Physics of Fluids

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Table of Contents for Physics of Fluids. List of articles from both the latest and ahead of print issues.
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Direct methods for solving the Boltzmann equations: Comparisons with direct simulation Monte Carlo and possibilities

Mon, 09/23/2019 - 03:37
Physics of Fluids, Volume 31, Issue 9, September 2019.
The possibilities of direct methods for solving the Boltzmann equation in comparison with direct simulation Monte Carlo are discussed. The general features of these different methods are considered, in particular, from the point of view of application of different variants of discretization in phase space. The advantages and disadvantages of both approaches are clarified. Comparative solutions of some simple problems are given. An important issue concerns anomalous heat transfer and validation of the effect by calculations based on these two methods. The solutions of the stationary one-dimensional heat transfer problem between two infinite plates with nonclassical nonequilibrium reflection from the surface are obtained; the anomalous heat transfer with a temperature gradient and a heat flux having the same sign is observed. One-dimensional and two-dimensional (in the square domain) problems with nonequilibrium “membranelike” boundary conditions are solved numerically; the anomalous heat transfer for all the considered cases is demonstrated.

The first and second laws of thermodynamics

Fri, 09/20/2019 - 06:21
Physics of Fluids, Volume 31, Issue 9, September 2019.
This article summarizes an alternative approach to the formulation of the laws of thermodynamics by making use of the conservation equations of transport phenomena. The principal novel element is the inclusion of the law of conservation of momentum in the derivations. This enables one to make the statement of the first and second laws more complete by including terms describing viscous dissipation of energy.

Symmetric and asymmetric coalescence of droplets on a solid surface in the inertia-dominated regime

Fri, 09/20/2019 - 02:09
Physics of Fluids, Volume 31, Issue 9, September 2019.
We present an investigation of symmetric and asymmetric coalescence of two droplets of equal and unequal size on a solid surface in the inertia-dominated regime. Asymmetric coalescence can result due to the coalescence of two unequal-sized droplets or coalescence of two droplets having different contact angles with the surface due to a step gradient in wettability. Based on the solution of an analytical model and lattice Boltzmann simulations, we analyze symmetric and asymmetric coalescence of two droplets on a solid surface. The analysis of coalescence of identical droplets show that the liquid bridge height grows with time as [math] for θ = 90° and [math] for θ < 90°, where t* is dimensionless time. Our analysis also yields the same scaling law for the coalescence of two unequal-sized droplets on a surface with homogeneous wettability. We also discuss the coalescence of two droplets having different contact angles with the surface due to a step gradient in wettability. We show that the prediction of bridge height with time scales as [math] irrespective of contact angles of droplet with the surface.

Surface rheological measurements of isolated food foam systems

Fri, 09/20/2019 - 02:09
Physics of Fluids, Volume 31, Issue 9, September 2019.
Liquid foams represent a key component to a vast range of food industry products, from ice creams to the crema on coffee. Longevity of these foams is a highly desirable attribute; however, in order for foam stability to be effectively controlled, a better understanding of the interdependence of the bulk liquid and air-liquid interfacial rheologies is required. This study follows an increasing trend in experimental investigations made of isolated foam structures at the microscale, where the bulk and surface dynamics of a single foam liquid channel can be accurately assessed. Isolated foam channels with adjoining nodes were studied for aqueous solutions of four food grade surfactants. Existing observations of distortions to sodium dodecyl sulphate channel geometries were confirmed for solutions of Tween 20 (T20) and Tween 80 (T80) and were well described by the theory presented here. Moreover, previously unseen distortions to liquid channels were observed for polymeric surfactant systems (hydroxypropyl methylcellulose and hydrolyzed pea protein blend), which were proposed to result from their high surface viscosities. The apparent surface viscosities, μs, of surfactants tested here ranged from high (10 g/s < μs < 10−3 g/s) for polymeric surfactants to very low (10−10 g/s < μs < 10−8 g/s) for Tweens, clearly demarking the regimes of viscous and inertial dominant flows, respectively. It is recommended that further work seeks to investigate the finding of a strong correlation between μs and channel surface tension, γ, for soluble surfactant systems, which could explain the apparent non-Newtonian values of μs that were consistently measured here.

