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Electric current oscillations in low-conducting liquids
Journal of Electrostatics 48 (1999) 27}41
Electric current oscillations in low-conducting liquids V.A. Polyansky*, I.L. Pankratieva Moscow State Lomonosov University, Institute of Mechanics, 1 Michurinskij Pr., 119899 Moscow, Russia Received 24 September 1998; received in revised form 8 April 1999; accepted 11 June 1999
Abstract An oscillatory mode for the electric current #ow between two electrodes immersed into a low-conducting liquid is studied. It is shown that such a mode is due to the "eld threshold of the ion injection from the electrode surface. The oscillatory mode is accompanied by the propagation of the waves of the volume charge density and electric "eld strength over the interelectrode space. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Low-conducting liquid; Electric "eld; Ion injection; Threshold phenomena; Current oscillation
1. Introduction Current oscillations in a cylindrical cell with #at electrodes to which a constant voltage is applied [1,2] arise when the "eld applied is greater than some threshold value and a steady hydrostatic state of liquid loses its stability. Measurements revealed that immediately after the formation of the secondary #ow, chaotic #uctuations of the current arise around its mean value corresponding to the new regular liquid motion in the form of a set of ellipsoidal Hill vortices. The authors reckon [1,2] that the current #uctuations are caused by #uctuations of the convective velocity of the liquid. They emphasize that, contrary to usual hydrodynamics, a #ow turbulization occurs even when the threshold value of the instability parameter exceeds to a small extent. Electrohydrodynamical theory of these phenomena has been developed in Ref. [3], where a model problem on the transfer of the injected charge by the cylindrical liquid * Corresponding author. E-mail address: [email protected] (V.A. Polyansky) 0304-3886/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 8 6 ( 9 9 ) 0 0 0 4 6 - 7
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vortices between two in"nite plate electrodes was examined. Theoretical [3] and experimental [1,2] spectral properties of the current #uctuations agree closely with each other in spite of essential di!erence in the #ow geometry. The mechanism of #ow turbulization was not discussed in Ref. [3]. Most likely there are several causes for synchronous rising in chaotic #uctuations of the current and convective velocity of the liquid. We point out some of them. An essential common property of the #ows examined in experiments [1,2] and theory [3] is the fact that in both cases the #ows have domains where the convective velocity of liquid is of the same order as the drift velocity of charged particles and these velocities are oppositely directed. At the electrode surface some constant concentration of ions is generated due to the injection. These ions are being #ung away permanently into a near-electrode boundary layer of liquid by the "eld since the liquid velocity on the electrode is equal to zero because of the viscosity. In the local domains, where the drift and convective velocities of ions have the same order and opposite directions, a nonstationary accumulation of the volume charge is possible which results in an increase of the Coulomb action on the liquid. But the volume charge causes at the same time some variation of the "eld strength, consequently, of the ion drift velocity. As a result, the local increase of the volume charge dissipates. Then this process is repeated in other places probably in a chaotic way. A possibility for the generation of narrow regions with a high density of the volume charge in electrogasdynamical #ows with opposite directions of the convective and drift velocities of ions has been studied in Ref. [4]. The presence of chaotically located domains with increased charge density can be clearly seen in Fig. 4 in Ref. [3]. It seems likely that the process described is one of the main causes for the generation of chaotic #uctuations of the current and velocity in the regular secondary #ow that appears on account of the loss of stability of the steady hydrostatic state of the liquid. In this paper we indicate one more possible cause of the current oscillations even in the case of no unsteady convective motion. We consider a mechanism connected with the threshold for the ion injection, when the surface ionization can occur only for the "eld strength exceeding some critical value. In particular, this value may be equal to zero. The mechanism considered is similar to that of the Trichel pulsation in the gas corona discharge for electronegative gases [5]. In the theory of Trichel current pulsations [6], a strong dependence of the rates of the bulk electron-impact ionization and electron attachment on the "eld is very essential. The electron concentration on the cathode surface is also assumed to be dependent on the "eld. Below, we show that the "eld dependence of the surface ionization is su$cient for generation in liquids of self-sustaining current oscillations. The range of the "eld strength for the existence of such oscillations is limited like that for gases.
2. Formulation of the problem We consider a multicomponent medium consisting of a carrying dielectric liquid and a small admixture of neutral electrolyte molecules of various kinds which are
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always present in the technical media under consideration. When dissolving the impurities, the weakly bonded ionic pairs are formed in the liquid. The ionic pairs dissociate and produce free ions which provide a residual conductivity of the liquid. In theoretical model of medium [7,8] used here these ions are combined in two kinds of intrinsic positive and negative ions. The third kind of ions included in the model is formed on the electrode as a result of a surface electrochemical ionization of neutral molecules of liquid mixture considered. In particular, this may be the process of dissociation of the ionic pairs adsorbed in the liquid}metal interface. The ions with the same sign of charge as the sign of the electrode are injected into the liquid. We suppose that the injection occurs only if a "eld of certain intensity is present. It is assumed that all kinds of ions may have di!erent kinetic and transport properties. The problem on calculation of the motion of the medium components under the action of the "eld is solved in the one-dimensional nonstationary setting. The liquid is placed between two electrodes with a distance ¸ between them. Choosing the coordinate axis x directed normal to the electrode surface, we write the equations of electrohydrodynamics [8] for the ion concentrations and the "eld in the form Ln 1 L 1# (xpn ; )"w!an n , 1 1 1 2 Lt xp Lx Ln 1 L 2# (xpn ; )"w!an n !bn n , 2 2 1 2 2 3 Lt xp Lx Ln 1 L 3# (xpn ; )"!bn n , 3 3 2 3 Lt xp Lx
(1)
Ln LF n ; "!D m!sign(e ) b n , m"1}3, m m m Lx m m m Lx
A B
LF 1 L xp "!q, ee 0xp Lx Lx
LF E"! , Lx
