Gravity-capillary waves on the free surface of a liquid dielectric in a tangential electric field

Transcription

1 Gravity-capillary waves on the free srface of a liqid dielectric in a tangential electric field Evgeny A. Kochrin Institte of Electrophysics, Ural Branch of Rssian Academy of Sciences 66, 6 Amndsen str., Yekaterinbrg, Rssia and Nikolay M. Zbarev Institte of Electrophysics, Ural Branch of Rssian Academy of Sciences 66, 6 Amndsen str., Yekaterinbrg, Rssia P.N. Lebedev Physical Institte, Rssian Academy of Sciences 999, 3 Leninskij prospect, Moscow, Rssia ABSTRACT Processes of propagation and interaction of nonlinear gravity-capillary waves on the free srface of a deep non-condcting liqid with high dielectric constant nder the action of a tangential electric field are nmerically simlated. The comptational method is based on the time-dependent conformal transformation of the region occpied by the flid into a half-plane. In the limit of a strong electric field, when the gravitational and capillary forces are negligibly small, there eists an eact analytical soltion of the electro-hydrodynamic eqations describing propagation withot distortions of nonlinear srface waves along (or against) the electric field direction. In the sitation where gravity and capillarity are taken into accont, the reslts of nmerical simlations indeed show that, for large eternal field, the waves traveling in a given direction tend to preserve their shape. In the limit of a strong electric field, the interaction of conter-propagating waves leads to the formation of regions, where the electrostatic and dynamic pressres ndergo a discontinity, and the crvatre of the srface increases infinitely. The Forier spectrm of the srface pertrbations tends to the power-law distribtion (k - ). In the case of a finite electric field, the wave interaction reslts in a radiation of massive cascade of small-scale capillary waves that cases the chaotic behavior of the system. The investigated mechanism of interaction between oppositely-traveling waves can enhance development of the capillary trblence of the flid srface. Inde Terms electro-hydrodynamics, free srface, nonlinear waves, capillary wave trblence, conformal transforms. INTRODUCTION IT is well known [] that an eternal electric field directed tangentially to the npertrbed free or contact srfaces of dielectric liqids has a stabilizing effect on the bondary. On the contrary, the normal field reslts in an aperiodic growth of the bondary pertrbations [-4]. In recent years, new eperimental reslts concerning the inflence of electric field on dynamics of capillary waves on the srface of dielectric liqids have been obtained [,6]. The applied interest in stdying the liqid srface dynamics in eternal electric field is related to the possibility of controlling the behavior of liqid srfaces and sppressing hydrodynamic instabilities [7-]. The featres of the nonlinear evoltion of capillary waves at the liqid interfaces in the presence of the horizontal field were analyzed in [-3]. It has been shown in [4-6] that, in the case of a strong electric field (the effects of gravitational and capillary forces are negligibly small), nonlinear waves on the srface of a liqid with high dielectric constant can propagate withot distortions along (or against) the field direction. As a rle, the description of nonlinear traveling waves preserving their form (the so-called progressive waves) imposes significant restrictions on the shape of pertrbations. Solitary waves [7] and periodic Crapper s waves [8] are the eamples of sch nonlinear pertrbations. In this connection, the soltions obtained in [4-6] are not typical: they describe Manscript received on X Month, in final form XX Month.