Homogenized model with memory for two-phase compressible flow in double-porosity media

Fri, 09/20/2019 - 02:09
Physics of Fluids, Volume 31, Issue 9, September 2019.
A completely averaged model of two-phase flow of compressible fluids in a medium with double porosity is developed. The variational asymptotic two-scale averaging method with splitting the nonlocality and nonlinearity is presented. Several mechanisms of delay are detected, as the nonequilibrium capillary redistribution of phases, pressure field relaxation caused by the compressibility, and the cross effects of fluid extrusion from pores due to rock compaction and fluid expansion. A generalized nonequilibrium capillary equation is obtained. All characteristic times of delay are explicitly defined as functions of saturation.

Bifurcations and pattern evolutions of thermo-solutocapillary flow in rotating cylinder with a top disk

Fri, 09/20/2019 - 02:09
Physics of Fluids, Volume 31, Issue 9, September 2019.
The characteristics of thermosolutocapillary flow bifurcations and pattern evolutions of binary fluid in a rotating cylinder with a top disk on the free surface are investigated through three-dimensional numerical simulations. The mixture of silicon-germanium is employed as the working fluid. For the special case of the capillary ratio equal to minus one, the total thermo and solutocapillary forces are balanced. Once rotation is introduced, the balance among the driving forces is broken, and a wide variety of flow structures are presented as meridional circulations rolling in different directions. When a threshold value of the thermocapillary Reynolds number is exceeded, the stability of capillary flow is destroyed. The two-dimensional steady flow transits to the three-dimensional oscillatory state. The critical conditions for flow bifurcations are explored, and the pattern transitions are mapped. The rotation of the cylinder can suppress the flow instabilities effectively. When the disk counter-rotates with the cylinder, the critical value for the formation of instabilities increases first and then decreases. For the oscillatory flow, various patterns appear with different combinations of the thermocapillary Reynolds number, disk, and/or pool rotation rate. Without rotation, the surface concentration pattern is shown as rosebudlike wave holding still in time but oscillating in space. With the increasing disk rotation rate, the surface pattern transits from hydrosolutal waves to spiral waves, rotating waves, and superimposition of rotating and annular waves propagating in the radial direction. For counter-rotation of the disk and cylinder, a new pattern with coexistence of hydrosolutal and spiral waves traveling in opposite directions is observed.

Significance of non-Oberbeck-Boussinesq effects augmented by power-law rheology in natural convection studies around fins

Fri, 09/20/2019 - 02:09
Physics of Fluids, Volume 31, Issue 9, September 2019.
The augmentation and diminution of non-Oberbeck-Boussinesq (NOB) effects due to power-law rheology cause significant changes in the results and associated implications of natural convection studies. This study focuses on the combined effect of spatial arrangement with NOB and power-law effects. Non-intuitive changes in heat transfer trends are caused by the additional effect on the shear rate distribution due to spatial arrangement of objects, represented here by an array of fins. An order of magnitude analysis was used to derive Oberbeck-Boussinesq type equations for a class of power-law fluids with all properties considered as linear functions of temperature and pressure. Significant temperature dependent properties were identified, and an explicit criterion to neglect viscous dissipation for power-law fluids in pure natural convection was derived. The identified temperature dependencies were incorporated into NOB equations and solved numerically to investigate their effect on flow field and heat transfer trends. Shear thinning significantly augmented (more than doubled) the accelerating NOB effect, while shear thickening diminished (nearly halved) it. The tendency of power-law rheology to augment or diminish NOB effects was demonstrated to considerably increase the sensitivity of results to temperature dependent properties, over and above that for the Newtonian case. Investigations to note their practical implications revealed that optimization results without NOB effects could be quite misleading for the fin array problem, due to the differing cumulative extents of augmentation. Additionally, correlation studies may be inaccurate as the nature of trends was changed fundamentally due to NOB augmentation.

Holmboe instability beyond the Boussinesq approximation revisited

Fri, 09/20/2019 - 02:08
Physics of Fluids, Volume 31, Issue 9, September 2019.
With no use of the Boussinesq approximation, the Holmboe instability is studied in sharply stratified shear flows with arbitrary monotonic and bounded velocity profiles without inflection points, including a piecewise-linear one. Particular attention is given to the separation and analysis of contributions to instability made by the wave–wave and wave–particle interactions which are the main physical mechanisms responsible for the loss of stability. It is shown that both mechanisms are equally important for understanding the instability properties and therefore should be taken into consideration simultaneously.

