Finite-Amplitude Surface Waves on a Thin Film Flow Subject to a Unipolar-Charge Injection

Transcription

1 Korean J Chem Eng () (005) Finite-Amplitude Surfae Waves on a Thin Film Flow Subjet to a Unipolar-Charge Injetion Hyo Kim Department of Chemial Engineering University of Seoul 90 Jeonnong-dong Dongdaemun-gu Seoul 0- Korea (eived January 005 aepted Marh 005) Abstrat The interation of an eletri field with a dieletri liquid film is investigated as it drains under gravity down an inlined plane eletrode emitting uniform positive ions into the liquid region By applying long-wave approximation to the governing equations the evolution equation for the free surfae is derived up to the first order of a thin film parameter ξ To investigate the spae harge effet on the development of a finite-amplitude surfae wave a neutral stability ondition is obtained as a ritial ynolds number through a linear stability analysis the amplitude veloity of a periodi disturbane are also alulated within a superritially stable flow region The presene of a unipolar spae harge in the fluid makes a steady surfae wave take on even higher amplitude faster wave speed ompared with the ase of no spae harge Key words: Finite-Amplitude Surfae Waves Unipolar-Charge Injetion Dieletri Liquid Film Critial ynolds Number INTRODUCTION To whom orrespondene should be addressed hkim@uosakr The hydrodynami study under an effet of eletri fore has reently attrated muh attention also extensively studied for many new proesses suh as ion-drag pump turbulent mixer heat onveting system et by using a dieletri liquid ontaining spae harges as a working fluid The transfer rates of momentum heat are known to be muh inreased owing to the unipolar harges injeted into the liquid medium [Atten 996] Here as a new basi problem dealing with the effets of free harges on the fluid motion the surfae wave behavior of a dieletri liquid layer will be onsidered when it is drained by gravity down an inlined plane eletrode at the same time it is subjeted to a unipolar injetion from this bottom eletrode with some eletri potential Another plane eletrode with zero potential is mounted above the liquid film at some distane parallel to the bottom eletrode Air is oupied between the liquid layer surfae the top eletrode In somewhat different appliations from the present stellanos et al [99] studied the behavior of the unipolar injetion indued instabilities in plane Poiseuille Couette flows where they determined the resulting onvetive patterns the instability thresholds to get the insight of the physial mehanisms In the absene of the spae harge effet as far as the nonlinear eletrohydrodynami film flows have been onerned Kim et al [99] González et al [996] Kim [99 00] have examined analyzed many aspets of a thin liquid layer flowing down an inlined plane under an applied eletri field Kim et al [99] systematially srutinized the effet of an eletrostati fore on the film flows by driving analytial equations of motion in the limits of small large ynolds numbers González et al [996] observed the stability of the nonlinear eletrohydrodynami waves based on the nonlinear evolution equation for the film height using the long-wave approximation In order to fous on the linear nonlinear stabilities they onsidered the fluid was a perfet eletri ondutor the eletri field was uniformly applied from infinity Kim [99] examined the nonlinear surfae-wave instabilities numerially with a Fourierspetral method he also onfirmed the existene of pulse-like solitary waves with the help of a global bifuration theory demonstrated their developing proesses numerially in his work [00] The main purpose of present study is to derive the equations proper boundary onditions governing the eletrohydrodynamis of a thin film flow into whih unipolar-harged ions are injeting Utilizing