2 propagation of waves of an arbitrary configration. The main physical restriction for their applicability is related to the vale of the eternal electric field: it mst be large enogh to neglect the effects of gravity and capillarity. In the present work, processes of propagation and interaction of nonlinear srface waves in the framework of the fll system of hydrodynamic eqations will be nmerically investigated. In particlar, it will be shown that the limit of a strong field [4-6] can be realized. For the description of nonlinear dynamics of the flid srface, we will se the method of time-dependent conformal transform (the region occpied by the liqid is mapped into a half-plane). This approach was developed in [9,] in stdying nonlinear waves on the srface of liqids in the absence of an eternal electric field. The method was sed in [3,,] for the stdy of electro-hydrodynamics of liqid dielectrics with free srfaces. At present time, the comptational techniqes based on the conformal transformations develop intensively; see, for eample, [3, 4]. The main advantage of this approach in comparison with the classical finite-difference schemes is in redcing the nmber of spatial variables. Together with the se of spectral methods for calclating the spatial derivatives, the approach allows condcting direct nmerical simlation of electrohydrodynamics of liqids with high efficiency and accracy. LINEAR ANALYSIS OF THE PROBLEM The dispersion relation for linear waves on the srface of a perfect dielectric liqid (free charges are absent) nder the action of the horizontal electric field has the following form []: ε ( ε ) E σ 3 ω = gk + k + k, () ρ( ε + ) ρ where is the freqency, k is the absolte vale of wavevector, g is the acceleration of gravity, E is the vale of eternal electric field strength, is the dielectric constant of vacm, and are the relative dielectric constant and, respectively, the mass density of the liqid, and is the srface tension coefficient. Here and below in the paper, we consider plane symmetric waves propagating along the direction of the eternal electric field. It shold be noted that this approimation is jstified, since there is anisotropy in the problem related to the distingished direction of the eternal field, see for more details []. In the absence of the eternal field, E=, the relation () describes the propagation of plane gravity-capillary waves. It is easy to show that for the wavenmber k =(g / ) /, the phase speed reaches the minimm vale, V =(4 g/ ) /4, i.e., waves on the free srface of a liqid cannot propagate with the velocity less than V. As an eample, the minimm phase speed of the gravity-capillary waves on the srface of water is V 3 cm/s; it corresponds to the wavelength =π/k.7 cm. In the presence of the electric field, the phase speed V p of srface waves also has a minimm at the same wavelength. In this case, the minimm vale of the speed depends on the electric field strength. It is easy to show that / σ g εε E Vp ( k ) = +, () ρ ρ where we take into accont that, in the problem nder stdy, ε. The epression () shows that the eternal electric field leads to increase in the velocity of waves propagation. The relation () allows to define the notion of a strong electric field in the problem. It is convenient to introdce the vale of the electric field strength E, for which the electrostatic forces are comparable with the gravitational and capillary ones, E = σ gρ / ε ε. The limit of a strong field is realized in the case β = ( E / E ), where is a dimensionless parameter defining the ratio of the eternal field to the specific vale E. Let s discss the feasibility of this limit. The estimations of E for water and ethyl alcohol (these liqids have relatively high permittivities) give.9 kv/cm and.3 kv/cm, respectively. In the case of liqid - atmospheric gas interface the parameter is bonded above by Paschen`s law. It is known that the breakdown voltage in air nder standard conditions is of the order of 33 kv/cm, so the parameter shold not eceed abot -3 dimensionless nits. As we will show frther, the limit of a strong field can be realized with high accracy for =, i.e., for the field less than the breakdown threshold. In the conclsion of this section, let s discss the important featre of the limit of a strong field. For β, the terms responsible for the inflence of gravity and capillarity can be neglected, and the relation () will describe the propagation of linear waves withot dispersion. As it was noted above, in sch a sitation, eact soltions of the fll electrohydrodynamic eqations have been fond [4-6]. According to them, the nonlinear srface waves of an arbitrary shape can propagate withot distortions along (or against) the direction of the eternal electric field. The interaction is possible only between conter-propagating waves (in [], it was shown that sch interaction is elastic). It shold be noted that this sitation is similar to that for the Alfven waves in an ideal condcting flid. The wave packets of arbitrary forms can travel nondispersively with the Alfven speed in or against the direction of the eternal magnetic field [6] and their interaction is elastic too. /

