studies of the motion generated by a two-frequency magnetic field at the free surface of a gallium pool. Metall Mater Trans B 34, 669–678 (2003). https://doi.org/10.1007/s11663-003-0038-5

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  • DOI: https://doi.org/10.1007/s11663-003-0038-5

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