a long-wave approximation to the flow system makes it possible to deouple the dominant terms of eletri potentials spae harge from the fluid dynamis whih pushes the system to be exposed to analytial solutions After obtaining a surfae-wave evolution equation of the flow region as in other general hydrodynami problems the eletrohydrodynami instability is arried out on a linearly perturbed surfae wave to determine the onset onditions at whih the layer will be suseptible to the development to turbulene MATHEMATICAL FORMULATION Assuming the liquid is isothermally inompressible visous dieletri the thin layer runs down an inlined plane under the ation of gravity g The plane supporting the fluid layer ats as an injetor with an eletri potential V a positive-ion harge density q i The spae harges will be spreaded out through the liquid phase exert the subsequent Coulomb fore on the flow It is assumed that the migration of the harged partiles relative to the fluid motion is insignifiant beause of their relatively small relaxation time (defined by the ratio of eletri permittivity to ondutivity of the liquid) thus the harge is frozen to the liquid the surfae harge density on the free surfae is also assumed inonsequential In order to solve the governing field equations with proper boundary onditions it is onvenient to take their dimensionless forms Only two-dimensional ase will be onsidered Hene the bottom 95

2 96 H Kim Φ q are the eletrial potential spae harge density in the liquid respetively the subsripts st for the partial derivatives ε f K m denote the eletri permittivity ion mobility in the liquid C measures the relative importane of spae harge to the applied eletri potential R is similar to the Prtl number in the hydrodynamis For the fluid dynamis the ontinuity equation beomes u x v y =0 (5) while the x y omponents of the momentum are respetively ξ( u t uu x vu y ) = ξp x ( ξ u xx u yy ) ξqneφ x sinβ Fr (6) ξ ξ ( v t uv x vv y ) = p y ξ osβ ( v xx v yy ) qneφ y () Fr where the ynolds number the Froude number Fr the eletrial Newton number Ne have the following definitions: Fig The physial onfiguration of the plane flow subjeted to a unipolar-ion injetion ρu = 0 d µ (8) plane eletrode is assumed to make an angle β with the horizontal the oordinate system is taken suh that the x axis is parallel to the plane while the y axis is perpendiular to it as shown in Fig Above the liquid layer there is an air whih is assumed free of harge the density visosity of air will be taken no aount of here in omparison with those of liquid Within the air region at a distane H from the bottom plane of length L loates an unharged eletrode with the same dimension of the bottom one Suppose that d is defined as the harateristi thikness of the primary film flow L is the length sale of the disturbane in the x diretion then the parameter ξ=d/l will appear in the dimensionless equations Letting V be the unit of eletri potential q i the unit of spae harge density d the unit of length in the y diretion U 0 the unit of veloity for u in the x diretion (the speifi U 0 will be hosen later) ξu 0 the unit of veloity for v in the y diretion L/U 0 the unit of time t ρu 0 the unit of pressure p the dimensionless field equations boundary onditions an be determined First for the eletrial field in the liquid regime the Gauss law onservation of spae harge due to the onvetion of harge with the fluid motion the ondution urrent [Atten 996] yield respetively the following equations: ξ Φ xx Φ yy = Cq () ξq t ξ(uq x vq y )R[(ξ q x Φ x q y Φ y )q(ξ Φ xx Φ yy )]=0 () where the dimensionless numbers C R are defined by C = q d i ε f V K R = m V U 0 d () () (9) (0) Here the density visosity of the liquid are noted as ρ µ respetively In the air there is only one field equation for the eletri potential Ψ ie ξ Ψ xx Ψ yy =0 () The dimensionless boundary onditions are still to be speified Along the injetor y=0 the eletri potential spae harge onditions the no-slip ondition are respetively Φ = q= () u=v=0 () On the fluid interfae y=h(t x) there are two eletrial boundary onditions ie the ontinuities of potential normal displaement field like U Fr = gd q Ne = i V ρu 0 Φ =Ψ () ξ h x Ψ x Ψ y =ε( ξ h x Φ x Φ y ) (5) while for the fluid dynamis there are jump mass normal tangential stress onditions respetively: h t uh x =v (6) h xx ξ = ( p ξ ( h x ) / a p) K ξ h -- ξ x ε {( h x Ψ x Ψ y ) ξ εψ ( x h x Ψ y ) } July 005

3 Finite-Amplitude Surfae Waves on a Thin Film Flow Subjet to a Unipolar-Charge Injetion 9 () ( ξ h x )(u y ξ v x )ξ h x (v y u x )=0 (8) Here p a is the air pressure ε is the dieletri onstant of liquid defined by ε f /ε 0 (ε 0 is the permittivity of free spae) the apillary number the dimensionless onstant K are introdued suh as µu = σ ε K = 0 V d µu 0 ξ{ ξ h x u x h x ( u y ξ v x ) v y } ξ ( h x ) (9) (0) where σ is denoted as the surfae tension of the liquid in (0) the MKS eletrostati units are employed The last dimensionless boundary ondition along y=h is Ψ=0 () SOLUTIONS AT LEADING ORDER IN ξ The thin film limit by taking ξ has been onsidered to get appropriate solutions from the nondimensionalized Eqs ()-() Assuming C R Fr Ne have order-one values in ξ exept whih is assumed order of ξ to inlude the stabilizing effets of surfae tension the solution desribing the interfae behavior upto order ξ ould be sought through a regular perturbation method by exping the dependent variables in ξ like Φ =Φ 0 ξφ q=q 0 ξq Ψ =Ψ 0 ξψ u=u 0 ξu v=v 0 ξv p=p 0 ξp From () () () the leading-order solutions in ξ for the eletri potentials harge density are obtained as Φ 0 = -- ( y ) / for 0<y<h Ψ 0 = y 5 for h<y<h q 0 = / for 0<y<h () y where n (t x) for n= 5 an be found by solving the following equations whih are given from the boundary onditions = C / = -- / = ε( h ) / 5 = H h = -- ( h ) / 5 () Here it is known that 5 are positive whereas is negative By hoosing the harateristi unit of veloity as the average veloity of the steady parallel flow ie U 0 =ρgd sin(β)/µ the leadingorder veloity omponents pressure are obtained as u 0 = h y h y h -- () v 0 = --h x y osβ p 0 = p a ( h y) -- Ne {( Φ Fr 0y ) y = h C ( Φ 0y ) } K ( Ψ0y ) h ξ ε xx (5) (6) From these leading-order solutions we have to note that the solutions for the eletri fields are independent of the hydrodynami ones beause the solutions are produed from a quasi-steady state in the thin film limit FILM EVOLUTION EQUATION A nonlinear evolution equation desribing the film deformation has to be derived to determine the stability of liquid layer subjeted to the spae harge In order to inlude the effets of gravity visosity surfae tension eletri field spae harge on the film flow it is still neessary to find the veloity omponents one step further to the first order in ξ However as long as the parameters stated in the previous setion are keeping order unity the sought solutions for Φ Ψ q at the leading order are still enough for the aounts of the eletri fore effets on the film flow that the proedures to get their next-order solutions of these are not neessary By plugging the leading-order solutions ()-(6) into (5) (6) the veloity omponents next to the leading order ie u v an be gained with the onditions () (8) The results are u = hh x --y --hy h y 8 Ne ot ( β) C hx y --- hy Ne x y hy --- h y C ξ h xxx K -- ε x ( y hy) Ne x ( y ) { ( h ) y} 8 C Ne x ( y hy) C Ne x --y ( ) / -- / ( h ) / y () C v = hh x -----y 5 --hy --h y 0 8 Ne ot ( β ) y C hx hy Ne x --hy y --h y C ξ h xxx K -- ε x y --- hy Ne ( y ) x y --h ( ) y 8 C Ne x y --- hy C Ne x -----y ( ) 5/ 5/ C 5 5 Korean J Chem Eng(Vol No )

