Hydrodynamic and hydromagnetic stability

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.

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Evolution, stability and equilibrium shapes of rotating drops which are charged or subject to electric fields

On the stability of a charged rotating droplet in a stream

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.

Capillary oscillations and stability of a charged drop rotating about the axis of symmetry

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

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R. R. Randall

Papers

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