Linear stability analysis of axisymmetric perturbations in imperfectly conducting liquid jets

Linear stability analysis of axisymmetric perturbations in imperfectly conducting liquid jets J. M. López-Herrera,a兲P. Riesco-Chueca,b兲and A. M. Gañán-Calvoc兲 Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, Sevilla 41092, Spain 共Received 7 June 2004; accepted 7 January 2005; published online 1 March 2005兲 A discussion is presented on the role of limited conductivity and permittivity on the behavior of electrified jets. Under certain conditions, significant departures with respect to the perfect-conductor limit are to be expected. In addition, an exploration is undertaken concerning the validity of one-dimensional average models in the description of charged jets. To that end, a temporal linear modal stability analysis is carried out of poor-conductor viscous liquid jets flowing relatively to a steady radial electric field. Only axisymmetric perturbations, leading to highest quality aerosols, are considered. A grounded coaxial electrode is located at variable distance. Most available studies in the literature are restricted to the perfect-conductor limit, while the present contribution is an extension to moderate and low electrical conductivity and permittivity jets, in an effort to describe a situation increasingly prevalent in the sector of small-scale free-surface flows. The influence of the electrode distance b, a parameter ␣ defined as the ratio of the electric relaxation time scale to the capillary time scale, and the relative permittivity ␤ on the growth rate has been explored yielding results on the stability spectrum. In addition, arbitrary viscosity and electrification parameters are contemplated. In a wide variety of situations, the perfect-conductor limit provides a good approximation; however, the influence of ␣ and ␤ on the growth rate and most unstable wavelength cannot be neglected in the general case. An interfacial boundary layer in the axial velocity profile occurs in the low-viscosity limit, but this boundary layer tends to disappear when ␣ or ␤ are large enough. The use of a one-dimensional 共1D兲averaged model as an alternative to the 3D approach provides a helpful shortcut and a complementary insight on the nature of the jet’s perturbative behavior. Lowest-order 1D approximations 共average model兲, of widespread application in the literature of electrified jets, are shown to be inaccurate in low-viscosity imperfect-conductor jets. ©2005 American Institute of Physics.关DOI: 10.1063/1.1863285兴 I. INTRODUCTION Electrification of free-surface flows is guided by three principal aims: first, increasing the surface-to-volume ratio by means of a reduction of the liquid’s apparent surface tension; second, controlling the stability range of the flow; third, directing the flow towards a target. Multiple applications derive from these achievements: atomization, particle sorting, ink jet printing, fuel injection, and fiber spinning. For instance, jet stimulation by electric fields is increasingly used in a variety of ink jet printing technologies as a method for charging and steering the ink drops resulting from jet disintegration. Alternatively, continuous deviated ink jet technology is based on direct deflection of electrified jets. Electric charges modulate the jet’s response to electric fields, random noise, or specific excitation wavelengths. In many applications, a grounded electrode located at the vicinity of the jet surface establishes a radial field; the quasielectrostatic pressure jump competes with convective, capillary, and viscous effects to determine the selected wavelength. Another research field involving electrified jets is electrospinning, a process in which solid fibers are produced from a polymeric fluid stream delivered through a millimeter-scale nozzle. Additional achievements are presently reported in pharmaceutical and material sciences. These applications are leading to a growing interest of researchers in electrified jets. In the search for diverse applications, poor conductivity liquids are becoming increasingly attractive, because food and pharmacy technologies, among others, depend on their efficient handling. An important motivation source for the study of electrohydrodynamics is the search for high quality sprays from liquid microjets, eagerly demanded in technological applications requiring small and homogeneous droplet size. The axisymmetric capillary breakup 共Rayleigh breakup兲of small and steady liquid ligaments into droplets gives rise to controllable monodisperse aerosols, unattainable through other high-yield processes. The foundations for the study of conducting jets in a radial electric field were set by Melcher,1assuming inviscid flow and constant electric potential. Saville2generalized the analysis for arbitrary viscosity liquids. Huebner and Chu3 studied the stability of a charged jet in the inviscid limit assuming arbitrary electrode distance b. Setiawan and Heister4extended this description to the nonlinear growth phase. The influence of the surrounding atmosphere on the stability of the jet was considered by Baudry et al.5An ela兲Electronic mail: [email protected] b兲Electronic mail: [email protected] c兲Electronic mail: [email protected] PHYSICS OF FLUIDS 17, 034106 共2005兲 1070-6631/2005/17共3兲/034106/22/$22.50 © 2005 American Institute of Physics17, 034106-1 egant overview of the linear stability problem was presented by García,6who explored the perturbative spectrum for a variety of parameters. This work has been recently extended to ac electric fields by González et al.;7the authors provide an attractive formulation of the dc problems, whose notation we follow here. The electrical behavior of a moving liquid follows from the ratio of the characteristic hydrodynamic time versus the electrical relaxation time te⬃␧i/K, where ␧iand Kare the electrical permittivity and conductivity of the liquid, respectively. In the absence of other forcing agents, the time scale on which perturbations grow and cause the jet’s breakup is given by a balance of surface tension and inertia, summed up by the capillary time tc⬃共 ␳ A3/ ␥ 兲1/2, where ␳ , ␥ , and Aare the liquid density, surface tension, and the jet radius, respectively. If the ratio te/tcis strictly zero the liquid can be considered as a perfect conductor even in the last stages of pinch-off. In the perfect-conductor limit the calculation of electrical stresses on the jet surface is simplified since all charges are interfacial, the external electric field is normal to it, and the electric field in the jet bulk is zero. In this limit, provided the breakup is axisymmetric, electrification stabilizes long-wavelength perturbations while destabilizing short ones.1,8 In the present work we aim to study the stability of electrified liquid jets when the ratio te/tcis not strictly zero. Our model is directly borrowed from the general formulation Saville provided in 1997 by expanding previous results from Taylor and Melcher.9–11 The analysis is applicable to situations where the relaxation time for free charges is short compared with the time scale for fluid motion. In the survey published in 1997 by Saville,9the stability of fluid cylinders or free jets and pinned cylinders or liquid bridges is studied. These examples are given as an illustration of the powerful set of equations for the leaky dielectric model, which builds the core of the above survey. The discussion is mostly qualitative, and the results disclosed are restricted to some particular cases where a solution can be obtained with relative ease. Additional contributions in this field by Saville and co-workers focused on the effect of axial electric forcing.2,12–15 The starting point is provided by the linearized Navier– Stokes equations. A careful description is then made of the parametrical behavior of the resulting dispersion equation. Our study attempts to complete a systematic consideration of all the input variables 共wavelength, viscosity, surface tension, electrical