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Journal ArticleDOI

On the shape and stability of a conducting fluid drop rotating in an electric field

01 Sep 1974-Acta Mechanica (Kansas State University)-Vol. 20, Iss: 3, pp 167-186

Abstract: 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
Topics: Inviscid flow (52%), Angular momentum (52%), Fluid mechanics (52%)
Citations
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01 Jan 1961

113 citations


Journal ArticleDOI
Abstract: In this article, we study by means of the boundary element method the effect that rotation at constant angular momentum L has on the evolution of a conducting and viscous drop when it holds an amount of charge Q on its surface or is immersed in an external electric field of magnitude E∞ acting in the direction of the rotation axis. This droplet is considered to be contained in another viscous and insulating fluid. Our numerical simulations and stability analysis show that the Rayleigh fissibility ratio χ at which charged drops become unstable decreases with angular momentum. For neutral drops subject to an electric field, the critical value of the field which destabilizes the drop increases with rotation. Concerning equilibrium shapes, approximate spheroids and ellipsoids are obtained and the transition values between these two families of solutions is described. When the drop becomes unstable, a two-lobed structure forms where a pinch-off occurs in finite time or dynamic Taylor cones (in the sense of [Betelú et al., Phys. Fluids. 18 (2006)]) develop, whose semiangle, for small L, remains the same as if there was no rotation in the system.

2 citations


Cites background or methods from "On the shape and stability of a con..."

  • ...(5.6) Even though our numerical results do not match the Rosenkilde and Randall approximate theoretical solutions for large values of the parameters, they are still linearly related in the ( L2, E2∞ ) -space (see Fig....

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  • ...3 Comparison between our numerical experiments and theoretical solutions by Rosenkilde and Randall ratio, α = rpolar/requat ....

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  • ...We also present several methods to obtain approximate equilibrium solutions and compare them with those introduced by Rosenkilde and Randall (17, 18), who used an appropriate extension of Chandrasekhar’s virial method when rotation takes place at constant angular velocity....

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  • ...Therefore, we can obtain from numerical results representations of the functions h and g appearing in (5.5) that are more accurate than those given by Rosenkilde and Randall....

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  • ...The formula deduced in (18) that gives the relationship between the angular momentum and the electric field is:...

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Proceedings ArticleDOI
01 Mar 2020

Journal ArticleDOI
Abstract: The stability of a charged conductive liquid drop rotating about the axis of symmetry against the pressure of the self-charge electric field and inertial force pressure is investigated in an approximation linear in oscillation amplitude and square of the spheroidal drop deformation eccentricity. It is found that the axisymmetric modes of the rotating drop are stable. Only nonaxisymmetric modes with azimuthal numbers maximal for a given mode may be unstable. The Coriolis force plays a stabilizing role.

References
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Book
01 Jan 1961

11,409 citations


Journal ArticleDOI
Abstract: The disintegration of drops in strong electric fields is believed to play an important part in the formation of thunderstorms, at least in those parts of them where no ice crystals are present. Zeleny showed experimentally that disintegration begins as a hydrodynamical instability, but his ideas about the mechanics of the situation rest on the implicit assumption that instability occurs when the internal pressure is the same as that outside the drop. It is shown that this assumption is false and that instability of an elongated drop would not occur unless a pressure difference existed. When this error is corrected it is found that a drop, elongated by an electric field, becomes unstable when its length is 1.9 times its equatorial diameter, and the calculated critical electric field agrees with laboratory experiments to within 1%. When the drop becomes unstable the ends develop obtuse-angled conical points from which axial jets are projected but the stability calculations give no indication of the mechanics of this process. It is shown theoretically that a conical interface between two fluids can exist in equilibrium in an electric field, but only when the cone has a semi-vertical angle 49.3$^\circ$. Apparatus was constructed for producing the necessary field, and photographs show that conical oil/water interfaces and soap films can be produced at the caloulated voltage and that their semi-vertical angles are very close to 49.3$^\circ$. The photographs give an indication of how the axial jets are produced but no complete analytical description of the process is attempted.

2,792 citations


"On the shape and stability of a con..." refers background or methods or result in this paper

  • ...The configurations which comprise }he abscissa form the sequence of prolate spheroidal shapes that have been investigated by TaYnou [ 4 ], t~OS~NKILD~ [5], and BI~AZI~R-S~ITK [6]....

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  • ...At the other extreme near the unstable point E in Fig. 2 (x ~ 0.18), TAYLO~ [ 4 ] has argued that the elongated ends of a drop develop conical points prior to the appearance of a narrow jet....

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  • ...There also have been studies of isolated, non-rotating dielectric fluid drops in uniform eleetrie fields by TaYLoR [ 4 ], tgOS~XKILD~ [5], and BR~ZlE~-S~ITK [6], [7], [8]....

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  • ...This conclusion agrees with the results of earlier investigations of the limiting case with no rotation [ 4 ], [5], [6]....

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  • ...which was used by TAYLOIr [ 4 ] and ROS~KILDE [5], has the disadvantage of not being monotonic in the present case....

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Book
01 Jan 1969

1,397 citations


Journal ArticleDOI
Abstract: It is shown that a bubble of gas or liquid, immersed in a liquid medium and subjected to an electric field between parallel plate electrodes, assumes the shape of a prolate spheroid in the direction of the field. Expressions for interfacial traction between two fluid dielectrics, if derived by taking into account electrostriction (Stratton 1941; Smythe 1950), are shown to be in disagreement with experimental results and must therefore be considered incorrect. Using expressions for interfacial traction not involving electrostrictive terms, equations are derived for the dependence on electric stress of the elongation of compressible (gaseous) and of incompressible (liquid) bubbles immersed in an insulating liquid. These show that as the field strength is increased, conducting bubbles, and also non-conducting bubbles for which the permittivity of the bubble exceeds twenty times the permittivity of the medium, elongate until a critical shape is reached when the bubble becomes unstable. For conducting bubbles the critical shape corresponds to a ratio of the major to the minor semi-axis of 1*85. Bubbles of permittivity ratio lower than 20 have no critical shape, the axial ratio increasing indefinitely with increase of field strength. There is satisfactory agreement between theory and experiment. The implications of these results with regard to electrical breakdown of liquids are discussed.

201 citations


Journal ArticleDOI
Subrahmanyan Chandrasekhar1Institutions (1)
Abstract: In this paper, the stability of a rotating drop held together by surface tension is investigated by an appropriate extension of the method of the tensor virial. Consideration is restricted to axisymmetric figures of equilibrium which enclose the origin. These figures form a one parameter sequence; and a convenient parameter for distinguishing the members of the sequence is Σ = ρΩ 2 a 3 /8 T , where Ω is the angular velocity of rotation, a is the equatorial radius of the drop, ρ is its density, and T is the interfacial surface tension. It is shown that Σ ⩽ 2.32911 ( not 1 + √2 as is sometimes supposed) if the drop is to enclose the origin. It is further shown that with respect to stability, the axisym metric sequence of rotating drops bears a remarkable similarity to the Maclaurin sequence of rotating liquid masses held together by their own gravitation. Thus, at a point along the sequence (where Σ = 0.4587) a neutral mode of oscillation occurs without in stability setting in at that point (i.e. provided no dissipative mechanism is present); and the in stability actually sets in at a subsequent point (where Σ = 0.8440) by overstable oscillations with a frequency Ω. The dependence on Σ of the six characteristic frequencies, belonging to the second harmonics, is determined (tables 3 and 4) and exhibited (figures 3 and 4).

140 citations