LE , q"+ e n , J "+ (e n ; )#ee m m 4 m m m 0 Lt m m e '0, e (0, e '0. 1 2 3 Here t is the time, p is the index of symmetry, p"0, 1 and 2 matches to a plane, cylindrical and spherical symmetry, n is the concentration, ; is the velocity, D and m m m b are coe$cients of di!usion and mobility of the ions of mth kind with the charge e , m m F is the electric potential, E is the "eld strength, q is the volume charge density, ee is 0 the absolute dielectric permeability of the medium, J is the density of the total current 4 including the displacement current. Note that in the one-dimensional setting the quantity J xp depends on the time only. It is assumed that the injected ions are 4 positive. On the right-hand side of the "rst three equations of system (1) there are source terms which describe the bulk dissociation of the ionic pairs with rate w, recombination
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between the intrinsic ions with recombination coe$cient a and recombination between the injected and intrinsic ions with recombination coe$cient b. We use for the concentrations the boundary conditions obtained from the balance of the #uxes of charged particles near the interface boundaries under the hypothesis that the surface electrochemical processes occur [7]. For de"niteness, we assume that the ions of all kinds recombine on the electrode surfaces, but the surface ionization occurs only at one of the electrodes, at the anode for instance. Under the action of the repulsive forces the positive ions are rejected from the anode into the liquid. Obviously, the intensity of the #ux of injected ions depends on the strength of the "eld that is generated near the electrode surface due to the external source and volume charge of the interelectrode gap. A detailed analysis of literature data on the surface ionization mechanism in #uids in the presence of electric "eld is beyond the scope of the present paper. Let us only note that in many works quite various phenomenological presentations for injected ions #ux are used. Validity of these presentations is estimated only indirectly, for example, by the shape of current}voltage characteristic [9], by the volume charge structure in the near-electrode region [8], by the value of the electrization current in a metal pipe with #owing dielectric liquid [10], etc. For the purposes of this study it is su$cient to accept a linear relation between the injected ion #ux and the "eld strength at the electrode surface. We make an additional assumption that the injection ceases when the "eld strength becomes lower than some threshold value E . # With regard to the aforementioned suppositions the boundary conditions at the electrode surfaces can be presented in the form n ; "!K n , m"1, 2, x"x , m m m m 0 n ; "K n , m"1!3, x"x #¸, (2) m m m m 0 n ; "A E!K n , x"x , 3 3 3 3 3 0 A "0, E(E , A "const, E*E , (3) 3 # 3 # F(x )"F "const'0, F(x #¸)"0. (4) o w 0 Here x is the anode coordinate and K , A are the parameters of the super"cial 0 m 3 electrochemical processes. All constants present in the equations and boundary conditions are supposed to be speci"ed. The technique of evaluating these constants by the experimental data is discussed in Ref. [8]. The threshold value E of the "eld for # the ion injection is a parameter of the model. The in#uence of the variation of this parameter on the state of the medium is investigated below. The initial distributions of the ion concentrations should be speci"ed for the "rst three relations of Eq. (1). We assume that before supplying the external voltage the medium contains only intrinsic ions. Their concentrations are uniform everywhere except for the narrow near-electrode di!usive layers where n and n are changed 1 2 according to the balance equations (2). When K "K and the transport properties of 1 2 the intrinsic ions are identical, the initial volume charge is equal to zero everywhere. The integration of Eq. (1) with the boundary conditions (2)}(4) and the initial conditions allows to "nd the distributions of the ion concentrations and the "eld strength at any instant and calculate the total current.
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3. The method of numerical solution of the problem We have developed the software package CELL for solving a wide range of problems concerning one-dimensional nonstationary #ows in the framework of the medium model [8,9]. The package is highly e$cient for the experimental data treatment and for performing computational experiments like those described below. A "nite-di!erence scheme implicit in time with nonuniform mesh points along the x-coordinate and constant time-steps is used in calculations. A nonlinear system of the "nite-di!erence equations is solved by the iterative method for each time step. All equations are integrated simultaneously. The scheme accuracy was tested with standard methods by increasing the number of space steps and decreasing the time-step. The integral xpJ "f (t) of Eq. (1) was used for checking the precision on each 4 time-step. Calculations start with "nding the initial distributions of the intrinsic ion concentrations for zero voltage. Then we apply the potential di!erence at the initial time instant (t"0) and "nd the distributions of the parameters for each time-step. The equations and boundary conditions are written in the dimensionless form. In the results below the characteristic parameters, which are indicated by subscripting zero, correspond to the dielectric liquid with the residual conductivity p "10~11 S/m and ion di!usion coe$cient D "10~9 m2/s. The numerical values 0 0 of the other characteristic parameters can be calculated with the help of the following relations: the ion mobility b "eD /k¹, where k is the Boltzmann constant and e is 0 0 the elementary charge, ¹"300 K; the equilibrium concentration of intrinsic ions is n "p /2eb ; the coe$cients of recombination are a "b "2eb /ee ; the dissocia0 0 0 0 0 0 0 tion rate is w "a n2. The x-coordinate is scaled by the distance between electrodes 0 0 0 ¸"0.002 m. For the time scaling, it is convenient to choose the charge of the relaxation time t "ee /p , though in the case of polar liquids this quantity has 3 0 0 a rather formal meaning. The relations D /¸ and D n /E ¸ with E "k¹/e¸ are used for scaling of the 0 0 0 t t surface reaction parameters K and A . As starting values for K , we used the data of m 3 m the experimental measurements of the electrization current in metal pipes with #owing dielectric liquids [7,10]. The parameters K and A are varied in a wide range m 3 to estimate their in#uence on the results.
4. Polar liquids: mechanism of appearing of the current oscillations * waves of the 5eld strength and volume charge First, we consider the case of the polar liquids. In these liquids the molecules of the electrolyte impurities dissociate almost completely and in the presence of the "eld the electrical puri"cation from the intrinsic ions occurs. Polar liquids are simulated by the Eq. (1) with the quantities w"a"b"0. The solution of the problem for this case allows to understand the mechanism of the generation of the self-maintaining current oscillations most clearly. We consider the solution for the case of plane symmetry (p"0).
