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PHYSICAL REVIEW E 82, 016322 !2010" Dynamics of an electrostatically modified Kuramoto–Sivashinsky–Korteweg–de Vries equation arising in falling film flows D. Tseluiko* School of Mathematics, Loughborough University, Leicestershire LE11 3TU, United Kingdom D. T. Papageorgiou† Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom !Received 13 January 2010; revised manuscript received 4 May 2010; published 30 July 2010" The gravity-driven flow of a liquid film down a vertical flat plate in the presence of an electric field acting in a direction perpendicular to the wall is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. A strongly nonlinear long-wave evolution equation is developed for flow parameters near the critical instability conditions. The equation retains terms up to second order in the slenderness parameter in order to incorporate the effects of shear-induced growth, short-wave damping due to surface tension, electric stress effects, and dispersive effects. In the additional asymptotic limit of small but finite interfacial perturbations the dynamics are shown to be governed by a Kuramoto–Sivashinsky !KS" equation with Korteweg–de Vries dispersion, also known as the Kawahara or the generalized Kuramoto–Sivashinsky !gKS" equation, which also includes a nonlocal energy growth term that arises from the electrostatics. Extensive numerical experiments are carried out to characterize solutions to this equation. Using perturbation theory and numerical solutions, it is shown that the electric field alters the far-field decay characteristics of bound states of the gKS equation from exponential to algebraic behavior. In addition, it is demonstrated numerically that chaotic solutions of the KS equation that are regularized into traveling-wave pulses when sufficient dispersion is added can in turn become chaotic by applying a sufficiently strong electric field. It is suggested, therefore, that electric fields can be utilized to enhance interfacial turbulence and in turn increase heat or mass transfer in applications. A physical example involving electrified falling film flows of ethelene glycol fluids is furnished and shows that the theory is within reach of experiments. DOI: 10.1103/PhysRevE.82.016322 PACS number!s": 47.15.gm, 47.15.Fe, 47.20.Ky I. INTRODUCTION Falling film flows of viscous liquids on vertical walls are unstable at all nonzero Reynolds numbers and produce a rich variety of complex two- and three-dimensional dynamics; see the pioneering experiments of Kapitza and Kapitza #1$. More recent experiments have established the presence of spatiotemporal chaos and coherent structures; see Liu and Gollub #2,3$, Vlachogiannis and Bontozoglou #4$, and Argyriadi et al. #5$. There are two modes of linear instability: a long-wave surface mode at moderate Reynolds numbers associated with the moving free surface and a shorter shear mode operating at high Reynolds numbers #6$. The stability characteristics depend strongly on the angle of inclination of the plate with the horizontal, and when the plate is vertical the interfacial mode is unstable at all Reynolds numbers while the shear mode enters at much higher values. For example, for an angle of inclination of 4° and a wide range of liquids, Floryan et al. #6$ found that the interfacial mode becomes unstable at a Reynolds number of 18, approximately, while the shear mode does not enter until a Reynolds number of approximately 3 ! 103. The Kuramoto– Sivashinsky !KS" equation !e.g., Ref. #7$" correctly captures the weakly nonlinear interfacial dynamics in the vicinity of the critical Reynolds number and is fully consistent with the *[email protected] † [email protected] 1539-3755/2010/82!1"/016322!12" linear stability characteristics. This equation has its roots in the modeling proposed by Shkadov #8$ and has been demonstrated to hold at higher Reynolds numbers !see, for example, Ref. #9$ and numerous references therein". We also refer to the monograph by Chang and Demekhin #10$ for an extensive discussion of falling films and evolution equations that describe such phenomena. Here, we are mostly concerned with the effects of electric fields on falling film flows down vertical plates. The experiments of Melcher #11$ !see also Refs. #12,13$" showed the existence of electrically induced surface waves; typically the electric field induces instability when the field is normal to the free surface but introduces dispersive effects when the field is in the plane of the undisturbed free surface. More recent experimental and theoretical work by Griffing et al. #14$ studied the effect of localized electric fields on nonuniform steady-state profiles in falling film flows. The present theoretical study is mostly concerned with flow regimes that support complex dynamics of electrified falling film flows, in particular we derive and study numerically and analytically a generalized Kuramoto–Sivashinsky !gKS" equation incorporating electrostatic effects. The gKS was derived by Topper and Kawahara #15$ in the context of falling film flows, and Kawahara #16$ first established numerically that a sufficient amount of dispersion can regularize complex dynamical behavior into pulses of traveling waves. The equation was also derived by Hooper and Grimshaw #17$ for two-fluid flows in channels and in the context of core-annular flows by Papageorgiou et al. #18$. The dispersive term in the latter study is 016322-1 ©2010 The American Physical Society PHYSICAL REVIEW E 82, 016322 !2010" D. TSELUIKO AND D. T. PAPAGEORGIOU a pseudodifferential operator arising from the physics of the problem and which localizes to the gKS for small wave numbers; numerical experiments in Ref. #18$ showed dispersive regularization by the pseudodifferential operator also. Smyrlis and Papageorgiou #19$ showed numerically that solutions to the nonlocal and local dispersion equations are very close in the energy norm for several attractors including timeperiodic ones. Numerous studies have ensued including the construction of traveling waves and their stability by Renardy #20$ and Chang et al. #21,22$, and