Three-dimensional flow simulations for polymer extrudate swell out of slit dies from low to high aspect ratios

Thu, 09/19/2019 - 13:43
Physics of Fluids, Volume 31, Issue 9, September 2019.
The impact of the slit die geometry and the polymer melt flow characteristics on the extrudate swell behavior, which is a key extrusion operating parameter, is highlighted. Three-dimensional (3D) numerical simulations based on the finite element method are compared with their conventional two-dimensional (2D) counterparts at the same apparent shear rates using ANSYS Polyflow software. The rheological behavior is described by the differential multimode Phan-Thien-Tanner constitutive model, with polypropylene as a reference. It is shown that increasing the aspect ratio of the die geometry (width/height ratio variation from 1 to 20) contributes to a significant change in the 3D extrudate deformation (relative changes of 10% in several directions; absolute changes up to 30%) and delays the equilibrium axial position (up to a factor 10). High aspect ratios induce a switch to contract flow (swell ratio <1) for the edge height swell. The 3D extrudate swell strongly deviates from the 2D simplified case due to the die effect near the wall, even for higher aspect ratios. Also a different relation with the material parameters is recorded. The initially large swell behavior is followed by a small shrinkage flow in the middle height direction which cannot be captured by the 2D counterpart. The findings are supported by a comprehensive analysis of the velocity and stress fields in and out of the slit dies.

Front dynamics in exchange flow of two immiscible iso-viscous fluids in two-dimensional straight and curved plane channels

Wed, 09/18/2019 - 02:16
Physics of Fluids, Volume 31, Issue 9, September 2019.
Exchange flow of two immiscible fluids at a low Atwood number in a straight and curved plane channel is considered in this analytical study. The fluids are considered immiscible, but practically, the results can be applied to miscible fluids for short times and in nearly horizontal channels where mixing is negligible due to strong segregation. The exchange flow and displacement flow in pipes at different inclinations with respect to vertical have been extensively studied and have many applications in industry or environmental settings. For the case of plane two-dimensional channels, however, because of the simpler geometry, it is more convenient to understand the physics of the problem and formulate the physical phenomena mathematically. An equation has been derived that describes the transient front velocity in exchange flow in a straight plane channel. The steady state front velocity in straight channels is estimated. The exchange flow in curved channels demonstrates an unstable front or a separated trail because of the curvature of the path. In the case of curved channels, some of the general behavior of the interface is predicted and validated against some experimental observations in curved pipes but quantitative analysis of the interface and the flow requires more advanced mathematical formulation and more detailed experiments for validation.

Spectral energy transfer in a viscoelastic homogeneous isotropic turbulence

Tue, 09/17/2019 - 04:20
Physics of Fluids, Volume 31, Issue 9, September 2019.
Energy dynamics in elastoinertial turbulence is investigated by performing different direct numerical simulations of stationary, homogeneous isotropic turbulence for the range of Weissenberg numbers 0 ≤ Wi ≤ 9. Viscoelastic effects are described by the finite extensibility nonlinear elastic-Peterlin model. It is found that the presence of the polymer additives can nontrivially modify the kinetic energy dynamics by suppressing the rate of the kinetic energy transfer and altering the locality nature of this energy transfer. Spectral representation of the elastic field revealed that the elastic energy is also transferred locally through different elastic degrees of freedom via a dominantly forward energy cascade. Moreover, the elastic energy spectrum can display a power-law behavior, k−m, with the possibility of different scaling exponents depending on the Wi number. It is observed that the energy exchange between macro- and microstructures is a two-directional process: there is a dominant energy transfer from the solvent large-scale structures to the polymers alongside a weak energy transfer from polymers to the solvent small-scale structures. This energy exchange consists of three different fluxes. Two of these fluxes equally transfer a small fraction of the kinetic energy into the mean and fluctuating elastic fields. However, the main energy conversion takes place between fluctuating kinetic and elastic fields through a completely nonlocal energy transfer process.