conductivity, permittivity and electrification levels, surrounding gas dynamics兲. Previous works do not include all these terms or do not weigh their relative influence. The ambient influence is explored by means of a sketchy model, which does however allow a preliminary glimpse into the stability of electrified coflowing liquid-gas streams. The simultaneous forcing of a jet by hydrodynamic focusing 共flowfocusing technique16兲and by electrification is a different field of research, with a variety of presumed applications, e.g., ultrafine atomization; the present contribution may provide a stepping stone for the joint analysis of electrification and flow focusing. In addition, under sufficiently general conditions, a onedimensional model following López-Herrera et al.17 yields excellent results 关for a comparison on the suitability of either a three-dimensional 共3D兲axisymmetric or a 1D model see for example Yildirim and Basaran18兴, which allows the detailed and quite inexpensive exploration of extense parametrical ranges of interest. It is our aim to determine under which conditions the lowest-order 1D approximation 共sliceaverage model兲, widely used in the literature, is to be trusted. It can be anticipated that the assumption of a flat axial velocity profile is inaccurate when either of two situations occur: 共a兲the velocity profile is markedly convex; 共b兲an axial velocity boundary layer develops, due to the action of shear, at the interface. In both cases, the flat-profile approximation fails to provide a trustworthy description. It will be shown that jets with moderate conductivity and permittivity often display sharply convex or boundary-layer velocity profiles. Indeed, nonuniform radial profiles are favored by the combination of small viscosity, poor conductivity and permittivity, and a shear agent. Tangential stress may be generated by a variety of mechanisms, such as surfactant diffusion or, in our case, an electric field. Axially oriented electric fields are the most efficient shear agents and they lead to the conversion of electric potential into kinetic energy; but even radial fields as studied here can give rise to significant shear provided ␣ and ␤ are small or moderate. It is important, therefore, to be cautious when modeling electrified jets: slice-average models may turn out to be inaccurate, so that a 3D model or higher-order expansions 共parabolic兲become unavoidable. Moderate liquid conductivity and permittivity is readily observed in many technological fields, in particular whenever oils or other organic compounds are manipulated. Other practical applications, such as electrospinning or jet printing, may involve imperfect conductors. In the following section, once the main dimensionless groups are introduced, reference will be made to experimental and technological applications involving low conductivity/permittivity liquids. II. FORMULATION OF THE PROBLEM A. Context and assumptions The results presented here arise from some previous work on the conditions leading to jet breakup, under a diversity of geometrical and electrification contexts. López- Herrera and Gañán-Calvo19 presented a detailed experimental procedure concerning charged capillary jet breakup, in the assumption of perfect-conductor behavior and radial electric forcing. In the course of experimentation, an extended exploration was attempted of the poor-conductor range. As stated above, food, pharmacy, carburation, and other applications at hand are increasingly pushing researchers toward the study of poor-conductors’ stability and breakup. Preliminary experimental results provided a wide diversity of data leading to considerable trouble in the effort to calibrate and sort out raw data from a theoretical point of view. Therefore, a detailed theoretical description of poorconductor jet behavior appeared to be desirable, both to provide a solid basis for the experimental program, and to orientate the production of an efficient numerical code able to include breakup. As an intermediate goal, the stability analy- 034106-2 López-Herrera, Riesco-Chueca, and Gañán-Calvo Phys. Fluids 17, 034106 共2005兲 sis of a liquid jet is hereby described: it yields inference on 共a兲the size of drops after breakup and the stimulation required therefore; 共b兲the conditions required in the opposite case, i.e., when breakup is to be avoided and the integrity and longevity of the jet is to be preserved. Both objectives are technologically relevant. We limit the scope of our study to the axisymmetric stability of capillary cylindrical jets of a liquid with density ␳ , permittivity ␧i, viscosity ␮ , conductivity K, and interfacial surface tension ␥ . Our interest will be focused on the axisymmetric capillary jet instability. This choice is dictated by two main reasons: first, the prevalence of axisymmetric instability in most applications where a high quality spray is required 共e.g., when electrospraying small liquid flow rates兲. Second, axisymmetric modes are dominant in most dc estimulation regimes provided the Weber number is moderate. Accordingly, we restrict our study to moderate We numbers and ignore sinuous or asymmetric perturbation modes. The surrounding atmosphere is assumed inviscid and its influence on the dynamics of the jet is modeled with the help of a simplified approach, which will be justified later by invoking some restrictions on the Weber numbers. Aand Uo are the jet radius, assumed constant, and the average axial velocity. The equilibrium velocity profile is assumed flat; its value Uois usually scaled with the capillary wave velocity vc=共 ␥ / ␳ A兲1/2 by means of the Weber number, We =共 ␳ AUo 2兲/ ␥ . In the gas-at-rest limit, where U⬁=0, and assuming WeⰇ1, Keller et al.20 show that the perturbation is rapidly convected downstream with the jet velocity, thus avoiding the simultaneous propagation of disturbances in the upstream and downstream directions 共absolute instability兲. See also Artana et al.21 for a discussion on the role of absolute and convective instabilities in the evolution of electrified jets, as well as O’Donnell et al.22 and Chauhan et al.23 for an experimental investigation of the convective instability in uncharged jets. Kalaaji et al.24 presented an experimental investigation of the breakup length of a jet and showed that provided the Weber number is large enough, temporal and spatial linear perturbation theories coincide. We therefore restrict our study to moderate or large Weber numbers 关approximately above 4, see Eq. 共5兲in Lin and Reitz25兴: this restriction ensures that the jet velocity is sufficiently higher than the group velocity of perturbations so that a temporal evolution of the perturbation is observed from a Galilean reference frame moving with the jet. However, when the Weber number exceeds a critical value, either nonsymmetric perturbations grow faster than axisymmetric perturbations or viscous effects in the surrounding atmosphere can no longer be neglected. The above references do not consider the coflowing regime, where both the jet and the surrounding gas flow 共with different mean velocity兲in the axial direction. This flow configuration is increasingly attractive as a source of potential technical applications. Unfortunately, the parametric limits of axisymmetric breakup in coflowing jets have not been explored in detail in terms of the liquid and gas Weber numbers. A mapping of the breakup regimes as sketched by Lin and Reitz25 in the gas-at-rest limit is not available in the coflowing case. It is known that interfacial shear promotes Taylor’s instability, where droplets much smaller than the jet diameter are produced,26 but a systematic exploration of the breakup regimes prevailing under different gas and liquid Weber numbers is not available. In the following, however, we narrow the scope of our attention to flow configurations located in the Rayleigh and first wind-induced regimes, where shear effects from the gas side can be ignored. Accordingly, aerodynamic effects 共gas– liquid interaction兲are modeled with a simplified theory; the influence of the gas is accounted for in terms of its pressure fluctuation, but gas viscosity is neglected. The surrounding gas is assumed to flow coaxially with an uniform speed U⬁ which, depending on the envisaged application, may range from 0 共ambient gas at rest兲to an amount one or two orders of magnitude above the jet velocity Uo. With this choice, we leave the door open to the study of electrified flow focusing configurations 共EF3兲, a field of considerable interest. The flow focusing technology is based on the acceleration of a liquid jet surrounded by a coflowing gas stream, whereby a favorable pressure gradient is created, allowing an enhanced control of the problem’s hydrodynamics.16 Three main simplifications are introduced in our gas– liquid interaction model. 