NONLINEAR DYNAMICS OF ELECTRIFIED THIN LIQUID FILMS 1 ...

SIAM J. APPL. MATH. Vol. 67, No. 5, pp. 1310–1329

c 2007 Society for Industrial and Applied Mathematics

NONLINEAR DYNAMICS OF ELECTRIFIED THIN LIQUID FILMS∗ DMITRI TSELUIKO† AND DEMETRIOS T. PAPAGEORGIOU† Abstract. We study a nonlinear nonlocal evolution equation describing the hydrodynamics of thin films in the presence of normal electric fields. The liquid film is assumed to be perfectly conducting and to completely wet the upper or lower surface of a horizontal flat plate. The flat plate is held at constant voltage, and a vertical electric field is generated by a second parallel electrode kept at a different constant voltage and placed at a large vertical distance from the bottom plate. The fluid is viscous, and gravity and surface tension act. The equation is derived using lubrication theory and contains an additional nonlinear nonlocal term representing the electric field. The electric field is linearly destabilizing and is particularly important in producing nontrivial dynamics in the case when the film rests on the upper side of the plate. We give rigorous results on the global boundedness of positive periodic smooth solutions, using an appropriate energy functional. We also implement a fully implicit numerical scheme and perform extensive numerical experiments. Through a combination of analysis and numerical experiments we present evidence for the global existence of positive smooth solutions. This means, in turn, that the film does not touch the wall in finite time but asymptotically at infinite time. Numerical solutions are presented to support such phenomena, which are also observed in hanging films when electric fields are absent. Key words. thin film, electrohydrodynamics, nonlocal evolution equation AMS subject classifications. 76D03, 76D08, 76D27, 76E17 DOI. 10.1137/060663532

1. Introduction. A viscous liquid film wetting the upper surface of a flat horizontal substrate is expected to be stable, under normal conditions, and eventually returns to its uniform undisturbed value of the film thickness. If the film is hanging (i.e., it wets the underside of the substrate), then gravity is destabilizing. This paper is concerned with the addition of electric fields normal to the substrate. Using experiments and linear theory, Taylor and McEwan [26] observed that a sufficiently strong field can overcome viscous forces in overlying films and induce wavy perturbations. We aim to model and analyze the nonlinear stages of this phenomenon for overlying and hanging films. In the absence of electric fields the problem was studied by Ehrhard and Davis [9], who considered spreading of viscous drops on smooth horizontal surfaces which are uniformly heated or cooled. Their isothermal evolution equation for the interface coincides with ours when there is no electric field. Yiantsios and Higgins [30] considered the behavior of a viscous film bounded below by a wall and above by an unbounded second heavier immiscible fluid. For the case when the Bond number B 1 (it measures the ratio between gravitational and interfacial forces) and the viscosity ratio m = μ1 /μ2 is O(1), where μ2 is the film viscosity, they obtained the same evolution equation for the interface as did Ehrhard and Davis [9]. Ehrhard [8] used the model derived by Erhard and Davis [9] to describe the quasi-steady evolution of a viscous drop hanging on the earth-facing side of a smooth horizontal plate, which is either uni∗ Received by the editors June 22, 2006; accepted for publication (in revised form) March 15, 2007; published electronically June 21, 2007. http://www.siam.org/journals/siap/67-5/66353.html † Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102 ([email protected], [email protected]). The first author acknowledges support from NJIT through its Presidential Research Initiative. The work of the second author was supported by National Science Foundation grant number DMS-0072228.

