Dynamics of EHD Laminar Plumes: scaling analysis and integral model

Dynamics of EHD Laminar Plumes: scaling analysis and integral model P.A. V´azquez1,2,A.T.P´erez2, A. Castellanos2and P. Atten3 1Departamento de F´ısica Aplicada, E.S.I., Avda. de los Descubrimientos, s/n, 41092 Sevilla, Spain, e-mail:[email protected], 00-34-5-4486192, 00-34-5-4486003 (Fax) 2Departamento de Electr´onica y Electromagnetismo, Facultad de F´ isica, Avda. Reina Mercedes s/n, 41012, Sevilla, Spain 3Laboratoire d’Electrostatique et Materiaux di´electriques (C.N.R.S), Grenoble, France In this paper electrohydrodynamic (EHD) plumes are examined in the region far from the injecting electrode and the collector plate, for both two-dimensional and axisymmetric geometries. The relative importance of the conduction mechanisms (convection, drift and diffusion of electric charge) is analyzed. Diffusion turns out to be negligible compared to convection and drift for the experimental conditions. But the transverse drift (Coulomb repulsion) is of the same order of magnitude than convection. We find a set of three differential equations giving the evolution of the velocity at the center of the plume and the widths of the plume and the charged core inside. I. INTRODUCTION Injection of charge from an electrode into an insulating fluid defines a class of EHD flows. The Coulomb force, f=qE, that the electric field, E, exerts on the space charge, q, induces a fluid motion which contributes to convect the charge. There is a drastic difference between gases and liquids. In gases the ion drift velocity is much higher than the gas flow induced by the Coulomb force, so the ions practically move along the electric field lines. The converse occurs in liquids where the typical fluid velocity is higher or much higher than the ion drift velocity1. The flow induced by the Coulomb force drastically affects the charge density distribution which in turn modifies the velocity. As a consequence there is a strong coupling between the charge density distribution and the velocity field. We consider here the case of electrode geometries where the injection of charge arises from sharp electrodes. In these cases the contrast between gases and liquids is even more marked. In liquids the localized injection induces a flow structure similar to those of thermal plumes with a tendency to confine the charge. Several geometries can be considered, but in this paper we focus our attention on the two-dimensional and axisymmetric cases. The former arises when the sharp electrode is a blade or a wire, while the latter appears when the injecting electrode is a point or a needle. The resulting flow in the region far from the electrodes (we will call this zone the asymptotic region) is similar to that of thermal plumes arising from line or point sources of heat. Here only the case of laminar flow in liquids is examined. Thermal plumes have been thoroughly studied2,3. The flow in the asymptotic region, for Prandtl numbers, Pr 1, (Pr =ν/κ is the Prandtl number, νthe kinematic viscosity of the fluid and κthe thermal diffusivity) has a double boundary layer structure, with an inner thermal layer, of width δT(x), and an outer hydrodynamic layer, of width δl(x), representing the volume of fluid put into motion (xis the coordinate along the flow main direction). These thermal plumes can be described in terms of self-similar solutions. The physical quantities of the plume scale with x, and the partial differential equations can be converted into a set of ordinary differential equations in a self-similar variable, η=y/δl(x), with ythe transverse coordinate. Several attempts have been made to apply this technique to the EHD problem. The first study was carried out by Zhakin4. In his paper, the author neglects the contribution of Coulomb repulsion in the evolution of the charged zone, retaining the diffusive term. In this way, the problem turns out