Electrohydrodynamic instability of the interface

PHYSICS OF FLUIDS 17, 084104 共2005兲

Electrohydrodynamic instability of the interface between two fluids confined in a channel R. M. Thaokar and V. Kumarana兲 Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India

共Received 9 August 2004; accepted 4 May 2005; published online 2 August 2005兲 The stability of the interface between two dielectric fluids confined between parallel plates subjected to a normal electric field in the zero Reynolds number limit is studied analytically using linear and weakly nonlinear analyses, and numerically using a thin-layer approximation for long waves and the boundary element technique for waves with wavelength comparable to the fluid thickness. Both the perfect dielectric and leaky dielectric models are studied. The perfect dielectric model is applicable for nonconducting fluids, whereas the leaky dielectric fluid model is applicable to fluids where the time scale for charge relaxation, 共⑀⑀0 / ␴兲, is small compared to the fluid time scale 共␮R / ⌫兲, where ⑀0 is the dielectric permittivity of the free space, ⑀ and ␴ are the dielectric constant and the conductivity of the fluid, ␮ and ⌫ are the fluid viscosity and surface tension, and R is the characteristic length scale. The linear stability analysis shows that the interface becomes unstable when the applied potential exceeds a critical value, and the critical potential depends on the ratio of dielectric constants, electrical conductivities, thicknesses of the two fluids, and surface tension. The critical potential is found to be lower for leaky dielectrics than for perfect dielectrics. The weakly nonlinear analysis shows that the bifurcation is supercritical in a small range of ratio of dielectric constants when the wavelength is comparable to the film thickness, and subcritical for all other values of dielectric constant ratio in the long-wave limit. The thin-film and boundary integral calculations are in agreement with the weakly nonlinear analysis, and the boundary integral calculation indicates the presence of a secondary subcritical bifurcation at a potential slightly larger than the critical potential when the instability is supercritical. When a mean shear flow is applied to the fluids, the critical potential for the instability increases, and the flow tends to alter the nature of the bifurcation from subcritical to supercritical. © 2005 American Institute of Physics. 关DOI: 10.1063/1.1979522兴 I. INTRODUCTION

Electrohydrodynamics, which involves a complex interplay of inertial, viscous, and electric forces, can lead to interesting phenomena which have found technological applications in ink jet printing, lithography, drop spraying, etc. The instability of a charged droplet was first analyzed by Rayleigh more than a century ago, and the electrohydrodynamics has been an active field ever since. In a review article, Saville1 summarized the work in this area to present a unified picture of electrohydrodynamics and electrokinetics. The present analysis is concerned with the stability of the interface between layered dielectrics under the influence of an applied potential as well as imposed flow fields. The low Reynolds number limit is becoming increasingly important in microfluidics applications, where the use of electrokinetic flows for the transport of fluids is an area of active research.2 For example, for fluids with viscosity of 10−2 kg/ m / s flowing in microchannels with a width of 1 ␮m, the Reynolds number 共assuming a fluid density of 103 kg/ m3兲 is about 10−4 for a flow velocity of the order of 1 mm/s. Since microchannels and tubes have a high surface area to volume ratio, the power required for pumping a given volume of a兲

Author to whom correspondence should be addressed; electronic mail: [email protected]

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fluid by conventional pumping techniques is larger than that for macroscopic applications. The preferred mode of transport of fluids in macroscale applications is by applying a pressure gradient across the ends of the tube. However, the ratio of the pressure gradient and the flow rate in a microchannel is proportional to the inverse fourth power of the radius, while the ratio of the pressure gradient and the average velocity is proportional to the inverse second power of the radius. Therefore, pressure gradients are not an efficient means for transporting fluids at the microscale, and alternative mechanisms of fluids transport, such as flows due to electric fields, are preferred. The instability of the interface between layered dielectrics is of particular importance in the lithography-induced self-assembly 共LISA兲 process, which is an alternative to conventional lithography and can be used to pattern polymeric and other nonconventional materials. Here, the objective is to create a well-defined array of polymeric pillars on the micron scale that bridge the gap between the substrate and the mask,3–6 and this array is created by triggering an electrohydrodynamic instability in an initially uniform thin polymer film. Conventional lithography techniques are not economically feasible for features thinner than about 100 nm, and LISA has been explored as an alternative technique which can make smaller features economically feasible.

17, 084104-1

© 2005 American Institute of Physics

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R. Thaokar and V. Kumaran

In the present analysis, attention is restricted to the electrohydrodynamic instability between two fluids in the zero Reynolds number limit. Two models are used for the dielectric fluid. The first is the “perfect dielectric” model,7,8 in which there are no free charges, and the electric field is caused by the polarization of bound charges. Perfect dielectrics have zero conductivity due to the absence of free charges. Studies on drop deformation using the perfect dielectric model7,8 showed that the drop deforms to a prolate shape, and the electric field caused a normal stress at the interface which was balanced by the product of surface tension and the curvature of the interface. However, experiments9 indicated that drops do deform into an oblate shape. In order to explain this, the “leaky dielectric” model was proposed by Taylor,10,11 in which the conductivity of the liquids is assumed to be uniform and nonzero. Thus, the leaky dielectric assumes the presence of free charge carriers, and under the influence of an electric field, these form induced charge layers at the interfaces. The leaky dielectric model does permit a high-conductivity limit, where the fluid behaves like a conductor. A more detailed model for dielectric fluids is the electrokinetic model, in which the motion of the charges is explicitly incorporated in the charge and momentum balance equations.4,12,13 There have been different approaches to understanding the approximations made in the leaky dielectric model. The electrokinetic models4,12,13 incorporate a balance equation for the charge density at the surface, in which the jump in the normal current at the interface 共which is the product of the bulk conductivity and the normal component of the electric field兲 is incorporated into a surface charge balance equation. It should be noted that the charge balance equations do not typically include surface diffusion of charges, which tend to damp out variations in the charge density. The time scale for ␶c for charge relaxation in a conducting medium is given by 共⑀⑀0 / ␴兲, where ⑀ and ⑀0 are the dielectric constant and the permittivity of free space, and ␴ is the conductivity.1 The leaky dielectric model applies when this time scale is small compared to other time scales in the problem. The flow time scale in the problem is 共R␮ / ⌫兲, since viscous and interfacial stresses are the dominant stress in the low Reynolds number regime, where ␮ and ⌫ are the fluid viscosity and surface tension, and R is the characteristic length scale which is the fluid thickness. 关Note that the critical potential is scaled by 共⌫R / ⑀0兲, so that the time scale obtained from the Maxwell stress is equivalent to 共R␮ / ⌫兲兴. Therefore, the charge relaxation is fast compared to the other time scales in the problem for 关␶c / 共R␮ / ⌫兲兴 Ⰶ 1, or for ␴ Ⰷ 关⑀⑀0⌫ / 共R␮兲兴. For low viscosity fluids with ␮ ⬃ 10−2 kg/ m / s and surface tension ⌫ ⬃ 10−1 kg/ s2, this corresponds to ␴ Ⰷ 10−4 S / m for micron-sized channels, which is not typically encountered in nonmetallic fluids, though metallic fluids have conductivities of order of 104 S / m. However, for high viscosity fluids with viscosity of 1 kg/m/s, the condition for fast charge relaxation corresponds to ␴ Ⰷ 10−6 S / m, and conductivities of this magnitude are encountered even in nonmetallic fluids.1 When the time scale for conduction is small compared to the time scale for the charge distribution along the surface due to surface flow, the discontinuity in the normal current at the surface is set equal

Phys. Fluids 17, 084104 共2005兲

to zero in the leading approximation, and the normal electrical displacement condition for leaky dielectrics is obtained. Another approach to understanding the leaky dielectric model is that of Zholkovskij, Masliyah, and Czarnecki14 who showed that the electrokinetic model reduces to the leaky dielectric model if the Debye screening length for the charges is small compared to the characteristic flow length scale, which is the fluid thickness in the present case. In this case, the charges are confined to thin regions on either side of the interface, and are approximated as charges localized at the interface. The approximation made is that the time scale for the formation of the Debye layer at the interface is small compared to process time scales, so that the charge redistribution in the Debye layer is instantaneous compared to the time scale for the lateral redistribution of charges across the Debye layer. Zholkovskij, Masliyah, and Czarnecki14 showed that the leaky dielectric results for drop deformation are recovered in this limit as well. Schaffer, Thurn-Albercht, Russel, and Steiner6 carried out experiments on a thin film with a thickness of the order of hundreds of nanometers. The system was found to show various patterns depending on the strength of the electric field and the height of the waves. The destabilizing wavelengths were of the order of micrometers, while the gap width was of the order of hundreds of nanometers, giving rise to the possibility of capturing the instability theoretically using a long-wave, thin-film analysis, although the thickness of the films does suggest the possibility of dispersive forces being important for film rupture. The authors carried out a linear stability analysis, and their experimental results showed good agreement with the wavelength of patterns predicted by the theory. However, the theory could not explain the columnar structures and patterns obtained which can be strongly influenced by the nonlinearities. Shankar and Sharma15 considered the linear stability of the interface between two perfect and leaky dielectric fluids, and the effect of viscosity on the fastest growing mode. The earlier study of Pease and Russel4 then turns out to be a special case for the model parameters of Shankar and Sharma.15 Both these studies carried out the linear stability analysis in the long-wave limit and the nonlinear effect was not considered. Baygents and Baldessari16 used a linear stability analysis to examine the instability due to an electric field across a fluid in which the electrical conductivity varied linearly with height. The variation in the conductivity was due to a gradient in the concentration of charge carriers, and the parameter that characterized the onset of instability was the electrical Rayleigh number which contained the relative concentration gradient and the diffusivity of the charged species instead of the temperature gradient and the thermal diffusivity. Wu and Chou3 carried out a numerical analysis of the formation of nanostructures, using a Poiseuille velocity approximation for the tangential velocity in the thin-film limit, and incorporated dispersion forces while calculating the pressure boundary condition at the interface. In a subsequent paper, Pease and Russel4 analyzed the validity of the lubrication approximation by incorporating the effect of lateral gradients which vary over length scales comparable to the electrocapillary length. They concluded that the lubrica-