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NONLINEAR DYNAMICS OF ELECTRIFIED THIN LIQUID FILMS

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formly heated or cooled. Also, other similar equations arising in the modeling of thin liquid films have been derived and studied by Bertozzi [1], Dussan [7], Greenspan [13], Haley and Miksis [14], Hocking [16], Myers [18], and Oron, Davis, and Bankoff [19]. Here we consider the problem of a perfectly conducting liquid film on a horizontal plane with the upper electrode placed far from the grounded substrate. There has been considerable interest in electrically induced instabilities and their use in pattern formation and transfer in photolithographic applications. Demonstrations of such instabilities in this context have been made by Schaffer et al. [23], [24] and Lin et al. [17], for example. Theoretical works have focused on linear theory, as in Pease and Russel [21] and references therein, as well as long wave theories in the thin film and small gap approximation (i.e., the second electrode is placed close to the grounded bottom plate) in Shankar and Sharma [25], and the more recent leaky dielectric study of Craster and Matar [5]. The present work is related to but different from those cited above, but we expect that the mathematical tools developed here can be used in those problems also. A long wave theory leads to a nonlocal nonlinear evolution equation at the leading order, which by a change of the sign of the gravitational parameter also describes hanging films. (Nonlocal equations have also been derived in related problems by Papageorgiou and Vanden-Broeck [20], Savettaseranee et al. [22], and Tilley, Petropoulos, and Papageorgiou [27].) If H(x, t) denotes the scaled interfacial position, then the equation takes the form 1 1 Hxxx − GHx + 2We H[Hxx ] (1.1) Ht + H 3 = 0, (x, t) ∈ R × R+ , 3 C x H(x, t) = H(x + 2L, t), where C > 0, We > 0, and G can be positive or negative for overlying or hanging films, respectively. In the absence of an electric field the film is linearly stable or unstable, depending on whether G > 0 or G < 0. The addition of an electric field can always make the film unstable irrespective of the sign of G. We prove that positive smooth solutions of (1.1) do not blow up and are uniformly bounded for all time in the Sobolev H 1 -norm. This is done by constructing an appropriate energy functional E[H] having the steady-state solutions as extrema. Our analysis extends that of Bertozzi and Pugh [3] to the nonlocal equation (1.1). We also note that Hocherman and Rosenau [15] considered a class of thin film equations with the coefficients in front of the spatial derivatives being polynomials of higher or lower degree of the unknown function (it is unclear whether such equations arise in physical applications). They were interested in identifying equations whose solutions blow up in finite time, and they made a conjecture regarding this. This possibility was also recently studied by Bertozzi and Pugh [3], [4], and Witelski, Bernoff, and Bertozzi [29] based on both rigorous analysis and numerical computations. For (1.1) we also establish analytically that for positive solutions the integral L (1/H)dx is bounded on each time interval. Extensive numerical experiments −L indicate that max |Hxx | is bounded on each time interval; if we assume this, then we L can use the observation on the evolution of −L (1/H)dx to prove global existence of positive smooth solutions (i.e., that the film does not touch down in finite time). A rigorous proof of boundedness of max |Hxx | is under investigation. Results of extensive computations and their relation to the analytical results are also reported. The outline of the paper is as follows. In section 2 we describe the physical problem and give the governing equations. Section 3 develops the asymptotic long

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DMITRI TSELUIKO AND DEMETRIOS T. PAPAGEORGIOU

wave theory that leads to the scaled evolution equation (1.1). Section 4 is devoted to rigorous analytical results, and section 5 describes the numerical method used to solve (1.1). Section 6 contains the numerical results, and finally, in section 7 we give our conclusions. 2. Physical model and governing equations. Consider a viscous liquid film completely wetting a solid horizontal substrate. Two related configurations are of interest: overlying two-dimensional films with the liquid layer resting on the substrate, and overhanging two-dimensional films with the liquid layer wetting the underside of the horizontal substrate. A schematic is provided in Figure 1 for the overlying film with a normal electric field present.

E0 Region II

Permittivity ea

s

z x

Region I

h(x, t)

h0

Perfectly conducting viscous fluid

g

l Fig. 1. Schematic of the problem.

The fluid is Newtonian of a constant density ρ and dynamic viscosity μ and is assumed to be a perfect conductor. The surface tension coefficient between the liquid and the surrounding medium is σ. We denote by h(x, t) the local film thickness, which is a function of space and time, and the unperturbed thickness of the liquid layer is h0 . The gravitational acceleration g acts in the vertical direction. The plate is a grounded infinite electrode held at zero voltage. Another flat parallel electrode is placed infinitely far from the wetted substrate, so that a uniform vertical electric field is set up at infinity; i.e., at infinity the electric field E approaches a constant value E 0 which is normal to the plate. The surrounding medium is assumed to be a perfect dielectric with permittivity εa , and the corresponding voltage potential in it is denoted by V . Since the liquid is a perfect electric conductor, the potential is zero on the interface, and there is no electric field inside the liquid layer. We use a rectangular coordinate system (x, z) with the x-axis pointing along the plate and the z-axis pointing up and being perpendicular to the plate. The associated velocity field is denoted by u = (u, v), the liquid layer is denoted by Region I, and the surrounding medium by Region II. The hydrodynamics in Region I is governed by the Navier–Stokes equations. In Region II the electric field can be written as E = −∇V , with V satisfying the Laplace equation. The equations are made dimensionless by scaling lengths with h0 , velocities with a typical velocity U0 , time with h0 /U0 , pressure with μU0 /h0 , and voltages by E0 h0 . This leads to the following equations in Region I: 1 ut + uux + vuz = (−px + uxx + uzz ), (2.1) R 1 vt + uvx + vvz = (−pz + vxx + vzz − G), (2.2) R ux + vz = 0, (2.3)