共a兲The gas is incompressible: results can be extrapolated to the case where the surrounding stream is also liquid, under the stringent condition that it be perfectly insulating. 共b兲The gas is inviscid, and a uniform flow is assumed 共flat axial velocity profile U⬁兲. 共c兲The surrounding stream is an insulator with vacuum permittivity: it removes all space charges and is perfectly unelectrified. Therefore, the ensuing description will not undertake the analysis of the velocity boundary layer at the ambient side. A detailed investigation of the gas velocity profile is performed by Gordillo et al.,27 but their results 共the liquid jet dynamics is assumed inviscid; no electric effects are considered兲cannot be extrapolated here. Additional assumptions in our model9are the following: magnetic fields generated by moving charges can be neglected 共electrostatic assumption兲; gravitational acceleration is ignored. Figure 1 shows the geometric outline of the problem. Initially, a potential difference Vois applied between the jet and a coaxial cylindrical electrode of radius R⬁=bA surrounding it. In this geometrical configuration, the axial electric field is negligible compared to the normal electrical field, as is the case in jets emitted by electrospray.28,29 The radial electric field creates, in the unperturbed state, a surface charge density located at the interface, the liquid bulk being electrically neutral. In most of the applications, charged jets issue from a liquid source, where a voltage Vois applied, and end up at the breakup region. A transition from conduction to convection, as the relevant charge transport mechanism, takes place in the vicinity of the needle. Charges relax from the bulk to 034106-3 Linear stability analysis Phys. Fluids 17, 034106 共2005兲 the surface in a characteristic time te; this process taking place in a region of length Lr⬃Uotemuch shorter than the total jet length Lj. The jet length is generally of the order of Uotc, but in many practical situations, the longevity of the jet can be extended by avoiding all instability factors. Most of the voltage drop along the jet ⌬Vis located in the charge relaxation region, which can be modeled as a resistor of order ⍀R⬃Lr/共K ␲ A2兲; therefore, the characteristic relative voltage drop is ⌬V Vo⬃⍀RI Vo⬃⍀R␧oU⬃We ␤ 冉 te tc 冊 2 Ⰶ1, 共1兲 where Iis the current transported by the jet. Thus, the axial voltage drop in the relaxation stretch is negligible, and the potential at the jet surface can be estimated, in first approximation, as the potential applied at the needle Vo. Further insight on the structure of the potential drop along the axial coordinate of a jet is provided by Gamero-Castaño and Hruby30 and Higuera.31 The equations of the problem are made dimensionless with the radius of the unperturbed jet A, the density ␳ , surface tension ␥ , and the characteristic surface charge density ␧oEo,␧obeing the vacuum permittivity and Eo =Vo/关Aln共b兲兴 the characteristic radial electric field, while b=R⬁/A. Our choice of Eorather than Voas a scaling factor has the following implication: at the interface, the normalized electrostatic pressure is of O共1兲. This is unrealistic when b→⬁, because an infinite potential difference Vois required in order to preserve the scaling; however, the results are consistent, and can be applied safely for any bⰇ1. The advantage in choosing Eorather than Vois that the normalized electrostatic pressure ␧oEo 2does not become vanishingly small in the b→⬁limit. Setiawan and Heister,4who used the Voscaling, alluded to the shortcomings of this choice: “as the ground location is moved far from the jet, the effect of electrostatic contributions vanishes.” Note, however, that Eo must be bounded in our problem; otherwise, dielectric rupture would lead to corona discharge effects. The above scaling leads to a set of nondimensional parameters which characterize each particular jet and perturbation. 共1兲The half wavelength of the perturbation ␭=L/Aor the dimensionless wavenumber k= ␲ /␭. 共2兲The radial position of the ground electrode b=R⬁/A. 共3兲The Weber number We= ␳ AUo 2/ ␥ =共Uo/vc兲2, where vc =共 ␥ / ␳ A兲1/2 is the capillary velocity, measuring the relative importance of the jet’s absolute inertia with respect to the capillary forces. Some authors prefer to use its inverse, which is referred to as a Euler number. 共4兲An external flow Weber number We⬁= ␳ AU⬁ 2/ ␥ =共U⬁/vc兲2, measuring the relative importance of the ambient fluid’s absolute inertia with respect to the capillary forces. Note that the liquid’s density 共and not the gas’兲is used. In many applications, the ambient gas is at rest, so that We⬁=0. An additional parameter of interest in describing the liquid-gas interaction is the ratio between the gas and liquid densities ␳ ¯ = ␳ g/ ␳ . 共5兲The Ohnesorge number C= ␮ /共 ␳ ␥ A兲1/2= ␯ tc/A2weigh- ing viscous forces against capillary forces. It is the inverse of a Reynolds number based on the capillary velocity; it can also be understood as the ratio between the capillary time tcand the viscous radial diffusion time tv⬃A2/ ␯ . The usual or convective Reynolds number, Re=UoA/ ␯ , can be expressed as Re=We1/2/C. 共6兲The electric number ␹ =A␧oEo 2/ ␥ , also known as Taylor number or electrical Bond number, comparing the electric pressure with the capillary pressure. It can be conceived as the ratio between the capillary time scale and a time scale obtained by geometric averaging of the viscous diffusion time tvand the shear or electrohydrodynamic time scale ts= ␮ /共␧oEo 2兲, i.e., ␹ ⬃tc 2/共tvts兲 ⬃Ctc/ts. Saville9introduced a Reynolds number based on the electrohydrodynamic velocity us=A/ts; it can be written as Res= ␳ ␧oA2Eo 2/ ␮ 2= ␹ /C2. 共7兲The liquid’s relative permittivity ␤ =␧i/␧o. The ambient fluid is supposed to have vacuum permittivity. 共8兲The relaxation parameter ␣ =关 ␳ A3K2/共 ␥ ␧i 2兲兴1/2=tc/te.It coincides with the ratio of the electrical relaxation to the capillary time 共generally, the shortest hydrodynamic time of the process兲. The combination ␣␤ will be shown later to describe the relative importance of conductive versus convective charge transport. The perfectconductor limit is recovered when ␣␤ Ⰷ1共generally simplified to ␣ Ⰷ1 by taking into account customary ␤ values兲: this limit implies that “relaxation is quicker than deformation,”32 so that the jet remains isopotential at any time, and its internal electric field is zero. In the opposite situation ␣␤ Ⰶ1, charges are glued to the interface 共“honey-bubble” limit兲, and are therefore passively convected and stretched with it. Each perturbative situation is therefore characterized by a set of eight free parameters: ␣ , ␤ ,C, ␹ ,b,We,We ⬁, and ␭, whose influence will be explored next. It is important to note that our hypothesis about the electric charge being relaxed to the jet surface does not necessarily imply ␣ Ⰷ1. Indeed, provided that the jet is long enough, the relaxation process can be considered to have taken place at an earlier location. Therefore, the influence of ␣ when it is of O共1兲or even smaller can be explored under the assumption that all FIG. 1. Geometrical sketch of the problem. 