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The relation between the threshold "eld E and the applied one E "F /¸ # w w determines the existence of three di!erent modes of the current #ow. The "rst mode is trivial: the complete electrical puri"cation of the liquid from the ions and the dumping of the current. This mode corresponds to the case when the applied "eld E (E and, w # hence, there is no ion injection from the anode. After switching on the voltage the process of separating the charged particles starts in the liquid. As a result, during the several periods of the charge relaxation most part of the positive ions is accumulated in the narrow Debye layer near the cathode, whereas the negative ions are being accumulated near the anode. Since the ion recombination proceeds on the electrodes and there is no volume source of ions, the concentrations of the intrinsic ions vanish rapidly. Another mode is established in the case when the applied "eld E
Fig. 1. Dependence of the density of the total current J/J on time t/t . 0 3
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voltage, a nonstationary process of the total current alteration takes place at the expense of electrical puri"cation of the liquid from the intrinsic ions and spreading of the injected ions onto the whole interelectrode gap. Then there arises the pulsating periodic mode of the alteration of the current and the total volume charge. The character of the current oscillations is shown in Fig. 1, where the ordinate represents the dimensionless total current density J/J with J "104p k¹/e¸. The applied 0 0 0 voltage F "104k¹/e (260 V). w Fig. 2 illustrates the periodic alteration of the dimensionless total volume charge in the gap Q/q ¸ with q "n e. The ratio of the threshold "eld to the applied one is 0 0 0 E /E "0.7. The "gures demonstrate a gradual regularization of the oscillations. One # w can also see that oscillation amplitudes of both the current and the total charge are "nite and have the same order as their mean values. The mechanism of the generation of the self-maintaining oscillations is illustrated by Figs. 3 and 4, which show the distributions of the volume charge density q/q 0 (Fig. 3) and the "eld strength E/E (Fig. 4) at di!erent time instants. Successive time w instants cover approximately one period of the oscillations. Time increases with the curve number. The injection of ions results in forming a positive volume charge in a narrow layer near the anode. This charge reduces the "eld on the electrode surface to
Fig. 2. Dependence of the total volume charge Q/q ¸ on time t/t . 0 3
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Fig. 3. Distributions of the volume charge density q/q along the interelectrode gap for di!erent time 0 instants. Consecutive time instants cover approximately one period of oscillations. Time increases with a rise of the curve number.
a value less than the threshold one (curve 1). As a consequence, the injection ceases and the layer of the charge under the action of the Coulomb forces begins to move with drift velocity b E towards the cathode (curves 2 and 3). Since the "eld increases 3 over the whole gap, the forefront of the layer moves faster than the backfront and the layer spreads gradually. The di!usion also contributes this process. When the charge layer moves o! from the anode, the "eld strength on the anode surface begins to grow. Curve 4 corresponds to the instant, when the "eld at the anode reaches the threshold value and the injection switches on. Further, a new layer of the positive charge is formed near the anode. Then the process described repeats again. Curve 1 in Fig. 3 shows that by the moment of the forming of a new near-anode layer, the preceding layers continue to move in the space between the electrodes towards the cathode #attening gradually. The presence of the moving layers of the positive volume charge results in the formation of waves of the "eld strength. These waves are spreading from the anode to the cathode along a very nonuniform distribution of the "eld. Fig. 4 shows that the "eld strength at the cathode E changes slightly. The quantity , E is determined by the time average value of the total volume charge, which remains ,
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Fig. 4. Distributions of the "eld strength E /E along the interelectrode gap for the same instants as in # w Fig. 3.
constant with su$ciently high accuracy during the oscillations. After the oscillatory mode has been established the maximum "eld strength at the injecting anode is close to E and always less than the applied "eld E . # w It is clear from the above considerations that the oscillation frequency f should be determined by the velocity of the motion of the charge layers, i.e. by the mobility of the ions and the threshold "eld value. The in#uence of these parameters on the frequency has been studied numerically. Fig. 5 presents the dependence of the frequency f/f with f "1/t on the mobility of 0 0 3 ions b /b . The curves di!er from each other by the threshold "eld E . One can see 3 0 # that, starting from a certain value b /b (1, these curves are close to straight lines, 3 0 whose slope decreases with the growth of E . The analysis of calculations shows that # in the range of existence of the current oscillatory mode the frequency is a linear function of the ratio b E /b E . 3 w 0 # The parameter A , characterizing the injection intensity, slightly a!ects the 3 frequency of the oscillations. The variation of A over su$ciently wide range 3 changes the ion concentration on the anode at the injection instant, when the "eld strength exceeds the threshold strength. The amplitudes of oscillations of the charge density and "eld strength change correspondingly in the space between the electrodes. For the data represented in the "gures, the dimensionless parameter A is equal 3 to 5]105.
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Fig. 5. Dependence of the frequency f/f on the mobility of ions b /b for di!erent values of the threshold 0 3 0 "eld E /E (1: 0.3, 2: 0.4, 3: 0.5, 4: 0.7, 5: 0.9). Curve 6 corresponds to the case p"1 with E /E "0.12. # w # w
The parameters of the surface recombination K de"ne the surface values and m distributions of the ion concentrations in the narrow near-electrode di!usive layers and have no in#uence on the frequency. The same dimensionless quantity of K "5]103 was used in calculations for all components. m 5. Current oscillations in nonpolar liquids The bulk nonequilibrium processes of the dissociation of electrolyte molecules of the impurities and recombination of the ions proceed in the case of nonpolar liquids. The solution of Eq. (1) with nonzero right-hand side shows that the above-described mechanism of the generation of self-maintaining oscillations does not change from the conceptual point of view. The presence of two additional charged components in the medium decreases the amplitudes of oscillations of the charge density and "eld strength when moving away from the injecting electrode. The distributions of the charge density and "eld strength in the interelectrode space are shown in Fig. 6 (curves 1 and 2) for dimensionless quantities w/w "a/a " 0 0 b/b "0.1 and for equal mobilities b "b (m"1}3). The voltage applied is 0 m 0
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Fig. 6. Distributions of the "eld strength E/EH (curve 1: p"0, curves 3, 4: p"1) with EH" 104F e¸/k¹ and volume charge density q/q (curve 2: p"0) along the interelectrode gap in the case of w 0 nonpolar liquids.