3 3 3 EQUATIONS OF MOTION We consider a potential flow of an incompressible ideal dielectric liqid of infinite depth with a free srface in an eternal niform horizontal electric field. The bondary of the liqid in the npertrbed state is the horizontal plane y= (the ais of the Cartesian coordinate system lies in this plane and the y ais is perpendiclar to it). Let the fnction (,t) specify the deviation of the bondary from the plane; i.e., the eqation y= determines the profile of the srface. Let the electric field be directed along the ais and be E in magnitde. As was shown in [4-6], the normal component of the electric field in the liqid, in the case of large permittivity, ε, is mch smaller than the tangential component. This means that field lines inside the liqid are directed along a tangent to its srface. In this case, the field distribtion in the liqid can be determined disregarding the field distribtion above it. This sitation is described by the following eqations of motion. The velocity potential of the liqid and electric field potential satisfy the Laplace eqations ϕ =, φ =, y < η(, y). They shold be solved together with the dynamic bondary condition (non-stationary Bernolli eqation) φ + φ ρ η + σ ρ η t ( ) / = ( P PE ) / g K /, y =, (3) where P E is the electrostatic pressre, P is the energy density of the eternal electric field in the liqid, K = η /( ) 3/ + η is the crvatre of the flid srface. In the sitation nder stdy, the qantities P E and P are defined by the following way: P = ε ε ( ϕ) / and P = ε ε E / (for more details, E see [4]). The dynamics of the srface is described by the kinematic bondary condition η = φ η φ, = η. (4) t y y The potentials satisfy the condition ϕ η ϕ = at the y bondary y= (,t), and the conditions φ, ϕ E. at infinity y. Together, the above relations are a closed eqations system describing the motion of a deep dielectric liqid with free srface nder the action of gravity, capillarity, and electrostatic forces cased by the eternal horizontal electric field. In the linear approimation, where the amplitde of srface pertrbations is infinitesimal, this system redces to the dispersion relation (). It is convenient now to switch to dimensionless notations as follows: y y/k, /k, t t, / k, E E, E /k, where we introdced the qantity τ / V k, determining = the characteristic time scale of the system in the absence of electric field, ~. s. 4 EQUATIONS IN CONFORMAL VARIABLES For complete description of the dynamics of the system, it is necessary to obtain distribtions of the velocity field and electric field inside the liqid. Ths, the eqations of motion given in the previos section have the dimension (+). It trns ot that there is an effective way to redce the dimension by transition to new spatial (conformal) variables. Sch a transition allows to significantly increase the speed and accracy of calclations. Similar to [9-], let s make the conformal transformation of the region occpied by the liqid into the half-plane: {, y} {, v}, where v<. In terms of the comple variables z=+iy and w=+iv, the conformality of the transformation implies that z is an analytic fnction of w. The ailiary variables and v in the problem nder stdy have clear physical meaning: coincides with the field potential ecept for the sign and the condition v=const specifies the electric field lines. In the new variables, the Laplace eqations for the electric field and velocity potentials can be solved analytically. As a reslt, the initial problem of motion of the liqid can be redced to the problem of motion of its free srface, which has a lower dimension of (+). Since the fnction y(w) is an analytic, the srface of the liqid in the new variables is specified by the parametric epressions y = Y (, t), = X (, t) = HY ˆ, where Ĥ is the Hilbert transform defined in Forier-space as Hˆ k = i sign( k ). Ths, the relation between the fnctions (,t) and Y(, t) is given in the implicit form Y (, t) = η( HY ˆ, t). The fnction Ψ(,t) specifying to the vale of velocity potential at the bondary v= is introdced by the same way. The procedre of obtaining the eqations of motion in conformal variables is well known; see [9-]. For this reason, we will not give a detailed derivation of these eqations, and jst write them down in the final form. Let s introdce the comple fnctions Z = X + iy, Φ = Ψ + ihˆ Ψ, which can be analytically contined into the lower comple half-plane. It is convenient to se the projection operator P ˆ = (+ ihˆ )/, which transforms a real fnction to an analytic comple fnction in the lower half-plane of the comple variable, i.e., Z = + ipˆ Y and Φ = P ˆ Ψ. Let s now introdce the Dyachenko variables [9]

