Nonlinear oscillations of an uncharged conducting drop in an external uniform electrostatic field

The problem of nonlinear oscillations of the finite amplitude of an uncharged drop of an ideal incompressible conducting liquid in an external uniform electrostatic field is solved for the first time by analytical asymptotic methods. The problem is solved in an approximation quadratic in amplitude of the initial deformation of the equilibrium shape of the drop and in eccentricity of its equilibrium spheroidal deformation. Compared with the case of nonlinear oscillations of charged drops in the absence of the field, the curvature of the vertices of uncharged drops nonlinearly oscillating in the field is noticeably higher, whereas the number of resonant situations (in the sense of internal resonant interaction of modes) is much smaller.     

Nonlinear vibrations of a charged drop in an electrostatic suspension

The problem of nonlinear vibrations of a charged drop of an ideal incompressible conducting fluid in an electrostatic suspension is analytically solved in an approximation quadratic in two small parameters: vibration amplitude and equilibrium deformation of the shape of the drop in an electrostatic field. To solve the problem analytically, the desired quantities are expanded in semiinteger powers of the small parameters. It is shown that the charge of the drop and the gravitational field influence the shape of the drop, nonlinear corrections to the vibration frequencies, and critical conditions for instability of the drop against the surface charge. At near-critical values of the charge, the shape of the nonlinearly vibrating drop falls far short of being a sphere or a spheroid, which should be taken into account in treating experimental data.     

Nonlinear analysis of the equilibrium shape of a charged conducting drop in an electrostatic suspension

An analytical asymptotic expression is derived that describes the equilibrium shape of a charged drop of an ideal incompressible conducting liquid suspended in superposed collinear uniform electrostatic and gravitational fields. The expression is obtained in an approximation quadratic in the small amplitude of deviation of the equilibrium drop from a sphere, with the electrostatic field dimensionless strength taken as a measure of the deviation amplitude. With allowance for the gravitational and electrostatic fields and interaction between the drop self-charge and external electrostatic field, the equilibrium shape of the drop is found to be very close to a spheroid when the charge and the electrostatic field strength are far from their critical values. The analysis is carried out with a refined procedure of calculation of the equilibrium shape of drops placed in external force fields.     

Equilibrium shape and stability of a charged drop in an air flow under an electric field

The pressure balance on the surface of a charged liquid drop moving along a uniform electrostatic field is analyzed. The liquid is assumed to be nonviscous and incompressible. In the approximation linear in deformation amplitude, the equilibrium shape of the drop as a function of the charge, field strength, and velocity of travel can be both a prolate and an oblate spheroid. Critical conditions for the surface instability of such a drop are obtained analytically in the form of a relationship between the charge, field strength, and velocity of travel. An instability criterion is found by extrapolating to large Reynolds numbers. This makes it possible to fit the earlier model of a corona-initiated lightning in the vicinity of large charged water drops or hailstones to the charges of the drops, field strengths, and velocities of travel (relative to the medium) typical of thunderclouds.     

Deformation and instability of nematic drops in an external electric field

The behavior of nematic liquid-crystal drops freely suspended in an isotropic liquid polymer exposed to an external electric field was studied. A giant deformation was observed for the drop. As the field intensity increased, its equilibrium shape took the form of a prolate ellipsoid. The dependences of the shape and critical fields on the concentration of ions in the polymer liquid were established. A plausible theoretical explanation is suggested for the observed effect. The experimental dependence of drop size on the electric-field strength is analyzed, and the conditions for the loss of drop stability are determined.     

On the possibility of corona initiation near a conducting drop nonlinearly vibrating in an external electrostatic field

An analytical asymptotic expression for the field strength near an ideal incompressible electrically conducting liquid drop nonlinearly vibrating in external electrostatic field E ₀ is found in an order of 5/2 in a small parameter. The small parameter here is the amplitude of deformation of the spherical shape of the drop. It is found that the strength of the electric field resulting at the tops of the drop exceeds the corona-initiating field even if E ₀ is one order of magnitude lower than the value at which the drop becomes unstable against the induced charge (that is, at such values of E ₀ as are observed in storm clouds in full-scale experiments).     