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and in Region II we obtain (2.4)

Vxx + Vzz = 0.

The dimensionless boundary conditions of no slip at the wall and the far field condition in Region II for the voltage become (2.5)

u|z=0 = 0,

(2.6)

Vx → 0, Vz → −1

v|z=0 = 0, as z → ∞.

At the interface z = h(x, t) we have (2.7)

V = 0, v = ht + uhx ,

(2.8) (2.9) (2.10)

−

We (1 + 2

(1 − h2x )(uz + vx ) + 4hx vz = 0, 1 + h2x 1 hxx h2x )Vz2 + patm − p) = vz + (¯ , 1 − h2x 2 2C(1 + h2x )3/2

where p¯atm = patm h0 /μU0 , with patm a constant, is the nondimensional constant pressure in Region II. The boundary condition (2.7) reflects the fact that z = h(x, t) is an equipotential surface, (2.8) is the kinematic condition, and (2.9) and (2.10) follow from the balance of tangential and normal stresses at the interface. The parameters (2.11)

R=

U0 h0 , ν

C=

U0 μ , σ

We =

εa E02 h0 , 2μU0

G=

ρgh20 μU0

are a Reynolds number, a Capillary number measuring the ratio of inertial to capillary forces, an electric Weber number measuring the ratio of electrical to fluid pressures, and a gravity number G measuring the ratio of gravitational to viscous forces. It is useful to use the following exact solution to (2.1)–(2.10), (2.12)

u ¯ = 0,

v¯ = 0,

p¯ = p¯atm − We − G(z − 1),

V = 1 − z,

and introduce new unknown functions u ˜, v˜, p˜, V by u = u ¯+u ˜, etc., and drop tildes from the transformed equations and boundary conditions. The resulting fully nonlinear dimensionless system is exact and presents a formidable computational and analytical task. In what follows we make analytical progress by studying the physically relevant case of thin films using a long wave nonlinear theory. 3. Long wave asymptotics. In the asymptotic analysis presented next we assume that the typical length λ of the interface deformation is long compared to the undisturbed thickness; that is, δ = h0 /λ 1 is a small parameter. In Region I, we introduce the lubrication scalings (3.1)

x=

1 ξ, δ

t=

1 τ, δ

v = δw,

p=

1 P. δ

The conditions at the interface z = h(ξ, t) become w = hτ + uhξ ,

(3.2) (3.3)

(1 −

δ 2 h2ξ )(uz

+ δ 2 wξ ) + 4δ 2 hξ wz = 0,

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− (3.4)

We 1 + δ 2 h2x (Vz − 1)2 (1 + δ 2 h2ξ ) − 1 + δ wz 2 1 − δ 2 h2x P δ 2 hξξ G = − (1 − h) − . 2 2δ 2C(1 + δ 2 h2ξ )3/2

The last boundary condition contains a nonlocal contribution since V satisfies the Laplace equation in the potential region above the fluid layer. This is obtained by considering the problem in Region II. The analysis is the same as in Tseluiko and Papageorgiou [28] and results in the following expression for Vz in terms of the interfacial position (keeping the O(δ) term and dropping the higher order terms): (3.5)

Vz (ξ, 0) = −δH[hξ ],

where H is the Hilbert transform operator defined by

∞ 1 g(ξ ) (3.6) H[g](ξ) = P V dξ , π −∞ ξ − ξ where P V denotes the principal value. Using (3.5) in (3.4) yields (3.7)