F "104k¹/e, the threshold "eld is E "0.5E , the dielectric permittivity is e"2.2. w c w The curves 1 and 2 are pictured for an instant when the "eld strength at the anode is less than the threshold one and there is no ion injection. One can see that the volume charge alters the sign several times in the near-anode region. We note that the rise of nonstationary structures with alteration of the sign of the volume charge is also typical for the pulsing corona discharge in gases [6]. The distributions of the ion concentrations in the interelectrode space are shown in Fig. 7 at the same time instant. The amplitude of the oscillations of the negative ion concentration (curve 2) damps out quickly with the increase of the distance from the anode. It is interesting to note that an interaction between waves of the "eld and injected ion concentration initiates an excitation of waves of the concentration of the positive intrinsic ions (curve 1). The calculations show that the excitation has a resonant character with respect to the mobility of the injected ions b . The 3 amplitude of the waves of n decreases sharply when the ratio b /b deviates from 1 3 0 unity in any side. The dependence of the oscillation frequency on the parameters characterizing kinetic and transport properties of the medium is shown in Fig. 8. For the sake of simplicity we consider the case when the dimensionless quantities w/w , a/a and 0 0
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Fig. 7. Distributions of the ion concentrations along the interelectrode gap in the case of nonpolar liquids (curve 1: n /n , curve 2: n /n , curve 3: n /n ). 1 0 2 0 3 0
b/b are equal and denoted by A in Fig. 8. The value of A characterizes actually 0 the ratio between the characteristic time of the ion drift through the interelectrode space and the characteristic time of the relaxation to chemical equilibrium in the medium. The mobility of the intrinsic ions is identical (b "b "b ). The 1 2 0 mobility b is varied. One can see that the dependence of the frequency on the 3 parameter A is nonmonotonic. The calculated frequencies fall within the range of the experimental ones [1,2] as shown in Fig. 8 because f "0.57 s~1 for conditions 0 under consideration. The dependence of the frequency on the ratio b E /b E is qualitatively the same as 3 w 0 # that in the case of polar liquids.
6. In6uence of nonuniformity of the applied 5eld on the current oscillations It is known that the Trichel pulsations in the gas corona discharge are observed in a needle-plane geometry when the "eld applied is highly nonuniform. In the theoretical model [6] the geometrical factor is taken into account only when calculating the "eld. The concentrations of the charged particles are calculated from
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Fig. 8. Dependence of the frequency f/f on the dimensionless kinetic parameters of the medium 0 (A,w/w "a/a "b/b ). (curve 1: b /b "75, curve 2: b /b "50, curve 3: b /b "10, curve 4: b /b "1). 0 0 0 3 0 3 0 3 0 3 0
one-dimensional equations like the "rst three relations of Eq. (1) with p"0. The expansion of the discharge domain when moving away from the injecting electrode is ignored. We consider a wire-cylinder geometry to estimate the in#uence of the nonuniformity of the applied "eld and geometrical factor on the current oscillations in liquids. Let the electrodes have the form of two coaxial cylinders with dimensionless radii 0.1 and 1.1, respectively. In Eq. (1) we set the index p"1. The calculations show that the nonuniformity of the applied "eld localizes the domain of the oscillations with "nite amplitudes closely to the surface of the inner injecting electrode. The frequency increases under the same conditions as in the #at case. The amplitude of the oscillations decreases rapidly with the increase of the distance from the inner electrode mainly as a result of diverging geometry. The distribution of the "eld strength is shown in Fig. 6 (curve 3) for an instant when there is no injection. For p"1, the x-coordinate is the distance from the inner cylinder. The values of the parameters are the following: w/w "a/a "b/b "0.1, b " 0 0 0 m b (m"1}3), F "104 k¹/e, E "0.5F /¸. The applied "eld E¸/F is represented by 0 w # w w the curve 4.
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The dependence of the frequency of the current oscillations on the mobility b for 3 p"1 is shown in Fig. 5 (curve 6, the scale of the ordinate axis for this curve is f/f ]10~2). One can see that the character of the dependence is not changed in 0 comparison with the case p"0. The same is valid for the dependence of the frequency on the threshold "eld.
7. Conclusion The study demonstrates that in the low-conducting liquids, as well as in gaseous media, the Trichel selfmaintaining oscillations of the current can exist. The mechanism of the rise of these oscillations is connected with a nonstationary interaction between the "eld at the electrode surface and the volume charge of a near-electrode region under the condition that the surface ionization process has the threshold in the "eld value. The calculated oscillations of the current have a regular character in contrast with the chaotic current #uctuations registered in [1,2]. From Fig. 5 we deduce that the frequency depends on the value of the threshold "eld E . This quantity is preserved # constant during the computation of each variant. Probably it is possible to simulate the process of randomization if we take E in a random way for random periods of # time during the calculation of the variant. From the physical viewpoint, such approach could correspond to nonuniform injecting properties of the electrode surface, when di!erent parts of the surface have di!erent values E and, as a result, generate # di!erent frequencies. In the framework of this hypothesis the chaotic current oscillations could be presumably obtained without any correlation with the turbulization of liquid convective motion. Acknowledgements This work was supported by the Russian Foundation of Basic Researches, project No. 98-01-00109. References [1] P. Atten, J.C. Lacroix, B. Malraison, Chaotic motion in a Coulomb force driven instability: large aspect ratio experiments, Phys. Lett. A 79 (1980) 255}258. [2] B. Malraison, P. Atten, Chaotic behavior of instability due to unipolar ion injection in a dielectric liquid, Phys. Rev. Lett. 49 (1982) 723}726. [3] R. Chicon, A. Castellanos, E. Martin, Numerical modelling of Coulomb-driven convection in insulating liquids, J. Fluid Mech. 344 (1997) 43}66. [4] V.V. Gogosov, V.A. Polyansky, Shock waves in electrohydrodynamics, Prog. Aerospace Sci. 20 (1983) 125}216. [5] B. Loeb, Electrical coronas, University of California, Berkley, 1965. [6] R. Morrow, Theory of negative corona in oxygen, Phys. Rev. A 32 (1985) 1799}1809. [7] V.V. Gogosov, K.V. Polyansky, V.A. Polyansky, G.A. Shaposhnikova, A.A. Vartanyan, Modelling of nonstationary processes in channels of EHD pump, J. Electrostat. 34 (1995) 245}265.
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[8] I.L. Pankratieva, V.A. Polyansky, Modelling of electrohydrodynamical #ows in low-conducting liquids, J. Appl. Mech. and Techn. Phys. 36 (1995) 36}44. [9] A.A. Vartanyan, V.V. Gogosov, V.A. Polyansky, G.A. Shaposhnikova, A numerical simulation of non-stationary electrohydrodynamic processes in weakly conducting liquid, J. Electrostat. 23 (1989) 245}262. [10] V.N. Pribylov, Experimental study of electrization current of dielectric liquids in cylindrical pipe, Colloidal J. 58 (1996) 524}527.
