Leaky Dielectric Model Research Articles

A smoothed particle hydrodynamics method for two-phase electrohydrodynamics modeling with Nernst–Planck equations

The widely used leaky dielectric model often overlooks the rate of change in electric charges, leaving the impact of the charge conservation mechanism on two-phase electro-hydrodynamics (EHD) flows inadequately explored. In this study, we address this gap by introducing a charge-conservative model (CCM) for simulating such EHD systems within the framework of the smoothed particle hydrodynamics (SPH) method. Our methodology employs a fully explicit incompressible SPH (EISPH) approach to discretize the pressure Poisson, the electric potential Poisson, and the Nernst–Planck (N–P) equations. This work presents two notable contributions: (i) the introduction of the charge-conservative model into the incompressible SPH framework and (ii) the achievement of its discretization through a fully explicit methodology. To validate the proposed CCM, we conduct a comprehensive comparison with analytical solutions, as well as existing numerical and experimental results. The results affirm that the CCM consistently produces accurate outcomes across various test cases.

Lattice Boltzmann study of droplet dynamic behaviors impinging on textured surfaces under an external electric field

Textured surfaces contribute to enhancing the cooling effectiveness of electrostatic spray, while the droplet impacting dynamics on such substrates under the influence of electric field are crucial for cooling efficiency. This study utilized a multiphase lattice Boltzmann method combined with the leaky dielectric model to systematically examine the dynamics of droplet impingement on textured surfaces when exposed to electric field. The impact of Weber number, microstructural surface parameters, and electric field strength on droplet impact behavior was discussed in detail. Simulation outcomes reveal that, without the presence of an electric field, the impingement of droplets on textured surfaces results in three distinct deposition states: the Cassie state, partial penetration state, and Wenzel state, primarily contingent upon the surface solid fraction and the droplet impingement velocity. In the Cassie impact regime influenced by an applied electric field, the droplet spreading behaviors exhibit minimal sensitivity to the electric field, with surface tension and inertia primarily governing the spreading dynamics. Throughout the retraction stage, the droplet elongated the direction of the electric field as a result of electric field forces, and eventually, as the electric field strength grows, it bounces off the surface. In the Wenzel impact regime, as the strength of the electric field escalates, the droplet undergoes upward stretching and splits into satellite droplets during the retraction phase, attributed to the dynamic pressure and electrostatic pressure at the apex exceeding the capillary pressure and gravity. These findings could aid in advancing electrostatic spray cooling technology.

Exploring Electrohydrodynamic Flows Inside Leaky Dielectric Drops: A Laser Velocimetry Approach

A novel experimental methodology based on laser velocimetry for the investigation of electrohydrodynamic (EHD) flows inside a neutrally buoyant leaky dielectric drop of 2.25 mm of radius is proposed. Utilizing fluorescent particles, 2D and 3D Lagrangian Particle Tracking (LPT) measurements of the EHD flows inside an initially spherical drop under a constant electric field are reported. Two leaky dielectric fluids, namely Silicone and Castor oils, were used as either the drop or the medium phases, depending on the case investigated, thus covering oblate and prolate drops both with small and larger deformations. 2D measurements of the trajectories and velocities of the tracer particles in a Lagrangian referential enabled a direct comparison with the leaky dielectric model (LDM), a well-stablished theory covering drops with smaller deformations, showing a good agreement with our measurements. The symmetry inside a drop with increased deformation is investigated from an analysis of the mean 3D velocity field, whose measurements converge to the two-dimensional analysis at the plane of symmetry of the drop, suggesting symmetrical flow structures inside the drop.

Electrohydrodynamic flows inside a neutrally buoyant leaky dielectric drop

We present an experimental investigation of electrohydrodynamic (EHD) flows within a neutrally buoyant drop with a radius of 2.25 mm. Utilizing particle image velocimetry and high-speed shadowgraphy, we measure the internal circulation and reported velocity profiles in the bulk and at the interface of the drop. Two leaky dielectric liquids, silicone and castor oils, are employed as the drop and as the external phase, allowing the analysis of two shape configurations: oblate and prolate. The strength of the applied uniform electric field (from 0.125 to 1.75 kV/cm) enables the analysis covering both the small-deformation limit (CaE≪1) and drops with larger deformations. Our measurements show good agreement with the leaky dielectric model (LDM) for the small-deformation cases. The flows begin at the interface as a result of jump in the electric stresses, leading then to four counter-rotating vortices inside the drop. At a permanent regime, the analytical solutions adequately predict the radial and tangential velocity components within the drop. However, a nuanced behavior is noticed for larger deformations, where the LDM theory underpredicts the internal circulation. Moreover, due to the increased deformation, a non-uniform azimuthal profile is observed for the velocity at the interface, vθ. Transient measurements of this velocity component enlighten the dynamic response of the EHD flows of the drop. Following the available analytical solutions, the dynamic response is governed by the timescale of the deformation of the drop, τdef=μa/γ. We propose a critical value of CaE≈0.1 below which the LDM adequately describes the velocity field in both quasi steady-state and transitory regimes.

