Tangential Electric Stress Research Articles

Numerical study on jet and stretch behaviors of an impingement leaky-dielectric droplet under electric field

The droplet breakup and ejection after impact under electric fields has attracted increasing attention in industrial applications, such as spray cooling, 3D printing, anti-icing of powerlines, etc. We numerically investigated the leaky-dielectric droplet impact on the hydrophilic surface in perfect dielectric surrounding under vertical electric field with OpenFOAM. The coupled volume of fluid (VOF) and leaky dielectric model have been used to solve two-phase electrohydrodynamic (EHD) problems. Impact behaviors are most strongly influenced by the electric capillary number, dielectric permittivity and electrical conductivity. Four impact modes were registered: no-stretch-no-jet, single-droplet split, stretch-no-jet and stretch-jet. The mechanism of the four impingement modes is explained through the relationship between normal electric stress, tangential electric stress and charge relaxation. The stronger distribution of normal electric stress near the cylindrical tip of the droplet leads to a stretching behavior. Then, the stretched filament must be sustained by tangential electric stresses. The surface charge distribution influences the electric stresses, which depend on charge relaxation, charge convection and tip curvature. Furthermore, the relationship between the electric capillary number, dielectric permittivity, electrical conductivity and the breakup modes was obtained. These findings could facilitate the design of electrospray cooling, deicing for power transmission lines, and 3D printing.

Dynamic Electrophoresis of a Hydrophobic and Dielectric Fluid Droplet.

Dynamic electrophoresis is the foundation for electroacoustical measurements, in which the electroacoustical signals may be used to analyze the size and electrostatic charge of colloidal entities by means of the results for dynamic electrophoretic mobility. Thus, the electrophoresis under an alternating electric field is the key foundation for electroacoustic theory. In this article, we develop a tractable analytical theory for the dynamic electrophoresis of hydrophobic and dielectric fluid droplets possessing uniform surface charge density. The tiny fluid droplets possess charged mobile surfaces and have found widespread applications in our day-to-day life. For dielectric fluid droplets (e.g., oil-water emulsions), the tangential electric stress at the interface is nonzero, which significantly affects its electrohydrodynamics under an oscillatory electric field, which has, however, a negligible impact on the electrophoretic motion of conducting droplets (e.g., mercury droplets). Besides, the micro/nanoscale fluid droplets often show hydrophobicity when they are immersed in an aqueous medium, and the impact of the electric field on hydrophobic surfaces remains a research frontier in the chemical discipline. Whereas a number of approximate expressions for electrophoretic mobility have been derived for the conducting droplet, none of them are applicable to such generic hydrophobic fluid droplets with dielectric permittivity that is significantly lower than or comparable to that of an aqueous medium. In this work, within the Debye-Hückel electrostatic framework, we elaborate an original analytical expression of dynamic electrophoretic mobility for this generic dielectric fluid droplet with a hydrophobic surface considering that the droplet retains its spherical shape during its oscillatory motion. We further derived a set of simplified expressions for dynamic electrophoretic mobility deduced under several limiting cases. The results are further illustrated, indicating the impact of pertinent parameters.

On Surface Resistivity Measurements of Thin Insulation Materials

Surface resistivity measurements are frequently used for material characterization. However, the influence of sample thickness is not covered sufficiently in ASTM D257 and IEC 62631-3-2 standards. In this work, electric field simulations and polarization–depolarization current (PDC) measurements are used to study the thickness dependence of such measurements. It is found that measurements according to the standards are not reasonable for low thickness (<1 mm) and high surface resistivity (>1013– <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{{14}}~ {\Omega } $ </tex-math></inline-formula> ). The dependence on thickness is shown to be mainly due to the different tangential electric field stress along the measured surface. Furthermore, indications for charge injection at high electric field stress inside the sample volume are found by a positive polarity of the depolarization currents. A novel electrode configuration for more accurate (more than one order of magnitude) and much less thickness-dependent surface resistivity measurements is proposed in this work and its benefits are demonstrated. It is also shown how the parasitic influence of an electrode holder can be effectively reduced by more than one order of magnitude when measuring surface resistivities. In addition, it is found that polarization times for steady-state surface resistivity values should be at least several minutes instead of the suggested 60 s from standards.