Three-dimensional simulation of a rising bubble in the presence of spherical obstacles by the immersed boundary–lattice Boltzmann method

Tue, 09/17/2019 - 04:20
Physics of Fluids, Volume 31, Issue 9, September 2019.
The dynamics of a bubble bypassing or passing between spherical obstacles, which is associated with many industrial applications, is investigated numerically. A gas–liquid–solid interaction model is established by combining the lattice Boltzmann method and the immersed boundary method. The deformation and the surface velocity of the bubble, as well as the streamlines of the flow field, are studied as the bubble bypasses a single spherical obstacle or passes between a pair of such obstacles. It is found that for the case of a single sphere, the rise velocity reaches a minimum value at the moment at which an annular bubble forms and the whole sphere is enveloped by the bubble. The initial distance between the bubble and the sphere, as well as the ratio of their sizes, has distinct influences on bubble shape and rise velocity. For a pair of spherical obstacles, the rise velocity of the bubble reaches a minimum value twice as the bubble rises between the obstacles. The distance between the two obstacles has a stronger influence on bubble motion than does their size, although when the two obstacles are of different sizes, the bubble will deviate toward the smaller one.

Low swirl premixed methane-air flame dynamics under acoustic excitations

Tue, 09/17/2019 - 04:20
Physics of Fluids, Volume 31, Issue 9, September 2019.
In this study, simultaneous particle image velocimetry and planar laser induced fluorescence of hydroxyl radical, chemiluminescence imaging, and hot-wire measurements are utilized to study reacting low swirl flow dynamics under low to high amplitude acoustic excitations. Results show that a temporal weak recirculation zone exists downstream of the flame, which is enlarged in size under acoustic excitations. Investigations show that temporal behaviors of this recirculation zone play a significant role in flame movements and instabilities. As the acoustic wave amplitude increases, the flame lift-off distance changes drastically, resulting in flame instabilities (flashback and blowout) during the excitations. Prior to the flame blowout, although the flame lift-off distance responds periodically to the acoustic perturbations, heat release fluctuations display an aperiodic response. Flame dynamics are further studied by calculated power spectra of acoustic velocity and heat release fluctuations and reconstructed phase portraits of heat release fluctuations. Investigations show that increasing the forcing amplitude results in more deterministic features in the flame dynamics and amplification of the higher harmonic modes in the heat release fluctuations. However, such regular patterns become scattered prior to the flame blowout due to the existence of nonlinearities induced by high amplitude excitations. It is speculated that flame blowout can be a symptom of such nonlinearities. The Rayleigh index is measured to study thermoacoustic couplings. At low amplitude excitations, various coupling patterns occur at the flame. However, such complex patterns are replaced by simple coherent patterns, when the flame is excited by high amplitude acoustic perturbations.

Drag reduction in turbulent flow along a cylinder by circumferential oscillating Lorentz force

Mon, 09/16/2019 - 06:28
Physics of Fluids, Volume 31, Issue 9, September 2019.
Direct numerical simulations are performed to study the drag reduction effect in turbulent flow along a cylinder by the circumferential oscillating Lorentz force at the Reynolds number Reτ = 272 based on the reference friction velocity and the thickness of the boundary layer. The maximum drag reduction rate obtained in the present work is 42.6%. The intensity, penetration thickness, distribution (idealized or realistic), and oscillation period of the Lorentz force are all crucial in determining the drag reduction rate. As the Lorentz force is intensified or its penetration thickness and oscillation period increase, the wall friction drag will prominently decrease as long as the circumferential flow is stable. The Stokes layer, introduced by the circumferential oscillating Lorentz force, effectively manipulated the near-wall coherent structures, leading to the decrease of the wall friction drag. However, the occurrence of the force-induced vortices in the near-wall region can also lead to significant drag increase by enhancing the radial momentum transportation due to centrifugal instability. By estimating the energy consumption rate, it is clear that the extra power to implement the Lorentz force is far more than the power saved due to drag reduction, which is the result of the low conductivity of the fluid media. Taking the coupling between the electromagnetic field and the flow field into consideration, the wall friction drag is nearly zero and the turbulence intensity in the near-wall region is very low when the induced Lorentz force is high. But the induced Lorentz drag is greatly increased and the turbulence fluctuations are enhanced in the outer region.