Data-driven surrogate modelling of multistage Taylor cone–jet dynamics

The Taylor cone jet is an electrohydrodynamic flow typically induced by applying an external electric field to a liquid within a capillary, commonly utilized in colloidal thrusters. This flow generation involves a complex multiphase and multiphysics process, with stability contingent upon specific operational parameters. The operational window is intrinsically linked to flow rate and applied electric voltage magnitude. High voltages can induce atomization instabilities, resulting in the production of an electrospray. Our study presents initially a numerical investigation into the atomization process of a Taylor cone jet using computational fluid dynamics. Implemented within OpenFOAM, our numerical model utilizes a volume-of-fluid approach coupled with Maxwell's equations to incorporate electric body forces into the incompressible Navier–Stokes equations. We employ the leaky-dielectric model, subjecting the interface between phases to hydrodynamic surface tension and electric stress (Maxwell stress). With this model, we studied the droplet breakup of a heptane liquid jet, for a range of operation of 1.53–7.0 nL s−1 and 2.4–4.5 kV of extraction. First, the developed high-fidelity numerical solution is studied for the jet breakup and acceleration of the droplets. Second, we integrate a machine learning model capable of extending the parametric windows of operation. Additionally, we explore the influence of extractor and acceleration plates on colloidal propulsion systems. This work offers a numerical exploration of the Taylor cone–jet transition and droplet acceleration using novel, numerically accurate approaches. Subsequently, we integrate machine learning models, specifically an artificial neural network and a one-dimensional convolutional neural network, to predict the jet's performance under conditions not previously evaluated by computationally heavy numerical models. Notably, we demonstrate that the convolutional neural network outperforms the artificial neural network for this type of application data, achieving a 2% droplet size prediction accuracy.

Effect of uniform electric field on the droplet deformation in uniform flow field

In this article, we theoretically investigate the transient electric field-driven deformation of a droplet. Notably, the droplet is subjected to both a uniform flow and an electric field simultaneously. We explore two cases: one with a perpendicular arrangement of uniform flow and electric field, where the electric field is not stronger than the uniform flow, and another with an aligned arrangement of both. The leaky dielectric model is used and the droplet shape is assumed to remain prolate spheroidal at all times. The droplet behavior was modeled and the corresponding program was written in MATLAB software. According to the selected equations, the outputs of the program are logical and consistent with the available data and experiments. Among the advantages of the theoretical solution, one can mention the possibly tracing the origin of desired events and system reaction and the lower computational cost. The solution method can be extended to investigate similar problems.

Dynamics of a viscoelastic droplet migrating in a ratchet microchannel under AC electric field

Droplet-based microfluidic devices can be powered or manipulated by applying an external electric field, and the ability to precisely control the flow in such devices is essential for various engineering and biomedical applications. In this numerical study, we investigate the deformation dynamics of a viscoelastic droplet in a ratchet microchannel under the influence of an AC electric field. We employ the leaky-dielectric electrohydrodynamic model for both the immiscible fluid phases coupled with the Oldroyd-B model for the droplet fluid. The effect of geometrical parameters such as the type of ratchet and the wavenumber of the ratchets along with the flow parameters such as the electrocapillary number, Weissenberg number and the capillary number significantly affect the droplet shape dynamics and the polymer chain extension. For the parameters considered in this work, the electric force tends to stretch the droplet in the streamwise direction and enhances the droplet deformation and polymer extension. Several interesting effects arise as a result of the coupling of the periodic hydrodynamic forcing of the ratchet walls and the electric field. Specifically, an exponential rise in the polymer chain extension for higher ratchet wavenumbers is observed, along with the cross-stream migration of the droplet for higher electrocapillary numbers when it reaches the outlet of the ratchet constriction.