Electric current oscillations in low-conducting liquids V.A. Polyansky*, I.L. Pankratieva Moscow State Lomonosov University, Institute of Mechanics, 1 Michurinskij Pr., 119899 Moscow, Russia Received 24 September 1998; received in revised form 8 April 1999; accepted 11 June 1999
Abstract An oscillatory mode for the electric current #ow between two electrodes immersed into a low-conducting liquid is studied. It is shown that such a mode is due to the "eld threshold of the ion injection from the electrode surface. The oscillatory mode is accompanied by the propagation of the waves of the volume charge density and electric "eld strength over the interelectrode space. ( 1999 Elsevier Science B.V. All rights reserved. Keywords: Low-conducting liquid; Electric "eld; Ion injection; Threshold phenomena; Current oscillation
1. Introduction Current oscillations in a cylindrical cell with #at electrodes to which a constant voltage is applied [1,2] arise when the "eld applied is greater than some threshold value and a steady hydrostatic state of liquid loses its stability. Measurements revealed that immediately after the formation of the secondary #ow, chaotic #uctuations of the current arise around its mean value corresponding to the new regular liquid motion in the form of a set of ellipsoidal Hill vortices. The authors reckon [1,2] that the current #uctuations are caused by #uctuations of the convective velocity of the liquid. They emphasize that, contrary to usual hydrodynamics, a #ow turbulization occurs even when the threshold value of the instability parameter exceeds to a small extent. Electrohydrodynamical theory of these phenomena has been developed in Ref. [3], where a model problem on the transfer of the injected charge by the cylindrical liquid * Corresponding author. E-mail address: [email protected] (V.A. Polyansky) 0304-3886/99/$ - see front matter ( 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 8 6 ( 9 9 ) 0 0 0 4 6 - 7
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vortices between two in"nite plate electrodes was examined. Theoretical [3] and experimental [1,2] spectral properties of the current #uctuations agree closely with each other in spite of essential di!erence in the #ow geometry. The mechanism of #ow turbulization was not discussed in Ref. [3]. Most likely there are several causes for synchronous rising in chaotic #uctuations of the current and convective velocity of the liquid. We point out some of them. An essential common property of the #ows examined in experiments [1,2] and theory [3] is the fact that in both cases the #ows have domains where the convective velocity of liquid is of the same order as the drift velocity of charged particles and these velocities are oppositely directed. At the electrode surface some constant concentration of ions is generated due to the injection. These ions are being #ung away permanently into a near-electrode boundary layer of liquid by the "eld since the liquid velocity on the electrode is equal to zero because of the viscosity. In the local domains, where the drift and convective velocities of ions have the same order and opposite directions, a nonstationary accumulation of the volume charge is possible which results in an increase of the Coulomb action on the liquid. But the volume charge causes at the same time some variation of the "eld strength, consequently, of the ion drift velocity. As a result, the local increase of the volume charge dissipates. Then this process is repeated in other places probably in a chaotic way. A possibility for the generation of narrow regions with a high density of the volume charge in electrogasdynamical #ows with opposite directions of the convective and drift velocities of ions has been studied in Ref. [4]. The presence of chaotically located domains with increased charge density can be clearly seen in Fig. 4 in Ref. [3]. It seems likely that the process described is one of the main causes for the generation of chaotic #uctuations of the current and velocity in the regular secondary #ow that appears on account of the loss of stability of the steady hydrostatic state of the liquid. In this paper we indicate one more possible cause of the current oscillations even in the case of no unsteady convective motion. We consider a mechanism connected with the threshold for the ion injection, when the surface ionization can occur only for the "eld strength exceeding some critical value. In particular, this value may be equal to zero. The mechanism considered is similar to that of the Trichel pulsation in the gas corona discharge for electronegative gases [5]. In the theory of Trichel current pulsations [6], a strong dependence of the rates of the bulk electron-impact ionization and electron attachment on the "eld is very essential. The electron concentration on the cathode surface is also assumed to be dependent on the "eld. Below, we show that the "eld dependence of the surface ionization is su$cient for generation in liquids of self-sustaining current oscillations. The range of the "eld strength for the existence of such oscillations is limited like that for gases.
2. Formulation of the problem We consider a multicomponent medium consisting of a carrying dielectric liquid and a small admixture of neutral electrolyte molecules of various kinds which are
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always present in the technical media under consideration. When dissolving the impurities, the weakly bonded ionic pairs are formed in the liquid. The ionic pairs dissociate and produce free ions which provide a residual conductivity of the liquid. In theoretical model of medium [7,8] used here these ions are combined in two kinds of intrinsic positive and negative ions. The third kind of ions included in the model is formed on the electrode as a result of a surface electrochemical ionization of neutral molecules of liquid mixture considered. In particular, this may be the process of dissociation of the ionic pairs adsorbed in the liquid}metal interface. The ions with the same sign of charge as the sign of the electrode are injected into the liquid. We suppose that the injection occurs only if a "eld of certain intensity is present. It is assumed that all kinds of ions may have di!erent kinetic and transport properties. The problem on calculation of the motion of the medium components under the action of the "eld is solved in the one-dimensional nonstationary setting. The liquid is placed between two electrodes with a distance ¸ between them. Choosing the coordinate axis x directed normal to the electrode surface, we write the equations of electrohydrodynamics [8] for the ion concentrations and the "eld in the form Ln 1 L 1# (xpn ; )"w!an n , 1 1 1 2 Lt xp Lx Ln 1 L 2# (xpn ; )"w!an n !bn n , 2 2 1 2 2 3 Lt xp Lx Ln 1 L 3# (xpn ; )"!bn n , 3 3 2 3 Lt xp Lx
(1)
Ln LF n ; "!D m!sign(e ) b n , m"1}3, m m m Lx m m m Lx
A B
LF 1 L xp "!q, ee 0xp Lx Lx
LF E"! , Lx