−δWe H[hξ ] + δwz −

P δ 2 hξξ G (1 − h) − = + O(δ 2 ), 2 2δ 2C

from which we deduce the following canonical scalings that retain the effects of the electric field, gravity, and surface tension (bar quantities are of order one): (3.8)

C = δ 3 C,

We =

We , δ2

G=

1 G. δ

The Reynolds number R is assumed to be o(δ −1 ) throughout. Expanding in powers of δ, e.g., u = u0 + δu1 + · · · , etc., gives the following leading order solutions: (3.9) (3.10) (3.11)

P0ξ 2 z − P0ξ H0 z, 2 P0ξξ H0 z 2 P0ξ H0ξ z 2 P0ξξ z 3 + + , w0 = − 6 2 2 1 P0 = −2W e H[H0ξ ] − G(1 − H0 ) − H0ξξ . C u0 =

Using the velocities (3.9) and (3.10) in the kinematic condition (3.2) gives H0τ = 1 3 3 [H0 P0ξ ]ξ ; using (3.11) for P0 , the evolution equation becomes 1 1 Hxxx − GHx + 2We H[Hxx ] = 0, (3.12) Ht + H 3 3 C x where for simplicity we write t and x for τ and ξ, H for H0 , and drop the bars from C, G, W e . There are several noteworthy features of (3.12). The electric field enters through a nonlocal term and is destabilizing as in falling films; see Tseluiko and Papageorgiou [28]. Gravity is present, and if we allow G to be negative, we obtain the long wave thin film dynamics of hanging films. In the absence of an electric field (We = 0) and if G > 0, the flow is linearly stable; instability is possible if G < 0, as is intuitive for hanging films (this case has been considered by Bertozzi and Pugh [3],

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0.5

0.5 C=1, G=1

C=1, G=−1

s(k)

0

s(k)

0

−0.5

−0.5

W =0

W =0

e

e

W =1

W =0.25

We=1.25

We=0.5

e

−1

0

0.5

e

1 k

1.5

2

−1

0

0.5

1 k

1.5

2

Fig. 2. The effect of the electric field on linear growth rates s(k). The left panel has G = 1 (overlying film) and the right panel G = −1 (under-hanging film). In both cases C = 1, and the values of We are shown in the figures.

Ehrhard [8], Ehrhard and Davis [9], Yiantsios and Higgins [30], [31]). The electric field, however, can be utilized to destabilize liquid films lying on top of a substrate electrode (G > 0), and the novel equation (3.12) enables a quantitative study of such phenomena. To quantify some of these observations we perform a linear stability analysis to identify stable and unstable regimes when We = 0. Writing H = 1 + η, linearizing with respect to , and seeking solutions proportional to η = ηˆ exp(st+ikx), where ηˆ is a complex constant, leads to the following linear dispersion relation: G 1 4 2We 2 k + k |k| − k 2 . 3C 3 3 (We have used the Fourier transform property F H[u] (k) = −i sign(Rek)ˆ u(k).) When G > 0 (the film is resting on a substrate), we see that for We < (G/C)1/2 all modes are stable; i.e., s(k) < 0 for all k. For We > (G/C)1/2 , however, there is a band of unstable waveswith wavenumbers extending from kL = CWe − (CWe )2 − CG to kR = CWe + (CWe )2 − CG, and the electric field is destabilizing. The most unstable mode has wavenumber k = (3CWe + 9(CWe )2 − 8CG)/4. Also, kL > 0 for all We , and hence all waves in the immediate vicinity of k = 0 are stable—for large We , for example, all waves longer that 4πWe /G are stable. Typical results are depicted in Figure 2 (the left panel), for the particular values C = 1, G = 1 for which We > 1 yields instability, as is clear from the figure. When G < 0,there is always a band of unstable waves extending from kL = 0 to kR = CWe + (CWe )2 − CG; see Figure 2 (the right panel) for typical results for the case C = 1, G = 1. For large We we have kR ∼ 2CWe , and so increasingly shorter wavelengths become linearly unstable as We increases. Damping of sufficiently high wavenumbers (and, hence, well-posedness) is provided by the presence of surface tension which is extremely important in this case. The linear results set the stage for nonlinear computations and analysis, and we can expect nontrivial behavior in parameter regimes that support unstable linear waves. We concentrate on such calculations next. (3.13)

s(k) = −

4. Analytical results. In this section we prove the global boundedness of positive classical solutions of the evolution equation (3.12). This is achieved by constructing an energy functional whose extrema are steady state solutions of (3.12) and using it to estimate the H 1 -norm of the solution and show that it is uniformly bounded.