034106-4 López-Herrera, Riesco-Chueca, and Gañán-Calvo Phys. Fluids 17, 034106 共2005兲 charges are superficial. This is a realistic situation, frequently observed in experimental work. Electrospray can be used to illustrate the above. This is an apt technique for the generation of electrified capillary microjets nanojets and sprays. It gives rise to a great variety of jet configurations and atomization modes; among them, the cone-jet mode is preferred because of its steadiness and spray characteristics.33 Cone-jet operation can be attained with a wide range of liquids34 exhibiting enormous diversity in the relevant physical constants 共conductivity K, surface tension ␴ , density ␳ , relative permittivity ␤ , and viscosity ␮ 兲. Conductivity and, to a lesser extent, viscosity coefficients, may exhibit radically different orders of magnitude. Cone jets have been obtained with toluene 共K=10−11 S/m兲, intensely acid water solutions 共K=1 S/m兲, hexane 共 ␮ =3 ⫻10−4 Pa s兲or glycerol 共 ␮ =1.3 Pa s兲. Experimental and theoretical investigations28,31,35–37 have led to the introduction of the following scaling magnitudes: Qo= ␥ ␧o ␳ K,Io= 冉 ␧o ␥ 2 ␳ 冊 1/2,do= 冉 ␧o 2 ␥ ␲ 2 ␳ K2 冊 1/3 .共2兲 The above magnitudes set the scale for the injected flow rate Q, the electric current transported by the jet I, and the jet diameter A. With their help, a universal ratio is established: I/Io⬃共Q/Qo兲1/2 and A/do⬃共Q/Qo兲1/2. Provided secondorder effects such as the needle geometry are ignored, the above scaling is accurate in a wide range of situations involving diverse liquids and geometries. On the basis of electrospray scaling, ␣ and Ccan be estimated in typical liquid jets issuing from a cone-jet source. Table I shows ␣ and Cvalues drawing on experimental data from Gañán-Calvo;36 different organic liquids are used. Some additional data from Gañán-Calvo et al.38 were obtained by electrospraying low-viscosity liquids 共heptane, dioxane, as well as dodecanol兲. As can be gathered from the new data, Ohnesorge numbers well below unity can be found by electrospraying nonpolar liquids such as toluene, cyclohexane, or heptane, characterized by their low conductivity and viscosity. Such low Ohnesorge numbers are called to play an increasing role in multiple technological applications involving organic liquids. These results indicate that low ␣ and ␤ are met in realistic experimental explorations. Electrical polarization, the key to dielectric energy storage, is the result of a wide variety of processes, including distortion or reorientation of molecules and orbitals as well as electrochemical reactions. Low values of ␤ , implying weak polarizability, are frequently observed in connection with organic or long-chain molecules, particularly when water is absent from the composition of the liquid. Ionic charge transport is reduced in low- ␤ liquids because of their reluctance to dissolve ions, so that low conductivities are frequent in such cases. Accordingly, the routine assumption that jets are perfect conductors cannot be made without risk. In addition, very low Ohnesorge numbers are frequently observed in low-viscosity jets. In particular, López-Herrera and Gañán-Calvo19 carried out a set of experiments with precisely the same electric field and configuration proposed here. In the above paper, the electrification of the jet tries to achieve the condition of zerotangential stress. Experiments leading to breakup were carried out and compared with a theoretical model assuming perfect-conductor behavior. Three mixtures of water and glycerine were used in the study. Agreement between experiments and theory was in general good, a logical consequence of the fact that all the jets tested were water solutions; and a small amount of water is all that is needed for the permittivity to become high 共so that ␣␤ Ⰷ1, close to the perfectconductor limit兲. Should the experimental range be expanded to include purely organic liquids, the perfect-conductor hypothesis would no longer hold. The need for a stability theory of imperfect conductors follows from such considerations. B. Equations Under the conditions and assumptions described above, the nondimensional, cylindrical coordinates equations in a reference frame moving with the jet velocity Uoare TABLE I. Estimated values in electrospray jets: diameter A, relaxation parameter ␣ , and Ohnesorge number C; experimental data and scaling law from Gañán-Calvo 共Ref. 36兲and Gañán-Calvo et al. 共Ref. 38兲. Liquid do共 ␮ m兲Qo共ml/min兲 ␤ Q/QoA共 ␮ m兲 ␣ C Dodecanola1.67 2.34 6.5 2 2.36 0.08 1.70 30 9.16 0.62 0.86 1-Octanola1.52 1.88 10 2 2.15 0.05 1.25 30 8.33 0.41 0.63 Propylenglycola1.24 1.54 31.2 2 1.76 0.02 5.18 30 6.82 0.13 2.63 Dioxaneb5.82 14.5 2.5 2 4.62 0.09 0.11 30 20.72 0.86 0.05 Heptaneb7.43 21.2 1.9 2 5.89 0.12 0.04 30 26.46 1.13 0.02 aReference 36. bReference 38. 034106-5 Linear stability analysis Phys. Fluids 17, 034106 共2005兲 共rU兲r r+Wz=0, 共3兲 Ut+UUr+WUz=−Pr+C共Uzz −Wzr兲,共4兲 and Wt+UWr+WWz=−Pz+C 冉 Wrr +Wr r+Wzz 冊 ,共5兲 where rand zare the radial and the axial coordinates, tis the time, Uand Ware the radial and axial velocity, and Pis the pressure. Subscripts denote partial derivatives. In the chosen frame, neglecting the gas viscosity, we can approximate the axial speed of the gas as U⬁−Uo, which in our scaling amounts to We⬁ 1/2−We1/2. Therefore, the equations for the gas phase are 共rUg兲r r+共Wg兲z=0, 共6兲 ␳ ¯ 关共Ug兲t+共We⬁ 1/2 −We 1/2兲共Ug兲z兴=−共Pg兲r,共7兲 and ␳ ¯ 关共Wg兲t+共We⬁ 1/2 −We 1/2兲共Wg兲z兴=−共Pg兲z.共8兲 The hydrodynamic variables are subject to periodicity conditions in the axial direction Fz兩z=0 =Fz兩z=␭=0, 共9兲 W兩z=0 =W兩z=␭=0, 共10兲 Uz兩z=0 =Uz兩z=␭=0, 共11兲 where r=F共z,t兲is the interface equation. In addition, the regularity conditions at the axis demand U兩r=0 =Wr兩r=0 =Pr兩r=0 =0. 共12兲 The electric problem is modeled by the Laplace equation for both the inner potential ␾ iand the outer potential ␾ o ⵜ2 ␾ i=ⵜ2 ␾ o=0, 共13兲 subject to the radial boundary conditions ␾ o兩r=b=0 共14兲 and ␾ r i兩r=0 =0, 共15兲 and periodicity in the axial direction ␾ z i共r,z兲兩z=0,␭= ␾ z o共r,z兲兩z=0,␭=0 共16兲 for both the inner and outer potentials. At the interface r=F共z,t兲, the electrical boundary conditions are 共En o− ␤ En i兲兩r=F共z,t兲= ␴ e共17兲 and ␾ o兩r=F共z,t兲= ␾ i兩r=F共z,t兲,共18兲 where ␴ eis the surface charge density and En oand En iare the normal component of the electric field for the outer and inner domain, respectively. The normal and tangential components of the electric field at the interface can be written in terms of the potential as En o,i=1 共1+Fz 2兲1/2共− ␾ r o,i+Fz ␾ z o,i兲兩r=F共z,t兲共19兲 and Et=1 共1+Fz 2兲1/2共−Fz ␾ r o− ␾ z o兲兩r=F共z,t兲.共20兲 In addition, the kinematic conditions for the liquid and for the gas, the continuity equation of the surface charge density, and the stress balance in the normal and tangential direction read 共Ft−U+FzW兲兩r=F共z,t兲=0, 共21兲 兵Ug−U+共We1/2 −We ⬁ 1/2兲Fz其兩r=F共z,t兲=0, 共22兲 共 ␴ e兲t+FzU+W 1+Fz 2共 ␴ e兲z− ␣␤ En i − ␴ e 1+Fz 2兵Ur+Fz 2Wz−Fz共Uz+Wr兲其兩r=F共z,t兲=0, 共23兲 共P−Pg兲兩r=F共z,t兲−pce −2C 1+Fz 2兵Ur+Fz 2Wz−Fz共Wr+Uz兲其兩r=F共z,t兲=0, 共24兲 and C 1+Fz 2兵2Fz共Ur−Wz兲+共1−Fz 2兲共Wr+Uz兲其兩r=F共z,t兲 − ␹␴ eEt=0. 共25兲 The capillary-electric pressure term is defined as pce =1 共1+Fz 2兲1/2 冉 1 F−Fzz 1+Fz 2 冊 − ␹ 2关共En o兲2− ␤ 共En i兲2+共 ␤ −1兲共Et兲2兴.共26兲 Note that the electrification number ␹ weighs the relative importance of the capillary pressure jump 共destabilizing for long wavelengths兲against the normal component of 034106-6 López-Herrera, Riesco-Chueca, and Gañán-Calvo Phys. Fluids 17, 034106 共2005兲 Maxwell’s stress, thereby creating the appearance of an altered surface tension. It will be shown later that provided ␹ ⬎1 and b⬎2.718 28, a wavelength range emerges in the vicinity of k=0 where axisymmetric disturbances are stable. In our linear analysis, initial conditions are only used to define the unperturbed solution, as will be done in Eq. 