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Electrohydrodynamic instability of the interface

tion approximation is valid only when the channel height is smaller than the electrocapillary length. A detailed comparison of the numerical and experimental values of the critical potential required for destabilizing the thin film was made by Pease and Russel,5 who found a good agreement between the predictions of the linear stability analysis and experimental results. They concluded that the feasibility of obtaining nanometer scale features could be limited by dielectric breakdown in the gap, as well as by the dynamical slowing down at higher viscosity. Lin, Storey, Oddy, Chen, and Santiago17 carried out experimental and linear stability studies of the stability of the interface between regions with different conductivities separated in the vorticity direction when the electric field is applied along the flow direction. The results of the linear stability analysis for the onset of instability were in good agreement with the experimental results, and it was found that viscous stresses on the lateral walls of the channel have a strong stabilizing effect on the instability. Numerical investigations of the most unstable modes were also carried out. More recently, Storey, Tilley, Lin, and Santiago18 developed a thin channel flow model for the electrokinetic flow in microchannels which is in good agreement with threedimensional models. It should be noted that the electric field, which is in the flow direction, as well as the conductivity gradient, which is in the spanwise direction, as considered by Lin, Storey, Oddy, Chen, and Santiago17 are in a direction different from that in the present study. As summarized above, the study of the viscous limit has been restricted to the linear stability to small perturbations in the absence of flow. In the present study, the instability of the interface between two perfect or leaky dielectric fluids, with the interface stressed by a normal electric field, is analyzed in the limit of zero Reynolds number. The zero Reynolds number regime is becoming increasingly important in microfluidic applications, where inertial effects are not important in the flow of fluids through channels and tubes of micron size. In applications where the transport of mass or heat is desired in systems of micron size, the transport coefficients are much lower than those encountered in macroscopic applications since the flow is laminar. One way to increase the transport coefficients is to induce secondary flows, and various strategies have been explored including wall roughness and electro-osmotic flows.2 In this context, the control of the electrohydrodynamic instability due to shear flow can be extremely important, especially in chip microdevices where aqueous liquids are driven through microchannels under the influence of electric field. The instability can be important in lithography, where creation of lateral patterns with wavelengths of nanometers to micrometers is desired.6 The analysis done here provides a detailed picture of the behavior of the instability in different parameter regimes, and the effect of flow on the instability. The leaky dielectric model in the present study is considered in the limit of instantaneous charge relaxation, which considerably simplifies the model. The present analysis is different in many respects from the earlier studies.3,4 Here, we consider the interface between two fluids of equal viscosity, in contrast with the earlier studies of Wu and Chou3 and Pease and Russel,4 which consider


the interface between a polymer film and air, where the viscosity of air is considered to be zero. It is necessary to examine two fluids of finite viscosity, since we are considering the effect of mean flow, and a linear velocity profile cannot be generated in a fluid if there is a zero tangential stress condition at the top surface. The viscosities are set equal for simplicity; different values of the viscosity ratios will affect the results quantitatively, but not qualitatively. The analysis of Wu and Chou3 is simpler than the present analysis, because a Poiseuille velocity profile is assumed for the tangential velocity, whereas the present analysis considers the detailed velocity profiles. However, we do not incorporate dispersion forces which have been considered by Wu and Chou. The analyses of Pease and Russel4 use a more sophisticated charge conservation condition at the interface, where the charge density and velocity at the interface are explicitly incorporated in the analysis. The present study differs from earlier linear stability studies because we focus on the nature of the initial bifurcation 共supercritical or subcritical兲 which is not accessible from linear stability analysis, and also on the numerical evolution of the interface after the initial bifurcation using the thin-film analysis and boundary element technique. We also focus on examining the effect of flow on the evolution of the interface, and indicate the magnitude of the velocity required to significantly alter the electrohydrodynamic instability. A numerical comparison between linear stability analysis and experimental results has been carried out by Pease and Russel,5 so we do not pursue this further. The nonlinear analysis also enables us to examine the effect of nonlinear interactions on the evolution of the initial perturbation. Pease and Russel5 have, for example, found that the wavelength of the patterns obtained in the experiments such as those of Schaffer, ThurnAlbercht, Russel, and Steiner is well predicted by the linear theory analysis. This is a surprising result since it implies that the nonlinear interactions which set in at finite perturbation amplitudes do not affect the wavelength of the perturbations, and there is no secondary bifurcation before the interface touches the top and bottom surfaces to form pillars. It should be noted that there have been two approaches used for the weakly nonlinear analysis. In the earlier approach formulated by Stuart19 and Watson,20 the amplitude A of the perturbation is assumed to be complex, and equations are written for the magnitude and the phase of the complex amplitude. This is also the standard procedure followed by, for example, Drazin and Reid.21 In the approach of Reynolds and Potter,22 the amplitude is assumed to be real, but the frequency ␻ is a function of the amplitude, and the variations to ␻ due to nonlinear effects are also calculated as a part of the perturbation procedure 共this is often referred to as the Poincaré eigenvalue-stretching procedure兲. The variation of the phase of A in the Stuart-Watson approach above is implicitly incorporated into the variation of the frequency due to nonlinear effects, which alters the phase of the product A exp共i␻t兲 in the Reynolds-Potter approach. The equivalence between the two has been shown by Reynolds and Potter.22 In the present analysis, we use the approach of Reynolds and Potter, since it is simpler to implement, but we have verified

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that both approaches provide the same result for the Landau constant. The nature of the initial bifurcation has important implications regarding the utility of the destabilization mechanism for practical applications. In applications such as LISA where the desired final state is a pillar from the mask to the substrate, it is preferable for the initial bifurcation to be subcritical, so that there is no nearby equilibrium state as the potential is increased. In addition, it is also necessary to ensure that there are no further bifurcations due to nonlinear interactions after the initial bifurcation, to ensure that the final spacing is set by the most unstable mode in the linear analysis. For other applications such as the conveying of fluids by generating instabilities, it is essential that the initial bifurcation should be supercritical, since it is necessary to generate a perturbation of controlled amplitude and wave speed. In addition, it is of interest to examine the effect of fluid velocity on the bifurcation, in order to examine whether fluid flow can be used to change the nature of the bifurcation and to obtain more control over the perturbed wave forms generated. A comprehensive study is carried out using linear and weakly nonlinear analyses for small amplitude perturbations, thin-film analysis to obtain the time evolution of long-wave disturbances, and boundary element analysis to determine the evolution of perturbations of finite wavelength. The effect of a linear shear flow on the stability of the interface is also examined. Previous studies in the zero Reynolds number limit have been restricted to the linear stability analysis of perfect and leaky dielectrics, though nonlinear studies have been carried out in the high Reynolds number limit for an inviscid flow. Thus, this analysis provides a complete picture of the linear and nonlinear stabilities of the interface both in the presence and absence of flow. Each type of analysis has its advantages and disadvantages. The linear stability analysis is the easiest to carry out, but it provides only the initial growth rate of the perturbations, and does not provide any information about the nonlinear evolution of perturbations. The weakly nonlinear analysis is more complicated since it incorporates terms that are one order higher in the amplitude expansion in comparison to the linear analysis. However, the additional information gained is that the nature of the initial bifurcation 共subcritical or supercritical兲 can be predicted, and in case the bifurcation is supercritical, the equilibrium amplitude for the perturbations can also be predicted. The assumption made in the long-wavelength analysis is that the wavelength of perturbations is large compared to the film thickness, so that an asymptotic expansion in the ratio of the film thickness and the wavelength of perturbations can be used. The advantage here is that a simplified set of nonlinear equations is obtained which depends on time and one spatial coordinate, but the analysis cannot be used when the wavelength is of the same magnitude as the film thickness. The boundary element method is used when the film thickness is comparable to the wavelength of perturbations. The boundary element method is applicable to zero Reynolds number flows, where the velocity field in a domain is completely specified if the forces acting on the boundaries are known. Though this is a numerical solution technique for partial dif-

FIG. 1. Configuration and coordinate systems for analyzing the electrohydrodynamic instability of the interface between two fluids.

ferential equations by inversion to integral equations, the advantage of this method is that it is not necessary to specify the velocity and pressure fields throughout the domain, but it is sufficient to specify these over the boundaries of the domain, thus reducing a three-dimensional problem to a twodimensional one. However, since this is a numerical technique, it is more computationally intensive than the weakly nonlinear analysis or the thin-film analysis. Each technique is used in the domain where it is applicable, in order to obtain a comprehensive picture of the stability characteristics and the nonlinear evolution of perturbations. The results for perfect dielectrics presented in this work are exact, although for leaky dielectrics the present analysis is restricted to instantaneous charge relaxation. The article is organized as follows. The problem configuration is discussed in Sec. II, and the various nondimensional quantities used are defined. The methods of analysis are discussed in Sec. III. The results for both perfect and leaky dielectrics are discussed in Sec. IV and the conclusions are given in Sec. V.

II. PROBLEM DEFINITION

The system consists of two perfect or leaky dielectric fluids, A and B, of thickness R and RH with conductivities ␴A and ␴B and dielectric constants ⑀A and ⑀B 共Fig. 1兲. Without loss of generality, the top and bottom plate velocities are chosen so that the mean velocity at the interface is equal to zero. The governing equations for the electric fields within the dielectrics are ⵜ · EA = ⵜ · EB = 0,

共1兲

where E = − ⵜ ␾ is the electric field, and ␾ is the electrical potential. The electric potential at the boundaries y = R and y = −RH are specified as V and 0, respectively. The boundary condition for the current continuity normal to the interface between the two fluids for a leaky dielectric is

␴AEA · n = ␴BEB · n,

共2兲

where n is the unit normal to the interface. For a perfect dielectric, the conductivities of the two fluids in Eq. 共2兲 are replaced by the respective dielectric constants,

⑀AEA · n = ⑀BEB · n.

共3兲

The condition for the continuity of the electric field tangential to the interface for both perfect and leaky dielectrics is

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共4兲

EA · t = EB · t,

where t is the unit tangent to the interface. The governing equations for fluid motion are the Stokes equations for an incompressible fluid in the limit of zero Reynolds number, which include the divergence of the Maxwell stress due to the electric field. ⵜ · u = 0,

共5兲

ⵜ · ␶ + ⵜ · T = 0,

共6兲

where u is the velocity field, ␶ = −pI + ␮关ⵜu + 共ⵜu兲T兴 is the stress tensor, and T, the Maxwell stress tensor, is given by ⑀⑀0关EE− 共1 / 2兲E2I兴, where I is the identity tensor, ⑀0 is the permittivity of free space, and ⑀ is the dielectric constant. The divergence of the Maxwell stress is zero for dielectrics in the absence of free charges, and so there is no electrical stress in the momentum conservation equation for the fluids. However, the Maxwell stress tensor does provide a contribution to the stress boundary conditions at the interface between the two fluids. The velocity continuity condition, which states that the normal and the tangential components of the fluid velocity are equal at the surface, 共7兲

uA = uB = uI ,

where uI is the velocity vector at the interface. In the linear and thin-film analyses, it is also necessary to relate the normal velocity at the interface to the interface displacement from the flat state. When the displacement of the interface is small compared to the wavelength of the fluctuations, the normal velocity is related to the height by the kinematic condition, time derivative of the interface position, uIy =

Dh , Dt

共8兲

where 共D / Dt兲 is the substantial derivative defined as 共D / Dt兲 ⬅ 共⳵ / ⳵t兲 + uIx共⳵ / ⳵x兲, and x and y are the directions tangential and normal to the flat interface, as shown in Fig. 1, and uIx is the x component of the velocity at the interface. The stress balance equation at the interface can be written in the following vector form: 共␶A − ␶B兲 · n = ⌫␬n − T · n,