the existence of traveling waves study of Ercolani et al. #23$. The presence of an electric field acting perpendicularly to the undisturbed liquid-layer free surface enhances the instability with short waves growing the fastest. It provides a mechanism which in the absence of other regularizing physical effects renders the problem ill posed with short waves growing at an exponential rate proportional to the inverse cube of their wavelength. The manifestation of this instability on the nonlinear spatiotemporal evolution of falling flows was considered by Tseluiko and Papageorgiou #24,25$ who studied an electrically modified Kuramoto–Sivashinsky equation both numerically and analytically, in the latter instance proving the existence of an inertial manifold and finding improved estimates for its dimension. In a series of papers, Tseluiko et al. #26–29$ considered modifications of the problem over two-dimensional topographically structured substrates and provided considerable evidence !by comparing direct simulations with asymptotic solutions" of the accuracy of long-wave theories in describing low-Reynoldsnumber dynamics. Staying with flat substrates, it is possible to retain shear-induced growth, surface tension stabilization, electric field growth, and dispersive effects by developing a long-wave evolution equation correct to second order in the small slenderness parameter. The resulting equation is onedimensional but strongly nonlinear in the sense that the freesurface deformation scales with the film thickness #this is Eq. !29" appearing in Sec. III$. In this study we specialize these solutions to the weakly nonlinear regime and derive the following nonlocal generalized Kuramoto–Sivashinsky !ngKS" equation: ut + uux + uxx + "uxxx + #H#uxxx$ + uxxxx = 0, !1" where u!x , t" is the scaled interfacial shape, x is the coordinate along the vertical wall, t is time, and " , # $ 0. In addition to the nonlinear inertial term the destabilizing term uxx originates from the shear at the free surface; the term "uxxx represents dispersive effects !the coefficient " is defined later"; the term uxxxx is a stabilizing term due to surface tension; and the nonlocal term #H#uxxx$, where H is the Hilbert transform operator, is due to the electric field with # being a dimensionless measure of the field’s strength. We study Eq. !1" on periodic domains x ! #−L , L$. The nonlocal Hilbert transform term arises from the electric field in the region outside the liquid layer and enters to modify the gKS equation by inducing additional short-wave instabilities as mentioned earlier. There are three parameters in the problem, ", #, and L, which measures the size of the system. In the absence of dispersion and electric fields, complex dynamics ensues as L increases with a transition to chaos through a Z X U κ<0 κ>0 h(X, T ) G E n t FIG. 1. Flow of a perfectly conducting liquid film down an inclined plane in the presence of an electric field. The region above the film is occupied by a passive dielectric gas. Feigenbaum period-doubling bifurcation scenario !see Refs. #30–32$". In the absence of an electric field but with dispersion present, it has been demonstrated that the small-time dynamics can be described by the interaction of a discrete number of bound-state solitons—see, for example, Ref. #33$. When # ! 0, we compute the bound-state solitons of Eq. !1" and analyze their far-field behavior showing that the decay is algebraic rather than exponential, something that can have a considerable influence on the soliton-soliton interaction dynamics. Finally, we compute the initial boundary value problem on periodic domains as ", #, and L vary in order to classify the most dominant attractors and characterize the long-time dynamics. We establish that, with all other parameters held fixed, sufficient dispersion regularizes the flow, whereas sufficiently large electric field induces spatiotemporal chaos. II. PROBLEM STATEMENT We consider the two-dimensional gravity-driven flow of a liquid film down a vertical flat plate when the film is subjected to an electric field applied perpendicularly to the plate as illustrated in Fig. 1. We choose a rectangular coordinate system !X , Z" such that the X axis points along the wall in the direction of gravity and the Z axis is perpendicular to it. The film surface is located at Z = h!X , T", where T denotes time. The wall is an electrode held !without loss of generality" at zero voltage potential and a uniform electric field of strength E0 is imposed as Z → %. The liquid is assumed to be a perfect conductor, so that there is no potential difference between the wall and the free surface and, consequently, no electric field in the film. The medium above the film is assumed to be a perfect dielectric with permittivity & p. The electrostatic limit is appropriate when the induced magnetic field can be ne- 016322-2 PHYSICAL REVIEW E 82, 016322 !2010" DYNAMICS OF AN ELECTROSTATICALLY MODIFIED… glected. This is the case in the present problem since there are no high-frequency ac fields imposed and no highfrequency free-surface oscillations. Charges reside on the free surface alone and their distribution can be determined from Gauss’s law a posteriori. For instance, a discussion of such limits can be found in work of Melcher and Taylor #34$. We nondimensionalize variables using the undisturbed film thickness h0 as the length scale and the Nusselt surface speed U0 = 'gh20 / 2# as the velocity scale, where ' and # are the density and the viscosity of the fluid and g is the acceleration due to gravity. The time scale is h0 / U0, the pressure is nondimensionalized by #U0 / h0, and the electric potential is nondimensionalized using E0h0. Standard dimensional analysis reveals the importance of three dimensionless parameters: a Reynolds number R, a Bond number B, and an electric Weber number We, which are defined as 2 R= gh30 2 , ' h 0U 0 ' = # 2# B= 'gh20 2#U0 = , ( ( & ph0E20 & pE20 We = = , 2#U0 'gh0 !3" where U = !U , V" is the fluid velocity, P is the pressure, and G = !2 , 0" is the dimensionless gravitational force. The noslip and no-penetration conditions at the wall require that U=0 !4" at Z = 0, and the kinematic condition at the film surface, Z = h!X , T", reads V = hT + UhX . !5" The dynamic balance of stress at the film surface demands that " · n = − !2)/B + Pa"n + M · n, where t is the unit tangent to the free surface depicted in Fig. 1. The electric potential * in the region Z + h!X , T" satisfies Laplace’s equation, "2* = 0. !9" Since the free surface is a null equipotential we demand that *=0 !10" !