On the structure of the velocity field under the free spheroidal surface of a viscous liquid drop oscillating in an electrostatic field

An asymptotic analytical solution to an initial boundary-value problem considering (i) the time evolution of the capillary oscillation amplitude as applied to a viscous spheroidal liquid drop placed in a uniform electrostatic field and (ii) the liquid flow velocity field inside the drop is found. The problem is solved in an approximation that is linear in two small parameters: the dimensionless oscillation amplitude and the dimensionless field-induced constant deformation of the equilibrium (spherical) shape of the drop. Terms proportional to the product of the small parameters are retained. In this approximation, interaction between oscillation modes is revealed. It is shown that the intensity of the eddy component of the oscillation-related velocity field depends on the liquid viscosity and the external uniform electrostatic field strength. The intensity of the eddy component decays rapidly with distance from the free surface. The depth to which the eddy flow (which is caused by periodical flows on the free surface) penetrates into the drop is a nonmonotonic function of the polar angle and increases with dimensionless viscosity and field strength.     

Analytical study of nonlinear vibrations of a charged viscous liquid drop

The generatrix of a nonlinearly vibrating charged drop of a viscous incompressible conducting liquid is found by directly expanding the equilibrium spherical shape of the drop in the amplitude of initial multimode deformation up to second-order terms. A fact previously unknown in the theory of nonlinear interaction is discovered: the energy of an initially excited vibration mode of a low-viscosity liquid drop is gradually (within several vibrations periods) transferred to the mode excited by only nonlinear interaction. Irrespectively of the form of the initial deformation, an unstable viscous drop bearing a charge slightly exceeding the critical Rayleigh value takes the shape of a prolate spheroid because of viscous damping of all the modes (except for the fundamental one) for a characteristic time depending on the damping rates of the initially excited modes and the further evolution of the drop is governed by the fundamental mode. In a high-viscosity drop, the rate of rise of the unstable fundamental mode amplitude does not increase continuously with time, contrary to the predictions of nonlinear analysis in terms of the ideal liquid model: it first decreases to a value slightly differing from zero (which depends on the extent of supercriticality of the charge and viscosity of the liquid), remains small for a while (the unstable mode amplitude remains virtually time-independent), and then starts growing.     

Nonlinear analysis of the equilibrium shape of a charged drop rotating around its symmetry axis

From the condition of pressure balance on the free surface of a charged rotating conducting-liquid drop, an analytical expression for the equilibrium shape of the drop is derived in the second-order approximation in a small parameter, the ratio of the deformation amplitude to the radius of the initial spherical shape. It is found that, in the linear approximation in the small parameter, the drop takes the form of an oblate spheroid, while in the quadratic approximation, the equilibrium shape of the drop differs from the spheroidal one.     

On the equilibrium shapes of a conducting drop in a uniform and nonuniform electric field

The equilibrium shape of a drop in the electrostatic field of a point charge and a point dipole is asymptotically calculated in terms of the dimensionless deformation of the shape and a ratio between the drop’s radius and the distance to the point charge (dipole). Irrespective of the degree of nonuniformity of the field, the prolate spheroidal deformation (typical of the uniform field) is shown to be the main reason for the change in the equilibrium shape of the spherical drop. When the nonuniformity of the field grows, the equilibrium shape becomes more and more asymmetric and different from the spheroidal one. This, all other things being equal, may influence the critical conditions for the instability of the drop’s surface against an induced charge. It follows from the aforesaid that the drop in the field of the dipole will be the first to undergo instability with the electrostatic pressure on the drop being the same.     

Nonlinear vibrations of a charged drop in a third-order approximation in the amplitude of multimode initial deformation

A solution to the problem of nonlinear surface vibration of a charged ideal liquid drop is found in a third-order approximation in initial multimode deformation of the equilibrium spherical shape by the method of many scales. It is shown that the spectrum of modes that are responsible for the shape of the drop at an arbitrary time instant depends considerably on the spectrum of modes governing the initial deformation of the drop. The latter spectrum also has an effect on nonlinear corrections to the vibration frequencies and, consequently, on a nonlinear correction to the critical Rayleigh parameter, which specifies the stability of the drop against self-charge.     

Nonlinear analysis of the equilibrium shape of a charged drop in the tornado funnel wall

From the pressure balance condition on the free surface of a conducting liquid charged drop, an expression is derived for the equilibrium shape of the drop placed in the field of centrifugal forces acting in the tornado wall. The analysis is carried out in an approximation quadratic in small parameter (the ratio of the deformation amplitude to the radius of the initially spherical drop). In the linear approximation, the drop is a spheroid oblate in the direction normal to the tornado axis. The eccentricity of the spheroid squared is proportional to the angular velocity squared and the radius of the drop cubed. In the quadratic approximation, the equilibrium shape of the drop is other than spheroidal.     