共9兲

where ⌫ is the surface tension of the interface and ␬ is the mean curvature, ␶A and ␶B are second-order fluid stress tensors, and T is the second-order Maxwell stress tensor. Equation 共9兲, when dotted with the local unit normal, gives the normal stress balance condition, and when dotted with a unit tangent to the surface, gives the tangential stress balance condition. Using the boundary conditions 共2兲–共4兲, the normal force due to the electric field for leaky dielectrics is T · n = ⑀0

冋

1 2 2 2 兲 − 共⑀AEAt − ⑀BEBt 兲兴n 关共⑀AE2n − ⑀BEBn 2

册

+ 共⑀AEAnEAt − ⑀BEBnEBt兲t ,

共10兲

where EAt,EBt and EAn,EBn are the components of the electric

fields tangential and normal to the surface, respectively, at the interface in fluids A and B, respectively. For perfect dielectrics, it is easily verified that ⑀AEAnEAt = ⑀BEBnEBt, and so the force at the interface acts in the normal direction, T · n = ⑀0

再

冎

1 2 2 2 2 关共⑀AEAn − ⑀BEBn 兲 − 共⑀AEAt − ⑀BEBt 兲兴n . 2

共11兲

The dimensional parameters affecting the fluid flow are the channel width R, the viscosities ␮ of the fluids 共which are assumed to be equal for simplicity兲, and the interfacial tension ⌫. The results for the evolution of the interface are presented in terms of nondimensional variables, which are obtained by scaling the velocity by 共⌫ / ␮兲, distances by R, time by 共R␮ / ⌫兲, and the pressure by 共⌫ / R兲. A balance between the Maxwell stresses and the viscous stresses indicates that the potential scales as 共⌫R / ⑀0兲1/2. Using these scalings, the Reynolds number for the fluid flow is Re= R⌫␳ / ␮,2 and inertial effects can be neglected when Re is small. It is of interest to examine the magnitude of the critical potential and the Reynolds number for systems of practical interest. Detailed calculations of the potential required to destabilize the interface between a polymeric film and air have been provided by Pease and Russel.4,5 It is known that the critical potential is independent of the fluid viscosity, though the growth rate of the perturbations depends on the fluid viscosity. For the present study, it is also of interest to determine the magnitude of the fluid velocity required for interference between flow and the electrohydrodynamic instability, and so we provide estimates of the critical potential and the fluid velocity for simple and polymeric liquids. Most fluids, simple and polymeric, have a density of the order of 103 kg/ m3, and the interfacial tension is in the range of 10−1–10−2 N / m. The viscosity varies from about 10−2 kg/ m / s for water to about 10 kg/m/s for polymers above the glass transition temperature. The critical potential required for destabilizing the interface, which is proportional to 共⌫R / ⑀0兲1/2, is on the order of 10–100 V for gaps with a thickness of 1 ␮m in the present case. This order of magnitude is in broad agreement with the critical potentials evaluated by Pease and Russel4,5 for polymeric films. A source of concern at such high potential gradients of 10 MV/m is the dielectric breakdown in the fluid. Though the dielectric breakdown potential for air is often quoted at 3 MV/m, Pease and Russel5 report that the dielectric breakdown for thicknesses less than 1 ␮m occurs by a mechanism called the Fowler-Nordheim emission which has much higher breakdown potentials, and cite literature results to indicate that the breakdown occurs at about 30 MV/m. One of the objectives of the present analysis is to examine the effect of shear in the fluid on the stability characteristics. The appropriate scale for the fluid velocity field does depend on the viscosity 共unlike the critical potential, which is independent of viscosity兲, and the velocity required to affect the stability characteristics of the flow scales as 共⌫ / ␮兲. This velocity is large 共about 0.1 m/s兲 for low viscosity fluids such as water, so it is difficult to alter the stability characteristics of the interface using flow in this case. However, the velocity required is quite small, 10−4 m / s, for polymeric liquids with a viscosity of

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10 N s / m2, and so changing the stability characteristics by fluid flow should be quite feasible in this case. The Reynolds number for these parameter values varies from about 10−2 for water to about 10−8 for polymer layers, and therefore inertia is not important in micron scale systems, and a zero Reynolds number analysis is appropriate.

and lower become unstable and generate higher harmonics due to nonlinear interactions. A dynamical variable 共velocity or potential兲 ␺共x , y , t兲 is expressed as an expansion in the harmonic series as well as the amplitude of the perturbations ⬁

␺ 共x,y,t兲 = 兺

⬁

兺关A1共␶兲兴n关E␣␾˜ 共␣,n兲共y兲 + E−␣␾˜ *共␣,n兲共y兲兴,

␣=0 n=␣

III. METHODS OF ANALYSIS

共15兲

A. Linear and weakly nonlinear stability analyses

In this section, we outline the linear and nonlinear stability analyses of the problem. In the base state, the interface is flat, and the velocity profile is a linear profile with equal gradients in the two fluids, since the viscosities of the two fluids are equal. The velocity profile in the flow 共x兲 direction is given by ¯uAx = ¯uBx =

关R共HUA + UB兲 + 共UA − UB兲y兴 , R共1 + H兲

共12兲

where UA and UB are top and bottom plate velocities, while the velocity in the y direction is identically equal to zero. The potential is linear, but the gradient of the potential is different in the two fluids since the dielectric constants 共for perfect dielectrics兲 and the conductivities 共for leaky dielectrics兲 are different. For leaky dielectrics, the mean potential is given by ¯␺ = 共HR + ␴Ry兲 V, A R共H + ␴R兲 ¯␺ = 共HR + y兲 V, B R共H + ␴R兲

A1共␶兲−1dtA1共␶兲 = sr共0兲 + A1共␶兲2sr共1兲 + ¯, 共13兲

where ␴R = 共␴B / ␴A兲 is the ratio of conductivities. For perfect dielectrics, the ratio of conductivities ⑀R is substituted for the ratio of conductivities in 13. Perturbations are imposed in the form of Fourier modes on the velocity and the potential fields ui共x,y,t兲 = ¯ui + ˜ui共y兲eikx+st,

where the integer superscript ␣ indicates the harmonic with wave number 共␣kc兲 and frequency 共␣␻兲, the integer superscript n indicates the order in powers of the amplitude of the perturbation, the superscript * is the complex conjugate, and the function E共x , t兲 is defined as E共x , t兲 = exp关i共kcx + ␻t兲兴. In ˜ 共0,0兲 refers to the potential and velocity the expansion 共15兲, ␾ ˜ 共1,1兲 are the perturbation fields for the mean flow, while ␾ fields imposed in the linear stability analysis. The results of the linear stability analysis are recovered by truncating the expansion 共15兲 at ␣ = 1,n = 1. To determine whether the linear instability is supercritical or subcritical, it is also necessary to consider equations for the perturbations at order ␣ = 0, n = 2, ␣ = 2, n = 2, and ␣ = 1, n = 3. The real amplitude A1共␶兲 is small compared to 1, since this is a small amplitude expansion, and ␶ is the “slow” time scale for the growth of perturbations near the neutral stability curve. The slow time scale ␶ arises for the following reason. In the vicinity of the transition point 共Vc , kc兲, the amplitude is governed by the Landau expansion

˜ 共y兲eikx+st , ␾共x,y,t兲 = ¯␺ + ␾ 共14兲

where k is the wave number and s is the complex growth rate. These are then substituted into the governing equations for momentum and the potential to obtain the eigenfunctions. The dispersion relation for the growth rate s is then obtained by substituting the eigenfunctions into the boundary conditions. The procedure is standard, and is not discussed in detail. The growth rate of perturbations predicted by the linear analysis is altered by the nonlinear terms in the boundary conditions which arise due to the Taylor expansion of the boundary conditions about the unperturbed state, as well as due to the variation of the surface normal along the perturbed interface. The effect of nonlinearities on the growth of these perturbations is analyzed in the weakly nonlinear analysis. In the weakly nonlinear theory, the value of the potential is assumed to be slightly higher than the potential on the neutral stability curves, Vc, in the linear theory. When V is slightly higher than Vc, perturbations with wave number kc

共16兲

where sr共0兲 is the real part of the linear growth rate s共0兲, and sr共1兲 is the real part of the first Landau coefficient s共1兲. Near the neutral curve, sr共0兲 can be written as sr共0兲 = 兩关dsr共0兲 / dV兴兩V=Vc共V − Vc兲. For 兩共V − Vc兲 / Vc兩 Ⰶ 1 near the neutral stability curve, the leading-order growth rate sr共0兲 in Eq. 共16兲 is small compared to 1, and we assume 兩关dsr共0兲 / dV兴兩V=Vc共V − Vc兲 ⬃ ␦2, where ␦ is the small parameter in the expansion. Since the growth of perturbations is driven by the linear instability for sr共0兲 ⬎ 0, it is expected that all terms in Eq. 共16兲 are of the same magnitude as sr共0兲. The slow time scale ␶ = 共t / ␦2兲 is obtained by balancing the term on the left with the first term on the right of Eq. 共16兲. If sr共1兲 is O共1兲, the magnitude of the amplitude A1共␶兲 ⬃ ␦ is obtained by balancing the first and second terms on the second term in the right side of 共16兲. For definiteness, let 共V − Vc兲 = V⬘␦2, and A1共␶兲 = ␦A共␶兲, where V⬘ and A共␶兲 are O共1兲 in an asymptotic expansion in ␦. The Landau equation, 共16兲, then becomes A−1d␶A = V⬘

冋册 dsr共0兲 dV

+ sr共1兲A2 .

共17兲

V=Vc

This is the “scaled” version of the Landau equation in the vicinity of the neutral curve. The objective of the weakly nonlinear analysis to determine sr共1兲, which determines whether the instability is subcritical or supercritical. The bifurcation is supercritical for sr共1兲 ⬍ 0, and the linearly unstable perturbations are stabilized by nonlinear effects. The bifurcation is subcritical for sr共1兲 ⬎ 0, and perturbations

084104-7



which are stable in the linear analysis become unstable when the amplitude of perturbations increases beyond 兵兩关V⬘ / sr共1兲兴关dsr共0兲 / dV兴兩V=Vc其1/2. The details of the weakly nonlinear analysis are given in Appendix A in Ref. 23. B. Thin-film analysis

The results of the linear stability analysis for a perfect dielectric, discussed in Sec. IV, indicate that the system shows an instability at all potentials in the small k limit, and it is useful to carry out the thin-film analysis of this system. The nondimensionalization appropriate for the thin-film analysis is different from that for the linear stability analysis when the wavelength is of the same magnitude as the film thickness, and so the appropriate scaled variables for the thin-film analysis are denoted by the superscript †. In the long-wave limit, the wavelength of perturbations, L, is considered to be large compared to the layer thickness R, and we define a small parameter ␧ = R / L. The scalings in the thinfilm analysis are discussed briefly, since they are standard in literature. The scaled coordinates are defined as x† = 共x / L兲 and y † = 共y / R兲. The scaled velocity u†x is defined as u†x = 共ux␮ / ⌫兲. From the mass conservation equation 共5兲, the appropriate scaling for the y velocity is u†y = ␧−1共uy␮ / ⌫兲, so that the mass conservation equation reduces to

⳵ u†x ⳵ u†y + = 0. ⳵ x† ⳵ y †

共18兲

The appropriate scaling for the pressure is p† = 共␧pR / ⌫兲, and the scaled momentum equations are

冋

册

⳵2u†x ⳵2u†x ⳵ p† 2 + ␧ = 0, 2 + 2 ⳵ x† ⳵ x† ⳵ y †

−

− ␧−1

冋

共19兲

册

⳵2u†y ⳵2u†y ⳵ p† 3 + ␧ 2 = 0. 2 † + ␧ ⳵y ⳵ y† ⳵ x†

共20兲

The appropriate scaled electrical potential is ␺† = 关␾ / 共⌫R / ⑀0兲1/2兴, and the scaled equation for the velocity potential is ␧2

⳵ 2␺ † ⳵x

†2

+

⳵ 2␺ † ⳵ y†

2

共21兲

= 0.