to be exactly analogous to the thermal one, with the charge density qplaying the role of temperature. Self-similar solutions for both two-dimensional and axisymmetric EHD plumes were given. P´erez et al5showed that Coulomb repulsion is dominant compared to diffusion, at least in the usual experimental conditions. V´azquez et al6analyzed both thermal and EHD plumes emphasazing their analogies and differences. They showed that the equations are similar to those of the thermal plumes in the limit Pr →∞, when diffusion of charge, drift and Coulomb repulsion are neglected compared to charge convection. From a physical point of view, the width of the inner charged layer (the heated layer for the thermal plumes) is zero in this limit. Obviously, this is not a complete solution of the problem, for in that case both Coulomb repulsion and diffusion of charge would spread the charged region. Even more, for the two-dimensional geometry the velocity of the fluid in the plane of symmetry tends to a finite value for Pr →∞ 7,8, but for the axisymmetric case the velocity in the axis diverges as √ln Pr 6. What really happens in practical situations is that the plume enters the similarity zone with a charged region of finite width, δq, due to the finite radius of the injecting electrode and to the effect of Coulomb repulsion in its vicinity9. To avoid this singularity, an effective Prandtl number was defined, Pr ef =(δl/δq)2,taking 1 into account that the Prandtl number gives the relation between the widths of the thermal and hydrodynamic layer, Pr =(δl/δT)2. Takashima et al.10 have done a numerical simulation for both two-dimensional and axisymmetric EHD plumes. They used finite differences to calculate the velocity field and the charge simulation method to determine the electric field. With the aim of reducing the computing time, they imposed a constant width of the charged layer, a rather drastic assumption not justified in the paper. In this way they obtained numerical values of the velocity and a relation between the potential difference and the current that they claim agree with their experiments. However, the underlying physical mechanisms are not clear in this numerical simulation, and the application of the calculations to other set of experimental data is not straightforward. An exact numerical solution of the whole problem is an extremely difficult task, because of the different length scales involved and the coupling between the electric field, the charge distribution and the velocity field. Integral methods can give valuable insights in this difficult problem. McCluskey and P´erez11 used these methods for the first time. Imposing several hypotheses obtained from the experimental observation of EHD plumes, and neglecting diffusion of charge and Coulomb repulsion, they obtained expressions for the velocity in the symmetry plane of the plume and the width of the hydrodynamic layer in the asymptotic region. It must be underlined that these expressions are similar to those obtained by Zhakin. An order of magnitude analysis has been performed by Malraison et al.12 for axisymmetric and turbulent plumes following the same approach. In the two-dimensional case, and for weak currents, the effect of space charge is negligible in comparison with convection under typical experimental conditions. In the axisymmetric case this is not true. Atten et al.13 considered the problem of EHD axisymmetric plumes including the effect of Coulomb repulsion with the use of integral methods. Making several hypotheses about the velocity profile, they obtained a set of differential equations for the velocity on the axis and the widths of the charged and hydrodynamic layers. However, their analysis is not complete and the results depend on an extra assumption concerning the ratio of the charged and hydrodynamic layers. In this paper, we focus our attention in the behavior of laminar EHD plumes, for