4 98 H Kim -- /y --h ( ) / y x (8) Finally substituting the leading-order veloity omponents of () (5) () (8) at the first order in ξ into the kinemati boundary ondition (6) the evolution equation for the surfae defletion h(t x) aurate to O(ξ ) is obtained suh as h t h h x ξ 6 h 5 --h6 x / ot ( β ) --Ne hx h h ξ xxx h --K x -- h ε / / -----Ne( x h x 6 / x )h Ne / x 8h ( ) 5/ 5/ [ 8 0 /h 5( h ) / h ] 5 =0 (9) The evolution equation is a paraboli partial differential equation the terms proportional to Ne desribe the effet of spae harges if there is no spae harge in the liquid phase the result has the same form as the one derived by Kim [99] x / P = ε( s ) ( H ) 8 s { / s ( s ) / } ( 5 s ) Q = s { ε( H s ) ( s ) } () Now it is possible to perform a linear stability about the system (9) by assuming the small disturbane h has a simple periodi form ie h =exp{iα(x t)} where α 0 is a wavenumber of the disturbane is the omplex wave speed set by = R i I After putting this harmoni form into (9) then setting the real part equal to zero with the imaginary wave speed I =0 the neutral stability ondition an be set up by defining the ritial ynolds number = ot ( β) -- s K -- ε where --α ξ /W W = 6 -- / -- s Ne ( s ) / -- s 8 -- s s -- s -- s s / () -- s s 5 () LINEAR STABILITY Linear stability for liquid layer running down an inlined plane was theoretially initiated studied by Benjamin [95] Yih [96] Gjevik [90] without the external effet of eletri field Kim et al [99] González et al [996] Kim [99] analyzed the stability of film flow linearly whih was being affeted under an applied eletri field with no embedded spae harges in the liquid The aim here is to obtain a general result inluding the spae harge effet on the film flow under an eletri field To ondut a linear stability analysis (9) has to be perturbed about the steady state solution ie h(t x)= h where h is a small disturbane of the free surfae And beause the values of i for i= 5 are dependent on h(t x) they also have to be exped about their steady solutions by setting i = is for i= i 5 (overbar values denote the small perturbations superimposed on the steady state solutions of is ) then s i have to be denoted with h to seure the hiness of linear stability task The relations between s i h are gained from the linearized equations of () The results are expressed in the linearized forms like = s h = s h = s h = s h 5 = 5s 5 h (0) where is s an be numerially alulated from () at h= i s are determined by using s as Q / Q = ---C s = --- = 8 -- Q P P ---C s P / εc = s Q --- ( s ) / s P where P Q st for 5 = H () Fig (a) Leading-order potential gradient profiles in the liquid for C=000 5 n n=0 with ε=0 h= H=5 (b) Leading-order spae harge profiles for C=000 5 n n=0 with ε=0 h= H=5 July 005

5 Finite-Amplitude Surfae Waves on a Thin Film Flow Subjet to a Unipolar-Charge Injetion 99 In the equations of () () it has to be noted the value s is positive usually gets smaller than one provided C beomes larger beause as the injetion is getting stronger the eletri field dereases at the injetor thus from the first equation of Φ 0 in () the ontribution of loses its weight When C it is referred to as a ase of spae-harge-limited urrent (SCLC) [Shneider et al 90; stellanos et al 99; Atten 996] To onfirm the validity of SCLC the steady-state solutions of Φ 0y q 0 are plotted in Fig with C inreasing for ε=0 V=0 kv/m As C it an be aknowledged that Φ 0y 0 that is the eletri field strength (= Φ 0y) 0 q 0 at y=0 The SCLC state is also well represented if substituting =0 into Φ 0 q 0 in () FINITE-AMPLITUDE SURFACE WAVES The linear stability analysis is only valid as long as the disturbed surfae waves are kept very small enough to have a single harmoni mode However beyond the neutral urve defined by () that is in the linearly unstable region around the (α-) domain it has been well known the flow beomes