Experimental study on electrohydrodynamic atomization (EHDA) in stable cone-jet with middle viscous and low conductive liquid

The cone-jet is one of the most interesting means in electrohydrodynamic atomization (EHDA) for wide applications because of highly charged and monodisperse drops. The pulsating and stable cone-jet modes in EHDA of diethyleneglycol (DEG) are observed by means of high-speed photography technology, as well as the breakup modes transition of electrified jet from varicose to kink instability. The conductive liquid is driven to form a liquid cone at the outlet of the capillary tube, and form a permanent jet from the apex of the cone by the tangential electric stress. With an increase in electric potential, the cone angle and jet length increase. The electrified meniscus and the cone may appear convex, linear sides and concave in cone jet regime, which depends on the static equilibrium of forces exists at each point of electrified liquid surface. The jet breakup modes with multi-jet, varicose and kink instabilities are successively recorded as liquid flow rates increasing. The jet breakup transition depends on flow rate. The drops size rises proportionally with an increase in liquid flow rate or a decrease in electric potential in stable cone-jet mode. The dimensionless diameter d/d0 as a function of the dimensionless flow rate qV/qm is in good agreement with the universal scaling laws.

Jet and progeny formation in the Rayleigh breakup of a charged viscous drop

Past experimental studies have indicated that the Rayleigh fission of a charged drop occurs via the formation of a jet followed by emission of progeny droplets. In order to understand this process, we model the evolution of a drop using an axisymmetric boundary element method in the viscous limit. In this work, the electrostatic model of a charged viscous liquid drop is modified by including surface charge dynamics. This model accounts for the finite charge relaxation time scales over which the drop surface is charged as well as the convection of charges by the interfacial flow. It is observed that, as the drop deforms with time, the generally applied assumption of an equipotential surface becomes invalid near the conical ends that experience singularly fast dynamics and the associated surface charge dynamics gives rise to tangential electric stresses. These tangential electric stresses exert an axial momentum on the fluid and are responsible for the formation of a jet and progeny droplets. Further, the progeny droplets are found to follow an inverse power-law scaling with the conductivity of the liquid and the smaller sized progenies carry a charge close to its Rayleigh limit.

The minimum flow rate of electrosprays in the cone-jet mode

Stable electrospraying in the cone-jet mode is restricted to flow rates above a minimum, and understanding the physics of this constraint is important to improve this atomization technique. We study this problem by measuring the minimum flow rate of electrosprays of tributyl phosphate and propylene carbonate at varying electrical conductivity $K$ (all other physical properties such as the density $\unicode[STIX]{x1D70C}$, surface tension $\unicode[STIX]{x1D6FE}$ and viscosity $\unicode[STIX]{x1D707}$ are kept constant and equal to those of the pure liquids), and through the analysis of numerical solutions. The experiments show that the dimensionless minimum flow rate is a function of both the dielectric constant $\unicode[STIX]{x1D700}$ of the liquid and its Reynolds number, $Re=(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700}_{o}\unicode[STIX]{x1D6FE}^{2}/\unicode[STIX]{x1D707}^{3}K)^{1/3}$. This result is unexpected in the light of existing theories which, for the conditions investigated, predict a minimum flow rate that depends only on $\unicode[STIX]{x1D700}$ and/or is marginally affected by $Re$. The experimental dependency on the Reynolds number requires the viscous stress to be a factor in the determination of the minimum flow rate. However, the numerical solutions suggest that a balance of opposing forces including the fixing viscous stress, which at decreasing flow rates may lower the acceleration of the flow to the point of making it unstable, is unlikely to be the cause. An alternative mechanism is the significant viscous dissipation taking place in the transition from cone to jet, and which at low flow rates cannot be supplied by the work done by the tangential electric stress in the same area. Instead, mechanical energy injected into the system farther downstream must be transferred upstream where dissipation predominantly takes place. This mechanism is supported by the balance between the energy dissipated and the work done by the electric stress in the transition from cone to jet, which yields a relationship between the minimum flow rate, the Reynolds number and the dielectric constant that compares well with experiments.

Breakup of a leaky dielectric drop in a uniform electric field.