Surfactant-laden droplet behavior on wetting solid wall with non-Newtonian fluid rheology

Mon, 09/16/2019 - 06:28
Physics of Fluids, Volume 31, Issue 9, September 2019.
We develop a coupled lattice-Boltzmann with finite-difference (LB-FD) method to simulate surfactant-laden droplet behaviors on wetting solid wall with non-Newtonian fluid rheology. The effects of the power-law exponent, wettability, force direction, and viscosity ratio on the droplet movement under the shear flow or body force are investigated. It is found that the surfactant-laden droplet moves faster and breaks up more easily than the clean droplet owing to the decreased local interfacial tension. During the initial period of the droplet movement, with the decrease of the power-law exponent of the matrix fluid, the unbalanced Young’s force plays a significant role in prompting droplet spreading along the hydrophilic wall whereas making the droplet recoil along the hydrophobic wall. Under the influence of the shear force, the droplet deformation is strengthened in the shear thickening matrix fluid due to high viscous stress from the external flow. However, under the influence of the body force, droplet deformation is strengthened in the shear thinning matrix fluid because the reduction of the matrix fluid apparent viscosity generates less viscous drag force. Furthermore, the shear thickening pendent droplet is more elongated and shows more flexible behavior than the shear thinning droplet during its falling in the Newtonian matrix fluid. The decrease of the viscosity ratio causes the shear thickening droplet to form the shape of a spherical cap, compared with the shear thinning droplet behaving like a rigid object. The present work not only demonstrates the capacity of the coupled LB-FD method but also sheds light on the mechanism of surfactant-laden droplet dynamics on wetting solid wall where non-Newtonian rheology is considered.

Vortex dynamics in low- and high-extent polymer drag reduction regimes revealed by vortex tracking and conformation analysis

Fri, 09/13/2019 - 09:31
Physics of Fluids, Volume 31, Issue 9, September 2019.
Turbulent flow profiles are known to change between low- (LDR) and high-extent drag reduction (HDR) regimes. It is however not until recently that the LDR-HDR transition is recognized as a fundamental change between two DR mechanisms. Although the onset of DR, which initiates the LDR stage, is explainable by a general argument of polymers suppressing vortices, the occurrence of HDR where flow statistics are qualitatively different and DR effects are observed across a much broader range of wall regions remains unexplained. Recent development of the vortex axis tracking by iterative propagation algorithm allows the detection and extraction of vortex axis-lines with various orientations and curvatures. This new tool is used in this study to analyze the vortex conformation and dynamics across the LDR-HDR transition. Polymer effects are shown to concentrate on vortices that are partially or completely attached to the wall. At LDR, this effect is an across-the-board weakening of vortices which lowers their intensity without shifting their distribution patterns. At HDR, polymers start to suppress the lift-up of streamwise vortices in the buffer layer and prevent their downstream heads from rising into the log-law layer and forming hairpins and other curved vortices. This interrupts the turbulent momentum transfer between the buffer and log-law layers, which offers a clear pathway for explaining the distinct mean flow profiles at HDR. The study depicts the first clear physical picture regarding the changing vortex dynamics between LDR and HDR, which is based on direct evidence from objective statistical analysis of vortex conformation and distribution.

Coupling effect on shocked double-gas cylinder evolution

Fri, 09/13/2019 - 05:18
Physics of Fluids, Volume 31, Issue 9, September 2019.
Interaction of a weak planar shock wave with double heavy gas cylinders has been investigated, focusing on coupling effect on the post-shock flow. In experiments, the ideal two-dimensional discontinuous double heavy gas cylinders with controllable initial conditions are generated by soap film technique, and the shocked flow is captured by a high-speed schlieren photography. Two different initial center spacings of cylinders are considered to highlight the coupling effect. As the center spacing reduces, the coupling effect occurs earlier and becomes more prominent. The coupling effect greatly promotes the inner vortex motions near the symmetry axis relative to the outer ones, resulting in the formation of the mushroom and twisted jets. The fusion of the inner vortices completely differs from the observation in previous experimental work in which the inner vortices separate from each other. Quantitatively, the motion of the upstream interface in streamwise direction is obtained, and can be predicted by a nonlinear model considering the coupling effect. Besides, a vortex model is proposed based on the induction equation of point vortex, and the effect of the mutual interferences among vortices on the vortex motions can be well evaluated.