The leading approximations for Eqs. 共19兲–共21兲 in the small ␧ limit are −

⳵ p† ⳵2u†x + = 0, ⳵ x † ⳵ y †2

⳵ p† = 0, ⳵ y†

⳵ 2␺ † ⳵ y†

2

= 0.

共22兲

The boundary conditions for the potential are 兩␺A† 兩y†=1 = V† ,

共23兲

兩␺B† 兩y†=−H = 0,

共24兲

兩␺A† 兩y†=h† = 兩␺B† 兩y†=h† ,

共25兲

冏冏 ⳵ ␺A† ⳵ y†

y †=h†

= ␴R

冏冏 ⳵ ␺B† ⳵ y†

. y †=h†

共26兲

The boundary conditions for the velocity field are expressed in terms of the interface height function h†共x† , t†兲, which provides the height of the interface at position x† and time t†, and which is set equal to zero in the base state. Since there is no penetration of the fluid across the interface, the normal fluid velocity at the interface is the convected derivative of the interface height by 8. † ⳵ h† † ⳵h + u = u†Iy , Ix ⳵ t† ⳵ x†

共27兲

† and u†Iy are the tangential and normal velocities at where uIx the interface in the thin-film approximation, and only the leading terms in the ␧ expansion are retained. The no-slip conditions for the velocity at the interface y † = h† are † † uAx = uBx ,

共28兲

u†Ay = u†By .

共29兲

The tangential and normal stress balance conditions across the interface for a leaky dielectric reduce to † † ⳵ ␺A† ⳵ ␺A† ⳵ uBx ⳵ ␺B† ⳵ ␺B† ⳵ uAx , † † + ⑀A † = † + ⑀B ⳵y ⳵x ⳵ y † ⳵ x† ⳵y ⳵y

− pA† +

冉冊

⑀A ⳵ ␺A† 2 ⳵ y†

2

= − pB† −

冉冊

⳵2h† ⑀B ⳵ ␺B† + ⳵ x†2 2 ⳵ y †

共30兲 2

,

共31兲

† † , u†Ay , pA† 兲 and 共uBx , u†By , pB† 兲 are the velocities and where 共uAx pressure at the interface in fluids A and B, respectively. The scaled tangential velocities at the top and the bottom plate are UA† and UB† , respectively, and the scaled potentials are V† and 0, respectively. The solution procedure is as follows. The equations for the velocity potential 共21兲 are first solved explicitly subject to the boundary conditions 共23兲–共26兲 to obtain explicit expressions for the velocity potentials. The velocities uAx and uBx are determined in terms of the pressure gradients using the no-slip conditions at the surfaces y = R and y = −HR, and the velocity continuity condition at the interfaces 共28兲 and 共29兲. The normal velocities uAy and uBy are determined in terms of the interfacial velocity uIy using the mass conservation equation. These are inserted into the stress conditions 共30兲 and 共31兲 to determine the pressure gradients in terms of the interface velocity uIy and the height perturbation h†. The expressions for the pressure gradients are then inserted back into those for the velocities, and these are substituted into the kinematic equation 共27兲 to obtain an equation for the height evolution. The details of the calculation are provided in Appendix B in Ref. 23.

C. Boundary integral method

The evolution of perturbations with wavelength comparable to the layer thickness can be best carried out by the boundary integral method, which is discussed briefly here because the details are provided by Pozrikidis.24 The boundary integral method is appropriate in the limit of zero Reynolds number because it is necessary to solve the Stokes equations for the fluid and the Laplace equation for the elec-

084104-8



tric potential. In this section, indicial notation is used to represent the components of vectors or tensors for convenience, and a repeated index denotes a dot product. The free-space Green’s function for the electric potential, which satisfies the Laplace equation ⵜ2␾ = 0, is G=

冉冊

1 1 log . 2␲ r

+

ui共x兲 =

冕

dx⬘Jij共x − x⬘兲F j共x⬘兲,

共33兲

p共x兲 =

冕

dx⬘Ki共x − x⬘兲Fi共x⬘兲,

共34兲

where Fi共x⬘兲 is the applied force density at position x⬘, and the Oseen tensors Jij and Ki are

冋冉冊册

Jij共x,x⬘兲 =

1 1 x ix j ␦ij log + 2 , 4␲␮ r r

共35兲

Ki共x,x⬘兲 =

xi , 2␲r2

共36兲

where r = 兩xi − xi⬘兩. In periodic systems, the free-space Green’s functions are replaced by periodic Green’s functions which are

冕

兺 G共X,Xn兲, n=−N

共37兲

where X共x , y兲 is the position of the observation point and Xn共x0 + nl , y 0兲 is the position of a periodic array of singular points, where n = 0 represents the principal singular point and l is the wavelength. The summation can be easily carried out using the series expression

⳵G 共x⬘,x0兲␾D共x⬘兲 ⳵n

⳵ ␾R 共x⬘,x0兲G P共x⬘兲 − ⳵n

dSA共x⬘兲

dSB共x⬘兲

⳵␾ ⳵n

⳵␾ ⳵n

⫻共x⬘,x0兲G P共x⬘兲 ,

共39兲

where SI, SA, and SB are the interfacial, top plate, and bottom plate areas, respectively, 共⳵ / ⳵n兲 represents the normal derivative, ␾D is the difference potential defined as ␾ − ␾R, where ␾ is the total potential, and ␾R is the reference potential which is assumed to be a linear function of the y coordinate between the two plates, given by

␾R =

HV Vy + , 1 + H 共1 + H兲R

共40兲

where G P is the periodic Green’s function, and the factor j is 共1 + ␴R兲 / 2 for the interface and 0 for the top and bottom plates. It should be noted that the reference potential is different from the mean potential, which is discontinuous at the interface, and it is necessary to identify a reference electric field which is continuous in order to avoid some singular integrals. The presence of singular integrals makes it difficult to analyze infinite or semiinfinite systems using this method. The governing equations are obtained by moving the singular point on to the interface, to the top plate, and the bottom plate. To obtain the governing equation for the normal component of the electric field E = −共⳵␾ / ⳵n兲, the potential equation is recast as

n=N

GN =

dSI共x⬘兲

⫻共x⬘,x0兲G P共x⬘兲 − ␴R

共32兲

In the zero Reynolds number limit, the velocity and pressure fields at a position x in the fluid can be expressed in terms of the applied force density as

冋冉冊冉冊册冋冕冉冊册冋冕冉冊册

j␾D共x兲 = 共1 − ␴R兲

␾D共x兲 = − 共1 − ␴R兲 − −

冋冕冋冕

冋冕

dSA共x⬘兲 dSB共x⬘兲

dSI共x⬘兲

冉冊冉冊

冉冊

⳵␾ 共x⬘兲G P共x0,x⬘兲 ⳵n

册册

册

⳵␾ 共x⬘兲G P共x0,x⬘兲 ⳵n

⳵␾ 共x⬘兲G P共x,x⬘兲 , ⳵n

共41兲

and the gradient of the above equation gives n=N

1

1

兺 log共rn兲 = 2 log关cosh共y − y0兲 − cos共x − x0兲兴 + 2 log共2兲 n=−N

ED共x兲 = + 共1 − ␴R兲

共38兲 for N → ⬁. The Green’s function for the velocity field which is zero at the two walls was given by Pozrikidis.24 This allows calculation of velocities by integrating only over the fluid interface, without the necessity of carrying out integrations over the boundaries. This is because the two integrals, one containing the product of the stress and the Green’s function, and the other containing the product of the velocity and stresslet, are zero at the walls 关Eq. 共44兲兴. For a singular point x inside the domain of the upper fluid, the boundary integral equation for the electric potential is

+ +

冋冕冋冕

冋冕

dSI共x⬘兲En ⵜ G P

dSA共x⬘兲EDn ⵜ G P

册册

dSB共x⬘兲EDn ⵜ G P ,

册

共42兲

where En = −共⳵␾ / ⳵n兲, and EDn is the difference between the normal components of the electric field and the reference electric field obtained by taking the normal derivative of 共40兲. The sign change after taking the spatial derivative is because the derivative is taken with respect to x. The normal electric field equation is then obtained by taking the inner product with the unit normal.

084104-9



The boundary integrals are obtained by moving the singular point on to the boundary, and using the jump property of the normal derivative of the Green’s function, 兩

冕

dS共x⬘兲En共x兲 ⵜ G共x,x⬘兲兩A =兩

冕

dS共x⬘兲En共x,x⬘兲 ⵜ G共x,x⬘兲兩B + 2␲En共x兲,

共43兲

where x on the left-hand side of Eq. 共43兲 is inside the fluid A, and x, on the right-hand side of the equation is on the boundary. The singular point is then moved to the interface of the two fluids to obtain the governing equation for normal electric field at the interface. The equation for the electric potential 共39兲 is solved for the values of potential at the interface and the electric fields at the top and bottom plates. The normal and tangential electric fields at the interface are obtained by solving Eq. 共42兲, and are inserted into the equation for the Maxwell stress in the momentum equation. u j共x兲 − uRj 共x兲 = −

1 2␲

冕

dS共x⬘兲

冋

册

1 ⌬f i共x⬘兲JijP共x,x⬘兲 . 2␮

共44兲

Here, ⌬f i is the force density which has contributions due the Maxwell stress tensor and the surface tension, as described earlier 关Eqs. 共10兲 and 共11兲兴, and uRj 共x兲 is the reference velocity which has no y component, and whose component in the x direction and is given by uRx =