both 2D and axisymmetric geometries. We discuss in detail the relative importance of the three mechanisms of transport of electric charge in plumes: diffusion, drift (including Coulomb repulsion) and convection. It is shown that in the charged core and along the transverse coordinate, Coulomb repulsion is of the same order of magnitude than charge convection and cannot be neglected. The inclusion of Coulomb repulsion in the equations prevents them from having self-similar solutions. We extend the integral methods, used by Atten et al., including now the global conservation of energy. In this way, a set of differential equations is found for the velocity at the center of the plumes and the widths of the charged and hydrodynamic layers. In these equations the explicit dependence of the longitudinal electric field with xappears. Therefore, we have to provide a function E(x), along with the corresponding initial conditions. We have solved numerically the obtained equations for an imposed electric field and different sets of initial conditions. Similar values are obtained for the magnitudes involved in all cases. II. FORMULATION OF THE PROBLEM A. Hydrodynamic equations We consider the steady electrohydrodynamic flow occurring between a blade or a needle and a plate a distance d apart. In the first case, the structure of the flow is 2D, and it is axisymmetric in the second case. The flow takes the form of a plume, originated at the point or the blade, that impinges upon the plate. The velocity in the zone of recirculation is much smaller than in the plume. For this reason one can analyze the motion considering a plume-like flow in a non-moving ambient liquid. This overall structure is sketched in figure 1. We examine the problem in the asymptotic region, that is, far from both the injector and the plate. In this asymptotic region the flow has a double boundary-layer structure, with an inner charged core of typical scale δq,and an outer hydrodynamic layer, of typical scale δl. This layer represents the volume of liquid put into motion by the electric forces. We have δq,δ ld. The usual approximations in boundary-layer analysis are applicable here. For the Navier-Stoke equation the transverse derivatives are much higher than longitudinal derivatives, and the ambient pressure, p0(x), is impressed in the boundary layer (in our case p0is independent of x, as a consequence of neglecting recirculation, and therefore the pressure gradient is negligible). As a consequence the mechanical equations reduce to ∂ ∂x yku+∂ ∂y ykv=0,(1a) u∂u ∂x +v∂u ∂y =ν yk ∂ ∂y yk∂u ∂y+qEx ρ,(1b) 2 with k= 0 for the 2D case and k= 1 for the axisymmetric one. Here, xand yare the longitudinal and transverse coordinates (in the case of axisymmetric plumes yis the radial coordinate), uand vare the longitudinal and transverse components of the liquid velocity, qis the density of charge, Exis the longitudinal electric field, and ρand νare the density and the kinematic viscosity of the liquid, respectively. As usual in boundary layer analysis, the conservation equation of the transverse component of momentum can be ignored14. Following the usual analysis for thermal plumes2, we may derive estimates for the longitudinal velocity, U,andδl by stating that the inertial, viscous and force terms are of the same order U∼qExx ρ1/2 ,δl x∼qExx3 ρν2−1/4 .(2) In analogy with the Grashof number defined for thermal plumes, we can define the electric Grashof number, Gr el = (Ux/ν)2=(qExx3/ρν2), which represents the ratio of electric and viscous forces. These expression are of formal rather than practical interest because qis not known a priori. In the following we turn toward more satisfactory expressions. B. Electric equations Two equations describe the electric aspects of the problem: the Poisson equation and the conservation of charge, in steady conditions, ∇·E=q ,(3a) ∇·j=0.