共27兲. III. LINEAR ANALYSIS: 3D APPROACH A. Perturbed equations and boundary conditions In this section we describe the linear analysis of the problem considered. A similar approach has been adopted by Mestel,39,40 who analyzed the linear dynamics of charged jets forced by an axial electric field in the quasi-inviscid and highly viscous limits. In both works the jet is charged and subject to an external uniform axial electric field, the electrical relaxation time being a free parameter. Gamero-Castaño and Hruby30 studied filaments emitted by cone jets assuming small wave axisymmetric disturbances. Their results are based on the perfect-conductor simplification 共Ei=0, Et=0兲, an approximation implicitly requiring that ␣␤ →⬁; therefore, the application of their results to the honey-bubble limit 共charge-convection dominant, i.e., ␣ =0兲demands special caution. Hartman et al.41 studied small wave disturbances 共both axisymmetric and helical modes兲in a jet issuing from a nozzle cone with the help of a lowest-order 1D model for the slice velocity. Their analysis is restricted to the perfectconductor limit. On the other hand, González et al.7provided a thorough stability analysis of conducting jets under ac radial electric fields. A brief description of the dc case is made drawing on previous work by García.6In their paper, González et al.7take into account surrounding gas dynamics effects using a semiempirical model. Azimuthal perturbation modes 共nonaxisymmetric fluctuations兲are included in their description, and modal competition is shown to take place under the stimulus of the imposed ac frequency. However, their equations are restricted to the perfect-conductor limit. In the present paper, as indicated in the Introduction, only axisymmetric modes will be considered, a hypothesis generally satisfactory when only dc forcing is considered. Most research on the subject agrees in considering nonaxisymmetric modes as evanescent, i.e., either damped or displaying slower growth as axisymmetric disturbances 共see for instance Lin and Webb42兲. Our study concentrates on the influence of arbitrary viscosity, permittivity ␤ , and conductivity Kin the presence of a dc radial electric field. The interplay of these parameters is shown to be a key factor in the emergence of an interfacial boundary layer. In addition, the effect of electrification ␹ and electrode distance bis discussed. The temporal linear analysis looks out for solutions in the form of a small sinusoidal spatial perturbation over the static or basic solution. We take the perfect cylinder to be our static reference; therefore, solutions as follows are examined: 冢 U共z,r,t兲 W共z,r,t兲 P共z,r,t兲 Pg共z,r,t兲 ␾ o共z,r,t兲 ␾ i共z,r,t兲 ␴ e共z,t兲 F共z,t兲 冣 = 冢 0 0 1− ␹ /2 P⬁ ln共b/r兲 ln共b兲 1 1 冣 + 冢 u ˜ 共r,z,t兲 w ˜ 共r,z,t兲 p ˜ 共r,z,t兲 p ˜ g共r,z,t兲 ␾ ˜ o共r,z,t兲 ␾ ˜ i共r,z,t兲 ␴ ˜ e共z,t兲 f ˜ 共z,t兲 冣 ,共27兲 and the perturbation is written as 冢 u ˜ 共r,z,t兲 w ˜ 共r,z,t兲 p ˜ 共r,z,t兲 p ˜ g共r,z,t兲 ␾ ˜ o共r,z,t兲 ␾ ˜ i共r,z,t兲 ␴ ˜ e共z,t兲 f ˜ 共z,t兲 冣 =Re 冤 冢 u ˆ共r兲 w ˆ共r兲 p ˆ共r兲 p ˆg共r兲 ␾ ˆo共r兲 ␾ ˆi共r兲 ␴ ˆe f ˆ 冣 ␨ 冥 ,共28兲 where ␨ is equal to e⍀t+ikz,kbeing the nondimensional wavenumber ␲ /␭共␭is the half wavelength兲, and ⍀= ␴ + ␻ i is the complex eigenvalue involving the growth rate ␴ and the oscillation frequency ␻ . Hatted variables denote small amplitudes. Neglecting nonlinear terms of the Navier–Stokes equations we obtain ⵱·v ˜ =0 共29兲 and v ˜ t=− ⵱p ˜ +Cⵜ2v ˜ .共30兲 A separate equation for the pressure can be obtained by applying the divergence operator to 共30兲and using 共29兲; the same holds for the gas pressure: ⵜ2p ˜ =0, ⵜ2p ˜ g=0. 共31兲 On the other hand, taking the Laplacian of 共30兲leads to ⵜ2 冉 ⵜ2−1 C ⳵ ⳵ t 冊 v ˜ =0. 共32兲 Due to the linear character of the operators in the latter equation we can split the velocity into two terms v ˜ =v ˜ v+v ˜ nv, where v ˜ nvis the inviscid contribution and v ˜ vthe viscous contribution; the following equations are satisfied: ⵜ2v ˜ nv=0, 冉 ⵜ2−1 C ⳵ ⳵ t 冊 v ˜ v=0, v ˜ nvt=− ⵱p ˜ .共33兲 The perturbations comply with the regularity condition at the axis p ˜ 共r,z,t兲兩r=0 and w ˜ 共r,z,t兲兩r=0 finite and u ˜ 共r,z,t兲兩r=0=0. By Taylor expansion around r=1 at the interface, the kinematic conditions and the stress balance yield u ˜ 兩r=1 =f ˜ t,共34兲 034106-7 Linear stability analysis Phys. Fluids 17, 034106 共2005兲 共u ˜ −u ˜ g兲兩r=1 =ik共−We ⬁ 1/2 +We 1/2兲f ˜ ,共35兲 共p ˜ −p ˜ g兲兩r=1 =−f ˜ −f ˜ zz +2Cu ˜ r兩r=1 + ␹ 共f ˜ + ␾ ˜ r o兲兩r=1,共36兲 and C共u ˜ z+w ˜ r兲兩r=1 + ␹ ␾ ˜ z o兩r=1 =0. 共37兲 The outer and the inner electrical potential perturbations satisfy the Laplace equation ⵜ2 ␾ ˜ o,i=0 共38兲 with regularity conditions, ␾ ˜ i共r,z,t兲兩r=0 finite, ␾ ˜ o共r,z,t兲兩r=b=0. 共39兲 At the interface the inner and the outer potential are coupled through the normal electric field jump and the continuity of the potential 共− ␾ ˜ r o+ ␤␾ ˜ r i兲兩r=1 = ␴ ˜ e共40兲 and ␾ ˜ o兩r=1 = ␾ ˜ i兩r=1 共41兲 together with the continuity equation for the charge surface density 共 ␴ ˜ e兲t−u ˜ r兩r=1 =− ␣␤␾ ˜ r i兩r=1.共42兲 Substituting the solution 共27兲in the bulk equations 共31兲–共33兲 and 共38兲we obtain the ODE system which governs the amplitudes of the variables ⌿⬙+⌿⬘ r−k2⌿=0, 共43兲 w ˆv ⬙+w ˆv ⬘ r−kv 2w ˆv=0, 共44兲 u ˆnv ⬙+u ˆnv ⬘ r− 冉 k2+1 r2 冊 u ˆnv=0, 共45兲 u ˆv ⬙+u ˆv ⬘ r− 冉 kv 2+1 r2 冊 u ˆv=0, 共46兲 where ⌿stands for p ˆ,p ˆg,w ˆnv, and ␾ ˆo,i. The primes denote differentiation with respect to rand kvis defined as kv 2=k2+⍀ C.共47兲 The additive term ⍀/C⬃⍀ ¯ A2/ ␯ ⬃tv/tdis occasionally identified as a Reynolds number based on the problem’s unsteady character, ⍀ ¯ being the physical magnitude of ⍀;39 it can also be viewed as the ratio between the viscous diffusion time scale tvand the disturbance time scale td. In addition, the continuity and momentum equations read u ˆ⬘+u ˆ r+ikw ˆ=0, p ˆ⬘=−⍀u ˆnv,ikp ˆ+⍀w ˆnv=0. 共48兲 The gas momentum equation in the radial direction is ␳ ¯ 关⍀+ik共We⬁ 1/2 −We 1/2兲兴u ˆg=−p ˆg ⬘.共49兲 The boundary conditions closing the problem are u ˆ共0兲=0,w ˆ共0兲,p ˆ共0兲and ␾ ˆi共0兲finite, p ˆg共⬁兲=0; 共50兲 u ˆ共1兲=⍀f ˆ,u ˆg共1兲=u ˆ共1兲+ik共We⬁ 1/2 −We 1/2兲f ˆ,共51兲 p ˆ共1兲−p ˆg共1兲=共−1+k2兲f ˆ+2Cu ˆ⬘共1兲+ ␹ 关f+共 ␾ ˆo兲⬘共1兲兴, 共52兲 C关iku ˆ共1兲+w ˆ⬘共1兲兴 + ␹ ik ␾ ˆi共1兲=0, 共53兲 −f ˆ+ ␾ ˆo共1兲= ␾ ˆi共1兲,共54兲 −f ˆ−共 ␾ ˆo兲⬘共1兲+ ␤ 共 ␾ ˆi兲⬘共1兲= ␴ ˆe,共55兲 ⍀ ␴ ˆe−u ˆ⬘共1兲=− ␣␤ 共 ␾ ˆi兲⬘共1兲,共56兲 ␾ ˆo共b兲=0. 共57兲 B. The dispersion equation The general solution of the ODE set 共43兲–共48兲consists of a combination of modified Bessel functions. Recalling 共50兲and 共57兲, the solution reads p ˆ共r兲=−A⍀Io共kr兲 kI1共k兲,p ˆg共r兲=AgKo共kr兲 kK1共k兲,共58兲 u ˆ共r兲=u ˆnv共kr兲+u ˆv共kr兲=AI1共kr兲 I1共k兲−BI1共kvr兲 I1共kv兲,共59兲 and w ˆ共r兲=w ˆnv共kr兲+w ˆv共kr兲=AiIo共kr兲 I1共k兲−BikvIo共kvr兲 kI1共kv兲共60兲 for the hydrodynamic variables, and ␾ ˆo共r兲=Ae关Ko共kb兲Io共kr兲−Io共kb兲Ko共kr兲兴 k关Ko共kb兲I1共k兲+Io共kb兲K1共k兲兴 共61兲 and ␾ ˆi共r兲=BeIo共kr兲 kI1共k兲共62兲 for the outer and inner electrical potential, where A,Ag,B,Ae, and Beare real constants. Note that, under adequate scaling, the radial profiles of the pressure and the electric potential are identical. The substitution of the previous solutions in the boundary conditions at the interface 共50兲–共57兲and the elimination of the constants A,Ag,B,Ae, and Beleads to the dispersion relation 034106-8 López-Herrera, Riesco-Chueca, and Gañán-Calvo Phys. Fluids 17, 034106 共2005兲 D=⍀2 ␰ 共k兲+2C⍀关2k2 ␰ 共k兲−1兴+4k2C2关k2 ␰ 共k兲 −kv 2 ␰ 共kv兲兴 +2⌶共k兲C ⍀k2关2+G共k兲兴关k2 ␰ 共k兲−kv 2 ␰ 共kv兲兴 +⌶共k兲 冉 2k2 ␰ 共k兲+k2 ␰ 共k兲kv 2 ␰ 共kv兲G共k兲+1 G共k兲 冊 + 冉 1+⌶共k兲k2G共k兲 ⍀2关kv 2 ␰ 共kv兲−k2 ␰ 共k兲兴 冊 T共k兲=0, 共63兲 where the driving term T共k兲sums up all the interfacial forcing 共surface tension, aerodynamic effects, electric charging兲 and is defined as T共k兲=k2−1+ ␹ 冉 1+ 1 G共k兲 冊 − ␳ ¯ 关⍀+ik共We⬁ 1/2 −We 1/2兲兴2⌫共k兲,共64兲 while the following auxiliary functions are introduced: G共k兲=Ko共kb兲Io共k兲−Io共kb兲Ko共k兲 k关Ko共kb兲I1共k兲+Io共kb兲K1共k兲兴, ␰ 共k兲=Io共k兲 kI1共k兲, 共65兲 ⌶共k兲= ␹␰ 共k兲 E共k兲,E共k兲=G共k兲 ␤ 冉 1+ ␣ ⍀ 冊 − ␰ 共k兲,共66兲 and ⌫共k兲=− Ko共k兲 kK1共k兲.