Effect of heating wires with electric potential on pool-boiling heat transfer under an electric field

In this study, we investigated the influence of heating wires with electric potential on a pool-boiling system under an electric field. We employed a numerical method that combines the pseudopotential lattice Boltzmann model, leaky dielectric model, and finite-difference model. Heating wires with different positions and electric potentials above a horizontal heater were explored. A detailed analysis of the boiling process occurring on the heater was performed at various electric-field intensities and for different boiling regimes, both with and without wires. For the pool-boiling system without heating wires, the results revealed that the critical heat flux (CHF) increased with the electric-field intensity, and the superheat required for the system to attain film boiling increased with electric field intensity. Placing heating wires on the heater surface resulted in the deterioration of heat transfer in the system without an electric field. However, the optimal distance between the heating wires and the heater remarkably promoted heat transfer during the film-boiling regime. The introduction of heating wires with electric potential into the system disrupted the original electric-field distribution, consequently impacting the boiling process and heat transfer. Compared with the system without both heating wires and an electric field, the system without heating wires exhibited a 7.9 % increase in the largest CHF at a low electric-field intensity of CaE = 0.2. However, at the same electric-field intensity, the largest CHF increased by up to 39.4 % when heating wires with an electric potential were employed. These findings provide a theoretical foundation for the practical application of effectively employing wires with electric potential to enhance boiling heat transfer.

Analysis of self-heating in electrosprays operating in the cone-jet mode

The electrohydrodynamic processes taking place in a cone jet cause ohmic and viscous dissipation, and ultimately self-heating of the liquid. Despite this, previous analyses have modelled cone jets as isothermal systems. To investigate the validity of this assumption, this work applies the leaky-dielectric model to cone jets, while also requiring conservation of energy to reproduce the variation of temperature caused by dissipation and temperature-dependent liquid properties. The main goals are to determine whether there exist electrospraying conditions for which the isothermal assumption is inaccurate, and quantify the temperature field under such conditions. The work confirms that self-heating and thermal effects are important in liquids with sufficiently high conductivities, which is a significant limit because these electrical conductivities are needed to produce jets and droplets with radii of tens of nanometres or smaller. The numerical solution provides accurate expressions for evaluating the dissipation and the temperature increase in cone jets, and confirms that thermal effects cause the apparent breakdown of the traditional scaling law for the current of cone jets of highly conducting liquids.

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.

Droplet dynamics: A phase-field model of mobile charges, polarization, and its leaky dielectric approximation

This paper presents a Poisson–Nernst–Planck–Navier–Stokes–Cahn–Hillard (PNP–NS–CH) model for an electrically charged droplet suspended in a viscous fluid under an external electric field. Our model incorporates spatial variations in electric permittivity and diffusion constants, as well as interfacial capacitance. Based on a time scale analysis, we derive two approximations of the original model: a dynamic model for the net charge (assuming unchanged conductance) and a leaky-dielectric model (assuming unchanged conductance and net charge). For the leaky-dielectric model, we perform a detailed asymptotic analysis to demonstrate the convergence of the diffusive-interface leaky-dielectric model to the sharp interface model as the interface thickness approaches zero. Numerical computations are conducted to validate the asymptotic analysis and demonstrate the model's effectiveness in handling topology changes, such as electro-coalescence. Our numerical results from these two approximation models reveal that the polarization force, induced by the spatial variation in electric permittivity perpendicular to the external electric field, consistently dominates the Lorentz force arising from the net charge. The equilibrium shape of droplets is determined by the interplay between these two forces along the direction of the electric field. Moreover, in the presence of interfacial capacitance, a local variation in effective permittivity results in the accumulation of counter-ions near the interface, leading to a reduction in droplet deformation. Our numerical solutions also confirm that the leaky-dielectric model is a reasonable approximation of the original PNP–NS–CH model when the electric relaxation time is sufficiently short. Both the Lorentz force and droplet deformation decrease significantly when the diffusion of net charge increases.

A transient solution for electric field driven orientation and deformation of the angled droplet

In this article, the transient electric field driven orientation and deformation of the droplet is investigated theoretically, with the innovation that the main axis of the droplet is not aligned with the electric field's direction. The leaky dielectric model is used, and the droplet shape is assumed to remain prolate spheroidal at all times. Separate equations for deformation and angular velocity equations were solved separately and then implemented step by step. According to the equations, the results are logical and consistent with the available data and experiments. The benefit of the theoretical solution is reducing the computational cost and the possibility of tracing the system's behavior back to its origin. The solution method can be extended to study similar problems.