4 4 R = / Z, V = iφ / Z. It shold be noted that these fnctions can be interpreted in terms of the real physical variables. The fnction V(,t) corresponds to the absolte vale of the flid velocity at the bondary, and the qantity R(,t) determines the local electric field strength on the srface in the case of. The fnction R(,t) has also a sense of the inverse Jacobian of t he conformal transformation: R J = ( X + Y ). = As a reslt, the dynamic and kinematic bondary conditions (3) and (4) are rewritten as follows: V = i( UV D R) ( R ) Q P( Q Q Q Q), () ˆ t + R = i( UR U R), (6) t where we introdced the notations U Pˆ V R V R D Pˆ VV RR Q = R / = ( + ), = ( β ),. The overline stands for comple conjgation. The eqations () and (6) constitte the system of integrodifferential eqations describing the flly-nonlinear evoltion of the free srface. It is completely eqivalent to the original system of partial differential eqations. For the nmerical soltion of eqations () and (6), we will se the eplicit Rnge-Ktta method of the forth order of accracy with respect to time. All spatial derivatives and Hilbert operators will be compted by means of the fast Forier transform, i.e., the bondary conditions will be periodic. For controlling calclation error, we can se the fact that the system nder consideration is conservative, and its total energy has the following form: + ˆ ˆ / = ( β [ ]). (7) Ψ Ψ + H H YHY Y X J X d Conclding this section, let s retrn to the strong field limit, where gravity and capillarity can be neglected. For the case β, the system () and (6) takes the compact form waves. SIMULATION RESULTS. TRAVELING WAVES First of all, let s consider the possibility of propagation of stationary nonlinear waves on the liqid bondary in the framework of the complete eqations () and (6). Since the gravity-capillary waves themselves are dispersive, let s set the initial conditions in sch a way that a wave wold be distorted only de to nonlinear effects (the linear dispersion is absent). The epression for the phase velocity in dimensionless variables has the form V k β ω k k β k (, ) = / = p + +. (9) Let s frther consider the evoltion of a periodic wave of the amplitde A and the wavelength π/k. In the linear approimation, its velocity is constant and eqal to V p (k, ) according to (9). The corresponding initial conditions for the fnctions R and V are the following: R(,) = + Aep( ik ), V (,) = AiVp ( k)ep( ik The deformation of this wave is intensified with an increase in the amplitde A. The figre shows the evoltion of the liqid srface for the wave amplitde A=. and different vales of. The vertical ais corresponds to time measred in the relative vales t/t, where T =π/v p (k, ) is the time period of the wave propagation. Since the wave speed depends on the field strength, it is convenient to compare the dynamics of the flid bondary for varios in the relative time scale. Figre shows that electric field really has a stabilizing effect on the dynamics of a nonlinear pertrbation of the srface. It can be seen that, for =, the deformation of the wave is very strong. At small field, =, the distortion of the wave occrs with less intensity, and for the relatively large electric field, =, the shape of the pertrbation almost does not change dring ten periods, t=t. The obtained reslts indicate that the limit of a strong electric field can indeed be realized for the waves propagating in a single direction. ). V = i( UV D R), R = i( UR U R). (8) t t It is important that this system admits a pair of eact ± particlar soltions, V ( βt) = iβ ( R ). These soltions correspond to waves of an arbitrary geometry that propagate at a constant velocity withot distortions along (pper signs) or against (lower signs) the direction of the eternal electric field. In the net section, we will nmerically investigate the possibility of realization of these soltions in the framework of the fll system of eqations () and (6). Also, we will consider the interaction between oppositely-propagating nonlinear Figre. Evoltion of the free srface for different vales of the electric field strength: (a) β =, (b) β =, (c) β =. It shold be noted that the investigated sitation is strongly nonlinear, since the amplitde of srface distrbance (A=.)