superritially stable there exist nearly sinusoidal surfae waves of finite amplitude [Gjevik 90; Nakaya 95] To investigate this behavior of finiteamplitude surfae waves developing on the thin film flow subjeted to a unipolar-ion injetion the amplitude wave-speed equations are derived by letting the surfae defletion h(t x) as a Fourier series h=a os(αϕ)a (b e iαϕ b 0 b * e iαϕ ) h=a (b e iαϕ b e iαϕ b * e iαϕ b * e iαϕ ) (5) Here b m n are arbitrary onstants the asterisk means the omplex onjugate the disturbane amplitude a(<) the phase ϕ are assumed as [Nakaya 95] da ---- = s a s a dt ϕ = x ϕ = w = w w a t (6) where s s whih is often alled Lau s seond oeffiient w w are some onstants these an be determined with b m n at the same time After substituting (5) into the linearized form of (9) around h= then using the relationships of (6) the film-depth evolution equation an be rearranged aording to the order of a From the equation for the first order of a s w are determined from the oeffiients of os(αϕ) sin(αϕ) respetively as s (α )=α ξ(a B) w = () where A B represent A = Ne s / s ( s ) / s s C / -- s -- s -- s s B = --K -- s ε --α ξ ot( β) (8) Here it has to be noted that if s =0 () gives the same result as () when C in (8) is replaed with the first relation in () That is the result from the first-order equation in a shows the linear stability ondition for a monohromati wave The oeffiients in a give U V b 0 = 0 b = i M N αξ( M N) where M N U V are defined as Ne M = s ( s ) / / { ( 8 s 6 s ) s s ( s 5) C 5 s ( 6 s s 6 s )} 6 N = 0K -- ε s 80α ξ 80ot( β) U = Ne ( s ) / 5 s s s C s ( 6 5 s ) s ( 9 0 s ) s s (9) (0) In addition from the analysis of seond harmoni omponent of exp(iαx) b has to meet the ondition s (α )<0 () for its onvergene [Nakaya 95] Within the region satisfying s (α )>0 () the flow system beomes superritially stable Hene there will exist some finite-amplitude surfae waves Finally to find out s w letting the oeffiients of os(αϕ) sin(αϕ) in a equal to zero provides the results: where / s { 5 s ( s ) 6 s ( s 5 5 s )} 5 --{ s ( s ) 0 s 8 s } 5 -- s ( 9 s 8 ) V = 60K -- ε ( s ) 90α ξ 5ot( β) 80 s = α ξ U V C D ξ( M N) M N α ξ U V E G I M N 5 F H J 9 -- w = -- 6 U V C -----U M N M N 5 C = --K -- ( ε s ) α ξ ot( β) D = K -- α ( ε s ) ξ -- ot( β) E = Ne / s ( s ) s s -- s -- s 8 -- s C s s -- s -- s s s ( s ) F = Ne / s ( s ) C 6 -- s -- s s -- s -- ( s ) s s () Korean J Chem Eng(Vol No )

6 500 H Kim s 5 -- s s -- s s ( s ) 6 G = Ne ( s ) / s -- s s -- s C s -- s s s s -- s s s H = Ne ( s ) / s 9 -- s -- s C s 6 -- s ( s ) s -- s s 8 -- s -- s ( s ) s s I = Ne ( s ) s -- s -- ( s ) -- ( C 9 s ) s s -- s 9 s s s s -- s 6 9 J = Ne ( s ) s -- s ( s ) C 8 s ( s ) s s -- s s s -- s s s -- s 8 () During the alulation we know the oeffiient b is to be zero beause otherwise it goes to infinity on the neutral urve s (α )= 0 The oeffiient b an be determined from the oeffiient equations of os(αϕ) sin(αϕ) whih has not been written here beause it is not neessary to find s s w w In the superritially stable flow region the initially growing surfae wave near the neutral urve will arrive at an equilibrium state of finite amplitude if the seond Lau onstant s is negative in that region From the first equation in (6) the finite amplitude of the surfae wave an be determined as To examine the interation of an eletri field with a dieletri liquid film flow into whih unipolar spae harges are steadily injeted from a bottom support plane the solution for the surfae defletion is derived aurate to the seond order of a thin parameter ξ Using this evolution equation linear weakly nonlinear stability analyses are performed In the linear analysis as both values of s are always negative as seen from () () where Q/P is negative its absolute value is very small ( ) ompared with s (this is always