共67兲 ⌶共k兲is a combined electrification number, barely sensitive to b, condensing all the influence of ␣ and ␤ ; its magnitude increase as the conductivity or the permittivity decrease. ⌶共k兲becomes zero in the perfect-conductor limit, when either ␣ , ␤ , or their combination tend to ⬁.E共k兲is always negative. In the limit where bⰇ1, G共k兲→⌫共k兲. The leading term in Dis ⍀2 ␰ 共k兲, representing the effect of inertia, i.e., forces caused by the unsteadiness of the disturbance. In lowviscosity jets 共CⰆ1兲, this term becomes dominant and is balanced by the driving term alone. In the opposite limit 共CⰇ1兲, the driving term is balanced by the viscous terms alone 共second and third terms in D兲. The calculations leading to the dispersion relation are detailed in the Appendix. It can be readily verified that ␣␤ →⬁implies that Be →0, so that the inner electric field becomes vanishingly small. Assuming the disturbances are small, it follows as well that the tangential field Etbecomes negligible. Therefore, the perfect-conductor case can be recovered by selecting ␣␤ Ⰷ1. Our description of the influence of the surrounding gas mirrors the procedure introduced by González et al.;7in their paper they suggest a simple semiempirical correction, based on Sterling and Sleicher,43 allowing the description of the gas viscosity influence. Additionally, concerning the validity range of the theory, the work of Lin and Reitz25 is cited to justify the parametric range where the above applies: We ⬎4, ␳ ¯ We⬍6.5. These limits are empirically consistent for unelectrified jets only, but can be used as orientative landmarks in our case. The first restriction is aimed at avoiding absolute instability 共dripping mode兲; it sets a criterion for Rayleigh breakup. The second restriction excludes situations belonging to the second wind-induced breakup regime, where gas shear becomes important. However, the range of validity is not well studied in the case where the outer fluid is flowing co-axially with a speed U⬁. In spite of the resulting uncertainty, the dispersion equation 共63兲is to be trusted whenever dripping or second wind-induced breakup can be excluded. Figure 2 shows a characteristic stability map; the growth rate ␴ and the oscillation frequency ␻ are plotted against the wavenumber k. The jet is characterized by an Ohnesorge number C=0.1, Taylor number ␹ =0.6, relative permittivity ␤ =10, relaxation parameter ␣ =1.0, and ␤ =⬁. Aerodynamic effects are ignored. A single unstable branch is observed, corresponding to values of kranging from 0 to 1.167 03, associated with aperiodic growth of perturbations. The other branches are stable, some of them being oscillatory stable 共I, II, and III兲. There are four complex solutions associated to each wavenumber. In Fig. 3 the velocity fields and external electric fields 共vectorial兲are depicted for each one of the solutions; k=0.5, a regime where all behaviors are aperiodic 共⍀is a real number兲. We observe that 共i兲the electric field is always symmetrical with respect to the wavelength; 共ii兲the only unstable mode is essentially capillary 共A兲while the others are stable. The main stabilizing force in mode B is the surface tension. Modes C and D are characterized by strong recirculations of the liquid where the viscous term balances the inertial term, the capillary term being negligible. An even number of roots are observed. The problem dealt with by Hohman et al.44 displays a distinctive difference compared to our problem: the presence of an axial, rather than radial, electric field. The dispersion relation obtained by them is cubic; accordingly, they find three modes; on the other hand, the radial geometry of our electric forcing leads to a fourth-order dispersion relation 共four modes兲. A salient physical difference between the two problems is that the first one displays antisymmetric electric charges in FIG. 2. Solution of the dispersion relation 共 ␣ =1, ␤ =10, C=0.1, and ␹ =0.6兲. 034106-9 Linear stability analysis Phys. Fluids 17, 034106 共2005兲 共b兲However, from a physical point of view it is not so obvious that a radial electric field will also give rise to a boundary layer. It is here shown that this boundary layer will only become conspicuous when the liquid is a poor conductor. This is a different result, and the following observation is added: the distance at which the electrode is placed gives rise to diverse velocity profiles, with the interface either running faster or slower than the core, while the core velocity profile is insensitive to conductivity and permittivity variables. 共c兲This type of boundary layer is based on the competition between the oscillatory acceleration and the viscous term, whereas the axial field boundary layer is based on the balancing of axial stress and viscous term. As such, there is no entrance length nor is there any downstream thickening of the layers; they do not merge. A uniform thickness will be observed in all the lengths of the jet. IV. 1D APPROACH: THE AVERAGE AND THE PARABOLIC APPROXIMATIONS One-dimensional models are based on the substitution of a simplified form of the velocity components 共a truncated Taylor series in the radial coordinate兲into the Navier–Stokes equations. To lowest order in r, such methods provide a simplified set of slice equations, widely used in the qualitative analysis of jets. As indicated above, the axial velocity profile tends to become flat as kapproaches zero. This is the conceptual basis allowing for an expansion in powers of r. However, higher-order approximations 共parabolic兲are reasonably accurate even for k⬃1 provided the real velocity profile does not exhibit singular patterns such as boundary layers. Therefore, 1D models fail to converge whenever a boundary layer is present. Such is the case of slightly viscous jets, under the condition that ␣␤ C1/2 is, at most, of order 1. The thickness of the boundary layer, assuming longwavelength disturbances, is approximately 关C/B1共b, ␹ 兲k兴1/2 when b⬎bo共 ␹ 兲and 关C/B2共b, ␹ 兲k2兴1/2 when b⬍bo共 ␹ 兲. This implies that, in realistic problems 共where Ccannot be equal to zero兲, given any value of CⰆ1, a range of kvalues close to the zero limit can be defined where the boundary layer vanishes and the 1D approximation holds. In spite of this, 1D models are scarcely useful in the CⰆ1 range, because, in the range of wavelengths where maximum growth is observed 共ca. ␲ /km兲, a distinct boundary-layer pattern is observed. García and Castellanos49 carried out a detailed examination of the equations and their range of validity in the 1D modeling of unelectrified jets. However, their results cannot be extrapolated to charged jets without a general reassessment of the problem. In this section we derive one-dimensional expressions for the conservation equations. We adopt the slice charge S =共1+Fz2兲1/2F ␴ erather than the surface charge density ␴ e.S represents the charge per unit of dimensionless length z. Rewritten in terms of S, the charge continuity equation 共23兲 becomes St+FzU+W 1+Fz 2Sz− ␣␤ F共1+Fz 2兲1/2En i −S F共1+Fz 2兲2共1+3Fz 2+2Fz 4+FFz 2Fzz兲U +SFz 3 F共1+Fz 2兲2共1+Fz 2+FFzz兲W −S 1+Fz 2共Ur−FzWr兲=0. 共83兲 The one-dimensional approximation is obtained by replacing the 3D velocities U,W, and pressure Pby W共r,z,t兲=wo共z,t兲+1 2r2w2共z,t兲+¯,共84兲 U共r,z,t兲=−1 2rwo ⬘共z,t兲−1 8r3w2 ⬘共z,t兲+¯,共85兲 P共r,z,t兲=po共z,t兲+1 2r2p2共z,t兲+¯,共86兲 where axial derivatives are denoted as primes; time derivatives will be indicated with dots. The tangential stress balance at the interface 共25兲is the key equation linking momentum and electric equations. It is from this equation that the convergence of the velocity expansion can be checked. To that end, we observe that axial derivatives are of order k: ⳵ / ⳵ z⬃␭−1⬃k. Therefore, Fw2−3wo ⬘F⬘−1 2Fwo ⬙+1 6F3w4 − 再 1 8F3w2 ⬙+7 4F2F⬘w2 ⬘+2FF⬘2w2−3F⬘3wo ⬘ −FF⬘2wo ⬙ 冎 +O共k3兲 = ␹ SEt FC共1+F⬘2兲1/2 .共87兲 To leading order, it follows that w2⬇ ␹ SEt/CF2. Note that the electric term in Eq. 共87兲is left intact while the hydrodynamic part is approximated. This uneven treatment is customary in the literature. By keeping the interfacial electric driving terms unexpanded, an improved accuracy is obtained. The same procedure will be applied to the capillaryelectrical pressure term in the momentum equation below. In addition, from the Laplace equation in the jet and under the assumption that the main electric forcing is radial, the tangential field 共physical magnitude兲can be estimated as E ¯ t⬃Eo/␭, so that Et⬃k. It can be inferred that w2/wo⬃k, w4/wo⬃k2, and so on. However, as will be shown later, p2/po⬃k2. This is a weaker convergence as in unelectrified problems, where it can be shown49 that w2/wo⬃k2. In fact, k2 convergence is recovered in our case provided the liquid is a perfect dielectric 共S=0兲or a perfect conductor 共 ␣␤ Ⰷ1兲.In both cases, the right-hand side of Eq. 