5 is close to the eistence threshold for srface waves. For larger amplitdes of pertrbations, the self-intersections of the bondary and the formation of air bbbles can arise at the liqid srface as shown in figre. η. temporal evoltion of the electric field pressre is plotted in figre 4. As one can see, at the initial stages of the system evoltion, the qantity R is smooth enogh. At some moment of time, the interaction of srface waves leads to the formation of narrow spatial regions (shock fronts), where electric field changes sharply. It is interesting to note that the shock fronts do not move with constant velocity see figre 4b. Let s eplain the mechanism of this phenomenon Figre. The calclated shape of the bondary is shown at the initial moment (ble crve ) and at the end of calclation interval t/t.76 (red crve ) for the parameters A=.6, k=, β =.. INTERACTION OF WAVES IN THE STRONG FIELD LIMIT As it was shown earlier, the limit of a strong field is realized =. Althogh nonlinear waves individally for propagating in the positive or negative direction of the ais behave as linear, this does not mean that oppositely propagating waves do not interact with each other. The qestion on the system dynamics in the case of wave collisions is of principal importance in the problem nder investigation. First of all, we consider elastic interaction of the waves in the framework of redced system (8). Let s set the initial conditions as follows: R(, ) = + A ep( ik ) + A ep( ik ), () V (, ) = Ai β ep( ik ) + A i β ep( ik ). The reslts of nmerical simlation are shown in figre 3; the parameters are chosen as A=., A=., k=, k=, =. The calclations were stopped at the moment when the relative error in comptation of energy (7) reached the vale -6. The nmber of sed Forier harmonics was eqal to, the spatial period of the problem was. The integration with respect to time was carried ot with the adaptive step, which minimm vale was of the order of -7. η β R. - (a) -. V - (b) - (c) Figre 3. (a) The srface of liqid, (b) the local electric field, and (с) the velocity of the flid srface are shown at the initial moment (ble crves ) and at the end of calclation interval t/t 4.6 (red crves ). It can be seen that the region with steep wave front has been formed on the liqid bondary at the end of calclation interval, see figre 3a. In this region, the electric field pressre ndergoes a discontinity, figre 3b, and the dynamic pressre increases almost an order of magnitde, figre 3c. The spatial- Figre 4. Evoltion (a) of the free srface and (b) of the qantity β R. In a sitation where the shock fronts have been formed, the spatial scales of oppositely traveling waves differ significantly, see figre 4. For the wave with the amplitde A and wavenmber k, the scale is defined by the width of the shock front d. For the other wave (with the parameters A and k) the spatial scale eqals to the wavelength π/k. Obviosly, we have π/k d. Sch a sitation was analyzed in [7], where it was shown that the velocity of a small-scale wave depends on geometry of the conter-propagating large-scale wave. From the physical point of view, this effect is related to the fact that the wave speed is proportional to the electric field strength in the limit β. The shock front velocity is defined not by the eternal (npertrbed) field, bt by the local (pertrbed) vale of the field R. The figre 4 demonstrates eactly this behavior: the velocity of the shock front is clearly correlated with the elevation and the local field R. It shold be noted that the velocity of the shock front does not depend on its shape and it is mostly defined by the geometry of large-scale wave, along which it propagates. The spectrm of the fnction Y(,t) presented in figre a gives an evidence of the singlar behavior of the system. It can be seen that dring the system evoltion the spectral fnctions of Y(,t) tend to a power-law distribtion. Its eponent is close to two, which indicates to the formation of a singlarity in the second spatial derivative of the fnction Y. From the physical point of view, sch a behavior can correspond to an infinite increase in the crvatre of the srface K. Figre b shows the time evoltion of the maimm of absolte vale of the srface crvatre. It can be seen that the crvatre increases jmpwise. Almost discrete jmps of the crvatre are associated with the interaction of shock fronts moving in the opposite directions.