Field trap effect on pool boiling enhancement in a non-uniform electric field: A numerical study

In this paper, the effect of a non-uniform electric field on pool boiling is investigated by using the pseudopotential lattice Boltzmann model and the leaky dielectric model. The focus of the study is on the field trap effect, which drives bubbles to migrate toward regions with a lower electric field intensity. Relevant results demonstrate that the non-uniform electric field effectively adjusts the distribution of dry and wet areas during boiling. Moreover, the field trap effect significantly delays the onset of film boiling by disrupting the vapor film through non-uniform electric field forces. Further, parameter analysis reveals a linear correlation between the critical heat flux and the electric capillary number (Cae). It is also determined that there exists an optimal electrode length that can achieve the best effect in delaying the Leidenfrost temperature (TL). To maximize the utilization of non-uniform effects in vapor film destruction, the hydrophilic strips are introduced to inhibit bubble coalescence and can further enhance the boiling heat transfer in this way.

Lattice Boltzmann finite-difference-based model for fully nonlinear electrohydrodynamic deformation of a liquid droplet.

Electric field-induced flows involving multiple fluid components with a range of different electrical properties are described by the coupled Taylor-Melcher leaky-dielectric model. We present a lattice Boltzmann (LB)-finite difference (FD) method-based hybrid framework to solve the complete Taylor-Melcher leaky-dielectric model considering the nonlinear surface charge convection effects. Unlike the existing LB-based models, we treat the interfacial discontinuities using direction-specific continuous gradients, which prevents the miscalculation arising due to volumetric gradients without directional derivatives, simultaneously maintaining the electroneutrality of the bulk. While fluid transport is recovered through the LB method using a multiple relaxation time (MRT) scheme, the FD method with a central difference scheme is applied to discretize the charge transport equationat the interface, in addition to the electric field governing equationsin the bulk and at the interface. We apply the developed numerical model to study the different regimes of droplet deformation due to an external electric field. Similar to the existing analytical and other numerical models, excluding the surface charge convection (SCC) term from the charge transport equation, the present methodology has shown excellent agreement with the existing literature. In addition, the effect of SCC in each of the regimes is analyzed. With the present numerical model, we observe a strong presence of SCC in the oblate deformation regime, contrary to the weak effect on prolate deformations. We further discuss the reason behind such differences in the magnitude of nonlinearity induced by the SCC in all the regimes of deformation.

Electrokinetic modelling of cone-jet electrosprays

The physics of electrospray has been subject to an intense debate for three decades regarding the ultimate electrokinetics that determines the electric current and the size of the emitted droplets in the steady Taylor cone-jet mode (TCJ). In order to solve with a high degree of accuracy the complete electrokinetic structure of the TCJ, in this work, we have used the full Poisson–Nernst–Planck model electrokinetic equations, which have been solved using a high accuracy numerical scheme. We consider a formulation with no interfacial adsorption of ions, as in Mori & Young (J. Fluid Mech., vol. 855, 2018, pp. 67–130). Our simulations corroborate Mori and Young's conclusion that the classical leaky dielectric model (LDM) recovers the electrodiffusion theory for weak electrolytes when disregarding ion adsorption at the interface. However, for strong electrolytes, our results differ drastically from those provided by the LDM. In this case, we observe that the ion distribution, and consequently the conductivity in the bulk, can be strongly non-homogeneous. Given the rather universal validity of the LDM experimentally observed so far, we postulate that ion interfacial adsorption must be considered in the case of strong, highly dissociated electrolytes to retrieve the LDM limit, mostly for a cone jet operating in the vicinity of the minimum flow rate.

A simplified lattice Boltzmann model for two-phase electro-hydrodynamics flows and its application to simulations of droplet deformation in electric field

This paper proposes a numerical scheme within the lattice Boltzmann framework for effective simulations of two-phase electro-hydrodynamic (EHD) flows, in which the simplified multiphase lattice Boltzmann method serves as the flow solver and a leaky dielectric model is utilized to recover electric potential equation. The resultant formulations present direct evolution of macroscopic variables, due to which the present method outperforms the conventional lattice Boltzmann method in terms of memory cost and boundary treatment. Numerical validations are first carried out under the same parametric conditions as the benchmarking experimental studies with the conductivity ratio ranging from 0.03 to 1000, demonstrating that the present method is a practical and accurate tool to study large-conductivity-ratio two-phase EHD flows in scientific applications. Through the comparison with previous theoretical and numerical studies, the performance of the present method is further evaluated by varying physical parameters such as electric capillary number, electric field strength, interfacial tension, and electrical properties of liquids. The good agreements with the reference data confirm the accuracy and robustness of the proposed method. Additional studies of the dynamics of a deformed droplet in uniform electric field are carried out using the validated method. Specifically, we conduct a close numerical examination on the effect of the electrical conductivity ratio and permittivity ratio on the electric forces, and highlight the role of the dielectric electrophoretic force and the Coulomb force on the electric forces the deformation of a leaky dielectric droplet in the electrical conductivity ratio and the permittivity ratio phase diagram.