Nonlinear secondary Raman resonances of oscillations of the fundamental mode of a charged drop moving in an environment

Second-order calculations show that, when a gas flows about a charged drop, the fundamental mode of the multimode initial deformation of its equilibrium shape builds up through nonlinear secondary Raman resonant interaction with higher modes if this mode is present in the mode spectrum specifying the initial deformation. This circumstance accounts for large-amplitude spheroidal oscillations of drops in natural liquid-drop systems and provides an insight into corona initiation in the vicinity of drops in thunderstorm clouds and into lightning initiation.     

Star-drops formed by periodic excitation and on an air cushion – A short review

When simply put on a solid, a liquid drop usually adopts the shape of a spherical cap or a puddle depending on its volume and on the wetting conditions. However, when the drop is subjected to a periodic field, a parametric excitation can induce a transition of shape and can break the drop’s initial axial symmetry, provided that the pinning forces at the contact-line are weak enough. Therefore, a standing wave appears at the drop interface and induces a periodic motion, with a frequency that equals half the excitation frequency. In the first part, we review the different situations where star drops can be generated from various types of periodic excitations. In the second part, we show that similar star drops can occur in a much less intuitive fashion when the drop is put on an air cushion, where no periodic motion is imposed a priori. Preliminary experiments as well as theoretical clues for a hydrodynamic interpretation, suggest that the periodic vibration is due to an inertial instability in the air layer below the drop.     

Nonlinear capillary vibrations of a charged drop placed in a dielectric medium: Single-mode initial deformation of the drop shape

The nonlinear vibrations of the equilibrium spherical shape of a charged drop placed in a perfect incompressible dielectric medium are asymptotically calculated in the second-order approximation in single-mode initial deformation of the drop surface. The drop is assumed to be a perfect incompressible liquid. It is shown that the nonlinear vibration amplitudes, as well as the energy distribution between nonlinearly excited modes, depend significantly on the parameter ρ, where ρ is the ratio of the environmental density to that of the drop. It is also demonstrated that an increase in ρ raises the amplitude of the highest of the vibration modes excited due to second-order nonlinear interaction. In the second order of smallness, the amplitude of the zeroth mode is independent of the density ratio. As ρ grows, the effect of the self-charge of the drop, the interfacial tension, and the permittivity of the environment on the nonlinear oscillations increases.     

Nonlinear analytical calculation of the equilibrium shape of a charged drop uniformly accelerated in an electrostatic field

An analytical asymptotic expression for the equilibrium shape of a charged drop of an ideal incompressible conducting liquid uniformly accelerated in collinear electrostatic and gravitational fields is derived in an approximation quadratic with respect to the deviation of the equilibrium shape of the drop from a sphere. It is found that the equilibrium shape of the drop is close to a prolate spheroid when its self-charge and the external electric field strength are far from their values critical in terms of instability against the self-charge and induced charge. This spheroid experiences an insignificant pear-shaped distortion even when the charge of the drop and the electrostatic field strength are high.     

On the calculation of the translational mode amplitude for a drop nonlinearly vibrating in an environment

The second-order amplitudes of the capillary vibration modes of a drop of an ideal incompressible liquid placed in an incompressible ideal medium are calculated. The approximation is quadratic in initial multimode deformation of the equilibrium spherical shape caused by nonlinear interaction. The mathematical statement of the problem is such that the immobility condition for the center-of-mass of the drop is met automatically. When the translational mode amplitude is calculated, a set of hydrodynamic boundary conditions at the interface, rather than the condition of center-of-mass immobility (which is usually applied for simplicity in the problems of drops vibration in a vacuum), should be used.     

The shape of a drop in a constant electric field

The variation of the shape of a drop immersed in an immiscible liquid under the action of an electric field is calculated. The charge is transferred both by ohmic current through the interface and by the convective component over the interface. A solution quadratic in the parameter that is the ratio of the electric pressure to the capillary pressure is analyzed. Conditions where the drop transforms into a spheroid that is prolate or oblate along the electric field vector are found. An experimental study of the drop deformation by electric forces is carried out.     

On nonlinear vibrations of a charged drop in the third-order approximation in the amplitude of initial single-mode excitation

Nonlinear axisymmetric motions of the free surface of a charged drop of an ideal liquid under the single-mode initial deformation of its equilibrium shape is investigated in the third-order approximation in the initial perturbation amplitude. An analytical expression for the drop shape generatrix is derived. Nonlinear corrections to the vibration frequencies for the initial perturbation of an arbitrary mode are found for the first time. The effect of vibration nonlinearity on the instability of the drop against its self-charge is studied.     

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

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.