Modulation of sound waves for flow past a rotary oscillating cylinder in a non-synchronous region

Thu, 09/12/2019 - 05:19
Physics of Fluids, Volume 31, Issue 9, September 2019.
Modulation of sound waves for the laminar flow past a rotary oscillating circular cylinder has been studied for a free-stream Reynolds number Re = 150 and Mach number M = 0.2. Modulation of sound waves has been observed if the combination of applied rotary oscillation frequency and amplitude belongs to the nonsynchronous region where the hydrodynamic and acoustic quantities vary with the vortex shedding frequency as well as the applied forcing frequency. Two-dimensional direct numerical simulations (DNS) are carried out on a highly refined grid using high resolution physical dispersion relation preserving schemes for a nondimensional forcing frequency-ratio range 0.1 ≤ fr ≤ 2.0 at a nondimensional surface speed A1 = 0.1. Both the synchronous and the nonsynchronous zones are identified based on the time-varying fluctuations in the lift and the drag coefficients. In the nonsynchronous zone, modulation phenomena of the lift and the drag coefficients are explained by plotting the stream-function contours over multiple vortex shedding cycles. The modulation periods associated with the fluctuating lift and the drag coefficients are different for some cases. This particular observation is in contrast with the observation expressed in the previous studies investigating similar problems. Disturbance pressure fields obtained from the present DNS data are used to analyze the characteristics of radiated sound fields, especially in the nonsynchronous zone. Information related to aerodynamic sound sources has been obtained using approximated Lighthill’s stress tensor, and it is shown that the aerodynamic sound sources also display the modulation phenomenon similar to that observed in the vortex shedding process. Sound fields related to the nonsynchronous zone also exhibit the modulation phenomenon and are governed by the shedding frequency, the forcing frequency, and their linear combinations. Radiated sound field characteristics are further related to the time-varying fluctuations of the lift and the drag coefficients using Curle’s acoustic analogy. Modulated sound waves observed along the upstream and the transverse directions have similar time variation as that of the drag and the lift coefficients, respectively. The phenomenon of beat formation has been observed for the ranges 0.9 ≤ fr ≤ 0.99 and 1.2 ≤ fr ≤ 1.4. Although the observed modulation of sound waves varies significantly with the forcing frequency-ratio, the net radiated sound power has almost remained constant in the nonbeating, nonsynchronous zone. Furthermore, it is confirmed that the dominant sound modes obtained during the proper orthogonal decomposition of disturbance pressure fields in the nonsynchronous zone are related to the shedding frequency-ratio, the forcing frequency-ratio, and their linear combinations.

An embedded boundary approach for efficient simulations of viscoplastic fluids in three dimensions

Thu, 09/12/2019 - 05:16
Physics of Fluids, Volume 31, Issue 9, September 2019.
We present a methodology for simulating three-dimensional flow of incompressible viscoplastic fluids modeled by generalized Newtonian rheological equations. It is implemented in a highly efficient framework for massively parallelizable computations on block-structured grids. In this context, geometric features are handled by the embedded boundary approach, which requires specialized treatment only in cells intersecting or adjacent to the boundary. This constitutes the first published implementation of an embedded boundary algorithm for simulating flow of viscoplastic fluids on structured grids. The underlying algorithm employs a two-stage Runge-Kutta method for temporal discretization, in which viscous terms are treated semi-implicitly and projection methods are utilized to enforce the incompressibility constraint. We augment the embedded boundary algorithm to deal with the variable apparent viscosity of the fluids. Since the viscosity depends strongly on the strain rate tensor, special care has been taken to approximate the components of the velocity gradients robustly near boundary cells, both for viscous wall fluxes in cut cells and for updates of apparent viscosity in cells adjacent to them. After performing convergence analysis and validating the code against standard test cases, we present the first ever fully three-dimensional simulations of creeping flow of Bingham plastics around translating objects. Our results shed new light on the flow fields around these objects.

A phase-field-based lattice Boltzmann modeling of two-phase electro-hydrodynamic flows

Tue, 09/10/2019 - 08:45
Physics of Fluids, Volume 31, Issue 9, September 2019.
In this paper, a simple and accurate lattice Boltzmann (LB) model based on phase-field theory is developed to study the two-phase electro-hydrodynamics flows. In this model, three LB equations are utilized to solve the Allen-Cahn equation for the phase field, the Poisson equation for the electric potential, and the Navier-Stokes equation for the flow field. To test the proposed model, the deformation of a single droplet under a uniform electric field is considered. It is found that under a small deformation, the results are in good agreement with the previous work. For a large deformation, however, the theoretical results would give a large deviation, while the present results are close to the available numerical work.

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