Summary
The equilibrium and stability of an isolated, inviscid, incompressible, neutral conducting fluid drop whose axis of uniform rotation coincides with the direction of a uniform applied electric field are examined by using an appropriate extension of the virial method developed byChandrasekhar. Rotating spherical, spheroidal, and ellipsoidal equilibrium shapes are shown to satisfy the first twelve moment equations. A linear, one-parameter (the elongation) family of equilibrium curves relates the electrostatic energy,x, to the square of the angular momentum,y, of a given spheroidal shape. Conditions for the onset of instability, obtained from a linearized normal-mode analysis associated with second-harmonic deformations, restrict stable spheroidal configurations to a closed region of thisx−y configuration plane. Genuine triaxial configurations are shown to bifurcate from these axisymmetric configurations in the same manner as the classical, self-gravitating Jacobi ellipsoids bifurcate from the Maclaurin spheroids.
Zusammenfassung
Gleichgewicht und Stabilität eines isolierten, reibungsfreien, inkompressiblen, neutralen, leitenden, gleichförmig rotierenden Flüssigkeitstropfens, dessen Rotationsachse mit der Richtung eines homogenen elektrischen Feldes übereinstimmt, werden mit Hilfe einer passenden Erweiterung der vonChandrasekhar entwickelten Virialmethode untersucht. Es wird gezeigt, daß die rotierenden sphärischen, sphäroidischen und elliptischen Gleichgewichtsformen den ersten zwölf Momentengleichungen genügen. Eine lineare, einparametrige (Parameter ist die Elongation, d. h. das Verhältnis der beiden Achsen) Familie von Gleichgewichtskurven verknüpft die elektrostatische Energie,x mit dem Quadrat des Drehimpulses,y einer gegebenen sphäroidischen Form. Bedingungen für den Beginn der Instabilität werden durch eine Untersuchung der charakteristischen Oszillationsfrequenzen (Störungsrechnung) erhalten. Sie beschränken stabile Konfigurationen auf einen geschlossenen Bereich dieserx−y-Konfigurationsebene. Es wird gezeigt, daß echte dreiachsige Konfigurationen von diesen axialsymmetrischen Konfigurationen in derselben Weise abzweigen wie die klassischen, durch Eigengravitation zusammengehaltenen Jacobischen Ellipsoide von den Maclaurinschen.
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Rosenkilde, C.E., Randall, R.R. On the shape and stability of a conducting fluid drop rotating in an electric field. Acta Mechanica 20, 167–186 (1974). https://doi.org/10.1007/BF01175922
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DOI: https://doi.org/10.1007/BF01175922