* → − k !11" &E&2 = !1 + hX2 "*Z2 !12" at Z = h!X , T", and as Z → %, where k is the unit vector pointing in the Z direction. We note that it can be easily verified that !6" Our goal is to derive a weakly nonlinear evolution equation that retains the fundamental physics of the electrified film flow, namely, instability and energy production, stability and energy dissipation, dispersion, and nonlinearity. We proceed with a long-wave analysis, that is, we assume that streamwise variations occur over a much longer length scale, say ,, than the typical film thickness h0, and introduce the small parameter - = h0 / ,. Our first aim is to derive a longwave evolution equation including O!-2" terms. We proceed as in Ref. #24$, where such an equation was derived up to order O!-". We note that for a given free-surface shape the electric field problem can be solved independently of the fluid problem, and then the electric field contribution in the normal stress balance condition !8" can be expressed in terms of the free-surface shape !implicitly in general but explicitly in the asymptotic analysis that follows", closing the fluid problem. The base solution for the electric potential for a flat un¯ = −Z. We introduce a perturdisturbed film, i.e., h ' 1, is * ˜ by writing bation potential * where " = −PI + !!U + !U " is the Newtonian stress tensor nondimensionalized by #U0 / h0, n is the unit normal to the free surface pointing into the film, ) is the curvature of the film surface taken to be positive when the surface is concave downward as indicated in Fig. 1, and Pa is the dimensionless air pressure. The Maxwell stress tensor M nondimensionalized by #U0 / h0 is given by !see Ref. #34$" !13" ˜ +* ˜ = 0, * XX ZZ !14" ˜ = h − 1 at Z = h!X , T", and with * ˜ → 0, * X ˜ →0 * Z !15" 1 X = ., - 1 Z = /, - !16" as Z → %. We introduce new independent variables . and /, !7" where E is the electric field and satisfies the relation E = −!* with * denoting the electric potential. It may be verified directly that the Maxwell stresses do not contribute to the tangential stress balance. In fact, the normal and tangential stress balance conditions at the free surface are ¯ +* ˜, *=* so that T M = We!EE − I&E&2/2", !8" III. WEAKLY NONLINEAR EVOLUTION !2" ! · U = 0, n · " · n = − !2)/B + Pa" + We&E&2 , at the free surface and this result is used below. where ( is the surface tension coefficient. Note that the Bond number B is the square of the ratio of the film thickness to the capillary length a = %( / 'g, and the electric Weber number We is the ratio of electric to gravitational forces. The flow in the liquid film is governed by the incompressible Navier–Stokes equations, R!UT + U · !U" = − !P + "2U + G, t · " · n = 0, and expand ˜ = * + -* + ¯ . * 0 1 It can be verified that the problem at leading order is 016322-3 !17" D. TSELUIKO AND D. T. PAPAGEORGIOU *0.. + *0// = 0, !18" with *0 = h − 1 at / = 0, and h1 + + , 2We! 3 2 3 2 3 8R 6 h +h h... + h H#h..$ h h. + 3 3 15 3B! * 0/ → 0 !19" + -2 *0/!.,0" = − H#h.$ !20" + *0. → 0, as / → %. It can be shown using complex variables that !see Refs. #24,35$, for instance", where H is the Hilbert transform operator defined by H#g$!." = 1 P.V. 0 ( % −% g!.!" d.! , . − .! *1. → 0, * 1/ → 0 as / → %. Solving as above, we obtain !23" *1/!.,0" = H#„h*0/!.,0"….$ = − H†!hH#h.$".‡. !24" ˜ = − H#h $ − -)H†!hH#h $" ‡ + hh * + O!-2" * / . . . .. !25" Using Eqs. !20" and !24", we find at the free surface whose position is / = -h in terms of the outer variables. The value of &E&2 at the free surface is readily determined using Eq. !12", ˜ "2 &E&2 = !1 + -2h.2"!− 1+ -* / = !1 + -2h.2"#1 + -E1 + -2E2 + O!-3"$2 , !26" where for brevity we have introduced the notations E1 = H#h.$, E2 = H†!hH#h.$".‡ + hh.. . !27" Next we proceed with the long-wave analysis for the hydrodynamic problem as in Ref. #24$. We assume that R = O!1", 1 / B = O!1 / -2", and We = O!1 / -" and introduce B! = B , -2 We! = -We . . 2R 16 5 2 8 9 2 + h6h.h... h h. h.. + h6h.. 5 4 B! 5 / 37 7 8 7 6 h h.H#h..$ − h H#h...$ + h5H#2hh.2 30 210 15 / 2We! 3 !2h h.h.. + h4h... + h3H#h.$H#h..$ 3 -0 = 0, . !29" where a new slow time scale 1 = -T has been introduced. It can be verified that the terms at the next order are at most of order O!-3" times derivatives of h and Hilbert transforms of the derivatives and therefore become of higher order when the weakly nonlinear regime is considered. The evolution equation !29" is strongly nonlinear in the sense that it describes free-surface deflections that scale with the undisturbed layer thickness. The existence of solutions for such classes of nonlinear equations is not guaranteed as the numerical results of Pumir et al. #36$ indicate in the absence of electric fields. In the present work, we proceed by deriving a canonical weakly nonlinear model that retains long-wave destabilization, short-wave dissipation, dispersion, and electric field effects. Our analysis shows that the consistent derivation of such a model from Eq. !29" is possible under the assumptions that R, 1 / B!, and We! are O!-". Note that this is not in contradiction with the earlier assumptions that R, 1 / B!, and We! are O!1" since any parameter that is O!