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The main result is Proposition 4.2 in section 4.2. The results extend those of Bertozzi and Pugh [3] to a class of physically meaningful nonlocal equations. 4.1. The energy functional. We consider the generalized equation (4.1) Ht + f (H)A[H]x x = 0, where f is a function which takes positive values for positive arguments and is zero only at zero, and A[H] is some integro-differential operator, which involves the function H, its first and second spatial derivatives, and the Hilbert transform operator. The additional condition that is satisfied by this operator will be given below. We consider (4.1) on a periodic interval [−L, L] with positive initial data H(x, 0) = H0 (x). Steady state solutions of (4.1) are found by integrating once to obtain (4.2)

f (H)A[H]x = C1 ,

where C1 is some constant. If H vanishes at some point, then C1 = 0. Otherwise L A[H]x = C1 /f (H); integration gives C1 −L (1/f (H))dx = 0, which in turn implies that C1 = 0. So, for steady state solutions, A[H]x = 0, i.e., A[H] = C2 ,

(4.3)

where C2 is some constant. Let E[H] be the energy functional having the form

E[H] =

(4.4)

L

−L

L H, Hx , H[H] dx,

and whose extrema are the steady state solutions of (4.1). More precisely, we assume that the following generalized Euler–Lagrange equation, d ∂L ∂L ∂L − (4.5) = 0, −H ∂H dx ∂Hx ∂H[H] coincides with the equation C2 − A[H] = 0. Then

L ∂L dE[H] ∂L ∂L = Ht + H[Ht ] dx. (4.6) Hxt + dt ∂Hx ∂H[H] −L ∂H Integrating the second term by parts and applying the property I u(x)H[v](x)dx = − I v(x)H[u](x)dx to the third term gives (4.7) (4.8)

d ∂L ∂L ∂L − Ht dx −H dx ∂Hx ∂H[H] −L ∂H

L

L =− (C2 − A[H]) f (H)A[H]x x dx = − f (H)A[H]2x dx.

dE[H] = dt

L

−L

−L

Therefore, dE[H]/dt ≤ 0 for nonnegative H i.e., E[H] is bounded above. For (3.12) we have f (H) = H 3 /3 and A[H] = (1/C)Hxx − GH + 2We H[Hx ]. The steady state solutions are determined by the equation (4.9)

1 Hxx − GH + 2We H[Hx ] = C2 , C


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which on integration yields G C2 = − 2L

(4.10)

L

Hdx. −L

It follows that the functional L H, Hx , H[H] can be chosen in the following form:

1 2 G 2 L H, Hx , H[H] = H + H + We Hx H[H] + C2 H. 2C x 2

(4.11)

Thus, the energy functional

E[H] =

(4.12)

L

−L

1 2 G 2 H + H + We Hx H[H] + C2 H dx 2C x 2

is a nonincreasing function of time for a nonnegative solution H. 4.2. Uniform boundedness of positive smooth solutions. In the previous section we have shown that the energy functional E[H] is bounded above by its initial value for a nonnegative solution H of (3.12). In this section we will show uniform boundedness of solutions. We will restrict our consideration to positive solutions, since, given upper and lower bounds for positive solutions, the equation is uniformly parabolic, which implies small time smoothness of the solutions (see Eidelman [10], Friedman [11]). We begin with the following lemma. Lemma 4.1. Let E[H] be the functional defined on H 1 (−L, L) by (4.12). Then there exist constants α > 0, β > 0, and γ such that (s.t.)