HUA + UB 共UA − UB兲y + . 1+H 共1 + H兲R

共45兲

The boundary integrals were solved by dividing the arc length into a number of elements n − 1, bounded by n marker points, and the physical variables were parametrized using the arc-length parametrization. Cubic splines were used to fit the variables as functions of arc length between two marker points,25 and periodic boundary conditions were used for the function values and the derivatives. The arc-length parametrization using cubic splines is valid if the interface does not turn back so that the y coordinate of the interface is a singlevalued function of the x coordinate. For problems where the interface bends back upon itself, so that the height function is a multiple-valued function of the horizontal distance, it is appropriate to use the method of circular arcs.24 In the circular arc method, the boundary is approximated by circular arcs passing through successive trios of marker points. The center of this circular arc is identified, and the curvature is the inverse of the radius of this circular arc, and is constant over the arc passing through these three points. In the present case, although the interface does not bend back upon itself, the circular arc method was used to verify the arc-length parametrization results, and to reproduce the results of Ref. 26 for two fluids sheared past each other. The circular arc method was also used for large curvatures, where the arclength parametrization did not give accurate results. All nonsingular integrations were carried out by a 14-point Gaussian quadrature method. The singular integrals were evaluated by subtracting the free-space Green’s function from the periodic Green’s function, and then integrating them analytically by

assuming linear elements. The periodic Green’s functions involve a summation running from N = 0 to N = −⬁. Numerically, it is found that the series for all the elements of the Green’s-function tensors converged very well, and the difference less than 0.1% when N is increased from 7 to 8. The value of N is fixed between 8 and 10 in the numerical calculations. The computational procedure is as follows. The initial interface position corresponding to a perturbation with a particular amplitude and wave number is first specified. The interface is discretized into elements with marker points. The boundary integral equations for the electric potential and electric field were then solved to obtain the normal and tangential electric fields at all marker points. These were spline fitted to get the values at all points along the interface. The boundary integral equation for the velocity of the marker points was then solved, with the electrical stresses calculated from the values of the electric fields obtained in the previous step. The marker points were then advanced by a Euler time integration and new interface position reached. The marker points tend to accumulate at the regions of high curvatures and at longer times and so the marker points were redistributed after each iteration. The procedure used for situations where the interface approaches the top or bottom surface is as follows. As the interface approaches the top or bottom surface, it might be expected that the flow in the gap approaches the lubrication flow limit, where the force is inversely proportional to the distance between the interface and the flat surface. In this case, it is necessary to redistribute the marker points along the interface in such a manner as to ensure that the distance between the marker points is smaller than the gap width, to resolve the forces accurately on this scale. In the arc-length parametrization technique, it was verified that the distance between the marker points was equal to half the distance between the interface and the solid surface, and the simulations were stopped when the distance between the interface and the solid surface was less than 5 ⫻ 10−3 times the channel width. It should be noted that as the interface approaches the solid surfaces, additional forces such as the van der Waals force become important, and the continuum approximation may not be valid. It is necessary to realistically account for these additional forces in order to obtain accurate results for the interaction between the interface and the top or bottom surfaces. This is beyond the scope of the boundary element technique, which is most efficient for solving the Laplace equation, and so we do not study this issue further. The code was verified for known results for a flat interface, and the results were found to be accurate to eight significant digits. The results for the electric potential for the sinusoidal interface were verified with the results obtained from MATLAB 6.3 PDE solver, which can solve the Laplace operators over any domain for electrostatic problem using the finite element method, and the agreement between the two solvers for high curvature interfaces was found to be accurate to eight significant digits. The calculation of the velocity field was verified using the results of Pozrikidis26 for the shear flow of two fluids past each other. Both the electric potential and the velocity fields calculated using the

084104-10



FIG. 3. The dimensionless parameter 关Vc共⑀0 / ⌫R兲1/2兴 as a function 共KR兲 for perfect and leaky dielectrics and for fluid layers of equal thickness 共H = 1兲. Here, Vc is the dimensional critical top plate potential, and k is the dimensional wave number. 共—兲 perfect dielectrics with ⑀B = 1.0 and ⑀R = 0.2; 共----兲 leaky dielectrics with ⑀R = 0.2, ⑀B = 1, and ␴R = 2; and 共¯兲 leaky dielectrics with ⑀R = 0.2, ⑀B = 1, and ␴R = 5.

Analytical results can be obtained for the growth rate in the limit k → 0. The solution for the growth rate is s=− FIG. 2. The product of the dimensional growth rate s and the characteristic time scale 共␮R / ⌫兲 as a function of 共kR兲 for 共a兲 perfect dielectrics with ⑀A = 1 and ⑀R = 5, and 共b兲 leaky dielectrics with ⑀A = 1, ⑀R = 5, and ␴R = 0.2. The heights of the two fluid are equal, H = 1.0, the top plate potential is given by V = 1.5共⌫R / ⑀0兲1/2, where V is the dimensional potential, and there is no fluid flow.

−

IV. RESULTS

The results of the linear and weakly nonlinear stability analyses are first presented together for perfect and leaky dielectrics, since analytical solutions can be obtained for the growth rates in the small wave-number limit. After this, the results the thin-film analysis and the boundary element technique are presented in different subsections for perfect and leaky dielectrics. The results of the linear stability analysis show that perturbations are unstable when the wave number is below a transition wave number for any nonzero potential. Figure 2 shows the growth rate for both perfect and leaky dielectrics for a specific set of fluid parameters and imposed electric field. The system is unstable in the low wave-number limit, and the growth rate reaches a maximum value before decreasing at a high wave number. In Fig. 3, the variation of the potential for transition from stable to unstable modes is shown as a function of the wave number. This figure shows that Vc ⬀ k in the k → 0 limit, while Vc ⬀ k1/2 for k Ⰷ 1 for both leaky and perfect dielectrics. The critical potential is lower for leaky dielectrics than for perfect dielectrics.

⌫ 共kR兲4 24R␮

共46兲

for leaky dielectrics, and s=

boundary element method were found to compare very well with the linear stability results in the small amplitude limit.

⑀A共1 − ␴R兲共⑀R − ␴R2 兲⑀0V2 2 i共UA + UB兲 k k+ 2 24␮共1 + ␴R兲3

共⑀A⑀R兲共1 − ⑀R兲2⑀0V2 2 i共UA + UB兲 ⌫ 共kR兲4 k − k+ 3 2 24␮共1 + ⑀R兲 24␮R 共47兲

for perfect dielectrics. Equation 共47兲 shows that perturbations are always unstable in the k → 0 limit for perfect dielectrics, but Eq. 共46兲 indicates that the system is stable for leaky dielectrics in the parameter regimes 共␴R ⬍ 1 ; ␴R2 ⬎ ⑀R兲 and 共␴R ⬎ 1 ; ␴R2 ⬍ ⑀R兲. The wave number of the fastest growing mode, kc, is given by kc =

冑

⑀A共1 − ␴R兲共⑀R − ␴R2 兲⑀0V2 2⌫R3共1 + ␴R兲3

共48兲

for leaky dielectrics, and by kc =

冑

共⑀A⑀R兲共1 − ⑀R兲2⑀0V2 2⌫R3共1 + ⑀R兲3

共49兲

for perfect dielectrics. The analytical solutions in the low wave-number limit, Eqs. 共46兲 and 共47兲, indicate that the destabilizing potential Vc should be proportional to the wave number in the limit k Ⰶ 1, as observed in Fig. 3. The first term on the right side of 共46兲 and 共47兲 represents the propagation of disturbances unamplified due to mean convection. The second and third terms on the right side are due to the normal stress due to the electrical force and the surfacetension force, respectively. The transition at larger wave

084104-11



TABLE I. Variation of the Landau coefficient sr共1兲, scaled by 共⌫ / R␮兲, with scaled wave number 共kR兲 for a perfect dielectric with ⑀R = 1. 共kR兲 0.1 0.5 1.0 5.0 10.0

FIG. 4. The discontinuity in the mean electric potential at the interface, which is in opposite directions at the crest and trough of the wave as shown by the arrows 共a兲, and the compensating discontinuity in the perturbation to the electric field in order to satisfy the potential continuity condition at the interface 共b兲.

numbers is influenced by the lateral variations in the potential and fluid, and the viscous dissipation due to fluid flow acts as an additional stabilizing mechanism. The physical reason for the electrohydrodynamic instability, and its dependence on the ratio of dielectric constants and conductivities, is as follows. Consider an interface between two perfect dielectrics with different dielectric constants, as shown in Fig. 4. We provide the reasoning for a leaky dielectric where the current continuity condition is given by 共2兲; the reasoning for a perfect dielectric with current continuity condition given by 共3兲 is obtained by substituting ⑀R for ␴R. The slope of the electrical potential on the two sides of the interface is different, due to the different dielectric constants, but the potential is continuous across the interface. When the interface is perturbed, the mean potential on the two sides is different at the perturbed surface, and the ˜ , which is difference in the mean potential is 共¯␺A − ¯␺B兲h ¯␺ 关共␴ − 1兲 / ␴ 兴h ˜ for leaky dielectrics from 共2兲. Note that this A R R difference is in opposite directions at the crest and the trough of the perturbation. In order to maintain the continuity of the potential and gradient, there has to be a perturbation to the potential which is of equal magnitude and opposite in direction to the mean potential, as shown by the arrows in Fig. 4共a兲. The gradient of the perturbation to the potential at the surface satisfies the field continuity condition 共4兲, which is ⑀Aik˜␺A = ⑀Aik˜␺B, so that the amplitudes of the potential on either side of the interface are related as ˜␺B = 共˜␺A / ⑀R兲. Since the discontinuity in the mean electric field at the interface is proportional to ¯␺A关共␴R − 1兲 / ␴R兴, that in the perturbation to the potential is proportional to ¯␺A共1 − ␴R兲 / ␴R. The difference in the normal Maxwell stress across the surface is given by 关⑀A共dz¯␺A兲共dz˜␺A兲 − ⑀B共dz¯␺B兲共dz˜␺B兲兴, which can be rewritten, from the current continuity condition 共2兲, as ⑀A共dz¯␺A兲共dz˜␺A兲共1 − ␴R2 / ⑀R兲. Since the perturbation to the elec-