Electrohydrodynamic (EHD) breakup phenomena for a leaky dielectric drop suspended in another immiscible viscous dielectric and subjected to a uniform electric field are examined using the leaky dielectric theory and the explicit forcing lattice Boltzmann method, by taking into account full nonlinear inertia effects. The breakup modes are first computed for varied conductivity of the drop fluid, as the viscosity ratio λ (=μ_{in}/μ_{out}) is momentarily set to unity, that is, for the slightly conducting (R=σ_{in}/σ_{out}<10), moderately conducting (10≤R≤20), and highly conducting (R>20) cases. For slightly conducting drops (R=5) only one breakup mode via two symmetrical necks persists for permittivity ratios 0.05<Q=ɛ_{in}/ɛ_{out}<3.0 and electric capillary number Ca_{E}>Ca_{E,critical} (ratio of electric and surface tension forces), despite significant length-scale variation of mother and daughter drops. At higher Q (for increased drop permittivity) two necks move closer to the bulbous midpart of the extended droplet, which helps enlarge two daughter drops. However, in the case of moderately conducting drops (10≤R≤20) the number of necks increased to four for increased Ca_{E}. Accordingly two pairs of symmetrical daughter drops are created because of recurrent fluctuations of the electrical shear stress and centerline momentum flux. For highly conducting cases of R>20, depending on Ca_{E}, three distinctly elongated droplet states are formed prior to breakup, which results in the onset of three different breakup modes, namely, via formations of lobed ends (Ca_{E}≤0.264), pointed ends (Ca_{E}≤0.68), and nonpointed ends (Ca_{E}>0.83). While being consistent with past measurements, here we precisely characterize the associated breakup mechanisms and physics in terms of the interactive electric pressure, electric shear stress, and hydrodynamic pressure plus velocity gradients. Since the EHD drop breakup is a dynamic process, on an elongated slender drop the activated locally distinct driving forces, i.e., electric pressure at the end regions and tangential electric stress in the midsection, effectively lead to neck formations by virtue of the created high centerline velocity gradient. Accordingly, resulting variations of local extension rate and net mass flux toward drop ends or into intermediate bulbous regions facilitate the multiple-mode drop breakup via the inertia effect, whereas the developed negative curvature around a neck encourages capillary breakup. We also explicitly reveal the effect of the viscosity contrast λ, which particularly influences the breakup characteristics over a broader range of conductivity ratios.

Visualization of periodic emission of drops with micro-dripping mode in electrohydrodynamic (EHD) atomization

Electrohydrodynamic (EHD) atomization could produce mono-dispersed drops in different spraying modes. The generation of drops of controllable size in the micrometer range in micro-dripping mode has been focused because of the important and broad practical applications. In present study, the periodic emission of drops from an electrified meniscus attached to the end of a capillary nozzle in micro-dripping mode and the dynamics of the ligament formation were investigated. The charge-to-mass ratio, the frequency of drops generation, dimensionless equivalent diameter, ligament length and width could be obtained under different dimensionless flow rates (q) and electric Bond numbers (BE), in which q is small but could be varied by a factor of around one hundred and BE is bound to a narrow range of order unity. The charge carried by each drop produced in present work is smaller than the fifty percent of critical value of the Rayleigh limit. Drop generation frequency with rang of a few to several hundred Hz obtained from offline analysis of time-resolved images captured by a high-speed digital camera with zoomed lens highly depends on q and BE, in which the drops produced are usually the smaller than one to ten times the diameter of the nozzle. The micro-dripping regime may be sustained and occur periodically within a certain ranges of flow rate and voltage, where the meniscus would be stretched by a tangential electric stress and further elongated by electric polarization stress and gravity, finally develops a drop ejected from the meniscus. The dimensionless equivalent diameter (dd) would increase as powers of dimensionless flow rate q, in which the correlation gives dd ∼ q0.32 for very small q and dd ∼ q0.45 for larger q. The dimensionless length ls and width ds would increase as powers of dimensionless flow rate q, in which the relation roughly could be given as ls ∼ q0.30–0.34 and ds ∼ q0.33–0.46 for very small q, while ls ∼ q0.43–0.45 and ds ∼ q0.52–0.62 for larger q. The results in present work were similar to previous studies and also existed some deviation, the possible reasons including electric field, liquid properties and viscosity and air resistance were discussed.