true beause it only appears in the perturbed terms) the numerator of () for the linear stability analysis has smaller value due to the presene of K for ε> when it is ompared with the noneletri ase K=0 The ritial value of the ynolds number beomes diminished by this amount (see Kim [99]) While the effet of the normal stress on the free surfae due to the applied eletri field appears in the numerator in () the effet of spae harges in the layer turns up in W Considering the terms in the urly brae in W it is known that the last parenthesized term is dominating over other ones beause s( 0) is muh greater than the small perturbations of Thus the value of W is always greater than 6/5 this auses the redution of the ritial ynolds number further more Just beyond the neutral stability urve defined by () or s (α )=0 it has been onfirmed there exist finite-amplitude surfae a = s --- / s () the speed of the finite-amplitude monohromati wave R is equal to R = w= w w a (5) The alulation results of a R in (α ) domain are shown in the next setion CONCLUSIONS Fig (a) Amplitude a of periodi wave in the (α-) domain when C=0 with K=0 for β=0 rad d=5 0 m ξ=00 = 0 5 (b) Veloity R of periodi wave in the (α-) domain when C=0 with K=0 for β=0 rad d=5 0 m ξ=00 = 0 5 July 005

7 Finite-Amplitude Surfae Waves on a Thin Film Flow Subjet to a Unipolar-Charge Injetion 50 waves developing from linearly unstable modes beause s is negative ranging from the neutral urve to the limiting urve s (α )= 0 Henethe weakly nonlinear analysis has provided formulations to alulate the amplitude wave speed of any surfae wave in the flow region To ompare the effet of spae harge on the finiteamplitude waves () (5) are plotted for the ases of C=0 with K=0 C=0 with K=5 (V=0 kv/m) respetively by setting β=0 rad d=5 0 m ξ=00 H=5 ε=0 = 0 5 taking the other physial parameters for water at 0 o C When C=0 K=0 ie there is no spae harge in the liquid as well as no eletrial potential some amplitudes wave speeds are plotted in (α ) domain as Fig a Fig b respetively Here the utoff ynolds number is orresponding to 5/6ot(β)5/9α ξ / Along a onstant wavenumber it is notied both of amplitude wave veloity beome larger as the ynolds number inreases (Fig a b) whih denotes the flow ould be easily unstable to a larger ynolds number As the spae-harged film flow is affeted by an eletrostati fore one an expet the system would be muh more unstable as shown in Fig a Fig b where C=0 K=5 The shapes of the amplitude wave speed with the same values as Fig a b are almost similar to the noneletri ase exept all of the urves are shifted to the lower region of ynolds number That is in a smaller ynolds number than the noneletri one the flow gets unstable along the same wavenumber From this result the weakly nonlinear analysis indiates the presenes of spae harge makes the value of the ynolds number smaller than the one in the absene of spae harge That is the film has more enhaned instability due to spae harge in the liquid Sine the harateristi time sale L/U 0 of the fluid motion is very large when it is ompared with the dieletri relaxation time (600 µse at 0 C for water) representative harge migration time o ( mse for hydronium ion at 0 C) based on a field of 0 kv/m o [Kunhardt et al 988] the film flow is governed limited by its own inertia effet thus the pseudo-steady-state eletrial solutions obtained in this paper are reasonably suitable to the stability analysis in the thin film limit It is noted that the major purpose of present study is foused on deriving an film evolution equation desribing a thin film flow down an inlined plane subjeted to a unipolar-harge injetion performing linear weakly nonlinear stability analyses based on the result Hene the outomes will be good guides to the experimenters who will need to determine the wave amplitude