6 6 6 Yk ma K k t t - t3 (a) k 3 3 (b) 4 t/t Figre. (a) The spectrm of the fnction Y(,t) is shown at sccessive instants of time t=, t/t=.3, t3/t=4.48, the solid line corresponds to the power-law fnction Yk ~ k-. (b) The maimm of the srface crvatre verss time. Ths, the nmerical simlation of the interaction between conter-propagating nonlinear waves in the framework of the redced system (8) demonstrates a tendency to the formation of singlarities at the flid srface; the regions appear where the pressre eerted by electric field ndergoes a discontinity and the dynamic pressre increases almost an order of magnitde. It is interesting to note that the interaction of the contrary-propagating Alfven waves leads to the acceleration of plasma particles [8]. It shold be noted that the observed behavior of the flid srface sfficiently differs from the simple wave breaking, which is shown in figre. For the breaking process, infinite gradients of pressre in the -projection arise only on the geometric reasons even in the case of their continos distribtions along the bondary. The evidence of the singlar behavior is the spectral fnctions of Y: they tend to the powerlaw distribtion, Yk ~ k-. A possible reason of the formation of these singlarities is the absence of terms responsible for capillarity and gravity in the eqations (8). Frther, we will consider the interaction of oppositely-traveling nonlinear waves in the framework of the complete system of the evoltion eqations () and (6)..3 INTERACTION OF WAVES IN THE CASE OF FINITE FIELD For nmerical simlation of the wave interaction on the basis of the complete eqations system () and (6), we take the initial condition for the fnction R in the form (). The initial condition for V is modified to the following: V (,) = AiV p (k ) ep( ik ) + AiV p (k ) ep( ik ) For a correct comparison of the reslts with the previos eperiment, we choose the calclation parameters A,, k,, to be the same as earlier. Figre 6 shows the evoltion of the liqid srface and local electric field at the bondary; the time period is chosen as T=π/Vp(k, ). One can immediately see the main difference in dynamics of the system from the case of a strong field: the tendency to the formation of a discontinity in the electric field strength is not observed now. The time interval in figre 6 was chosen to provide convenience of comparison with the figre 4. In fact, the calclation interval has reached a rather large vale t/t Figre 6. Evoltion (a) of the free srface and (b) of the qantity β R. One can see from the figre 6 that the region of the discontinity formation is smoothed. The reason is that the capillary waves are ecited in the region of shock formation. At the developed stages of the system evoltion, this process leads to the appearance of a massive cascade of small-scale waves. It can be seen from the figre 7 that, at large times, the dynamics of liqid fndamentally differs from the case of a strong field shown in the figre 3. The capillary waves generation reslts in the fact that dependencies plotted in the figre look very chaotic. Apparently, the pictre observed in the figre 7 evidences that the eternal tangential electric field can lead to the development and acceleration of the so-called capillary wave trblence [9]. η. V β R -. - (a) - (b) - (c) Figre 7. (a) The srface of liqid, (b) the local electric field, and (с) the velocity of the flid srface are shown at the initial moment (ble crves ) and at the end of calclation interval t/t (red crves ). Figre 8 shows the time dependence of the crvatre maimm and spectra of the fnction Y(,t) at sccessive instants of time. At the initial stages of the system evoltion, t/t<, it is possible to distingish discrete jmps of the crvatre as in the figre b. At large times, the dependence shown in the figre 8a becomes very complicated. The transition of energy to small scales leads to the error accmlation and, as a conseqence, to a bonded comptation interval. Indeed, the spectrm of the fnction Y(,t) ceases to be localized with time. From the physical point of view, the divergence of the algorithm is related to the absence of viscos forces in the problem nder consideration. For the fll description of the trblent motion of the srface, one shold introdce additional terms into the eqations () and (6), as it was done, for instance, in [3]. We plan to consider the trblent motion of gravity-capillary waves in a tangential electric field and also take into accont the terms responsible for the trblent viscosity in ftre.

7 7 ma K Y k t/ T k (a) (b) Figre 8. (a) Maimm vale of the srface crvatre verss time, (b) the spectrm of the fnction Y(,t) is shown at sccessive instants of time, t =, t /T =.3, t 3/T =4.48. The solid line corresponds to the power-law fnction Y k ~ k -. - t k - t t3 In conclsion, let s give an eample of the simlation of the interaction between conter-propagating periodic waves in the case, where the eternal electric field is zero, =. It trns ot that in this sitation, the generation of capillary waves becomes very slow (it is almost absent in comparison with the previos calclation). Figre 9 shows the evoltion of the free srface and of the qantity R, which does not have a sense of the electric field strength in the latter case. From the figre 9b, one can see that the interaction of waves is not