-" is automatically O!1". Besides, as it will be shown below, the terms that we neglected in Eq. !29" become accordingly of higher order and cannot appear in the weakly nonlinear model. We introduce rescaled order-unity constants !28" These canonical scalings allow us to retain the competing effects of nonlinearity, shear-induced instability, short-wave stabilization due to surface tension, and instability due to the electric field. It is important to note that any other scaling that promotes the size of the electric field effects relative to the other physical mechanisms leads to an evolution equation that is ill posed in the sense that it is short-wave unstable— the instability is similar to the Kelvin–Helmholtz one but with much larger growth rates, namely, proportional to k3 rather than k, where k is the wave number of the perturbation. The mathematical model that emerges, then, facilitates an extensive quantitative study of these competing physical mechanisms. After a lot of algebra which involves the determination of the velocity and pressure in terms of h!X , T", the kinematic condition provides the final equation correct to order O!-2", . / + h3H†h..H#h.$ + 2h.H#h..$ + hH#h...$‡" !22" with *1 = −h*0/!. , 0" at / = 0, and . - 14 3 2 596 9 2 2 10 h h. + 2h4h.. + R2 h h + h h.. 3 315 . 7 + h2h..$ + and P.V. denotes the principal value of the integral. At the next order we have *1.. + *1// = 0, , + RWe! !21" PHYSICAL REVIEW E 82, 016322 !2010" R̄ = R , - B̄ = -B! = B , - !30" noting that also We = O!1" if We! = O!-". A balance of terms in the weakly nonlinear limit is possible for amplitudes which are of size -2. Therefore, we substitute h!. , 1" = 1 + -22!. , 1" into Eq. !29" to obtain the following equation, correct to order O!-4": -221 + 2-22. + 4-422. + -4 + -4 2We H#2...$ = 0. 3 8R̄ -4 2.. + 2-42... + 2.... 15 3C̄ !31" Note that the terms that we neglected in Eq. !29" are O!-3&h.&" or smaller, and they become O!-5" under the assumption that h = 1 + O!-2" and hence should be neglected in 016322-4 PHYSICAL REVIEW E 82, 016322 !2010" DYNAMICS OF AN ELECTROSTATICALLY MODIFIED… the weakly nonlinear equation derived above. Utilizing the change of variables, . − 21 = . / 5 1/2 4R̄B̄ x, 1= . / 4 1 3 2= R̄ B̄ 15 5 75 2 2 32- R̄ B̄ t, 30 25 20 !32" u0 10 1/2 5 !33" u, 0 -5 -15 we arrive at the canonical equation ut + uux + uxx + "uxxx + #H#uxxx$ + uxxxx = 0, !34" where " and # are order-unity constants defined by "=3 ./ ./ 5B̄ 4R̄ 1/2 =3 5B 4R μ 15 1/2 , #= We ". 3 !35" Equation !34" is a ngKS equation and is considered on periodic domains of size 2L. The parameters " and # measure the relative importance of dispersion and the electric field, respectively. In the numerical work that follows, we treat " and # as independent parameters and in particular consider situations where " = 0 and # ! 0. In view of their definitions !35", it appears that this is not permissible. However, a canonical equation which is identical to Eq. !34" but with " = 0 and # ! 0 can be derived from the fully nonlinear Eq. !29" by looking for weakly nonlinear solutions of larger amplitude than before, namely, by writing h = 1 + -2 and proceeding in the usual fashion. This derivation has been carried out by Tseluiko and Papageorgiou #24$ who started with Eq. !29" correct to order -. In the sequel, then, no confusion should be caused by the independent treatment of " and #—where we take " = 0 we implicitly assume that # ! 0 with the understanding that this equation is canonical also. One of our objectives is to study mathematically the dynamics in the presence or absence of the different effects and this can be done by studying Eq. !34" with the parameters varying independently. IV. NUMERICAL SOLUTIONS Equation !34" is solved numerically on 2L-periodic domains with an initial condition of the form u!x , 0" = u0!x" L u0!x"dx = 0, i.e., zero-mean initial states. Note with !1 / 2L"1−L that in physical experiments one needs to consider the effects of confining walls, but in many situations periodic wavy structures emerge with minimal border effects, and these are the types of wave regimes we focus on in this study. Since L u!x , t"dx is conserved, the solution has zero mean for all 1−L subsequent times also, and from a Fourier decomposition viewpoint this implies that the k = 0 mode of the spectrum is zero. A pseudospectral method is used that treats the stiffness of the linear operator accurately—for details, see Ref. #24$. It is well established numerically that the KS equation #i.e., Eq. !34" with " = # = 0$ exhibits large-time chaotic dynamics if the period of the system is sufficiently large !see the Introduction". The addition of small amounts of dispersion does not affect this complex behavior, but it has been shown that sufficiently strong dispersion leads to a regularization of cha- -10 -5 0 y 5 10 15 FIG. 2. Dependence of the pulse solution of the ngKS equation on # for " = 0.5 and L = 60. Parameter # changes from 0 !the shape with the smallest amplitude" to 2 !the shape with the largest amplitude" with the step of 0.4. otic dynamics into traveling-wave pulses #18,19,33,37$. On the other hand, in the absence of dispersion the addition of electric field effects !# ! 0" enhances the chaotic behavior in the sense that chaotic dynamics emerge for smaller values of L #24,25$. The objective of the present study is in evaluating the competing effects of dispersion and electric fields on the spatiotemporal dynamics of the KS equation. We solve the boundary value problem to construct nonlinear traveling waves and also compute initial value problems to explore the large-time dynamics including the stability of traveling-wave pulses. V. TRAVELING WAVES As noted above, solutions of the KS equation !" = # = 0" considered on sufficiently large periodic domains are chaotic in both space and time. However, a sufficiently large dispersion parameter " arrests spatiotemporal chaos, and a solution to the gKS equation looks like a superposition of weakly interacting nonlinear pulses of approximately the same shape, i.e., close to the shape of the solitary-pulse solution !e.g., Ref. #16$". It is important, therefore, to study solitarypulse solutions of Eq. !34" and analyze the effect of the nonlocal term on such pulses. We rewrite Eq. !34" in a frame moving with the velocity c = c!" , #" of a stationary pulse, ut − cuy + uuy + uyy + "uyyy + #H#uyyy$ + uyyyy = 0, !36" where y = x − ct. A stationary pulse u0 = u0!y ; " , #" is a solution of the steady version of Eq. !36", i.e., − cu0y + u0u0y + u0yy + "u0yyy + #H#u0yyy$ + u0yyyy = 0. !37" Equation !37" is a nonlinear eigenvalue problem for the pulse