(3b) Here is the permittivity and jthe current density. Dielectric liquids of very poor conductivity are not ohmic. Space charge may appear in these liquids under certain restrictions1. Essentially, the space charge is observable when the transit time of ions between electrodes d2/KV (Kbeing the ionic mobility and Vthe applied voltage) is smaller than the space charge relaxation time /σ (σbeing the conductivity). Under this condition, the density current has three components: the drift of ions with respect to the liquid, KqE, the charge convection, quand the charge diffusion −D∇q.Thus,j=q(KE+u)−D∇q, and eq. (3b) writes, in 2D geometry, KEx ∂q ∂x +KEy ∂q ∂y +Kq2 +u∂q ∂x +v∂q ∂y − −D∂2q ∂x2−D∂2q ∂y2=0.(4) The first term is related to the longitudinal electric drift, the second and third terms are due to the field created by the space charge (the Coulomb repulsion), the fourth and fifth terms are related to the convection of electric charge due to the liquid motion and the last two terms are due to charge diffusion. C. Basic scales From the structure of EHD plumes a more satisfactory set of estimates can be obtained. If the electric charge remains confined into the plume the current is (current per unit length for 2D plumes) J≈¯q2δq(U+K¯ Ex) two-dimensional case, I≈¯qπδ2 q(U+K¯ Ex) axisymmetric case, (5) with ¯q,Uand ¯ Ex∼V/d the typical scales of the density of charge, the axial velocity and the axial electric field. From now on we assume δqto be smaller enough than δlfor the previous expressions to be valid. In most EHD flows, including plumes, the ion drift velocity is negligible compared to the liquid velocity15,K¯ ExU.Now,integrating (1b) across an x-constant plane, and stating that the inertial, viscous and electric terms are of the same order we obtain 3 U∼(J2¯ E2 xx/ρ2ν)1/5 δl∼(ρν3x2/J ¯ Ex)1/5two-dimensional case, (6a) U∼(I¯ Ex/ρν)1/2 δl∼(ρν3x2/I ¯ Ex)1/4axisymmetric case,(6b) For 2D plumes the typical experimental values5(J=10 −8−10−7Am −1,¯ Ex=10 6Vm −1,ρ=10 3Kg m−3,ν= 2×10−5m2s−1,x∼10−2m) give u∼3−9cm/s,δ l∼1.5−2.4 mm. For axisymmetric plumes12 (I=10 −8)weget u∼1m/s,δ l∼300 −530 μm. In the following, we will use these values for the estimates. D. Charge transport mechanisms We now discuss the relative importance, in both geometries, of the three mechanisms of transport of charge in equation(4), that is, diffusion, electric drift and convection by the liquid. 1. Diffusion vs. drift Longitudinal and transverse diffusion must be considered separately, as the length scales are quite different. In the asymptotic region the longitudinal scale of change of the density of charge is d, and the longitudinal electric field scale is Ex∼V/d. Comparing the terms of (4) corresponding to diffusion and drift in the xdirection we have D∂2q/∂x2 KEx∂q/∂x ∼D KV =κBT eV ,(7) with κBthe Boltzmann constant, Tthe absolute temperature and ethe electric charge of the ions. We have used the Einstein relation, D/K =kBT/e, to derive (7). In usual experimental conditions, (T= 300 K,V=10kV,ethe electron charge) the value of (7) is 2.5×10−61. This shows that longitudinal diffusion is fully negligible compared to drift. The transverse length scale for the charge density change is δq, the typical width of the charged core. The transverse electric field mainly arises from the spatial charge density. In order to estimate it, we model this charge density as an infinite layer of width 2δqand the typical uniform density of charge ¯q, in the 2D case, and as an infinite cylinder of radius δqin the axisymmetric case. Using the Gauss law we obtain Ey∼¯qδq/. From (5) the transverse electric field is Ey∼J/2U two-dimensional, I/πUδqaxisymmetric.(8) Comparing the transverse drift and diffusion in (4) we have D∂2¯q/∂y2 KEy∂¯q/∂y ∼2UkBT/eJδqtwo-dimensional, πUkBT/eI axisymmetric.(9) From the above estimates D∂¯q/∂y ¯qKEy∼10−6/δqtwo-dimensional, 3×10−5axisymmetric.