Transient electrohydrodynamics of a liquid drop in AC electric fields.

The transient behavior of a leaky dielectric liquid drop under a uniform AC electric field of small strength is investigated, using a closed form analytical solution. The drop settles to a quasi-steady state in a relaxation time that is set by the viscosities of the drop and the ambient fluid and the surface tension, and oscillates around a mean deformation with a frequency that is twice the electric field frequency. The mode of instantaneous deformation remains the same (oblate or prolate) or switches between oblate and prolate, depending on the relative importance of the time-periodic component of the deformation compared to that of the time-exponential. The structure of the flow field and its evolution is studied for representative fluid systems at a high and a low electric field frequency. The individual contribution of the net tangential and normal electric stresses, which are the driving forces of the problem, on the flow structure and drop deformation is characterized. On the basis of the mean (time-independent) and time-periodic components of the driving forces, the flow field is represented as the superposition of three different flow patterns. It is shown that the interplay of these flow patterns leads to formation and destruction of toroidal vortices, and that the residence time of these vortices correlates inversely with the field frequency.

Diffusiophoresis of a charged drop

Diffusiophoresis describes the motion of colloids in an electrolyte or non-electrolyte solution where there is a concentration gradient. While most of the studies of diffusiophoresis focus on the motion of solid particles, soft objects such as drops and bubbles are also known to experience diffusiophoresis. Here, we investigate the diffusiophoresis of charged drops in an electrolyte solution both analytically and experimentally. The drop is assumed to remain spherical. An analytical solution of the diffusiophoretic velocity of drops is obtained by perturbation methods. We find that the flow inside the drop is driven by the tangential electric stress at the interface and it directly influences the diffusiophoretic speed of the drop. Using charged oil droplets, we measure the drop speed under solute concentration gradients and find good agreement with the analytical solution. Our findings have potential applications for oil recovery and drug delivery.

Deformation and breakup of a leaky dielectric drop in a quadrupole electric field

AbstractThe deformation and breakup of a leaky dielectric drop suspended in a leaky dielectric medium subjected to a quadrupole electric field are studied. Analytical (linear and nonlinear asymptotic expansions in the electric capillary number, $C{a}_{Q} $, a ratio of electric to capillary stress) and numerical (boundary element) methods are used. A complete phase diagram for the drop deformation in the $R$–$Q$ plane is presented, where $R$ and $Q$ are the non-dimensional ratios of the resistivities and dielectric constants, respectively, of the drop and the medium phase. The prolate and oblate deformations are mapped in the phase diagram, and the flow contours are also shown. The large deformation and breakup of a drop at higher $C{a}_{Q} $ are analysed using the boundary element method. Several non-trivial shapes are observed at the onset of breakup of a drop. A prolate drop always breaks above a certain critical value of $C{a}_{Q} $. In the oblate deformation cases, breakup as well as steady shapes are observed at a higher value of $C{a}_{Q} $. A detailed study of prolate and oblate deformation tendencies due to the normal and tangential electric stresses and the countervailing role of viscous stresses is presented. The circulation inside a drop is found to be more intense for a quadrupole field as compared with a uniform electric field. More intense internal circulations can lead to enhanced mixing characteristics and will have implications in microfluidic devices.

The Study of Correlations among Parameters in Electrohydrodynamics Micro-Jet Printing System

Electrohydrodynamics (EHD) micro-jet printing technology has become a new method of micro/nano patterning and fabrication because of high resolution, good compatibility and simple structure. Firstly, stresses that Taylor cone meniscus subjected to under pulse voltage are discussed in EHD micro-jet printing system and tangential electric stress grows faster than normal electric stress, and tangential electric stress helps to “push” jet generation. Secondly, apply a pulse voltage superimposed over DC voltage, and investigate correlations between pulse frequency and drop spacing and then pulse frequency and drop diameter, and then compare and analyze the theoretic results with the experimental ones. Finally, as a result that it’s possible to obtain droplets with almost the same interval and desired size in the certain situation through controlling pulse frequency.