velocity c. We integrated Eq. !37" numerically on periodic domains by using a spectral fast Fourier transform !FFT" method and Newton iterations. The results for " = 0.5 and various values of # on the periodic interval #−60, 60$ are shown in Fig. 2. It can be seen that the pulse amplitude increases with # with the shape with the smallest amplitude 016322-5 PHYSICAL REVIEW E 82, 016322 !2010" D. TSELUIKO AND D. T. PAPAGEORGIOU -1.5 12 -2 log10(|u 0|) 10 -2.5 8 c 6 4 -3.5 2 0 0 -3 0.2 0.4 0.6 0.8 1 μ 1.2 1.4 1.6 1.8 -4 0.8 2 FIG. 3. Dependence of speed of the pulse of the ngKS equation on # when " = 0.5 and L = 60. corresponding to # = 0 and that with the largest amplitude corresponding to # = 2. The speed of the pulse is also a monotonically increasing function of #, as is seen in Fig. 3. In addition, as # increases, the amplitude of the oscillations downstream of the pulse grows and a depression forms upstream of the pulse. In contrast to the case when # = 0 for which the tails of the pulse tend to zero exponentially as y → % #37$, for nonzero # the tails of the pulse tend to zero as 1 / &y&3 as &y& → %. This is confirmed numerically in Fig. 4 which depicts a logarithmic plot of &u0& with y for y 3 1. The solid and the dashed lines correspond to the right and the left tails of the pulse, respectively, and the thick straight line included has a slope of −3 in agreement with the algebraic behavior mentioned above !the analysis for small # below also confirms this". As the period of the domain increases, the pulse shape and the speed of the pulse tend to limiting values. The dependence of c on the period of the domain is shown in Fig. 5 and we can conclude that for sufficiently large L the periodic solutions approximate well the solitary pulses on an infinite domain. Analysis for small # In this section, we carry out a perturbation expansion to analyze the effects of the electric field on the pulse solitons of the gKS equation. It was found numerically !see Fig. 4" that nonzero values of # alter the decay at infinity of the solitary pulses from exponential to algebraic and we seek to establish this analytically. Let us consider Eq. !37" on y ! !−% , %" and assume that # 4 1. We look for asymptotic solutions of the form u0 = f 0 + # f 1 + ¯ , c = c0 + #c1 + ¯ . !38" − c0 f 0y + f 0 f 0y + f 0yy + " f 0yyy + f 0yyyy = 0, !39" 1 1.1 1.2 log10(|y|) 1.3 1.4 1.5 FIG. 4. The right !solid line" and the left !dashed line" tails of the pulse solution of the ngKS equation for " = 0.5 and # = 2. Results are shown on a log-log plot, from which it is evident that the tails of the pulse tend to zero as 1 / y 3 as y increases !the thick solid line is a straight line of slope −3". L#f 1$ = c1 f 0 − H#f 0yy$, !41" where L is a linear operator defined by L#f$ ' − c0 f + f y + " f yy + f yyy + f 0 f . !42" The adjoint operator is defined by L!#f$ ' − c0 f − f y + " f yy − f yyy + f 0 f . !43" By inspection we see that L#f 0y$ = 0 and hence f 0y is in the null space of L—it has been verified numerically that the null space is one dimensional and therefore spanned by f 0y !this has been done by projecting the L2 space onto an N-dimensional space and finding the matrix representation of L in this space and verifying that the rank of the matrix is N − 1 by computing its singular value decomposition". Similarly we verified numerically that the null space of the adjoint operator L! is also one dimensional. Let 5 be in the null space of L! normalized, so that 2f 0y , 53 = 1, where 2· , ·3 denotes the usual inner product in LC2 . We can compute 5 numerically and the result for the typical case " = 0.5 is shown in Fig. 6. The solvability condition for Eq. !41" requires that the right-hand side of this equation is orthogonal to 5, which gives the following expression for c1: 2.5179 2.51785 2.5178 At leading order we find c i.e., f 0 is a pulse solution of the gKS equation for a given ". At first order, we obtain 2.51775 2.5177 2.51765 2.5176 − c0 f 1y + f 1yy + " f 1yyy + f 1yyyy + !f 0 f 1"y = c1 f 0y − H#f 0yyy$. 2.51755 16 20 24 28 32 36 40 44 48 52 56 60 !40" Integrating once and noting that the constant of integration is zero due to the decay to zero of the solution at infinity, we find 0.9 L FIG. 5. The variation of speed of traveling-wave pulse solutions of the ngKS equation on the half-period L when " = 0.5 and # = 0.4. 016322-6 PHYSICAL REVIEW E 82, 016322 !2010" DYNAMICS OF AN ELECTROSTATICALLY MODIFIED… 2.97 -2.97 500 0.3 u 0.2 0.1 -0.1 1 u0 300 t ψ -0.2 -0.3 -0.4 -1 200 -2 -30 -20 -10 0 100 -0.5 0 -20 150 -0.6 -0.7 -20 2 400 0 -15 -10 -5 0 y 5 10 15 20 F†H#f 0$‡!k" = − i sgn!k"F#f 0$!k", H#f 0$ 5 1 y 50 0 0 200 t 300 1 y3 H#f 0yy$ = H#f 0$yy 5 400 500 as y → 6 %, !47" and, therefore, from Eq. !41" it follows that f1 5 1 y3 !48" as y → 6 %. Thus, the asymptotic result for small # confirms the above observation that the tails of the ngKS pulse should tend to zero as 1 / y 3 as y → 6 %, in contrast to the gKS case. This observation is important for the development of a weak5.70 -2.77 500 u 5 4 3 2 u1 0 -1 -2 -30 -20 -10 0 10 20 30 400 !46" as y → 6 %. 100 FIG. 8. Solution of the KS equation #Eq. !34" with " = # = 0$ when L = 30. Top left: evolution of u!x , t". Top right: profile at the last computed time t = 500. Bottom: evolution of the energy norm. !45" % p!y"e−ikydy is the usual Fourier transwhere F#p!y"$!k" = 1−% form operation. By integrating Eq. !39" twice we obtain % % 2 % f 0dy = !1 / 2"1−% f 0dy, which shows that 1−% f 0dy ! 0. c01−% Therefore, F[H#f 0$]!k" has a jump discontinuity at zero !note that the zero Fourier mode is given by F#f 0!y"$!0" % f 0!y"dy" implying that !see Ref. #38$, Theorem 19, p. = 1−% 52" 20 ||u||2 !44" For example, when " = 0.5 we find c1 4 1.58. Next, we can solve Eq. !41" numerically with an additional requirement that 2f 1 , 53 = 0 !this is imposed to obtain uniqueness since any term proportional to f 0y can be added to f 1"; a numerical solution of Eq. !41" for " = 0.5 is shown in Fig. 7. It is well known that the tails of a solitary pulse of the gKS equation tend to zero exponentially as y → 6 %. Moreover, the left tail tends to zero monotonically as y → −%, while the right tail tends to zero either in an oscillatory manner or monotonically as y → % depending on whether " is below or above a threshold value, "! 