Modification of the boundary layer theory for calculating viscous drop oscillations in a uniform electrostatic field

Capillary oscillations on the free surface of a viscous conductive liquid drop placed in an electrostatic field are calculated. In an approximation linear in stationary deformation amplitude, the drop in this field has the shape of a spheroid extended along the field. The initial problem is modified and simplified in terms of the boundary layer theory by applying an approximation that is linear in the oscillation amplitude and quadratic in the eccentricity of the drop. The accuracy of the approximate solution relative to an exact one is estimated. It is shown that, with a rise in the electrostatic field strength (with an increase in the eccentricity of the drop) and in the viscosity of the liquid, the boundary layer at the free surface of the drop becomes thicker.     

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.     

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 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).     

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.     

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.     

Numerical analysis of decaying nonlinear oscillations of a viscous liquid drop

An adaptive grid numerical model is developed for simulating the dynamics of a viscous liquid drop whose initial shape is strongly disturbed by an external field. Simulated oscillations of a drop in microgravity and on a horizontal surface are compared with available numerical and experimental results.     

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.     

Some laws of polarization and dispersion of a drop in an electrostatic field

The laws of distribution among contributions in various interactions to the total polarization energy of a conductor in a uniform electrostatic field was analyzed. It is shown that in a closed system, spontaneous shape variations of a liquid conductor with a free surface in an external magnetic field are possible only if they are accompanied by an increase in the conductor dipole moment. Variations of the intrinsic energy of a conductor are studied by the example of a conductive liquid drop in the case where a drop affected by a polarization charge becomes unstable. Analytical expressions defining the sizes and charges of the droplets ejected out of the initial drop under the conditions of instability are derived.     

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.     

光诱导液晶中偶氮聚合物形成相光栅的研究

报道了一种新型偶氮聚合物掺杂液晶的复合体系.该体系中的偶氮聚合物在光诱导下发生顺反式异构形成与液晶分离的相光栅.通过建立多指数模型研究该相光栅在He-Ne光的辐照下折射率的变化情况,并且通过对聚合物掺杂液晶的样品加上连续变化的电压来研究其对电场的响应过程.同时研究了聚合物的单体掺杂液晶后随光场和电场的变化情况.结果表明,光诱导下聚合物掺杂液晶的样品折射率改变值经45s达到饱和,弛豫时间为10min;单体掺杂时,饱和时间与弛豫时间分别为0.17s和0.9s.外加电场在1.0V、1.4V、2.6V和4.0V时光栅的衍射效率会突然下降,而单体在变化的外电场下衍射无明显变化.     

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 oscillations of a drop moving at a constant velocity in an insulating medium subjected to an electrostatic field

Nonlinear asymptotic calculations of the second order of smallness in the amplitude of the initial deformation of an ideally conducting liquid drop show that the laminar flow of an ideal conducting incompressible dielectric liquid flowing about the drop in an external electrostatic field parallel to the flow causes oscillation mode’s interaction in the first and second orders of smallness. Both linear and nonlinear interactions between the oscillation modes of the drop excite modes that are absent in the spectrum of modes governing the initial deformation of the drop’s equilibrium shape. In the second order of smallness, the mode interaction decreases the electrostatic field strength, as well as the velocity and density of the environment, that are critical for development of instability of the drop against the polarization charge.     

Caustics and caustic-diffraction in laser shadowgraphy of a sessile drop and identification of profile near contact line

This paper presents an optical method based on the caustics and caustic-diffraction in laser shadowgraphy of a sessile drop to identify and estimate the drop profile near the contact line. A parallel laser beam passes through a liquid sessile drop placed on a transparent substrate to produce a shadowgraphic image of the drop on the screen far from the substrate. Along the inflection line of the drop the Gaussian curvature of the wavefront deformed by the drop vanishes, and therefore the inflection line gives caustics in the far field of the wave, which can be seen on the screen. The neighboring light rays at both sides of the inflection line interfere with each other to form interference fringes at the inner side of the caustics. According to the pattern of the caustics, the drop-profile shape can be identified and estimated.     

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 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.     

Magnetic fluid droplet deformation in electrostatic field

The stability of the ferrofluid (FF) subjected to the electric field is crucial for the application in high voltage (HV) technology. There are several cases where the fluid interacts with solid interfaces. We examined experimentally FF drop on a glass surface. The drop was exposed to the steady electric field. During tests the suspended particles started to aggregate. Changing drop's shape was recorded during a time period. It was observed that the further deformation development depends on aggregates shape and location. The results are compared with the behaviour of pure carrier fluid. Understanding the phenomenon associated with FF drop deformation can help more reliable HV component design.     

Peculiarities of motion of a ferrofluid drop in a rotating magnetic field

The motion of a ferrofluid drop in a liquid nonmagnetic medium under the action of a rotating magnetic field is analyzed experimentally. A deviation of the falling drop from the vertical direction in the fields exceeding a certain critical value is detected. An extremum of the dependence of the deviation of the drop on the frequency of the rotating field is also observed. The results are analyzed theoretically and the algorithm of the numerical solution of the problem is proposed.     

Optical switch based on variable aperture

We report on a pixel whose aperture can be varied electrically. The pixel is confined by a hole-patterned polymer wall and a dielectric liquid forms a ring shape around the wall surface. Without an electric field, the pixel has the largest aperture. The applied fringing field stretches the liquid surface, leading to a decrease in the aperture size. The switchable aperture ratio of the pixel is over 80% and the response time is ~10 ms. Such a device is useful for an optical attenuator, a light shutter, an adaptive iris, and an information display.     

Forces and charges on a slightly deformed droplet in the DC field of a plate condenser

A conducting drop in partial wetting on the lower electrode of a plate condenser and surrounded by a dielectric fluid is considered. When a DC field is applied the drop, acquiring electric charges, is subjected to an electrostatic force normal to the electrode. The force exerted on undeformable drops was previously calculated. In this paper, the distortion from a spherical shape is asymptotically calculated at low electric Bond numbers to generalize previous developments. A mechanism is proposed to explain the drop detachment, leading to an electrical field strength threshold. Some experiments were performed confirming the mechanism.