Current-Voltage Characteristic of Electrospray Processes in Microfluidics

We use a glass-based microfluidic device to study the electric current behavior of an electrospray process in the presence of a coflowing liquid. The current shows strong voltage dependence and weak flow rate dependence, in stark contrast to classical electrospray. By considering that the current is dominated by convection near the apex of the conical meniscus and driven by tangential electric stresses, we quantitatively capture the voltage and flow rate dependence of the current. Our results elucidate the influence of external field strength and open the way to achieve robust electric control of the current and of the drop size in microfluidics.

Capillary-dominated electrified jets of a viscous leaky dielectric liquid

The capillary-dominated regime of dynamics of electrified jets of a viscous leaky dielectric liquid is studied numerically. In this regime the effective force in the direction of an applied field due to tangential electric stresses is balanced by the gradient of liquid pressure governed by the surface-tension stresses. As is characteristic of this regime, the electric current and the characteristic jet radius are dependent on the surface-tension coefficient and not on viscosity. In the scope of this work, the conditions of the existence of this regime are determined. A qualitative order-of-magnitude analysis gives the power-law dependences of the jet radius and electric current on the parameters of the problem (conductivity, applied electric field, flow rate, and surface-tension coefficient). Numerical results are obtained for low conductive liquids for a large range of the dimensionless flow rate (capillary number, Ca). The order-of-magnitude estimations of electric current are in agreement with the numerical results given a small Ca. The corresponding numerically obtained jet shapes are discussed and explained.

Formation and droplet size of EHD drippings induced by superimposing an electric pulse to background voltage

The effect of electric voltages on droplet size was investigated. Results show that both background voltage and amplitude of electric pulse could affect the process of droplet formation and droplet size while the width of pulsed voltage had a slight effect on droplet size. Pulsed voltages introduced a tangential electric stress that led to relatively greater droplet formation while greater background voltage was associated with smaller droplets. Due to different components of the normal and tangential electric stress three dripping modes were created. Based on above observations, a simplified discussion is presented on the relationships between droplet size and above parameters.

EHD sprayings induced by the pulsed voltage superimposed to a bias voltage

Effects of pulsed voltage superimposed on dc bias voltage and meniscus height on electrohydrodynamic (EHD) spraying were investigated. Results show that greater pulsed voltages were associated with jet formation while a dripping mode was apt to appear with a lower pulsed voltage. This is because that with increasing pulsed voltages the energy gain per unit area of the liquid and the tangential electric stress at the meniscus lateral were increased more quickly than the normal electric stress at the apex of the meniscus. Additionally, the increment of meniscus height led to an unchanged tangential electric stress at the meniscus lateral, but a more quickly increased energy gain per unit area of the liquid than the normal electric stress at the apex of the meniscus. For the same pulsed voltage, spraying in Dripping I mode was produced from menisci of smaller heights due to the intensive normal electric stress. A much greater meniscus height, on the other hand, led to spraying in Dripping II mode when the pulsed voltage was insufficiently great. These various modes were determined by contributions of the tangential electric stress, the normal electric stress and the meniscus height.

Linear oscillations and stability of a liquid bridge in an axial electric field

Small amplitude oscillations of viscous, capillary bridges are studied in the presence of an electric dc field. The electric field is proposed as a means to maintain bridges longer than their perimeter and of uniform cylindrical shape. This is desired in the fabrication of semiconductor crystals. The material of the bridge and the surrounding medium is modeled either as a perfect or as a leaky dielectric. The frequency and the damping rate of the oscillations are calculated numerically by solving a generalized eigenvalue problem. It is shown that they depend on the ratios of the dielectric constants, ε=εin/εout, and conductivities, S=σin/σout, of the two materials, the aspect ratio of the bridge, Λ=πR̃/L̃, the ratio of viscous to the capillary force, Oh=Re−1, which can also be viewed as the inverse Reynolds number of the flow, and, finally, the electrical Bond number, Cel, which is the ratio of the electric stresses to the capillary force. The stability limit of an initially cylindrical bridge is determined with respect to varicose disturbances. In agreement with previous studies it is shown that, if both materials are perfect dielectrics, application of an electric field has a stabilizing effect on the bridge, in the sense that the minimum value, Λmin, of the aspect ratio for the bridge to remain stable drops below 0.5, irrespective of the specific value of the ratio ε. If both materials are leaky dielectrics, bridge stability is determined by the sign of (S−ε) and (S−1)(ε−1), with the positive sign indicating bridge stabilization. The factor (S−ε) arises due to the appearance of a tangential electric stress in the perturbed state for leaky dielectrics. For both cases of leaky and perfect dielectrics, the most unstable mode is the one leading to amphora shaped bridges. It was also found that, when application of an electric field stabilizes the bridge, leaky dielectrics require a lower field than perfect dielectrics and that a large enough field tends to stabilize the bridge for almost the entire range of values of the aspect ratio Λ. These findings concur with earlier analytical results for the stability of jets in longitudinal electric fields and, in conjunction with certain experimental observations, point to the usefulness of the leaky dielectric model pertaining to the stability of bridges.