H 2H 1 ≤ αE[H] + β H 21 + γ H 1

(4.13)

1 k for all nonnegative H ∈ Hper . (Here and everywhere else we denote by L2per , Hper , 2 k k = 1, 2, . . . , the subspaces of the Sobolev spaces L (−L, L), H (−L, L) consisting of periodic functions with period 2L.) Proof. First, using the Cauchy–Schwartz and Young’s inequalities and the property H[u] = u of the Hilbert transform gives

L

−L

Hx H[H]dx ≥ − Hx 2 H[H] 2 = − Hx 2 H 2 ≥−

(4.14)

ε1 1

Hx 22 −

H 22 , 2 2ε1

where ε1 is some positive number. Hence,

E[H] ≥

L

−L

1 = 2C 1 (4.15) = 2C

1 2 G 2 ε1 We We H + H + C2 H dx −

Hx 22 −

H 22 2C x 2 2 2ε1

L ε1 W e G We −

Hx 22 + −

H 22 + C2 Hdx 2 2 2ε1 −L

L ε1 W e ε1 W e G We 1 2 2 −

H H 1 + − +

H 2 + C2 − Hdx. 2 2 2ε1 2C 2 −L

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L Note that −L Hdx = H 1 for nonnegative H. We define A = 1/2C − ε1 We /2 and B = −G/2 + We /2ε1 + 1/2C − ε1 We /2. Choosing ε1 sufficiently small gives A > 0, B > 0. Also, using the interpolation inequality (4.16)

1/3

2/3

H 2 ≤ C3 H H 1 H 1

and applying Young’s inequality for the right-hand side of the expression above gives ε2 2 (4.17)

H 2 ≤ C3

H H 1 + 1/2 H 1 , 3 3ε2 where ε2 is some positive number. Therefore, (4.18)

H 22 ≤

2ε22 C32 8C32

H 21 ,

H 2H 1 + 9 9ε2

and we get (4.19) (4.20) (4.21)

E[H] ≥ A H 2H 1 − B H 22 + C2 H 1 2 2 2ε2 C3 8C32 2 2 2 ≥ A H H 1 − B

H 1 + C2 H 1

H H 1 + 9 9ε2 2ε2 BC32 8BC32

H 2H 1 − = A− 2

H 21 + C2 H 1 . 9 9ε2

˜ = 8BC 2 /9ε2 . Note that B ˜ is positive and We define A˜ = A − 2ε22 BC32 /9 and B 3 ˜ choosing ε2 small enough makes A also positive. Thus, (4.22)

2 2 ˜ ˜ E[H] ≥ A H

H 1 − B H 1 + C2 H 1 ,

i.e., (4.23)

H 2H 1 ≤ αE[H] + β H 21 + γ H 1 ,

˜ β = B/ ˜ A, ˜ γ = −C2 /A, ˜ as required. where α = 1/A, We can now prove uniform boundedness of positive smooth solutions to (3.12). Proposition 4.2. Let H(x, t) be a positive smooth solution of (3.12) with peri1 odic boundary conditions on some time interval [0, T ]. If H(x, 0) = H0 (x) ∈ Hper , then H H 1 is uniformly bounded. Proof. First, note that (3.12) is a conservation law. The spatial integral of the L solution is conserved. Indeed, integrating over [−L, L] gives (d/dt) −L Hdx = 0 and hence H 1 = H0 1 . Also, since E[H] is a nonincreasing function of time, (4.13) implies (4.24) (4.25) (4.26)

H 2H 1 ≤ αE[H] + β H 21 + γ H 1 = αE[H] + β H0 21 + γ H0 1 ≤ αE[H0 ] + β H0 21 + γ H0 1 ,

which was to be proved. Remark. Proposition 4.2 is essentially a no-blow-up theorem for the solution H. The result does not prevent a touchdown (the numerics predicts a touchdown in infinite time—see Figures 3–6).


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L 4.3. Evolution of −L H −1 dx. In this section we show that the spatial integral of H −1 is bounded on each finite time interval. Indeed,

L Ht d L dx =− dx. (4.27) 2 dt −L H H −L Substituting the expression for Ht from (3.12) into (4.27) gives

d L dx 1 L 1 3 1 H − GH + 2W H[H ] dx = H xxx x e xx dt −L H 3 −L H 2 C x

1 1 L 1 2 (3H Hx ) = Hxxx − GHx + 2We H[Hxx ] dx 3 −L H 2 C

1 L 1 3 1 H + H − GH + 2W H[H ] dx xxx x e xx 3 −L H 2 C x

L

L 1 L 2 Hx Hxxx dx − G Hx dx + 2We Hx H[Hxx ]dx = C −L −L −L

L

1 G L 2We L + (4.28) HHxxxx dx − HHxx dx + HH[Hxxx ]dx. 3C −L 3 −L 3 −L Integration by parts then gives

L

L d L dx 1 L 2 =− Hxx dx − G Hx2 dx + 2We Hx H[Hxx ]dx dt −L H C −L −L −L

L

1 G L 2 2We L 2 + H dx + (4.29) H dx − Hx H[Hxx ]dx. 3C −L xx 3 −L x 3 −L Therefore, (4.30)

d dt

L

−L

2 dx =− H 3C

L

−L

2 Hxx dx −

2G 3

L

−L

Hx2 dx +

4We 3

L

−L

Hx H[Hxx ]dx.