elastic. The wave with large amplitde moves in the positive direction of ais absorbing the conter-propagating wave with small amplitde. If the electric field is absent, the dynamics of liqid is very stable, the comptation interval reached the vale, t/t. 3. At the end of calclation interval, the relative error has not eceeded the small qantity -8. Figre shows the dependencies analogos to that shown in the figres 3 and 7. It can be seen that even after a long interval of time, the plotted fnctions remain smooth and do not have noticeable small-scale pertrbations. It is noteworthy that, dring the evoltion of the system, a critical increase in the crvatre of the srface was not observed, and the spectrm of the fnction Y(,t) remained localized. Ths, the mechanism of the formation of massive cascade of small-scale capillary waves, which was observed in the previos nmerical eperiment, is eactly related to the interaction of conterpropagating nonlinear waves in the presence of a strong tangential electric field. η R V (a) (b) ( c) Figre. (a) The srface of liqid, (b) the qantity R, and (с) the velocity of the flid srface are shown at the initial moment (ble crves ) and at the end of calclation interval t/t. 3 (red crves ). 6 CONCLUSION At the present work, the processes of propagation and interaction of nonlinear waves on the free srface of a noncondcting liqid nder the action of gravity, capillarity, and tangential electric field were nmerically simlated. The comptational algorithm was based on the time-dependent conformal transformation of the region occpied by the flid into a half-plane. This approach allows to redce the original spatially two-dimensional problem to the one-dimensional one withot loss of generality that essentially increases the efficiency of calclations. The reslts of or nmerical simlations show that, for sfficiently high electric field strength, the srface waves can separately propagate along, or against the field direction almost withot distortions. This reslt evidences that the limit of a strong field [4-6], in which the gravitational and capillary forces are neglected, indeed can be realized for the waves traveling in one direction (at least for a finite time). Nmerical simlations show that in the limit of a strong electric field, the elastic interaction of conter-propagating waves leads to the formation of singlarities, viz. points on the liqid srface where the electrostatic and dynamic pressres ndergo a discontinity. The spectrm of the srface tends to the power-law distribtion, Y k ~ k -. This indicates to an infinite increase in the crvatre of the bondary. In the case of a finite electric field, the gravitational and capillary forces lead to smoothing the region of pressre discontinity at the initial stages of the wave interaction. At the developed stages of the system evoltion, the small-scale capillary waves are intensively radiated from of the shock front region that cases the chaotic behavior of the system. It is shown that, in the absence of an eternal field, the wave radiation is very weak; it is not almost noticeable at the comparable time scales. Ths, the investigated mechanism of the conter-propagating waves interaction reslting in the formation of singlarities in the strong field limit can enhance and accelerate the development of capillary trblence of the liqid srface. Figre 9. The evoltion (a) of the free srface and (b) of the qantity R. ACKNOWLEDGMENT This work was spported by the Ministry of Edcation and Science of the Rssian Federation (state contract No ). The work of E.A.K. was spported jointly by RFBR project No _mol_a_dk and by the

8 8 Presidential Programs of Grants in Science (project No. SP3.6.). The work of N.M.Z. was spported by the RFBR (project Nos , ) and by the Presidim of UB, RAS (project Nos , 8---). [] [] REFERENCES [] [] [3] [4] [] [6] [7] [8] [9] [] [] [] [3] [4] [] [6] [7] [8] [9] [] J. R. Melcher and W. J. Schwarz, Interfacial relaation overstability in a tangential electric field, Phys. Flids, Vol., pp , 968. J. R. Melcher, Electrohydrodynamic and magnetohydrodynamic srface waves and instabilities, Phys. Flids, Vol. 4, pp , 96. E. A. Kochrin and N. M. Zbarev, Formation of crvatre singlarities on the interface between dielectric liqids in a strong vertical electric field, Phys. Rev. E, Vol. 88, 34, 3. N. M. Zbarev, Criteria for hard ecitation of electrohydrodynamic instability of the free srface of a condcting flid, Physica D, Vol., pp ,. D. Kolova, C. L. Loste, and H. 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He received the M.Sc. degree in applied mathematics and physics from Moscow Institte of Physics and Technology, Moscow, Rssia, in 994, the Cand.Sci. degree from the Institte of High Crrent Electronics, Rssian Academy of Sciences, Tomsk, Rssia, in 997, and Dr.Sci. degree from the Institte of Electrophysics, Rssian Academy of Sciences, Ekaterinbrg, Rssia, in 3. He is crrently with the Institte of Electrophysics, Rssian Academy of Sciences, where he is involved in theoretical stdying of nonlinear phenomena in liqids with free srface nder the action of an electric field and electrical discharges in gas and vacm. Prof. Zbarev is a Corresponding Member of the Rssian Academy of Sciences since 6. He was a recipient of the State Prize of the Rssian Federation in Science in 3.