Application of Galerkin finite-element computations in studying electrohydrodynamic problems

In studying electrohydrodynamic problems, nonlinearly coupled equations for both fluid flows and electric variables are usually considered. Often deformable fluid interfaces are also involved, leading to challenging free-boundary problems. This paper presents a systematic computational approach, based on the Galerkin finite-element method, to solve nonlinear electrohydrodynamic problems even with free fluid interfaces. Examples include electric force effects on the upstream meniscus of a liquid coating bead where only electric normal stress at the fluid interface is important, nonlinear deformations of a leaky dielectric drop in an electric field where both normal and tangential electric stresses are present at the drop interface, and space-charge induced electrohydrodynamic flows due to corona discharge from a wire enclosed in a rectangular shield where electric body force appears in the bulk fluid.

Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric Reynolds number

The electrohydrodynamic flow associated with a fluid drop in an electric field is a consequence of the tangential electric stress at the fluid interface. The tangential viscous stress due to the electrohydrodynamic flow arises to just balance the tangential electric stress at the fluid interface so that the traction boundary condition is satisfied. Influenced by both the local electric stress and viscous stress, the drop interface may exhibit various shapes. The presence of fluid flow also leads to charge convection phenomena. The relative significance of the charge convection effect is usually measured in terms of the electric Reynolds number, Re E , defined as the ratio of the timescales of charge convection by flow and that for charge relaxation by ohmic conduction. This work presents a quantitative analysis of the charge convection effects in a framework of the leaky dielectric model at finite Re E , which has not been considered in previous investigations. Axisymmetric steady flows driven by an applied uniform electric field about a deformable fluid drop suspended in an immiscible fluid are studied by computational means of the Galerkin finite–element method with supplementary asymptotic analysis. The results of finite–element computations are in general agreement with the prediction by the asymptotic analysis for spherical drops at vanishingly small Re E . A common effect of charge convection is found to reduce the intensity of electrohydrodynamic flow. As a consequence, oblate drops are predicted to be less deformed in an electric field when charge convection is taken into account. The prolate drops are often associated with an equator–to–pole flow, which convects charges toward the poles to form a charge distribution resembling that in a highly conducting drop immersed in an insulating medium. Therefore, charge convection tends to enhance the prolate drop deformation. In many cases, charge convection effects are found to be significant even at apparently small Re E , corresponding to the charge relaxation time–scale about 10 −3 s, suggesting that many experimental results reported in the previous publications could have been influenced by charge convection effects.

Electrohydrodynamics of a pair of liquid drops

In this paper we consider the flow field generated by a uniform electrostatic field in and about a pair of identical liquid drops immersed in a conducting fluid. It is assumed that the undisturbed electric field is parallel to the line joining the centres of the two drops. The flow field is due to the tangential electric stress over the surfaces of the drops and here this stress and the flow field are expressed in terms of bispherical harmonics. When the distance between the centres of the drops is many drop diameters the tangential electric stress and the flow field in and about one drop are unaffected by the presence of the other drop, as expected. When the distance between the centres of the drops is of the order of two drop diameters there is a substantial modification in the tangential electric stress at the surfaces of the drops and in the induced flow field, especially in the region between the planes through the drop centres perpendicular to the undisturbed electric field.