(10) The conclusion is that diffusion effects are negligible compared to drift if δqis greater than a few μminthe2Dcase, and for any value of δqin the axisymmetric case. Hence, diffusion will be neglected in the following derivation. Remark that, if there would be other physical situations for which, in the 2D case, charge diffusion would be dominant over drift the problem would be completely analogous to the thermal case. This situation was considered by Zhakin4. The electric Schmidt number, Scel =ν/D, then plays the role of the Pr number. Since Scel =ν/D ∼ 105−106, we could apply the self-similar solutions found elsewhere5,6. 4 2. Drift vs. convection Consider now the relative importance of drift and convection terms in (4) at the boundary of the charged region. Taking y∼δqas transverse length scale, from the continuity eq. (1a) the scale of transverse velocity is v∼(δq/x)U. For the electric field we deduced in II D 1, Ex∼V/d,Ey∼¯qδq/. Thus, the order of magnitude of the different terms is K¯qV/xd for the longitudinal drift, K¯q2/ for the Coulomb repulsion and U¯q/x for the convection. Comparing the longitudinal terms due to drift and convection we get KEx(∂q/∂x) u(∂q/∂x)∼KV/d U1,(11) in the usual experimental conditions. So longitudinal drift can be fully neglected. For the transverse components we have KEy(∂q/∂y) u(∂q/∂x)∼Kx¯q U .(12) An estimation of ¯qis needed. The density of charge can be expressed, in the absence of diffusion, in terms of the time during which Coulomb repulsion acts, i.e. the time required for the ions to go from the injector to the point of coordinates (x, y). Neglecting charge diffusion, the charge conservation equation in steady conditions can be written1: (KE+u)·∇q=−Kq2 ⇒dq dt =−Kq2 ,(13) Here d/dt is a total derivative associated to the resulting velocity (KE+u) of charge carriers. Eq. (13) is readily integrated to obtain q(t)= qi 1+t/τr .(14) Here, qiis the density of charge at the injector, τr=/Kqithe typical evolution time. If xδq,wehaveinthecore t(x, y)∼t(x, 0) ∼x/U, because KExU.Inthisway,¯qcan be taken as the density of charge at the center of the plume, ¯q(x)=q(x, y =0)=q0(x). We define Ω(x) as the quotient of the two characteristic times Ω=t(x)/τr∼Kqix/U. (15) If Ω 1, (14) gives q0(x)≈qiand (12) is very small. In this case, Coulomb repulsion is negligible in the spreading of the charged region and (4) writes u∂q ∂x +v∂q ∂y =0.(16) The problem is formally analogous to that of thermal plumes in the limit Pr →∞and self-similar solution are conceivable. If Ω 1, q0(x) takes the asymptotic value q0(x)∼/Kt ∼U/Kx. Therefore, (12) is of order 1, and Coulomb repulsion and convection are of the same order. In this case (4) writes Kq2 +u∂q ∂x +(KEy+v)∂q ∂y =0.(17) Eqs. (16) or (17), with (1a)-(1b), define the dynamics of EHD plumes. We must now estimate the value Ω in experimental conditions. 3. Estimation of Ω The crucial parameter is the initial density of charge, qi. It can be related to the current because, at the electrodes, the liquid velocity vanishes, and conduction is due only to drift. We consider first the 2D case. Modeling the electrode as an hyperbole, and in the absence of space charge, the electric field can be calculated by means of conformal mapping16.ItgivesEel =√2V/πd1/2r1/2 0,r0being the radius of curvature of the electrode. 5 The injection of charge occurs when Eel reaches a threshold value, Es. The space charge modifies the electric field on the injector and, similarly to what occurs for corona effect in gases17,forV>V Sthe steady electric field in the vicinity of the needle tip keeps nearly constant values (corresponding to the harmonic field for V=Vs) independently of the current. From experiments in mineral and silicone oils5we get Es∼0.1MVcm −1. In order to get the order of magnitude of the injected charge we supposed the blade to inject into an angle 0 <θ<θ 0(see figure 2). In the vicinity of the surface of the point, the liquid must be at rest, and only the drift can transport the electric charge. For θ0=π/2 the current per unit length is J=qiπr0KEs.Takingr0=5μm, and K=10 −9m2V−1s−1one obtains qi∼J πr0KEs∼6×10−2Cm −3J=10 −8Am −1, 6×10−1Cm −3J=10 −7Am −1.