⑀R = 0.5

⑀R = 0.2 −6

3.594⫻ 10 3.252⫻ 10−3 0.048 13 ⫺9.31 ⫺86.11

⑀R = 0.1 −5

1.39⫻ 10

−3

9.16⫻ 10 0.1353 9.3391 30.55

2.094⫻ 10−5 13.13⫻ 10−3 0.1941 21.925 109.297

tric field is related to the mean electric field by ˜␺A ⬃ ¯␺A共1 − ␴R兲 / ␴R, the difference in the Maxwell stress is proportional to ⑀A共dz¯␺A兲2共1 − ␴R2 / ⑀R兲共1 − ␴R兲. This stress is positive for 共␴R ⬎ 1 ; ␴R2 ⬎ ⑀R兲 and 共␴R ⬍ 1 ; ␴R2 ⬍ ⑀R兲, and the normal Maxwell stress acts in the direction opposite to the surface tension, thereby destabilizing perturbations for these parameter values. The stress is negative for 共␴R ⬍ 1 ; ␴R2 ⬎ ⑀R兲 and 共␴R ⬎ 1 ; ␴R2 ⬍ ⑀R兲, resulting in a Maxwell stress contribution in the same direction as surface tension, stabilizing perturbations. The result for perfect dielectrics is obtained by setting ␴R = ⑀R, and perturbations are always unstable in this case. The physical reason for the scaling of the growth rate with wave number in the low and high wave-number regimes is discussed in detail in Appendix C in Ref. 23. In the linear analysis, the imposition of fluid flow results in an additional imaginary contribution to the growth rate, resulting in the translation of the perturbations by a constant speed, but does not alter the magnitude of the growth rate. The nonlinear analysis carried out a little later indicates that the correction to the pressure generated by nonlinear interactions acts in the same direction as surface tension, thereby stabilizing perturbations. For a perfect dielectric, the fluid velocity field is zero on the neutral stability curve, since the real and imaginary part of the growth rate pass through simultaneously. The weakly nonlinear analysis for this problem shows that the bifurcation is supercritical for certain parameter ranges for wavelengths comparable to the channel thickness, as indicated by a negative Landau constant in Table I, and the supercritically stable state is characterized by zero velocity throughout the fluid. At higher ratios of dielectric constants for finite wave number and for all ratios of dielectric constants in the k → 0 limit, the Landau constants are positive, indicating a subcritical bifurcation. The nonlinear stabilization or destabilization is due to a two-mode interaction between the fundamental mode with wave number k and its harmonic with wave number 2k. This nonlinear interaction provides an additional contribution to the normal Maxwell stress at the interface which is always in the same direction as the linear contribution to the normal Maxwell stress in the long wave limit. For finite wavelengths, the nonlinear contribution to the normal Maxwell stress is in the same direction as the linear contribution for most parameter values, resulting in a subcritical bifurcation. There are, however, select parameter ranges where the nonlinear contribution is in the direction opposite to the linear contribution, resulting in a supercritical bifurcation. The parameter ranges can be obtained using a procedure similar to that for the

084104-12



TABLE II. Variation of Landau coefficient sr共1兲, scaled by 共⌫ / R␮兲, with scaled wave number 共kR兲 for a leaky dielectric with ⑀R = 1. 共kR兲 0.1 0.5 1.0 5.0 10.0

␴R = 0.5

␴R = 0.2 −6

−7.363⫻ 10 −2.138⫻ 10−3 ⫺0.028 83 ⫺37.131 ⫺308.33

␴R = 0.1 −6

1.304⫻ 10−5

−3

9.14⫻ 10−3 0.154

1.28⫻ 10 2.76⫻ 10 0.071 21.013 126.04

30.81 181.74

linear instability provided in the previous section. However, the algebra is very complicated since it is necessary to expand up to third order in the perturbation amplitudes and collect terms, as indicated in Appendix A in Ref. 23, and so we do not attempt this here. In the case of leaky dielectrics, the analysis shows that the system is supercritically stable for a larger range of ratios of dielectric constants and electrical conductivity than for perfect dielectrics, as indicated in Table II. The weakly nonlinear analysis indicates the presence of a circulatory flow at equilibrium due to the tangential stresses at the interface, so that the normal stress difference across the interface due to the electric field is balanced by the surface tension, and the shear stress generated by fluid flow is balanced by the difference in the electrical tangential stress across the interface. Although the weakly nonlinear theory can predict whether the bifurcation is subcritical or supercritical, the theory is valid only for small amplitudes as it involves an asymptotic expansion in the amplitude. The time evolution of the perturbations is studied using the thin-film analysis and the boundary element technique. In the thin-film analysis, the nonlinear governing equation for the height fluctuations was numerically solved using the forward time central space 共FTCS兲 finite difference method. The time evolution of an initial perturbation was then tracked till a steady state is reached or the interface reached the confining walls. The initial growth rate of perturbations from the thin-film analysis was compared with the growth rate from the linear stability analysis in the small wave-number limit, and the two were found to be identical.

FIG. 5. Comparison of numerical and linear theory results for the 共x / R兲 and 共y / R兲, where 共x , y兲 are the positions on the interface, as a function of time. The wavelength of the perturbation is k = 共␲ / 50R兲, the potential is V = 0.25共⌫R / ⑀0兲1/2, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 0.23共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 1.0 and ⑀B = 2.0. The thick lines show the results of the thin-film analysis, while the thin lines show the results of the linear stability analysis. The initial perturbation has amplitude 0.1 times the channel width at t = 0. The evolution of the interface is shown at times 关t = 0 . 5 ⫻ 106共R␮ / ⌫兲 , 10⫻ 106共R␮ / ⌫兲 , 15⫻ 106共R␮ / ⌫兲 , 20⫻ 106共R␮ / ⌫兲兴.

indicates that the system can admit traveling waves even in a reference frame moving with the mean velocity at the interface, and the additional wave speed of the traveling waves is due to nonlinear interactions. The effect of mean shear on the instability is shown in Fig. 7. The mean velocity at the interface is zero in all the cases studied 共UA = −UB = U , H = 1兲, and the linear stability theory predicts that the growth rate is real, and the wave speed is zero 关Eq. 共46兲兴. The numerical results indicate that the mean shear has a stabilizing effect on the instability, and the system admits traveling waves. The direction of the traveling waves depends upon the ratio of dielectric constants. Figure 8 shows the variation of the height of the interface after the subcritical bifurcation as a function of

A. Perfect dielectrics

Figure 5 shows the comparison between linear stability analysis and the numerical results of the thin-film analysis for the case of perfect dielectrics. The agreement between the theory and the numerical results is good even when the height fluctuation is as high as 0.2. Figure 6 shows the time evolution of an unstable mode, which grows till the interface touches the upper or the lower wall, depending upon the ratio of dielectric constants. The linear stability analysis in the presence of flow shows that the real part of the growth rate is not modified by the shear, and the system admits traveling waves with wave speed equal to the mean velocity at the interface 关Eqs. 共46兲 and 共47兲兴. This implies that the waves are stationary in the reference frame moving with the mean velocity of the interface. The weakly nonlinear analysis of this system, however,

FIG. 6. Time evolution of the 共x / R兲 and 共y / R兲 where 共x , y兲 are the interface coordinates, after a subcritical bifurcation obtained from the thin-film analysis for perfect dielectrics and for two fluids of equal thickness H = 1. The wavelength of the perturbation is k = 共␲ / 50R兲, the potential is V = 0.3共⌫R / ⑀0兲1/2, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 0.23共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 1.0 and ⑀B = 2.0. The initial perturbation has amplitude 0.1 times the channel width at t = 0. Circles, t = 0; solid line, t = 2 ⫻ 106共R␮ / ⌫兲; dotted line, t = 4 ⫻ 106共R␮ / ⌫兲; dot-dash line, t = 6 ⫻ 106共R␮ / ⌫兲; dashed line, t = 9 ⫻ 106共R␮ / ⌫兲; and ⫻, t = 1 ⫻ 107共R␮ / ⌫兲.

084104-13



FIG. 7. Effect of flow on the evolution of 共x / R兲 and 共y / R兲 where x and y are the dimensional interface coordinates, for an instability in a perfect dielectric using the thin-film approximation for H = 1. The wavelength of the perturbation is k = 共␲ / 50R兲, the potential is V = 0.12共⌫R / ⑀0兲1/2, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 0.103共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 1.0 and ⑀B = 5.0. The velocities of the top and bottom plates are equal in magnitude and opposite in direction, UA = −UB = U. The x and y positions of the interface are shown at different times, thick dashed line is the initial profile, and circle is the final profile. 共a兲 U = 0, 共b兲 U = 0.000 05共⌫ / ␮兲, 共c兲 U = 0.0001共⌫ / ␮兲, 共d兲 U = 0.0005共⌫ / ␮兲, and 共e兲 U = 0.001共⌫ / ␮兲. Thick solid line, t = 0; solid line, t = 5 ⫻ 106共␮R / ⌫兲; dotted line, t = 6.5⫻ 106共␮R / ⌫兲; dot-dash line, t = 8.5⫻ 106共␮R / ⌫兲, and dashed line, t = 1.5⫻ 107共␮R / ⌫兲.

the fluid velocity. It is found that fluid flow changes the nature of the bifurcation from subcritical to supercritical, and the equilibrium interface height decreases proportional to U−1, where U is the magnitude of the top plate velocity. Though there is good qualitative agreement between the thin film and the weakly nonlinear analysis, we find that there is a difference between the numerical values. We carried out a more detailed comparison of the interface stresses predicted

by the thin film and the weakly nonlinear analysis. From the comparison, we found that the coefficient of the third correction to the amplitude equation is large, so that this correction is of the same magnitude as the second correction even when the amplitude is about 10−2 times the channel thickness. Due to this, the contribution of the third and higher corrections was not small, and this resulted in a significant difference between the nonlinear analysis and the thin-film analysis.

084104-14


FIG. 8. Effect of flow on the evolution of 共h / R兲 where h is the dimensional height of the interface from its equilibrium position, as a function of 共t⌫ / ␮R兲, where t is the dimensional time, for different values of 共UA␮ / ⌫兲 = −共UB␮ / ⌫兲 = 共U␮ / ⌫兲, where UA and UB are the dimensional top and bottom plate velocities, obtained using the thin-film approximation for perfect and leaky dielectrics for H = 1. The initial bifurcation, which was subcritical in the absence of flow, has been altered to a supercritical bifurcation due to the imposed flow. The wavelength of the perturbation is k = 共␲ / 50R兲. The solid lines show the results of the thin-film analysis, and the dashed lines show the results of the weakly nonlinear analysis. The diamonds are the results for perfect dielectrics with ⑀A = 1 and ⑀B = 5 for which the potential is V = 0.12共⌫R / ⑀0兲1/2, and the potential for the bifurcation is Vc = 0.103共⌫R / ⑀0兲1/2. The triangles are the results for leaky dielectrics with dielectric constants ⑀A = 1.0 and ⑀B = 2.0, conductivity ratio ␴R = 0.5, and potential V = 0.124共⌫R / ⑀0兲1/2, for which the initial subcritical bifurcation in the absence of flow takes place at Vc = 0.12共⌫R / ⑀0兲1/2.