共87兲becomes zero and the tangential stress balance is the same as in the unelectrified limit. In all other cases, kconvergence will be observed, and this explains why, in some situations involving poor con- 034106-16 López-Herrera, Riesco-Chueca, and Gañán-Calvo Phys. Fluids 17, 034106 共2005兲 ductors, lowest-order 1D models fail to give accurate approximations and at least second-order truncations are required. Note that the above equation includes three terms in curly brackets which can be neglected to order 共k3兲. These terms have been retained just for the sake of comparing results with available higher-order expressions in the unelectrified limit.49 Indeed, when ␹ =0, Eq. 共87兲is accurate with errors O共k6兲. A. Parabolic or O „ k 3…approximation In this section, the 1D model is taken to order k3by keeping the first two terms in the velocity expansion. This amounts to modeling the axial velocity as a parabolic profile. The continuity equation of the surface charge, the stress balance in the normal direction, and the kinematic condition read S ˙+ 关共 wo+1 2F2w2 兲 S 兴 ⬘= ␣␤ F共1+F⬘2兲1/2En i+O共k3兲, 共88兲 po+1 2F2p2=pce −C 共 wo ⬘+3 4F2w2 ⬘+2FF⬘w2 兲 +兵C共FF⬘wo ⬙+3F⬘2wo ⬘兲其 +O共k3兲,共89兲 F ˙+ 共 wo+1 2F2w2 兲 F⬘+1 2Fwo ⬘+1 8F3w2 ⬘+O共k3兲;共90兲 the pressure equation includes O共k3兲terms in curly brackets, which are retained with the same purpose as before. These equations are accurate to O共k4兲in the unelectrified limit. The capillary-electrical pressure term is defined as in Eq. 共26兲. The momentum equations follow from inserting the velocity r-expansions and equating terms with the same power in r. To order O共k2兲only three equations are required 共r0,r2 terms in the axial direction; leading term in the radial direction兲: w ˙o+wowo ⬘=−po ⬘+C共wo ⬙+2w2兲,共91兲 w ˙2+wow2 ⬘=−p2 ⬘+C 共 w2 ⬙+4 3w4 兲 ,共92兲 w ˙o ⬘+wowo ⬙−1 2wo ⬘2=2p2+C共wo ⵮+2w2 ⬘兲.共93兲 The interfacial stress balance equations are fed into the momentum equations and the pressure components po,p2,as well as w4, are eliminated. As a result, two condensedmomentum equations are obtained in replacement of Eqs. 共87兲,共89兲, and 共91兲–共93兲: w ˙o+wowo ⬘− 兵 1 4F2 关 w ˙o ⬘+wowo ⬙−1 2wo ⬘2 兴其 ⬘ =−pce ⬘+C 兵 共2−4F⬘2−FF⬙兲wo ⬙ +2共1+F⬘2+FF⬙兲w2−1 4F2wo iv−3 2FF⬘wo ⵮ +1 4F2w2 ⬙+5 2FF⬘w2 ⬘−6F⬘F⬙wo ⬘ 其 ,共94兲 w ˙2+wow2 ⬘+1 2w ˙o ⬙+1 2wowo ⵮ =8 ␹ SEt F4共1+F⬘2兲1/2 +C F4 再 4F2共1−2F⬘2兲wo ⬙ +24FF⬘共1−F⬘2兲wo ⬘+1 2F4wo iv 冎 +C F4兵−8F2共1−2F⬘2兲w2+3F4w2 ⬙+14F3F⬘w2 ⬘其. 共95兲 Up to this point, the equations hold for arbitrary electric field, i.e., axial field, radial field, or other situations are encompassed in the model. Therefore, the parabolic 1D equations above can be used in a wide variety of electrohydrodynamic problems. Next, we assume small departures of the electric variables with respect to the static solution 共woand w2are intrinsically small兲; in addition, the radial electric field assumption is introduced: 冢 ␾ o共z,r,t兲 ␾ i共z,r,t兲 S共z,t兲 F共z,t兲 冣 = 冢 ln共b/r兲 ln共b兲 1 1 冣 + 冢 ␾ ˜ o共r,z,t兲 ␾ ˜ i共r,z,t兲 S ˜ 共z,t兲 f ˜ 共z,t兲 冣 ,共96兲 and the perturbation is written as 冢 wo共z,t兲 w2共z,t兲 ␾ ˜ o共r,z,t兲 ␾ ˜ i共r,z,t兲 S ˜ 共z,t兲 f ˜ 共z,t兲 冣 =Re 冤 冢 −iw ˆo −iw ˆ2 ␾ ˆo共r兲 ␾ ˆi共r兲 S ˆ f ˆ 冣 ␨ 冥 ,共97兲 where ␨ is equal to e⍀t+ikz. In addition, the electric boundary conditions are imposed 共here, primes denote radial derivatives兲: −f ˆ+ ␾ ˆo共1兲= ␾ ˆi共1兲,共98兲 −共 ␾ ˆo兲⬘共1兲+ ␤ 共 ␾ ˆi兲⬘共1兲=S ˆ,共99兲 ␾ ˆo共b兲=0. 共100兲 The inner and outer electric fields are exactly modeled as in the 3D description 关see Eqs. 共61兲and 共62兲兴. In addition, the condensed-momentum equations, the charge conservation equation, and the kinematic condition are linearized. The result is a linear system M2·x=0, where x =共w ˆo,w ˆ2,f ˆ,Ae,Be,S ˆ兲Tand M2isa6⫻6 matrix: 034106-17 Linear stability analysis Phys. Fluids 17, 034106 共2005兲 冢 4kk8⍀000 Ck2共8+k2兲+⍀共4+k2兲C共−8+k2兲4k共1−k2− ␹ 兲−4k ␹ 00 C共8k2−k4兲−k2⍀2关C共8+3k2兲+⍀兴16k ␹ −16kG ␹ 00 000−1 ␤ −1 00−1 G− ␰ 0 kk/2 00 ␣␤ ⍀ 冣 ,共101兲 the rows of M2containing the kinematic condition, the wo and w2momentum equations, the jump in the normal electric field at the interface, the continuity of potential at the interface and the charge conservation equation. The dispersion equation, D2=0, is readily obtained by setting the determinant of M2equal to zero. B. Slice-average approximation A more simplified 1D model is widespread in the literature; in this model, the tangential stress balance is truncated with errors O共k2兲: Fw2− 再 3wo ⬘F⬘+1 2Fwo ⬙ 冎 +O共k2兲= ␹ SEt FC共1+F⬘2兲1/2 ,共102兲 implying that w2=O共k兲. Again, higher-order terms, in curly brackets, have been kept with the purpose of preserving a consistent equation valid with errors O共k4兲in the unelectrified case, where w2=O共k2兲. A single condensed-momentum equation is obtained: w ˙o+wowo ⬘=−pce ⬘+3C共F2wo ⬘兲⬘ F2+2 ␹ SEt F2共1+F⬘2兲1/2 +O共k2兲. 共103兲 The kinematic condition is truncated at O共k2兲. The remaining equations, boundary conditions, and linearization model are identical as in the parabolic case. The average linear system is now M1·y=0, where y=共w ˆo,f ˆ,Ae,Be,S ˆ兲T and M1isa5⫻5 matrix: 冢 k2⍀000 −共⍀+3Ck2兲k共−1+k2+ ␹ 兲k ␹ 2k ␹␰ 0 00−1 ␤ −1 0−1 G− ␰ 0 k00 ␣␤ + ␹ k2 ␰ /2C⍀ 冣 ,共104兲 the rows of M1containing the kinematic condition, the wo momentum equation, the jump in the normal electric field at the interface, the continuity of potential at the interface and the charge conservation equation. The dispersion equation is D1=det共M1兲=0. This equation can be written in analogy to the 3D- dispersion equation. D1=2 k2⍀2+6C⍀+⌶1共k兲 冉 4+4G共k兲+1 G共k兲 冊 +T共k兲=0, 共105兲 where the auxiliary functions G共k兲and ␰ 共k兲and the driving term T共k兲are defined as in the 3D-dispersion equation, and ⌶1共k兲= ␹␰ 共k兲 E1共k兲,共106兲 E1共k兲=G共k兲 ␤ 冉 1+ ␣ ⍀ 冊 − ␰ 共k兲+ ␹ G共k兲 ␰ 共k兲k2 2⍀C.共107兲 Apart from the divergent ⌶definition, the average dispersion equation, D1=0, can be easily deduced from the general one, Eq. 共63兲, by assuming kⰆ1, kvⰆ1, and observing that k2 ␰ 共k兲→2 when k→0. The same simplifications as in the 3D case can be obtained in different parametric extremes. For instance, assuming low viscosity 共CⰆ1兲, D1=2 k2⍀2+T共k兲=0; 共108兲 an identical expression is obtained when the jet is a perfect conductor 共 ␣␤ C1/2Ⰷ1兲. In the opposite limit 共CⰇ1兲, the positive real root of the dispersion equation is ␴ =−T共k兲 6C.共109兲 034106-18 López-Herrera, Riesco-Chueca, and Gañán-Calvo Phys. Fluids 17, 034106 共2005兲 C. Assessment of the validity of 1D models The simplifications inherent to 1D models tend to become unacceptable whenever the axial velocity profile departs significantly from the flat or parabolic curve. Such is the case of low-Cjets, particularly when ␣ and ␤ are small or moderate. Figure 12 shows the scaled difference between the 3D and 1D axial velocity 关w3D共r兲−w1D共r兲兴/w3D共r兲at the interface 共r=1兲and at the axis 共r=0兲in the average and parabolic approximation, for different values of kand ␣ 共 ␤ =3, C=0.1, ␹ =0.6, bⰇ1兲. Provided ␣ is small enough, significant errors can be noticed in the average model. These errors are smoothed out with the parabolic approximation. The maximum relative error Max兵关w3D共r兲 −w1D共r兲兴/w3D共r兲其 of the 1D approximation at r=0 and r=1, for arbitrary kand different values of Cand ␣ C1/2 共 ␤ =3, ␹ =0.6, bⰇ1兲is shown in Fig. 13. Considerable errors can be observed in the low-Climit, but the parabolic approximation achieves a better fit. Figure 14 shows different radial velocity profiles for variable ␣ C1/2. The 3D solution exhibits a sharp boundarylayer pattern, which becomes abortive as ␣ C1/2→⬁. The parabolic approximation tries to adhere to the interfacial velocity, and fails to describe intermediate values of r. Nevertheless, since the key physical phenomena take place at the interface, the parabolic model will be substantially more accurate than the average model, given its