(18) Similar considerations can be made for axisymmetric plumes. The harmonic field is calculated approximating the needle by an hyperboloid( Laplace equation between an hyperboloid and a plate16,18). The field at the tip of the needle is Eel =2V/(r0ln(4d/r0)). From experiments in mineral oils13,Es∼4MVcm −1. This value of Esis similar to that obtained from experiments in cyclohexane and liquid argon19,20. In order to get the order of magnitude of qi, we simplify further the geometry modeling the point by a hemisphere of radius r0.Takingθ0=π/2, the current is I∼2πr2 0KEsqi.Weget 13 (K=10 −9m2V−1s−1) qi∼I 2πr2 0KEs∼160 C m−3I=10 −8A, 1600 C m−3I=10 −7A.(19) From (19) and (18) the value of the parameter Ω can be estimated for both geometries, planar (Ωp) and axisymmetric (Ωt). Taking the typical values estimated in II C we get, for x=1cmandr=2, Ωp∼0.3J=10 −8Am −1, 3J=10 −7Am −1.Ωt∼90 I=10 −8A, 900 I=10 −7A.(20) We see that only in the 2D case, and for low values of the intensity of the current, the effect of Coulomb repulsion can be neglected. E. Boundary conditions The boundary conditions are given by the geometry of the problem. At the center line of the plumes, the distributions of longitudinal velocity and charge have a maximum and the transverse velocity must vanish. On the other hand, the longitudinal velocity far from the plume vanishes. We have, for both geometries, ∂u ∂yy=0 =∂q ∂yy=0 =v|y=0 = 0; (21a) u|y→∞ = 0 (21b) At steady state, the intensity of current per unit length J, in two-dimensional plumes, and the intensity current Iin axisymmetric plumes, must be a constant at any plane normal to the axis of the plume. Using again that KExU, J=+∞ −∞ qudy, (22a) I=2π+∞ 0 quy dy (22b) 6 III. INTEGRAL ASYMPTOTIC SOLUTIONS When Coulomb repulsion is negligible, self-similar solutions to (1a)-(1b) in both geometries have been found in the case of constant electric field, the charge remaining confined into an infinitely thin layer or cylinder, so that δqis taken to be null6. In the 2D case we obtain u(x, y)=4J2E2 ρ2ν1/5 x1/5f(η),(23a) δl(x)=16ρν3 JE 1/5 x2/5.(23b) Here η=y/δl(x) is the self-similar variable and f(η) is the self-similar stream function. In the axisymmetric case u(x, y)=IE 2πρν 1/2f η,(24a) δl(x)=2πρν3 IE 1/4 x1/2.(24b) These solutions cannot describe the internal structure of the charged core. Even more, in the axisymmetric case, in the limit δq→0, the velocity tends to infinity6. The effect of Coulomb repulsion must be considered to fully understand the EHD plumes structure. Self-similar solutions are no longer possible when the Coulomb repulsion term in (17) is not neglected. Now we present an integral method analysis, generalizing the previous works by McCluskey and P´erez11 andAttenetal. 13, that can describe the behavior of the principal magnitudes of EHD plumes. A. 2D plumes If δq<δ l, we can define an average velocity um(x)sothat J=+∞ −∞ qudy =Qp(x)um(x),(25) with Qp(x)=∞ −∞ qdy the charge per unit length. So umis the average value of the axial velocity of the liquid into the charged core. When δqδl,umcoincides with the velocity at the center of the plume. Our aim is to find the equations relating δq,δland um. Integrating (1b), with k= 0, with respect to yfrom y=−∞ to y=∞, using (1a) and (21a), after some manipulations, leads to ∞ −∞ ∂u2 ∂x dy =∞ −∞ qEx ρdy. (26) If δqis small enough, the variations of the longitudinal electric field as a function of the coordinate ycan be neglected so that, to a first approximation, Ex(y,x)=E0(x)for|y|≤δq. Then, the right-hand side term in (26) can be expressed in terms of Qp(x). Multiplying (26) by umwe get um d dx ∞ −∞ u2dy =JE0 ρ.