Twist-bend nematic drops as colloidal particles: Electric instabilities.

The mesogen 1,''7''-bis(4-cyanobiphenyl-4'-yl)heptane (CB7CB), doped with a small quantity of an amphiphilic compound, is examined in its biphasic state in which twist-bend nematic (N_{TB}) drops are dispersed in the isotropic fluid. Various flexoelectric and electrokinetic responses of small drops in their escaped-radial-like (ER) geometry, and also of larger ones with parabolic focal conic defects, are discussed. A pair of confocal parabolas with their axes along the applied low-frequency electric field undergo periodic dimensional changes so as to contribute flexoelectrically to free-energy reduction. In an ER droplet, the same result is achieved by periodic relocations of the hedgehog core. Sine-wave fields of low frequency and high voltage excite patterned states near zero-voltage crossings and homeotropic alignment at peak voltages. ER drops also exhibit electrohydrodynamic effects; in relatively weak fields, they undergo translatory motion with a velocity that is a quadratic in the field strength; the drift, which occurs over a very wide frequency range, extending from dc to MHz region, is enabled by radial symmetry breaking that their off-centered geometry entails; and the drift direction reverses across a critical frequency. In high fields, vortical flows occurring within an ER N_{TB} drop become discernible. The hydrodynamic effects are discussed based on the Taylor-Melcher leaky dielectric model.

Binary coalescence of non-Newtonian droplets under an electric field: A numerical study

We numerically investigate the effect of electrohydrodynamics on a non-Newtonian droplet pair suspended in a Newtonian medium. The leaky dielectric model is implemented to study the response of emulsion drops in an externally applied electric field. Subsequently, the non-Newtonian drop behavior is incorporated using the power law model, whereby three different fluid behaviors are considered for the drops: Newtonian, Shear thinning, and Shear thickening. We validated our numerical model with the available literature data, and the results are in good agreement. The droplets' deformation and net motion are investigated for a range of electrical permittivity ratios of the droplet medium with respect to the surrounding fluid. In this study, four distinct regimes are identified based on the net drop pair motion and the circulation pattern that develops due to the electric stresses inside and around the drops. Furthermore, it is observed that the droplet deformation and their net motion are fastest for the pseudo-plastic drops and slowest for dilatant drops. We devised a simple ratio-based model to understand this behavior. The inferences drawn from this study will help contribute to a better understanding of the behavior of nonlinear fluids under an electric field.

Simulation of electrospray emission processes for low to moderate conductivity liquids

The leaky-dielectric model is incorporated in the Finite Volume Method (FVM) code, OpenFOAM, to investigate the electrospray emission behavior of low to moderate conductivity liquids. This work extends FVM modeling to moderate conductivities by employing a new interface interpolation scheme that is devised in the volume of fluid method to ensure charge conservation for accurate reproduction of charge accumulation and resulting meniscus shape in the cone-to-jet region and jet breakup. The model results agree well with experiments and scaling laws for droplet diameter and total current for low and moderate conductivity fluids, i.e., heptane and tributyl phosphate, respectively. The droplet diameter is shown to increase as the dimensionless flow rate increases or the electric Reynolds number decreases. The results are also consistent with a parametric investigation of the meniscus shape and the maximum charge density for key operating conditions (flow rate and extraction potential) and liquid properties (conductivity, surface tension, viscosity, and relative permittivity). These results show that the new interface interpolation scheme provides accurate results for a wide range of conductivities, fluid properties, and operating conditions. The results also provide valuable physical insight for varying liquid conductivity in the electrospray emission process. In particular, low dimensionless flow rate or high electric Reynolds number leads to the emergence of convex-outward menisci associated with a high charge density in the cone-to-jet region, resulting in high jetting velocity and high specific charge droplets.

Instability of Viscoelastic Liquid Sheets in a Transverse Electric Field

The temporal linear instability of a viscoelastic liquid sheet moving around an inviscid gas in a transverse electrical field is analyzed. The fluid is described by the leaky dielectric model, which is more complex than existing models and enables a characterization of the liquid electrical properties. In addition, the liquid is assumed to be viscoelastic, and the dimensionless dispersion relation of the sinuous and varicose modes between the wavenumber and the temporal growth rate can be derived as a 3 × 3 matrix. According to this relationship, the effects of the liquid properties on the sheet instability are determined. The results suggest that, as the electrical Euler number and the elasticity number increase and the time constant ratio decreases, the sheet becomes more unstable. Finally, an energy budget approach is adopted to investigate the instability mechanism for the sinuous mode.