fidelity to the interfacial speed. Finally, Fig. 15 shows growth rate curves as obtained with the 3D equations and compares them to the average and parabolic 1D approximations. In the absence of a boundary layer 共C=1兲, the parabolic approximation ensures improved accuracy, particularly in the long-wavelength range. However, very low Ohnesorge numbers give rise to a double peak axial velocity profile, and both orders of approximation fail to describe accurately the growth behavior. V. CONCLUSIONS The present paper can be considered as an investigation on the shortcomings of the perfect-conductor hypothesis in electrified jets. This is a widely used assumption in most of the electrospray literature; with perfect conductors, the electric stress is perpendicular to the interface, a hypothesis that FIG. 12. Scaled difference between the 3D and 1D axial velocity 关w共r兲−w1D共r兲兴/w共r兲at the interface 共r=1兲and at the axis 共r=0兲in the average and parabolic 共dashed line兲approximation, for different values of kand ␣ 共 ␤ =3, C=0.1, ␹ =0.6, bⰇ1兲. FIG. 13. Maximum relative error of the parabolic 1D approximation at the axis 共r=0兲and at the interface 共r =1, dashed line兲Max兵关w3D共r兲−w1D共r兲兴/w3D共r兲其, arbitrary k, plotted for different values of ␣ and ␣ C1/2 共 ␤ =3, ␹ =0.6, bⰇ1兲. 034106-19 Linear stability analysis Phys. Fluids 17, 034106 共2005兲 cannot be preserved when dealing with leaky dielectrics.9In addition, lowest-order 1D models, assuming a flat axial velocity profile, are assessed and found defective whenever C is sufficiently small 共an interfacial boundary layer emerges兲 or in cases where the real velocity profile is sharply convex. This latter situation is enhanced by finite permittivity and conductivity liquids. Moderate and low ␣ and ␤ are shown to be abundant in the literature and applications of electrified jets. In such cases, the routine adoption of an average model may be misleading, and parabolic or 3D models become mandatory. The following main conclusions are drawn. 共a兲A general dispersion equation has been obtained for axisymmetric perturbations modeling the influence of conductivity, permittivity, electric field, viscosity, surface tension, electrode distance, and ambient velocity. Different parametric limits are explored: longwavelength, high and low viscosity, charge-convection dominant or negligible. The growth factor is insensitive to ␣ and ␤ in the low-Climit. In the general case, as ␣ and ␤ grow, the perfect-conductor limit is recovered. 共1兲When the viscosity is sufficiently small, CⰆ⍀, i.e., the viscous diffusion time scale tvis much longer than the disturbance time scale td, an interfacial boundary layer, driven by electric shear, is observed. At the inside, the boundary layer borders with the inviscid jet core, where the axial velocity profile follows a self-similar trend, insensitive to all electric parameters ␣ , ␤ , and ␹ . In spite of the increased velocity gradient at the boundary layer, viscous stresses become vanishingly small in the C→0 limit. The occurrence of a boundary layer is a key event in interface-driven jets, and it is only through finite conductivity and permittivity that the boundary layer comes to be fully deployed. 共2兲Depending on the location of the ground electrode, the boundary layer either runs faster than the core 共b⬎b1 ⬇e1/2兲or lags behind it 共b⬍b1兲. Substantial velocity differences can be observed, particularly for shortwavelength disturbances: therefore caution is required when modeling surface charge transport. 共3兲Liquids with high conductivity or high dielectric constant 共 ␣␤ C1/2Ⰷ1兲give rise to an electric field perpendicular to the interface 共zero-tangential field兲and therefore weaken the boundary layer. In the perfect-conductor limit, the boundary-layer velocity jump drops to zero. This may explain why the boundary layer has been disregarded in most of the state-of-the-art studies on the subject. 共4兲Inviscid jets cannot balance electric shear stresses at the interface and therefore behave as perfect conductors. The emergence of a boundary layer gives rise to an axial velocity profile that cannot be modeled accurately with low-order expansions in terms of the radius. 共5兲Increasing convexity in the axial velocity profile is observed when ␣ or ␤ decrease, inviting adoption of higher-order expansions rather than average models. 共b兲Slice equations, based on the approximation of the axial velocity profile as an expansion in powers of the radius, are obtained to lowest order 共average 1D兲and second order 共parabolic 1D兲. The expressions obtained hold for arbitrary electric field. 共1兲Significant errors introduced by the lowest-order 1D approximation 共widely used in the literature兲may be masked by the prevalence of large conductivity and per- FIG. 14. Radial profiles of the axial velocity, scaled with the average axial velocity, for several values of ␣ C1/2. The 3D solution, exhibiting a clear boundary-layer pattern for moderate or small ␣ C1/2, ranging from 0.001 to 10, is shown together with the parabolic 1D approximation 共dashed line兲: bⰇ1, ␤ =3, k=0.7, ␹ =0.6, and C=0.0001. FIG. 15. Growth rate ␴ vs the wavenumber kfor a jet with bⰇ1, ␣ =0.001, ␤ =3, ␹ =0.6, and several Ohnesorge numbers C=1, C=0.0001. The average and parabolic approximations are plotted as dot curves. 034106-20 López-Herrera, Riesco-Chueca, and Gañán-Calvo Phys. Fluids 17, 034106 共2005兲 mittivity liquids in most reported studies. A realistic assessment of ␣ and ␤ leads to questioning the validity of average approximations, especially in their description of interfacial magnitudes. In effect, the accuracy of the interfacial velocity as described by the 1D average model is poor in a wide parametric range provided ␣ and ␤ are moderate. ACKNOWLEDGMENTS This work was partially supported by the Spanish Ministry of Science and Technology under Grant No. DPI2002- 04305-C02-02. One of the authors 共P.R.-C.兲wishes to thank the regional Government of Andalusia, for the support received through a Return Grant No. 72/2003. APPENDIX: DERIVATION OF THE DISPERSION RELATION The dispersion relation can be routinely obtained by equating to zero the 7⫻7 determinant of the algebraic equations corresponding to the boundary conditions at the interface 共50兲–共57兲. However, arriving at the simplified expression reported in this paper is not straightforward. A brief note on the procedure followed 共in the simplified case where the gas influence can be neglected兲is therefore appended for reference: −f ˆ+AeG共k兲=Be ␰ 共k兲,共A1兲 ␴ ˆe=−共f ˆ+Ae兲+ ␤ Be,共A2兲 ⍀ ␴ ˆe=− ␣␤ Be+A关k2 ␰ 共k兲−1兴−B关kv 2 ␰ 共kv兲−1兴,共A3兲 A−B−⍀f ˆ=0, 共A4兲 A兵⍀ ␰ 共k兲+2C关k2 ␰ 共k兲−1兴其 −B兵2C关kv 2 ␰ 共kv兲−1兴其 +共−1+k2兲f ˆ+ ␹ 共f ˆ+Ae兲=0, 共A5兲 and A−B 冉 1+ ⍀ 2k2C 冊 + ␹ 2C ␰ 共k兲Be=0, 共A6兲 where the auxiliary functions Gand ␰ are defined as in 共65兲. Using Eqs. 共A1兲–共A3兲,Aeand Becan be expressed as a function of Aand B Ae=f ˆ E 冋 ␤ 冉 1+ ␣ ⍀ 冊 + ␰ 共k兲 册 + ␰ 共k兲 E⍀兵A关k2 ␰ 共k兲−1兴 −B关kv 2 ␰ 共kv兲−1兴其 共A7兲 and Be=f ˆ E关1+G共k兲兴 +G共k兲 E⍀ ⫻兵A关k2 ␰ 共k兲−1兴−B关kv 2 ␰ 共kv兲−1兴其,共A8兲 where Eis defined as in 共66兲. Substituting these expressions in 共A5兲and 共A6兲we get A 冋 ⍀ ␰ 共k兲+ 冉 2C+⌶共k兲 ⍀ 冊 关k2 ␰ 共k兲−1兴 册 −B 冋 冉 2C+⌶共k兲 ⍀ 冊 关kv 2 ␰ 共kv兲−1兴 册 +f ˆ 冋 k2−1+关 ␹ +⌶共k兲兴 冉 1+ 1 G共k兲 冊 册 =0 共A9兲 and A 冋 1+⌶共k兲 2C⍀G共k兲关k2 ␰ 共k兲−1兴 册 −B 冋 1+ ⍀ 2k2C+⌶共k兲 2C⍀G共k兲关kv 2 ␰ 共kv兲−1兴 册 +f ˆ⌶共k兲 2C关G共k兲+1兴=0, 共A10兲 where ⌶is defined as in 共66兲. Aand Bcan be expressed in terms of f ˆusing 共A10兲and 共A4兲 A=f ˆ⍀+2k2C+⌶共k兲 ⍀k2关1+G共k兲kv 2 ␰ 共kv兲兴 1+⌶共k兲 ⍀2k2G共k兲关kv 2 ␰ 共kv兲−k2 ␰ 共k兲兴 共A11兲 and B=f ˆ2k2C+⌶共k兲 ⍀k2关1+G共k兲k2 ␰ 共k兲兴 1+⌶共k兲 ⍀2k2G共k兲关kv 2 ␰ 共kv兲−k2 ␰ 共k兲兴 ,共A12兲 and substituting both 共A11兲and 共A12兲in 共A9兲, the dispersion relation 共63兲is obtained. 1J. R. Melcher, Field-Coupled Surface Waves 共MIT Press, Cambridge, MA, 1963兲. 2D. A. Saville, “Stability of electrically charged viscous cylinders,” Phys. Fluids 14, 1095 共1971兲. 3A. L. Huebner and H. N. Chu, “Instability and break up of charged liquid jets,” J. Fluid Mech. 49, 361 共1971兲. 4E. R. Setiawan and S. D. 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