(27) and expresses that the variation of total momentum in a plane transverse to the flow equates the total force per unit length exerted by the imposed electric field on the liquid. Note that the viscous term does not contribute to the balance because it is an internal force. Conservation of kinetic energy allows us to take the viscous effects into account. Multiplying (1b) by uand manipulating it in the same way we obtain 7 d dx ∞ −∞ 1 2u3dy =−ν∞ −∞ ∂u ∂y2 dy +JE0 ρ.(28) This equation expresses the balance between the kinetic energy of the liquid, the viscous losses and the power injected by the electric force. Finally, let us reconsider equation (25), J=Qum≈2¯q(x)δq(x)um(x)⇒δqum≈J 2¯q(x).(29) For y= 0 eq. (17) gives um ∂¯q ∂x =−K¯q2 ⇒um ∂ ∂x 1 ¯q=K ,(30) because KExum.For¯q(x), the partial derivative can be substituted by the total derivative. Hence, applying umd/dx to (29) we get um d dx (δqum)=KJ 2.(31) Eqs. (31), (27) and (28) define the problem. In order to obtain a set of differential equations we have to express the integrals in (27) and (28) in terms of umand δl. This can be performed by imposing a velocity profile. If δqis clearly lower than δlthe electric force remains confined into a very narrow region inside the plume, so that we can approximate the distribution u(x, y) by a pseudo self-similar velocity profile u(x, y)=um(x)f(η), with η=y/δl(x), f(0) = 1 and f(η)≤0∀η. Therefore, we obtain three differential equations for um(x), δl(x)andδq(x), um d dx um2δl=1 B1 JE0(x) ρ,(32a) B2 d dx um3δl=−νB3 um2 δl +JE0(x) ρ,(32b) um d dx (umδq)=JK 2.(32c) The constants are B1=∞ −∞ f2(η)dη, (33a) B2=∞ −∞ 1 2f3(η)dη, (33b) B3=∞ −∞ (f)2(η)dη. (33c) In order to estimate the values of B1,B2and B3, a velocity profile must be given. Choosing f(η)=exp(−η)asa plausible profile leads to B1=1,B2=1/3, B3= 1. Any other choice satisfying the boundary conditions gives similar values for the coefficients. 1. Solutions for some particular dependences of the electric field If we assume that the electric field depends on xas E0(x)= ¯ E(x/d)m,weobtain um=CuJ2¯ E2 ρ2νd2m1/5 x(1+2m)/5,(34a) 8 δl=Clρν3dm J¯ E1/5 x(2−m)/5,(34b) δq=Cq K Jρ4ν2d4m ¯ E41/5 x(3−4m)/5,(34c) δq δl =Cql K J2ρ3d3m ν¯ E31/5 x(1−3m)/5.(34d) The constants are Cu=B2 3 n1B1 (n1B1−n2B2)1/5 ,(35a) Cl=n1B1B3 3 (n1B1−n2B2)31/5 ,(35b) Cq=B2 3 32n1n5 3B1 (n1B1−n2B2)1/5 ,(35c) Cql =(n1B1−n2B2)4 32n2 1n5 3B2 1B31/5 ,(35d) with n1=(4−3m)/5,n 2=1+m, n3=(4−2m)/5.(36) In the case m= 0 (uniform electric field), the expressions (34a) and (34b) are quite analogous to those obtained neglecting Coulomb repulsion. Therefore, for 2D EHD plumes, including Coulomb repulsion terms does not affect the hydrodynamic boundary layer. But now the charge boundary layer is characterized. Note that δqvaries with xin a way clearly different from δland from the evolution predicted by Zhakin when only diffusion is retained. Taking the same values used in II C, with m=0,weobtainatx=1cm: um∼8cms −1,δ l∼2.2mm,δ q∼ 4μm,δ q/δl∼2×10−3. Recalling (10), we see that the order of magnitude of δqis big enough for the effect of charge diffusion being negligible. B. Integral solutions for axisymmetric EHD plumes The previous analysis for 2D EHD plumes can be easily extended to axisymmetric plumes. Here, the flow is considered again to have a double boundary layer structure, with an inner charged core of radius δq(x) and an outer hydrodynamic layer of radius δl(x), with δq<δ l. The total charge per unit length in a plane at constant xis Q(x)=2π+∞ 0yq dy, and the total current is I=2π+∞ 0yqudy ≈Q(x)um(x). From (1b) and (14), following the same steps than in §III A, we obtain the equations that describe the dynamics of axisymmetric EHD plumes um d dx ∞ 0 yu2dy =IE0(x) 2πρ ,(37a) d dx ∞ 0 1 2yu3dy =−ν∞ 0 y∂u ∂y2 dy +IE0(x) 2πρ ,(37b) um d dx umδ2 q=IK π .(37c) 9 FIG. 5. Evolution of the thickness of the hydrodynamic layer, δland the charged layer, δq,for different initial values of λ, with an imposed electric field. Linear and logarithmic scales in x/d are used. 16