wave veloity for their applying fields Sooner or later the fully nonlinear dynamis related to the present topi will be addressed ACKNOWLEDGMENT This study was performed by the supports from the University of Seoul in 00 the author gratefully aknowledges it NOMENCLATURE Fig (a) Amplitude a of periodi wave in the (α-) domain when C=0 with K=5 for β=0 rad ε=0 H=5 d=5 0 m ξ=00 = 0 5 (b) Veloity R of periodi wave in the (α-) domain when C=0 with K=5 for β= 0 rad ε=0 H=5 d=5 0 m ξ=00 = 0 5 a : dimensionless wave amplitude C : dimensionless number for a relative importane of spae harge : apillary number : dimensionless omplex wave speed i (i= 5) : dimensionless oeffiients defined in Eq () d : harateristi film thikness [m] Fr : Froude number g : gravity [m/se ] H : dimensionless distane between two eletrodes h : dimensionless free-surfae thikness K : dimensionless eletri fore onstant K m : ion mobility [C se/kg] L : harateristi length sale parallel to plane [m] Ne : eletrial Newton number p : dimensionless pressure q : dimensionless spae harge density q i : spae harge density at y=0 [C/m ] R : dimensionless number for a diffusivity ratio of ions to fluid partiles : ynolds number s s : dimensionless oeffiients for amplitude variane defined in Eq (6) Korean J Chem Eng(Vol No )

8 50 H Kim t : dimensionless time U 0 : harateristi veloity [m/se] u : dimensionless veloity omponent of x diretion V : eletri potential at y=0 [m kg/(c se )] v : dimensionless veloity omponent of y diretion w w : dimensionless oeffiients for phase variane defined in Eq (6) x : dimensionless distane oordinate parallel to plane y : dimensionless distane oordinate perpendiular to plane Greek Letters α : wavenumber β : inlination angle of plane with the horizontal ε 0 : permittivity of free spae [885 0 C se /(m kg)] ε f : permittivity of liquid [C se /(m kg)] ε : dimensionless permittivity of liquid µ : visosity [kg/(m se)] ξ :d/l ρ : fluid density [kg/m ] σ : surfae tension [N/m] Φ : dimensionless eletri potential for liquid ϕ : wave phase Ψ : dimensionless eletri potential for air Supersripts - : small disturbane * : omplex onjugate Subsripts 0 : leading-order term in ξ : first-order term in ξ a :air : ritial value I : imaginary part R : real part s : steady-state value t : partial derivative with t x : partial derivative with x y : partial derivative with y REFERENCES Atten P Eletrohydrodynami Instability Motion Indued by Injeted Spae Charge in Insulating Liquids IEEE Trans Diel Eletr Insul () (996) Benjamin T B Wave Formation in Laminar Flow Down an Inlined Plane J Fluid Meh 55 (95) Benney D J Long Waves on Liquid Films J Math Phys 5 50 (966) stellanos A Agrait N Unipolar Injetion Indued Instabilities in Plane Parallel Flows IEEE Trans Ind Appl 8() 5 (99) Gjevik B Ourrene of Finite-Amplitude Surfae Waves on Falling Liquid Films Phys Fluids 98 (90) González A stellanos A Nonlinear Eletrohydrodynami Waves on Films Falling down an Inlined Plane Physial view E 5() 5 (996) Kim H Bankoff S G Miksis M J The Effet of An Eletrostati Field on Film Flow Down an Inlined Plane Phys Fluids A (99) Kim H Charateristis of Solitary Waves on a Running Film down an Inlined Plane under an Eletrostati Field Korean J Chem Eng 0 80 (00) Kim H Long-Wave Instabilities of Film Flow under an Eletrostati Field : Two-Dimensional Disturbane Theory Korean J Chem Eng (99) Kunhardt E E Christophorou L G Luessen L H The Liquid State Its Eletrial Properties NATO ASI Series B: Physis Vol 9 Plenum Press New York (988) Nakaya C Long Waves on a Thin Fluid Layer Flowing Down an Inlined Plane Phys Fluids 8 0 (95) Shneider J M Watson P K Eletrohydrodynami Stability of Spae-Charge-Limited Currents in Dieletri Liquids I Theoretial Study Phys Fluids 98 (90) Yih C-S Stability of Liquid Flow Down an Inlined Plane Phys Fluids 5 (96) July 005