However, the scaling with the flow velocity was found to be the same even when the higher-order corrections to the amplitude equation were included. The analysis of perturbations with wavelength comparable to the film thickness is carried out using the boundary element method. The analysis could not be extended to very high wave numbers due to difficulty in the resolution of high curvatures associated with higher wave numbers 关共␭ / R兲 ⬍ 0.5兴, where ␭ is the wavelength. For 共␭ / R兲 = 2, the results for perfect dielectrics by the boundary integral method showed a supercritical state for ⑀R ⬃ O共2兲 for finite wave numbers. All higher ratios of dielectric constants showed a subcritical instability. This is in agreement with the results of weakly nonlinear analysis 共Table I兲. Figure 9 shows the time evolution of a small amplitude sinusoidal disturbance using the boundary integral method. The disturbance saturates to an equilibrium amplitude, which is assumed to be achieved when the absolute normal velocities for all the marker points is less than 10−4. Figure 10 shows the equilibrium interface position as the top plate potential is increased. It is found that the equilibrium amplitude increases with an increase in the applied potential. When the potential is further increased, the system undergoes further secondary subcritical bifurcation. For a configuration with ⑀R = 0.5, 共␭ / R兲 = 2, and Vc = 6.5共⌫R / ⑀0兲1/2, the supercritical state undergoes a subcritical bifurcation at V = 6.7共⌫R / ⑀0兲1/2. We denote the potential for the transition from stable to unstable modes in the linear analysis by Vc, and the potential at which the supercritical state undergoes further secondary subcritical bifurcation by Vc⬘. The numerical results indicate that there is a small parameter range of


FIG. 9. Time evolution of 共x / R兲 and 共y / R兲, where x and y are the dimensional coordinates on the interface, after a supercritical bifurcation obtained from the boundary element method for perfect dielectrics and for two fluids of equal thickness H = 1. The wavelength of the perturbation is k = 共␲ / R兲, the potential is V = 6.55共⌫R / ⑀0兲1/2, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 6.5共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 2.0 and ⑀B = 1.0. The initial perturbation has amplitude 0.1 times the channel width at t = 0. Thick solid line, t = 0; solid line t = 50共␮R / ⌫兲; dotted line, t = 100共␮R / ⌫兲; dot-dash line, t = 150共␮R / ⌫兲; and dashed line, t = 200共␮R / ⌫兲.

the ratios of dielectric constants, where the bifurcation is supercritical for potentials between Vc and Vc⬘. At these potentials, the amplitude becomes so large that the weakly nonlinear theory is no longer applicable and the supercritical stability predicted by the weakly nonlinear state undergoes another bifurcation. A typical case is shown in Fig. 11, where the interface evolves without stabilization at an equilibrium amplitude, until it touches the top or bottom wall. The effect of flow on the dielectric instability at finite wave numbers is similar to that for the low wave-number case. The weakly nonlinear analysis at finite wave numbers

FIG. 10. Equilibrium values of 共x / R兲 and 共y / R兲, where x and y are the coordinates of the interface, after a supercritical bifurcation obtained from the boundary element method for perfect dielectrics and for two fluids of equal thickness H = 1. The wavelength of the perturbation is k = 共␲ / R兲, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 6.5共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 2.0 and ⑀B = 1.0. Thick solid line, V = 6.52共⌫R / ⑀0兲1/2; solid line, V = 6.55共⌫R / ⑀0兲1/2; dotted line, V = 6.60共⌫R / ⑀0兲1/2; dot-dash line, V = 6.63共⌫R / ⑀0兲1/2; and dashed line, V = 6.65共⌫R / ⑀0兲1/2.

084104-15



FIG. 11. Time evolution of 共x / R兲 and 共y / R兲 where x and y are the dimensional coordinates of the interface locations, after a secondary subcritical bifurcation of the supercritical state, obtained from the boundary element method for perfect dielectrics and for two fluids of equal thickness H = 1. The wavelength of the perturbation is k = 共␲ / R兲, the potential is V = 6.7共⌫R / ⑀0兲1/2, the critical potential for the initial supercritical bifurcation at this wave number is Vc = 6.5共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 2.0 and ⑀B = 1.0. Thick solid line, t = 0; solid line, t = 20共␮R / ⌫兲; dotted line, t = 40共␮R / ⌫兲; dot-dash line, t = 60共␮R / ⌫兲; dashed line, t = 80共␮R / ⌫兲; and crosses, t = 100共␮R / ⌫兲.

indicates that the system can admit traveling waves even in a reference frame moving with the mean velocity at the interface due to nonlinear interactions. The effect of fluid flow is considered for a system which, in the absence of flow, has a supercritical bifurcation at V = Vc, and for which the supercritical state undergoes a further subcritical bifurcation at V = Vc⬘. The effect of fluid flow on the instability is considered in for potentials Vc ⬍ V ⬍ Vc⬘ and for V ⬎ Vc⬘. 共1兲 Figure 12 shows the time evolution of the interface for no mean flow in for Vc ⬍ V ⬍ Vc⬘. The numerical computations indicate that the equilibrium amplitude in the

FIG. 13. Effect of flow on the evolution of 共x / R兲 and 共y / R兲, where x and y are the dimensional coordinates of the interface obtained using the boundary element method after a supercritical bifurcation and a secondary subcritical bifurcation of the supercritical state for perfect dielectrics. The wavelength of the perturbation is k = 共␲ / R兲, the potential is V = 6.7共⌫R / ⑀0兲1/2, the critical potential for the first supercritical bifurcation at this wave number is Vc = 6.5共⌫R / ⑀0兲1/2, the potential for the subcritical bifurcation of the supercritical state is Vc = 6.7共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 1.0 and ⑀B = 2.0. The velocities of the top and bottom plates are equal in magnitude and opposite in direction, UA = −UB = U, where U = 0 in 共a兲, and U = 0.25共⌫ / ␮兲 in 共b兲. Circles, t = 0; solid line, t = 5共␮R / ⌫兲; dotted line, t = 25共␮R / ⌫兲; dot-dash line, t = 50共␮R / ⌫兲; and dashed line, t = 75共␮R / ⌫兲.

presence of flow is lower than the corresponding equilibrium amplitude in the absence of flow. This feature is observed for other parameter values also when the initial bifurcation is supercritical. 共2兲 The effect of fluid flow on the evolution of perturbations for V ⬎ Vc⬘ is shown in Fig. 13. It is found that flow stabilizes the supercritical state, and increases the potential required for the secondary subcritical instability. FIG. 12. Effect of flow on the evolution of 共x / R兲 and 共y / R兲 where x and y are the dimensional interface coordinates, obtained using the boundary element method after a supercritical bifurcation for perfect dielectrics. The wavelength of the perturbation is k = 共␲ / R兲, the potential is V = 6.6共⌫R / ⑀0兲1/2, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 6.5共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 1.0 and ⑀B = 2.0. The velocities of the top and bottom plates are equal in magnitude and opposite in direction, UA = −UB = U = 0.25共⌫ / ␮兲. Circles, t = 0; solid line, t = 5共␮R / ⌫兲; dotted line, t = 25共␮R / ⌫兲; dot-dash line, t = 50共␮R / ⌫兲; and dashed line, t = 75共␮R / ⌫兲.

The effect of flow on the stability in the parameter regime in which the primary bifurcation is subcritical, and which does not have an equilibrium amplitude, is analyzed next. Figure 14 shows the effect of flow for ⑀R = 0.2 which is predicted to be subcritically unstable by the weakly nonlinear theory. When the velocity is U = 0.25共⌫ / ␮兲 for the parameters chosen in Fig. 14, the system is unstable and admits traveling waves. However, when the velocity is U = 1.0共⌫ / ␮兲 for the parameters chosen in Fig. 14, the system reaches an equilib-

084104-16



FIG. 15. Time evolution of the 共x / R兲 and 共y / R兲, where x and y are the coordinates of the interface locations, after a subcritical bifurcation obtained from the thin-film analysis for leaky dielectrics and for two fluids of equal thickness H = 1. The wavelength of the perturbation is k = 共␲ / 50R兲, the potential is V = 0.125共⌫R / ⑀0兲1/2, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 0.123共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 1.0 and ⑀B = 2.0, and the conductivity ratio ␴R = 2. The initial perturbation has amplitude 0.1 times the channel width at t = 0. The solid line is the profile at t = 0, the dotted line is the profile at t = 1 ⫻ 106共R␮ / ⌫兲, the dot-dash line is the profile at t = 6 ⫻ 106共R␮ / ⌫兲, and the dashed line is the profile at t = 7 ⫻ 106共R␮ / ⌫兲.

negative with increasing top plate velocity, thereby indicating a change from subcritical to supercritical bifurcation. Thus, the imposition of a mean flow can cause a change in the type of bifurcation and can stabilize the instability. The frequency obtained from the nonlinear analysis is shown to increase with the shear velocity. These general features reported above on the effect of flow on the evolution of perturbations were observed in all other parameter regimes that were studied. B. Leaky dielectrics

FIG. 14. Effect of flow on the evolution of the interface obtained using the boundary element method after a subcritical bifurcation of the flat state of the interface for perfect dielectrics. The wavelength of the perturbation is k = 共␲ / R兲, the potential is V = 2.95共⌫R / ⑀0兲1/2, the critical potential for the first subcritical bifurcation at this wave number is Vc = 2.905共⌫R / ⑀0兲1/2, and the dielectric constants are ⑀A = 1.0 and ⑀B = 2.0. The velocities of the top and bottom plates are equal in magnitude and opposite in direction, UA = −UB = U. 共a兲 Velocity U = 0: solid line, t = 0: dotted line, t = 10共␮R / ⌫兲; dot-dash line, t = 20共␮R / ⌫兲; and dashed line, t = 30共␮R / ⌫兲; 共b兲 Velocity U = 0.25共⌫ / ␮兲: solid line, t = 0; dotted line, t = 10共␮R / ⌫兲; dot-dash line, t = 20共␮R / ⌫兲; and dashed line, t = 30共␮R / ⌫兲; 共c兲 U = 1.0共⌫ / ␮兲: solid line, t = 0, dotted line, t = 50共␮R / ⌫兲; dot-dash line, t = 100共␮R / ⌫兲; and dashed line, t = 150共␮R / ⌫兲.

rium state. Thus, the subcritical instability is modified into a supercritical bifurcation in the presence of flow. This is confirmed by the weakly nonlinear results, which indicate that the Landau constant can change from a positive value to

Figure 15 shows the time evolution for the unstable modes of a leaky dielectric at low wave numbers. The system shows a subcritical instability for all conductivity and dielectric constant ratios in the long-wave limit. The height increases with time and the interface finally touches the top or the bottom plate depending upon the conductivity and dielectric constant ratio. Leaky dielectric systems are characterized by a nonzero tangential stress and a nonzero tangential velocity at the interface. This leads to shapes which are qualitatively different from the perfect dielectric systems. Figure 16 shows the effect of flow on the electrohydrodynamic instability in leaky dielectrics in the long-wave limit. The mean velocity at the interface is zero in all the cases studied 共UA = −UB = U , H = 1兲 and the growth rate from the linear stability theory 关Eq. 共46兲兴 is real, indicating that there are no traveling waves. However, the numerical results indicate that flow has a stabilizing effect on the instability and the system admits traveling waves, the direction of which is dependent upon the dielectric constants and the conductivity ratio. Figure 8 shows the variation of the maximum height of the interface with the mean velocity for both perfect and leaky dielectric systems in the long-wave limit. The maximum height amplitude is seen to decrease with increasing

084104-17



FIG. 16. Effect of flow on the time evolution of 共x / R兲 and 共y / R兲, where x and y are the dimensional coordinates of the interface positions, for an instability in a leaky dielectric using the thin-film approximation for H = 1. The wavelength of the perturbation is k = 共␲ / 50R兲, the potential is V = 0.125共⌫R / ⑀0兲1/2, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 0.123共⌫R / ⑀0兲1/2, the dielectric constants are ⑀A = 1.0 and ⑀B = 2.0, and the ratio of conductivities ␴R = 0.5. The velocities of the top and bottom plates are equal in magnitude and opposite in direction, UA = −UB = U; 共a兲 U = 0, 共b兲 U = 0.000 25共⌫ / ␮兲, 共c兲 U = 0.0005共⌫ / ␮兲, 共d兲 U = 0.001共⌫ / ␮兲, and 共e兲 U = 0.005共⌫ / ␮兲. Solid line, t = 0; dotted line, t = 2.5⫻ 106共␮R / ⌫兲; dot-dash line, t = 5 ⫻ 106共␮R / ⌫兲; and dashed line, t = 1 ⫻ 107共␮R / ⌫兲.