LE , q"+ e n , J "+ (e n ; )#ee m m 4 m m m 0 Lt m m e '0, e (0, e '0. 1 2 3 Here t is the time, p is the index of symmetry, p"0, 1 and 2 matches to a plane, cylindrical and spherical symmetry, n is the concentration, ; is the velocity, D and m m m b are coe$cients of di!usion and mobility of the ions of mth kind with the charge e , m m F is the electric potential, E is the "eld strength, q is the volume charge density, ee is 0 the absolute dielectric permeability of the medium, J is the density of the total current 4 including the displacement current. Note that in the one-dimensional setting the quantity J xp depends on the time only. It is assumed that the injected ions are 4 positive. On the right-hand side of the "rst three equations of system (1) there are source terms which describe the bulk dissociation of the ionic pairs with rate w, recombination
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between the intrinsic ions with recombination coe$cient a and recombination between the injected and intrinsic ions with recombination coe$cient b. We use for the concentrations the boundary conditions obtained from the balance of the #uxes of charged particles near the interface boundaries under the hypothesis that the surface electrochemical processes occur [7]. For de"niteness, we assume that the ions of all kinds recombine on the electrode surfaces, but the surface ionization occurs only at one of the electrodes, at the anode for instance. Under the action of the repulsive forces the positive ions are rejected from the anode into the liquid. Obviously, the intensity of the #ux of injected ions depends on the strength of the "eld that is generated near the electrode surface due to the external source and volume charge of the interelectrode gap. A detailed analysis of literature data on the surface ionization mechanism in #uids in the presence of electric "eld is beyond the scope of the present paper. Let us only note that in many works quite various phenomenological presentations for injected ions #ux are used. Validity of these presentations is estimated only indirectly, for example, by the shape of current}voltage characteristic [9], by the volume charge structure in the near-electrode region [8], by the value of the electrization current in a metal pipe with #owing dielectric liquid [10], etc. For the purposes of this study it is su$cient to accept a linear relation between the injected ion #ux and the "eld strength at the electrode surface. We make an additional assumption that the injection ceases when the "eld strength becomes lower than some threshold value E . # With regard to the aforementioned suppositions the boundary conditions at the electrode surfaces can be presented in the form n ; "!K n , m"1, 2, x"x , m m m m 0 n ; "K n , m"1!3, x"x #¸, (2) m m m m 0 n ; "A E!K n , x"x , 3 3 3 3 3 0 A "0, E(E , A "const, E*E , (3) 3 # 3 # F(x )"F "const'0, F(x #¸)"0. (4) o w 0 Here x is the anode coordinate and K , A are the parameters of the super"cial 0 m 3 electrochemical processes. All constants present in the equations and boundary conditions are supposed to be speci"ed. The technique of evaluating these constants by the experimental data is discussed in Ref. [8]. The threshold value E of the "eld for # the ion injection is a parameter of the model. The in#uence of the variation of this parameter on the state of the medium is investigated below. The initial distributions of the ion concentrations should be speci"ed for the "rst three relations of Eq. (1). We assume that before supplying the external voltage the medium contains only intrinsic ions. Their concentrations are uniform everywhere except for the narrow near-electrode di!usive layers where n and n are changed 1 2 according to the balance equations (2). When K "K and the transport properties of 1 2 the intrinsic ions are identical, the initial volume charge is equal to zero everywhere. The integration of Eq. (1) with the boundary conditions (2)}(4) and the initial conditions allows to "nd the distributions of the ion concentrations and the "eld strength at any instant and calculate the total current.
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3. The method of numerical solution of the problem We have developed the software package CELL for solving a wide range of problems concerning one-dimensional nonstationary #ows in the framework of the medium model [8,9]. The package is highly e$cient for the experimental data treatment and for performing computational experiments like those described below. A "nite-di!erence scheme implicit in time with nonuniform mesh points along the x-coordinate and constant time-steps is used in calculations. A nonlinear system of the "nite-di!erence equations is solved by the iterative method for each time step. All equations are integrated simultaneously. The scheme accuracy was tested with standard methods by increasing the number of space steps and decreasing the time-step. The integral xpJ "f (t) of Eq. (1) was used for checking the precision on each 4 time-step. Calculations start with "nding the initial distributions of the intrinsic ion concentrations for zero voltage. Then we apply the potential di!erence at the initial time instant (t"0) and "nd the distributions of the parameters for each time-step. The equations and boundary conditions are written in the dimensionless form. In the results below the characteristic parameters, which are indicated by subscripting zero, correspond to the dielectric liquid with the residual conductivity p "10~11 S/m and ion di!usion coe$cient D "10~9 m2/s. The numerical values 0 0 of the other characteristic parameters can be calculated with the help of the following relations: the ion mobility b "eD /k¹, where k is the Boltzmann constant and e is 0 0 the elementary charge, ¹"300 K; the equilibrium concentration of intrinsic ions is n "p /2eb ; the coe$cients of recombination are a "b "2eb /ee ; the dissocia0 0 0 0 0 0 0 tion rate is w "a n2. The x-coordinate is scaled by the distance between electrodes 0 0 0 ¸"0.002 m. For the time scaling, it is convenient to choose the charge of the relaxation time t "ee /p , though in the case of polar liquids this quantity has 3 0 0 a rather formal meaning. The relations D /¸ and D n /E ¸ with E "k¹/e¸ are used for scaling of the 0 0 0 t t surface reaction parameters K and A . As starting values for K , we used the data of m 3 m the experimental measurements of the electrization current in metal pipes with #owing dielectric liquids [7,10]. The parameters K and A are varied in a wide range m 3 to estimate their in#uence on the results.
4. Polar liquids: mechanism of appearing of the current oscillations * waves of the 5eld strength and volume charge First, we consider the case of the polar liquids. In these liquids the molecules of the electrolyte impurities dissociate almost completely and in the presence of the "eld the electrical puri"cation from the intrinsic ions occurs. Polar liquids are simulated by the Eq. (1) with the quantities w"a"b"0. The solution of the problem for this case allows to understand the mechanism of the generation of the self-maintaining current oscillations most clearly. We consider the solution for the case of plane symmetry (p"0).