4 3.71 #37$. The Fourier transform of the Hilbert transform H#f 0!y"$ is given by x 10 20 30 100 FIG. 6. The null space eigenfunction 5 of the adjoint operator L given by Eq. !43", when " = 0.5. c1 = 2H#f 0yy$, 53. 0 x 300 t From this, we conclude that 200 100 5 0 -20 300 4 0 x x 20 3 f1 200 2 ||u||2 1 100 0 -1 -2 -20 0 0 -15 -10 -5 0 y 5 10 15 20 FIG. 7. The unique solution f 1 of the first-order perturbation problem !41", when " = 0.5. Uniqueness is fixed by imposing 2f 1 , 53 = 0 as described in the text. 100 200 t 300 400 500 FIG. 9. Solution of the gKS equation #Eq. !34" with " = 1, # = 0$ when L = 30. Top left: evolution of u!x , t". Top right: profile at the last computed time t = 500. Bottom: evolution of the energy norm. 016322-7 PHYSICAL REVIEW E 82, 016322 !2010" D. TSELUIKO AND D. T. PAPAGEORGIOU 3.00 -2.58 500 u 140 400 300 t u 100 ||u||2 60 200 0 x interaction theory of the pulses and this will be pursued elsewhere. VI. SPATIOTEMPORAL DYNAMICS In this section we solve the initial value problem !34" on periodic domains as the half-size of the domain L and the parameters " and # vary. The initial condition for most simulations presented is taken to be u!x , 0" = cos!0x / L"; random initial conditions have been used also where noted in the text. If the dispersion is sufficiently strong, then travelingwave pulses result and it is important to have an accurate way of computing the speed of steady-state traveling waves. A spectrally accurate method is possible by noting the formula c= ( −L ( L −L ( L −L 2 uxx dx !49" , u2x dx 100 80 60 ||u|| 40 20 0 1 7 8 9 3 4 5 6 Dispersion δ -20 0 20 20 10 0 -10 -10 -20 0 20 x x 2 1 0 0 10 1 2 3 1 2 3 4 5 6 7 8 9 10 4 5 6 7 8 9 10 Dispersion δ 60 ||u|| 2 20 x spectral accuracy using trapezoidal integration, for instance. We begin by exhibiting the regularizing effect of dispersion on the KS dynamics. Taking # = 0 !no electric field", L = 30, " = 0, and an initial condition u!x , 0" = cos!0x / L", we observe chaotic long-time dynamics !this has been noted by many authors and in fact the route to chaos has been shown to follow a Feigenbaum period-doubling scenario #30–32$". Results up to 500 time units are shown in Fig. 8 which depicts the evolution of u, the final computed profile u!x , 500", and the evolution of the energy 6u!· , t"62 of the solution. Analogous solutions starting from the same initial condition but having a dispersion parameter " = 1.0 are given in Fig. 9. The dispersion is sufficient to regularize the chaotic dynamics and produce traveling-wave pulses with computed speed c 4 0.79. It can be shown #39$ that the traveling wave-train is composed of the solitary pulses computed from Eq. !37" plus a small correction function. In the results of Fig. 9 the solution is almost exactly a traveling wave composed of four solitary pulses. If " is too small, then the large-time dynamics remains chaotic—an example is given in Fig. 10 where " = 0.1. The variation with " of the traveling-wave speed and the corresponding L2 norm 6u6!defined by 6u6 L u!x , t"2dx$1/2" for this branch of three-pulse waves are = #1−L given in Fig. 11 from which we observe a monotonic increase in both the speed and the norm, both scaling linearly 2 6 0 FIG. 12. Computed traveling-wave profiles for the three-pulse branch in Fig. 11 corresponding to the values " = 7 !top left", " = 8 !top right", " = 9 !bottom left", and " = 10 !bottom right". 4 5 -20 40 0 3 Dispersion δ x u 6 4 -10 20 10 4 3 10 30 8 2 0 20 Speed c Speed c which follows from multiplication of Eq. !37" by u0y and integration by parts over the domain. Note that we used u!x , t" in Eq. !49" above, so that if a traveling-wave state exists then the quantity c!t" will tend to a constant value which is the speed of the pulse. Given a periodic function u!x , t", the right-hand side of Eq. !49" can be computed with 0 1 20 0 -20 30 u uu2x dx − " u 10 40 t FIG. 10. Solution of the gKS equation #Eq. !34" with " = 0.1, # = 0$ when L = 30. Left: evolution of u!x , t". Right: evolution of the energy norm. L 30 20 -10 20 0 -20 40 0 20 0 100 200 300 400 500 100 40 30 7 8 9 40 20 0 0 10 FIG. 11. Variation of the speed c !top" and norm 6u6 !bottom" with " for traveling-wave solutions of the gKS equation #Eq. !34" with # = 0$ when L = 30. Three-pulse branch. Dispersion δ FIG. 13. Variation of the speed c !top" and norm 6u6 !bottom" with " for traveling-wave solutions of the gKS equation #Eq. !34" with # = 0$ when L = 30. Two-pulse branch. 016322-8 PHYSICAL REVIEW E 82, 016322 !2010" DYNAMICS OF AN ELECTROSTATICALLY MODIFIED… 24 Speed c 0 20 -2 -3 0 18 1 2 3 150 4 5 6 Dispersion δ 7 8 9 10 ||u|| 16 14 12 100 ||u|| 50 0 0 Bimodal Trimodal 22 -1 10 8 1 2 3 4 5 6 Dispersion δ 7 8 9 6 10 0 0.2 0.4 0.6 μ 0.8 1 1.2 1.4 FIG. 14. Variation of the speed c !top" and norm 6u6 !bottom" with " for traveling-wave solutions of the gKS equation #Eq. !34" with # = 0$ when L = 30. Heptamodal branch. FIG. 16. Bifurcation diagram: L2 norm of steady states as # increases. L = 7, " = 0. Diamonds: bimodal steady states. Circles: trimodal steady states. with " as first noted in Ref. #16$. Traveling-wave profiles from this branch at " = 7 , 8 , 9 , 10 are depicted in Fig. 12. We emphasize, however, that the solutions are not unique; there exist other branches with different modal behavior and we have followed the characteristics of two-pulse traveling waves and the results are presented in Fig. 13. Another branch can be constructed by using as initial condition the most unstable wave provided by linear theory. In this case the resulting traveling waves are heptamodal, i.e., they have period 2L / 7 and have negative speeds, relative to the traveling-wave frame used to derive the gKS equation, for all nonzero values of ". This branch emanates from a heptamodal steady-state solution of the KS equation for L = 30 with the speed being zero when " = 0. Characteristics of the traveling waves for this branch are given in Fig. 14 and typical profiles