Using the Cauchy–Schwartz and Young’s inequalities and the property H[u] = u

gives

L Hx H[Hxx ]dx ≤ Hx 2 H[Hxx ] 2 = Hx 2 Hxx ] 2 −L

(4.31)

≤

1 ε

Hx 22 + Hxx 22 , 2ε 2

where ε is some positive number. Hence,

d L dx 2 2G 4We 1 ε ≤−

Hxx 22 −

Hx 22 +

Hx 22 + Hxx 22 dt −L H 3C 3 3 2ε 2 (4.32)

= A Hxx 22 + B Hx 22 ,

where A = −2/3C + 2We ε/3, B = −2G/3 + 2We /3ε. Choosing ε small enough (s.t. A < 0) implies

d L dx ≤ B Hx 22 . (4.33) dt −L H

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Since the H 1 -norm of the positive solution H is uniformly bounded (as was shown in the previous section), we obtain the desired boundedness result since there exists a L constant D s.t. (d/dt) −L (1/H)dx ≤ D. It was proved in section 4.2 that the H 1 -norm of a positive solution is bounded for all time. Due to Agmon’s inequality this also implies boundedness of the maximum of the solution. Hence, if the solution is positive for all time, then it is also uniformly bounded above; i.e., it does not blow up in finite or infinite time. In this section we have shown that the spatial integral of 1/H is bounded on each finite time interval; i.e., it does not blow up in finite time (though this can happen in infinite time). This result can be used to show that as long as max |Hxx | remains bounded on each finite time interval, an initially positive solution will remain positive for all time; that is, the interface does not touch down in finite time. We prove this below. (The boundedness of max |Hxx | comes from extensive computations (see section 6) and is used as an assumption in what follows. A rigorous proof has not yet been found.) Proposition 4.3. Let H(x, t) be a positive smooth solution of (3.12) with periodic boundary conditions on some time interval [0, T ). In addition, assume that max |Hxx | is bounded above on each finite time interval. Then the solution H remains positive for all time, i.e., T = ∞. Proof. Suppose that T is finite and the solution becomes zero at some point x = x0 in finite time t = T , and seek a contradiction. This means that at t = T the solution obtains a minimum at x = x0 . Denote by ξ(t) the point at which the solution has a (local) minimum at a given time t s.t. ξ(t) is a continuous function of t and ξ(T ) = x0 . Also, let max |Hxx | = A, where A is a function of time which is bounded on [0, T ). At each time t < T we expand the function H into a Taylor series about x = ξ(t) and use the fact that Hx (ξ(t), t) = 0, (4.34)

1 H(x, t) = H(ξ(t), t) + (x − ξ(t))2 Hxx (ζ, t), 2

where ζ = ζ(x, t) is some point between x and ξ(t). We get

L

L dx dx = 1 H(x, t) H(ξ(t), t) + (x − ξ(t))2 Hxx (ζ, t) −L −L 2

L dx ≥ A 2 −L H(ξ(t), t) + 2 (x − ξ(t)) L 2 A −1 tan (ξ(t) − x) . (4.35) =− AH(ξ(t), t) 2H(ξ(t), t) −L

It follows that the right-hand side of (4.35) blows up when t → T , since then L H(ξ(t), t) → 0 by the assumption. This contradicts the boundedness of −L (1/H)dx on each finite time interval. Thus, if max |Hxx | is bounded on each finite time interval, then the solution cannot become zero in finite time; i.e., it remains positive for all time. 5. Numerical method. We use a fully implicit two-level scheme with Newton iterations for (3.12) on a finite periodic interval [−L, L]. The method was developed for the following more general equation (this also includes the falling film equation derived in Tseluiko and Papageorgiou [28]): (5.1) Ht + f1 (H)Hxxx x + f2 (H) xx + f3 (H)H[Hxx ] x + f4 (H) x = 0,