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The relation between the threshold "eld E and the applied one E "F /¸ # w w determines the existence of three di!erent modes of the current #ow. The "rst mode is trivial: the complete electrical puri"cation of the liquid from the ions and the dumping of the current. This mode corresponds to the case when the applied "eld E (E and, w # hence, there is no ion injection from the anode. After switching on the voltage the process of separating the charged particles starts in the liquid. As a result, during the several periods of the charge relaxation most part of the positive ions is accumulated in the narrow Debye layer near the cathode, whereas the negative ions are being accumulated near the anode. Since the ion recombination proceeds on the electrodes and there is no volume source of ions, the concentrations of the intrinsic ions vanish rapidly. Another mode is established in the case when the applied "eld E
Fig. 1. Dependence of the density of the total current J/J on time t/t . 0 3
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voltage, a nonstationary process of the total current alteration takes place at the expense of electrical puri"cation of the liquid from the intrinsic ions and spreading of the injected ions onto the whole interelectrode gap. Then there arises the pulsating periodic mode of the alteration of the current and the total volume charge. The character of the current oscillations is shown in Fig. 1, where the ordinate represents the dimensionless total current density J/J with J "104p k¹/e¸. The applied 0 0 0 voltage F "104k¹/e (260 V). w Fig. 2 illustrates the periodic alteration of the dimensionless total volume charge in the gap Q/q ¸ with q "n e. The ratio of the threshold "eld to the applied one is 0 0 0 E /E "0.7. The "gures demonstrate a gradual regularization of the oscillations. One # w can also see that oscillation amplitudes of both the current and the total charge are "nite and have the same order as their mean values. The mechanism of the generation of the self-maintaining oscillations is illustrated by Figs. 3 and 4, which show the distributions of the volume charge density q/q 0 (Fig. 3) and the "eld strength E/E (Fig. 4) at di!erent time instants. Successive time w instants cover approximately one period of the oscillations. Time increases with the curve number. The injection of ions results in forming a positive volume charge in a narrow layer near the anode. This charge reduces the "eld on the electrode surface to
Fig. 2. Dependence of the total volume charge Q/q ¸ on time t/t . 0 3
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Fig. 3. Distributions of the volume charge density q/q along the interelectrode gap for di!erent time 0 instants. Consecutive time instants cover approximately one period of oscillations. Time increases with a rise of the curve number.
a value less than the threshold one (curve 1). As a consequence, the injection ceases and the layer of the charge under the action of the Coulomb forces begins to move with drift velocity b E towards the cathode (curves 2 and 3). Since the "eld increases 3 over the whole gap, the forefront of the layer moves faster than the backfront and the layer spreads gradually. The di!usion also contributes this process. When the charge layer moves o! from the anode, the "eld strength on the anode surface begins to grow. Curve 4 corresponds to the instant, when the "eld at the anode reaches the threshold value and the injection switches on. Further, a new layer of the positive charge is formed near the anode. Then the process described repeats again. Curve 1 in Fig. 3 shows that by the moment of the forming of a new near-anode layer, the preceding layers continue to move in the space between the electrodes towards the cathode #attening gradually. The presence of the moving layers of the positive volume charge results in the formation of waves of the "eld strength. These waves are spreading from the anode to the cathode along a very nonuniform distribution of the "eld. Fig. 4 shows that the "eld strength at the cathode E changes slightly. The quantity , E is determined by the time average value of the total volume charge, which remains ,
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Fig. 4. Distributions of the "eld strength E /E along the interelectrode gap for the same instants as in # w Fig. 3.
constant with su$ciently high accuracy during the oscillations. After the oscillatory mode has been established the maximum "eld strength at the injecting anode is close to E and always less than the applied "eld E . # w It is clear from the above considerations that the oscillation frequency f should be determined by the velocity of the motion of the charge layers, i.e. by the mobility of the ions and the threshold "eld value. The in#uence of these parameters on the frequency has been studied numerically. Fig. 5 presents the dependence of the frequency f/f with f "1/t on the mobility of 0 0 3 ions b /b . The curves di!er from each other by the threshold "eld E . One can see 3 0 # that, starting from a certain value b /b (1, these curves are close to straight lines, 3 0 whose slope decreases with the growth of E . The analysis of calculations shows that # in the range of existence of the current oscillatory mode the frequency is a linear function of the ratio b E /b E . 3 w 0 # The parameter A , characterizing the injection intensity, slightly a!ects the 3 frequency of the oscillations. The variation of A over su$ciently wide range 3 changes the ion concentration on the anode at the injection instant, when the "eld strength exceeds the threshold strength. The amplitudes of oscillations of the charge density and "eld strength change correspondingly in the space between the electrodes. For the data represented in the "gures, the dimensionless parameter A is equal 3 to 5]105.
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Fig. 5. Dependence of the frequency f/f on the mobility of ions b /b for di!erent values of the threshold 0 3 0 "eld E /E (1: 0.3, 2: 0.4, 3: 0.5, 4: 0.7, 5: 0.9). Curve 6 corresponds to the case p"1 with E /E "0.12. # w # w
The parameters of the surface recombination K de"ne the surface values and m distributions of the ion concentrations in the narrow near-electrode di!usive layers and have no in#uence on the frequency. The same dimensionless quantity of K "5]103 was used in calculations for all components. m 5. Current oscillations in nonpolar liquids The bulk nonequilibrium processes of the dissociation of electrolyte molecules of the impurities and recombination of the ions proceed in the case of nonpolar liquids. The solution of Eq. (1) with nonzero right-hand side shows that the above-described mechanism of the generation of self-maintaining oscillations does not change from the conceptual point of view. The presence of two additional charged components in the medium decreases the amplitudes of oscillations of the charge density and "eld strength when moving away from the injecting electrode. The distributions of the charge density and "eld strength in the interelectrode space are shown in Fig. 6 (curves 1 and 2) for dimensionless quantities w/w "a/a " 0 0 b/b "0.1 and for equal mobilities b "b (m"1}3). The voltage applied is 0 m 0
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Fig. 6. Distributions of the "eld strength E/EH (curve 1: p"0, curves 3, 4: p"1) with EH" 104F e¸/k¹ and volume charge density q/q (curve 2: p"0) along the interelectrode gap in the case of w 0 nonpolar liquids.