for " = 1 , 4 , 7 , 10 are depicted in Fig. 15. We note that in all time-dependent computations, steady-state traveling waves with equally separated pulses must emerge when " is sufficiently large and the time of integration is also large. It has been shown by Tseluiko et al. #39$ that times of integration of 104 or larger may be required to obtain convergence with equal pulse separation. In the present numerical results our typical integration times are 100–500 and in all cases the L2 norm reaches to within less than 0.1% of its steady-state value. An interesting feature of the dynamics is that the relaxation to equally separated pulses is very slow and also different initial conditions result in different pulse separation distances at moderate times and hence require different integration times to converge to the ultimate equally separated pulse solutions—see Ref. #39$. The next set of numerical experiments considers the effect of the electric field on the nonlinear dynamics. We fix L = 7 which in the case of " = # = 0 supports a bimodal steady state. Introducing an electric field !but no dispersion for the moment" modifies the flow to produce larger energy steady states, initially, that maintain the bimodal spatial structure. In the vicinity of # = 1 there is a competition between the bimodal steady state, a much higher energy trimodal steady state, and a time oscillatory attractor !for example, when # = 1.03 computations with 256 modes and time step 7t = 20/ 1282 up to 500 time units indicate nonperiodic oscillatory behavior in time; this is supported on a very small interval of values of #". We emphasize that the results given here are large-time solutions of the initial value problem with the same cosine initial condition and constitute a description of the most attracting states for this set of initial conditions !typical maximum integration times are 100–200 time units". Figure 16 is a bifurcation diagram showing the variation of 6u6 with # when steady states emerge from the large-time computations. The disconnected branches represent steady 30 30 10 20 20 8 10 10 6 u u 0 -10 -20 -20 0 x 20 -20 -20 30 20 0 x 2 20 u0 -2 20 10 -4 10 0 u 0 -10 -20 4 -10 30 u 0 -20 0 x 20 μ =0 μ =1.0 μ =1.2 -6 -10 -8 -20 -10 -20 0 x 20 FIG. 15. Computed traveling-wave profiles for the heptamodal branch in Fig. 14 corresponding to the values " = 1 !top left", " = 4 !top right", " = 7 !bottom left", and " = 10 !bottom right". -6 -4 -2 0 x 2 4 6 FIG. 17. Representative steady-state profiles as # increases. L = 7, " = 0. Solid line: # = 0 !bimodal"; dashed line: # = 1.0 !bimodal"; dotted line: # = 1.2 !trimodal". 016322-9 PHYSICAL REVIEW E 82, 016322 !2010" D. TSELUIKO AND D. T. PAPAGEORGIOU 108.1 -107.9 100 u 150 100 50 u 0 -50 -100 -150 80 60 t 40 20 0 0 -5 x 4 7 x 10 -6 -4 -2 0 2 4 6 x 5 6 5 ||u|| 4 2 3 2 1 0 0 20 40 t 60 80 100 FIG. 18. Solution of the electrified KS equation #Eq. !34" with " = 0, # = 4.0$ when L = 7. Top left: evolution of u!x , t". Top right: profile at the last computed time t = 100. Bottom: evolution of the energy norm. states with decreasing periods as # increases. The branches are labeled 2M and 3M to denote bimodal and trimodal solutions, respectively. Typical profiles from the two branches are presented collectively in Fig. 17 for # = 0 , 1.0, 1.2. The modal nature of the solutions can be seen clearly in addition to the higher energy state of the trimodal branch. For values of # between 1.3 and 1.4, approximately, the large-time dynamics gets attracted to time oscillatory and chaotic states. Results for # = 4 are depicted in Fig. 18 which shows the evolution of the energy, the spatiotemporal evolution of the solution between 0 8 t 8 100, and the final computed profile at t = 100. The effect of introducing dispersion into the electrified KS equation is considered next. To fix matters we consider 4 4 x 10 4 6 x 10 3.5 5 3 4 2.5 ||u||2 2 ||u||2 3 1.5 2 1 1 0.5 0 0 10 5 t 15 20 0 0 25 4 7 x 10 20 25 30 35 40 45 50 10 15 20 25 30 35 40 45 50 t 8 6 4 ||u|| 3 2 5 4 3 2 2 1 0 0 10 15 7 5 ||u|| 5 4 9 x 10 6 2 three cases # = 0.5, # = 1.1, and # = 4.0 which, in the absence of dispersion, produce large-time states which are bimodal steady states, trimodal steady states, and chaotic states, respectively. For the dispersionless steady-state branches corresponding to # = 0.5 and # = 1.1, the effect of " is to produce traveling waves with negative speeds that maintain the bimodal and trimodal nature of the solutions; as " increases both the speed and norm 6u6 of the solutions behave linearly as established in earlier analogous numerical experiments. Uniqueness is not guaranteed and in fact when # = 1.1 both bimodal and trimodal traveling waves are found as " varies depending on the initial conditions. The trimodal branch for # = 1.1 was constructed by continuation, whereas the bimodal branch results for # = 0.5 emerged as the large-time solutions of a cos!0x / L" initial condition. In fact, using this initial condition for the case # = 1.1, " ! 0 can produce bimodal traveling waves which are connected to the bimodal branch in Fig. 16. We do not discuss these characteristics further because of their analogs with the results shown in Fig. 14, for example. We finally turn to the case having # = 4 which produce chaotic dynamics in the absence of dispersion as indicated in Fig. 18. It is found numerically that if " is larger than 1.2, approximately, the dynamics is regularized into a traveling wave while chaotic or quasiperiodic dynamics are found for weaker dispersion. To illustrate this we provide in Fig. 19 the evolution of the energy norm 6u62 for " = 0.5, 1.0, 1.5, 2.0; the results show that for " = 0.5 and 1.0 the dynamics is still oscillatory in time but with the amplitude of the oscillations decreasing with increasing " !for comparison see also Fig. 18 in the absence of dispersion". As " increases to 1.5 and beyond, the energy tends to a constant value reflecting the fact that the solution evolves to become a traveling wave. In addition, the energy increases with " as established previously. More detailed characteristics of the traveling-wave branch for the