top plate velocity and scales as h ⬃ U−1 for both cases, as shown in Fig. 8. The figure shows the numerical results and the weakly nonlinear theory are in disagreement by an order of magnitude, though the qualitative trends in both cases are the same. This may be attributed to the neglecting of higherorder terms in the Landau expansion. The leaky dielectric model is next studied for finite wave numbers using both weakly nonlinear theory and the boundary integral method. The weakly nonlinear theory 共Table II兲

indicates that the bifurcation is subcritical in the absence of flow for a range of wave numbers, and the boundary integral results are consistent with this result. Figure 17 shows the equilibrium interface position for a leaky dielectric at finite wave number. It is found that the shape of the interface depends on the relative conductivities and the relative dielectric constants of the two fluids. The results are in agreement with the predictions of the weakly nonlinear theory. When the top plate potential is further increased, the supercritical equilib-

084104-18



FIG. 17. Equilibrium values of 共x / R兲 and 共y / R兲, where x and y are the dimensional coordinates of the interface locations, after a supercritical bifurcation obtained from the boundary element method for leaky dielectrics and for two fluids of equal thickness H = 1. The wavelength of the perturbation is k = 共␲ / R兲, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 3.57共⌫R / ⑀0兲1/2, the dielectric constants are ⑀A = 2.0 and ⑀B = 1.0, and the ratio of conductivities is ␴R = 0.5. The solid line is the interface profile for V = 3.5共⌫R / ⑀0兲1/2, and the dashed line is the interface profile for V = 3.55共⌫R / ⑀0兲1/2.

rium state becomes unstable, as shown in Fig. 18, and the final state consists of alternate vertical columns of fluid spanning the distance between the two plates. Figure 19 shows the equilibrium profiles for a larger wavelength 关共␭ / R兲 = 5兴. The equilibrium height amplitude is found to be larger for larger wavelengths for the same two fluids. The height of the fluctuation in the supercritical state increases with an increase in the applied voltage, and then the supercritical state undergoes a further bifurcation which is found to be subcritical. The interface profile at long times shows columns of one fluid in the other 共Fig. 19兲.

FIG. 19. Equilibrium profiles of 共x / R兲 and 共y / R兲, where x and y are the dimensional coordinates of the interface positions, after a supercritical bifurcation obtained from the boundary element method for leaky dielectrics and for two fluids of equal thickness H = 1. The wavelength of the perturbation is k = 共␲ / 2R兲, the critical potential for the transition from stable to unstable perturbations at this wave number is Vc = 2.03共⌫R / ⑀0兲1/2, the dielectric constants are ⑀A = 2.0 and ⑀B = 1.0, and the ratio of conductivities is ␴R = 0.5. The solid line is the interface profile for V = 2.11共⌫R / ⑀0兲1/2, the dotted line is the profile for V = 2.15共⌫R / ⑀0兲1/2, the dot-dash line is the profile for V = 2.21共⌫R / ⑀0兲1/2, and the dashed line is the profile for V = 2.5共⌫R / ⑀0兲1/2.

Similar to the case of perfect dielectrics, the electrohydrodynamic instability in leaky dielectrics is also stabilized by shear flow at finite wave numbers. Figure 20 shows the effect of flow on the unstable interface between two leaky dielectric fluids. In the absence of flow, the bifurcation is subcritical and the interface evolves till it touches the confining wall. As the velocity is increased, the system admits traveling waves and there is a change from subcritical to supercritical bifurcation after a critical strain rate. This is confirmed by the weakly nonlinear analysis, which indicates that the mean flow could alter the nature of the initial bifurcation from subcritical to supercritical, and the supercritical steady state has traveling waves. V. CONCLUSIONS

FIG. 18. Time evolution of 共x / R兲 and 共y / R兲 where x and y are the dimensional coordinates of the interface positions, after a secondary subcritical bifurcation of the supercritical state, obtained from the boundary element method for leaky dielectrics and for two fluids of equal thickness H = 1. The wavelength of the perturbation is k = 共␲ / R兲, the potential is V = 3.6共⌫R / ⑀0兲1/2, the critical potential for the initial supercritical bifurcation at this wave number is Vc = 3.47共⌫R / ⑀0兲1/2, the dielectric constants are ⑀A = 2.0 and ⑀B = 1.0, and the ratio of conductivities ␴R = 0.5. The initial perturbation has amplitude 0.05 times the channel width at t = 0. The solid line is the profile at t = 0, the dotted line is the profile at t = 35共R␮ / ⌫兲, the dot-dash line is the profile at t = 45共R␮ / ⌫兲, and the dashed line is the profile at t = 47.5共R␮ / ⌫兲.

The stability of the interface between two dielectric fluids confined between parallel plates subjected to a normal electric field was studied using the linear and weakly nonlinear analyses in the limit of zero Reynolds number. Both perfect dielectrics and leaky dielectrics were considered. The linear stability analysis showed that the interface becomes unstable in the absence of flow when the applied potential exceeds a critical value, and this critical potential depends on the ratio of dielectric constants, ratio of viscosities, ratio of thicknesses, and surface tension. The critical potential increases proportional to k in the small wave-number limit, and proportional to k1/2 in the limit of large wave number. It is also found that the critical potential for leaky dielectrics is smaller than that for perfect dielectrics. The weakly nonlinear analysis predicts that the bifurcation is supercritical in a very narrow range of the ratio of dielectric constants, and subcritical for all other ratios of dielectric constants. The numerical computation of the full equations was carried out to verify the results of the weakly nonlinear analysis, and to examine the dynamics of the inter-

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FIG. 20. Effect of flow on the evolution of 共x / R兲 and 共y / R兲 along the interface obtained using the boundary element method after a subcritical bifurcation of the flat state of the interface for leaky dielectrics. Here, x and y are the dimensional interface coordinates. The wavelength of the perturbation is k = 共␲ / R兲, the potential is V = 1.17共⌫R / ⑀0兲1/2, the critical potential for the subcritical bifurcation at this wave number is Vc = 1.16共⌫R / ⑀0兲1/2, the dielectric constants are ⑀A = 5.0 and ⑀B = 1.0, and the ratio of conductivities is ␴R = 5. The velocities of the top and bottom plates are equal in magnitude and opposite in direction, UA = −UB = U. 共a兲 U = 0, 共b兲 U = 0.25共⌫ / ␮兲, and 共c兲 U = 1.0共⌫ / ␮兲. The solid line is the profile at t = 0, the dotted line is the profile at t = 50共R␮ / ⌫兲, the dot-dash line is the profile at t = 65共R␮ / ⌫兲, and the dashed line is the profile at t = 75共R␮ / ⌫兲.

face when the amplitude becomes large. Two numerical techniques were used, the thin-film analysis in the long-wave limit and the boundary integral technique for perturbations with wavelength comparable to the channel thickness. The


results of the weakly nonlinear analysis were found to be in agreement with the results of both these numerical studies. The thin-film analysis indicated that in the absence of mean flow, the low wave-number instability is supercritical for all the values of dielectric constants and conductivities for both the perfect and leaky dielectrics. The instability is stabilized in the presence of flow for both the cases and the equilibrium height amplitude scales as U−1 for both the perfect and leaky dielectric models, when the top and bottom plates are moved with velocities ±U. The boundary integral analysis showed that when the primary bifurcation is subcritical, the perturbation amplitudes grow until the interface reaches the top or bottom surface. Even when the primary bifurcation is supercritical, there is a secondary bifurcation at a potential slightly larger than the critical potential which tends to further destabilize the supercritical state, and this secondary bifurcation is subcritical. The effect of fluid flow on the electrohydrodynamic instability was studied using the weakly nonlinear theory and boundary element method when the wavelength is comparable to the channel width. For both perfect dielectric and the leaky dielectric fluids, the interface is stabilized by flow, and the critical potential in the presence of flow is higher than that in the absence of flow. Flow also tends to alter the nature of both the initial bifurcation 共in cases where it is subcritical兲 as well as the secondary bifurcation 共which sets in at a potential close to that for the initial bifurcation when the initial bifurcation is supercritical兲 from subcritical to supercritical. It is found that the system admits traveling waves in a reference frame moving with the mean velocity due to the breakdown of the up-down symmetry in the system under an electric field. The stabilization of the electrohydrodynamic instability and admittance of traveling waves because of shear are found to be due to the effect of nonlinear interactions. The results of the linear stability analysis are in agreement with the results of Pease and Russel.4 The increase of the critical potential proportional to the wave number at low wave number, and the increase proportional to the square root of the wave number at high wave number, are identical to those reported by Pease and Russel. We find numerical difference in the coefficients, however, because we have assumed that the two fluids have equal viscosity and density, whereas most of the earlier results are for the interface between a polymeric liquid and air. The results of the nonlinear analysis are also in broad agreement with the experimental results. The nonlinear analysis predicts that the initial bifurcation is subcritical in the long-wave limit, though it could be subcritical or supercritical for wavelengths comparable to the channel thickness. Even when the initial bifurcation is supercritical, it is followed by a subcritical bifurcation when the potential is increased by a few percent of the critical potential. Further, the thin-film and boundary element analyses show that the wavelength of the patterns is very close to that predicted by initial stability analysis, and there are no secondary bifurcations which alter the wavelength after the initial instability has set in. This validates the application of linear stability analysis for predicting the dimensions of the

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patterns produced by Schaffer, Thurn-Albercht, Russel, and Steiner.6 The implications of the present analysis for the flow control in micron scale devices, where it is advantageous to generate secondary flows to increase transport rates are as follows. The analysis indicates that in the absence of flow, the instability is supercritical in a very narrow range of dielectric constants, and subcritical for all other values of the ratio of dielectric constants. Since flow control is difficult if the instability is subcritical, this implies that the dielectric constants have to be adjusted carefully in order to take advantage of the secondary flows generated after a supercritical bifurcation. It is important to avoid operation in the regime where the bifurcation is subcritical, since this results in an increase in the amplitude of the interface until it touches the walls. Further, it is important to note that even after a supercritical bifurcation, there is the possibility of a secondary bifurcation at a potential which is not much more than that for the first bifurcation. Therefore, the normal potential has to be adjusted so that this secondary bifurcation does not take place. There is greater flexibility of operation in the presence of a shear flow, however, since the flow tends to stabilize the electrodynamic instability, and to convert it from a subcritical instability to a supercritical instability. This provides the option of operating the device at sufficiently high strain rates so that the instability is supercritical, and greater transport rates are generated by the secondary flow. 1



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