F "104k¹/e, the threshold "eld is E "0.5E , the dielectric permittivity is e"2.2. w c w The curves 1 and 2 are pictured for an instant when the "eld strength at the anode is less than the threshold one and there is no ion injection. One can see that the volume charge alters the sign several times in the near-anode region. We note that the rise of nonstationary structures with alteration of the sign of the volume charge is also typical for the pulsing corona discharge in gases [6]. The distributions of the ion concentrations in the interelectrode space are shown in Fig. 7 at the same time instant. The amplitude of the oscillations of the negative ion concentration (curve 2) damps out quickly with the increase of the distance from the anode. It is interesting to note that an interaction between waves of the "eld and injected ion concentration initiates an excitation of waves of the concentration of the positive intrinsic ions (curve 1). The calculations show that the excitation has a resonant character with respect to the mobility of the injected ions b . The 3 amplitude of the waves of n decreases sharply when the ratio b /b deviates from 1 3 0 unity in any side. The dependence of the oscillation frequency on the parameters characterizing kinetic and transport properties of the medium is shown in Fig. 8. For the sake of simplicity we consider the case when the dimensionless quantities w/w , a/a and 0 0
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Fig. 7. Distributions of the ion concentrations along the interelectrode gap in the case of nonpolar liquids (curve 1: n /n , curve 2: n /n , curve 3: n /n ). 1 0 2 0 3 0
b/b are equal and denoted by A in Fig. 8. The value of A characterizes actually 0 the ratio between the characteristic time of the ion drift through the interelectrode space and the characteristic time of the relaxation to chemical equilibrium in the medium. The mobility of the intrinsic ions is identical (b "b "b ). The 1 2 0 mobility b is varied. One can see that the dependence of the frequency on the 3 parameter A is nonmonotonic. The calculated frequencies fall within the range of the experimental ones [1,2] as shown in Fig. 8 because f "0.57 s~1 for conditions 0 under consideration. The dependence of the frequency on the ratio b E /b E is qualitatively the same as 3 w 0 # that in the case of polar liquids.
6. In6uence of nonuniformity of the applied 5eld on the current oscillations It is known that the Trichel pulsations in the gas corona discharge are observed in a needle-plane geometry when the "eld applied is highly nonuniform. In the theoretical model [6] the geometrical factor is taken into account only when calculating the "eld. The concentrations of the charged particles are calculated from
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Fig. 8. Dependence of the frequency f/f on the dimensionless kinetic parameters of the medium 0 (A,w/w "a/a "b/b ). (curve 1: b /b "75, curve 2: b /b "50, curve 3: b /b "10, curve 4: b /b "1). 0 0 0 3 0 3 0 3 0 3 0
one-dimensional equations like the "rst three relations of Eq. (1) with p"0. The expansion of the discharge domain when moving away from the injecting electrode is ignored. We consider a wire-cylinder geometry to estimate the in#uence of the nonuniformity of the applied "eld and geometrical factor on the current oscillations in liquids. Let the electrodes have the form of two coaxial cylinders with dimensionless radii 0.1 and 1.1, respectively. In Eq. (1) we set the index p"1. The calculations show that the nonuniformity of the applied "eld localizes the domain of the oscillations with "nite amplitudes closely to the surface of the inner injecting electrode. The frequency increases under the same conditions as in the #at case. The amplitude of the oscillations decreases rapidly with the increase of the distance from the inner electrode mainly as a result of diverging geometry. The distribution of the "eld strength is shown in Fig. 6 (curve 3) for an instant when there is no injection. For p"1, the x-coordinate is the distance from the inner cylinder. The values of the parameters are the following: w/w "a/a "b/b "0.1, b " 0 0 0 m b (m"1}3), F "104 k¹/e, E "0.5F /¸. The applied "eld E¸/F is represented by 0 w # w w the curve 4.
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The dependence of the frequency of the current oscillations on the mobility b for 3 p"1 is shown in Fig. 5 (curve 6, the scale of the ordinate axis for this curve is f/f ]10~2). One can see that the character of the dependence is not changed in 0 comparison with the case p"0. The same is valid for the dependence of the frequency on the threshold "eld.
7. Conclusion The study demonstrates that in the low-conducting liquids, as well as in gaseous media, the Trichel selfmaintaining oscillations of the current can exist. The mechanism of the rise of these oscillations is connected with a nonstationary interaction between the "eld at the electrode surface and the volume charge of a near-electrode region under the condition that the surface ionization process has the threshold in the "eld value. The calculated oscillations of the current have a regular character in contrast with the chaotic current #uctuations registered in [1,2]. From Fig. 5 we deduce that the frequency depends on the value of the threshold "eld E . This quantity is preserved # constant during the computation of each variant. Probably it is possible to simulate the process of randomization if we take E in a random way for random periods of # time during the calculation of the variant. From the physical viewpoint, such approach could correspond to nonuniform injecting properties of the electrode surface, when di!erent parts of the surface have di!erent values E and, as a result, generate # di!erent frequencies. In the framework of this hypothesis the chaotic current oscillations could be presumably obtained without any correlation with the turbulization of liquid convective motion. Acknowledgements This work was supported by the Russian Foundation of Basic Researches, project No. 98-01-00109. References [1] P. Atten, J.C. Lacroix, B. Malraison, Chaotic motion in a Coulomb force driven instability: large aspect ratio experiments, Phys. Lett. A 79 (1980) 255}258. [2] B. Malraison, P. Atten, Chaotic behavior of instability due to unipolar ion injection in a dielectric liquid, Phys. Rev. Lett. 49 (1982) 723}726. [3] R. Chicon, A. Castellanos, E. Martin, Numerical modelling of Coulomb-driven convection in insulating liquids, J. Fluid Mech. 344 (1997) 43}66. [4] V.V. Gogosov, V.A. Polyansky, Shock waves in electrohydrodynamics, Prog. Aerospace Sci. 20 (1983) 125}216. [5] B. Loeb, Electrical coronas, University of California, Berkley, 1965. [6] R. Morrow, Theory of negative corona in oxygen, Phys. Rev. A 32 (1985) 1799}1809. [7] V.V. Gogosov, K.V. Polyansky, V.A. Polyansky, G.A. Shaposhnikova, A.A. Vartanyan, Modelling of nonstationary processes in channels of EHD pump, J. Electrostat. 34 (1995) 245}265.
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[8] I.L. Pankratieva, V.A. Polyansky, Modelling of electrohydrodynamical #ows in low-conducting liquids, J. Appl. Mech. and Techn. Phys. 36 (1995) 36}44. [9] A.A. Vartanyan, V.V. Gogosov, V.A. Polyansky, G.A. Shaposhnikova, A numerical simulation of non-stationary electrohydrodynamic processes in weakly conducting liquid, J. Electrostat. 23 (1989) 245}262. [10] V.N. Pribylov, Experimental study of electrization current of dielectric liquids in cylindrical pipe, Colloidal J. 58 (1996) 524}527.