range 1.28 " 8 5 are given in Fig. 20 which shows the variation of the speed and the norm 6u6 with ". The speed is positive at the lowest " for which a traveling wave exists 1 5 10 15 20 25 30 t 35 40 45 50 0 0 5 t 016322-10 FIG. 19. Solution of the ngKS equation #Eq. !34" with # = 4.0 and increasing values of "$ when L = 7. Top: left, " = 0.5; right, " = 1.0. Bottom: left, " = 1.5; right, " = 2.0. PHYSICAL REVIEW E 82, 016322 !2010" DYNAMICS OF AN ELECTROSTATICALLY MODIFIED… 150 Speed c 5 0 100 -5 -10 -15 1 1.5 2 2.5 3 3.5 Dispersion δ 4 4.5 50 5 u 500 ||u|| 0 400 -50 300 200 1 1.5 2 2.5 3 3.5 Dispersion δ 4 4.5 -100 5 FIG. 20. Variation of the speed c !top" and norm 6u6 !bottom" with " for traveling-wave solutions of the ngKS equation #Eq. !34" with # = 4$ when L = 7. Hexamodal branch. but then decreases monotonically to negative values with a linear dependence for large "; at the same time the norm increases monotonically and again follows a linear dependence as expected by scaling arguments #16$. The solutions are hexamodal traveling waves #i.e., they have period !2L" / 6 = 7 / 3$ and the solution for " = 2 is shown in Fig. 21—as " increases the modal property is preserved but the amplitude increases linearly with ". VII. PHYSICAL CONTEXT The perfect-conductor-film model would be appropriate for liquid metals, e.g., mercury. Also, salted water acts as a near-perfect conductor. For our model to be valid we would need the Reynolds number R and the Bond number B to be small and of the same order. This is hardly achievable for liquid metals and water since for film thicknesses that would evolve dynamically over reasonable time scales the Reynolds number turns out to be significantly larger than the Bond number. This can be fixed by considering more viscous liquids. Appropriate choices would be, for example, solutions of ethylene glycols and NaCl. Ethylene glycols play a significant role in a wide range of industrial applications; for example, they are used in heat transfer fluids, in natural gas hydration, and treating applications to remove water and impurities, in production of a variety of products, e.g., antifreeze, coolants, coatings, emulsifiers, and lubricants, among others. The conductivity of a solution of an ethylene glycol and NaCl depends on the concentration of NaCl. For example, the conductivity of monoethylene glycol and NaCl can be 18.9 S m−1, which is approximately four times the conductivity of sea water !see Ref. #40$". As a putative experimental setup, let us consider tetraethylene glycol, the most viscous of ethylene glycols, at room temperature. The density, viscosity, and surface tension of tetraethylene glycol are ' = 1.1 ! 103 kg m−3, # = 5.8 ! 10−2 Pa s, ( = 4.5 ! 10−2 N m−1 . !50" For a film of thickness h0 = 0.55 mm, we obtain the following values of the Reynolds number, the Bond number, and the dispersion parameter: -6 -4 -2 0 x 2 4 6 FIG. 21. Typical hexamodal traveling-wave profile from the branch in Fig. 20, at " = 2 !# = 4 , L = 7". The hexamodal solution has period !2L" / 6 = 7 / 3. R 4 0.29, B 4 0.074, " 4 1.69, !51" which are reasonably suitable for the validity of the ngKS equation. For the validity of the model, we also need the electric Weber number We of order unity. For example, to get the value We = 1, which gives # 4 0.57, we would require the electric field E0 4 8.18! 105 V m−1. Although this value is large, it is significantly less than the critical value for the dielectric breakdown of air, 3 ! 106 V m−1, and can be achieved in experiments !note that in their experiments on electrified liquid films, Griffing et al. #14$ used a vacuum system and were able to achieve very strong electric fields in excess of 1.2! 107 V m−1". The velocity scale U0 turns out to be 0.028 m s−1. Finally, we note that the dimensional time t̃ is related to the time t in the ngKS equation through the formula t̃ = At, where the time scale A is defined by A= 75h0 . 64U0R2C !52" For a 0.55-mm-thick film of tetraethylene glycol, A turns out to be 7.38 s. VIII. CONCLUSIONS A weakly nonlinear equation has been derived and studied numerically and analytically to describe falling film flows at small Reynolds numbers in the presence of both dispersion and electrostatic effects. The model extends the known dispersive !or generalized" Kuramoto–Sivashinsky equation !denoted by gKS" to a nonlocal equation with an additional linear pseudodifferential part due to electric field effects. The electric field is energy supplying and causes linear short waves to grow the fastest. Surface tension dominates at large wave numbers and damps them, thus keeping the nonlinear problem well posed. We have carried out extensive numerical experiments in order to characterize the effect of the electric field on the gKS equation. The results can be summarized as follows: in the absence of dispersion and electric fields, chaotic dynamics result as long-time solutions of the model if the length L of the domain is sufficiently large !these are the familiar phenomena associated with the Kuramoto–Sivashinsky equation". Starting with such a cha- 016322-11 PHYSICAL REVIEW E 82, 016322 !2010" D. TSELUIKO AND D. T. PAPAGEORGIOU otic state of the KS equation, an increase in the dispersion coefficient beyond a certain threshold value that depends on the length L regularizes the dynamics into a traveling wave composed of pulse solitons of the gKS. On the other hand, steady-state solutions of the KS equation that are supported on sufficiently small lengths L can become chaotic if the electric field parameter is increased sufficiently. These chaotic states can in turn be regularized into a train of traveling waves by increasing the value of the dispersion parameter. 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A perturbation theory valid for small electric fields has also been carried out and analytical grounds for the algebraic decay have been given. The change in far-field behavior can be significant in theories using pulse soliton interactions to describe the dynamics with few degrees of freedom, and further investigations are left for future work. ACKNOWLEDGMENTS 016322-12