1321

with f1 , . . . , f4 polynomials in H. We incorporate the ideas of Bertozzi and Pugh [2], Diez, Kondic, and Bertozzi [6] into nonlocal problems. The equation is solved on a uniform spatial grid xm = (m − M )Δx, m = 1, 2, . . . , 2M , where Δx = L/M , and spatial derivatives are discretized using central differences; Hm denote the values of a 2L-periodic function H at the mesh points. We also set H0 = H2M , H−1 = H2M −1 , etc., and H2M +1 = H1 , H2M +2 = H2 , etc., which follow by the periodicity of H. For m = 1, 2, . . . , 2M − 1, we introduce the midpoints xm+1/2 = (xm + xm+1 )/2 with x1/2 = (−L + x1 )/2 and define Hm+1/2 = (Hm + Hm+1 )/2. Second order accurate central differences are used to approximate odd derivatives at xm+1/2 and even derivatives at xm . To approximate the Hilbert transform of Hxx at x = xm+1/2 we use trapezoidal quadrature in the periodic representation

L H[g](x) = (1/2L)P V −L g(ξ) cot π(x − ξ)/2L dξ, 2M π(xm+1/2 − xk ) 2 (H)]m+1/2 ≡ Δx (5.2) H[Hxx ](xm+1/2 ) ≈ H[∂ . ∂2 (H)k cot 2L 2L k=1

This leads to the system of ordinary differential equations for H1 , H2 , . . . , H2M : f1 (Hm+1/2 )∂3 (H)m+1/2 − f1 (Hm−1/2 )∂3 (H)m−1/2 dHm =− − ∂2 (f2 (H))m dt Δx 2 (H)]m+1/2 − f3 (Hm−1/2 )H[∂ 2 (H)]m−1/2 f3 (Hm+1/2 )H[∂ − Δx f4 (Hm+1/2 ) − f4 (Hm−1/2 ) − (5.3) , Δx where the grid operators ∂2 and ∂3 correspond to the second- and third order spatial derivatives, respectively. We write (5.3) in the following compact form: (5.4)

dH = F (H), dt

where H = (H1 , H2 , . . . , H2M )T , F (H) = (F1 (H), F2 (H), . . . , F2M (H))T are given by the right-hand side of (5.3). Note that, unlike similar thin film problems that have been studied previously, Fm (H) depends on all the components of H due to the presence of the nonlocal Hilbert transform. This semidiscrete scheme preserves the discrete approximation of the volume. We show this by multiplying (5.3) by Δx and summing over m = 1, 2, . . . , 2M 2M 2M to obtain m=1 (dHm /dt)Δx = 0. A time integration yields m=1 Hm (t)Δx = 2M H (0)Δx, as desired. m m=1 For the time discretization of (5.4) we use the usual implicit two-level scheme, (5.5)

H n+1 − H n = F (θH n+1 + (1 − θ)H n ), Δtn

n )T is the numerical solution for H at t = tn , Δtn = where H n = (H1n , H2n , . . . , H2M tn+1 −tn , and θ is some real number between 0 and 1 (the scheme is first order accurate in time). To advance from the time level n to the time level n + 1, the algebraic system of nonlinear equations (5.5) for H n+1 is solved iteratively using Newton’s method. The time step is chosen dynamically for each time level by requiring several constraints to be satisfied as described below (see also Bertozzi and Pugh [2] and Diez,

1322


Kondic, and Bertozzi [6]). If the numerical solution violates one of the constraints, then the time step is reduced and the calculation is repeated until all the constraints are met. On the other hand, if all the constraints are met after the first application of Newton’s method, the time step is increased at the next time level in order to prevent using unnecessarily small time steps. The constraints are the following: (a) the minimum of the solution should change by no more than 10%, (b) the local relative error should be small (10−3 , say). The local relative error em approximates n n )(d2 Hm /dt2 ) and is computed as follows (see [2], [6]): ((Δtn−1 )2 /Hm (5.6)

em =

n+1 n−1 n + Δtn−1 Hm − (Δtn−2 + Δtn−1 )Hm 2Δtn−1 Δtn−2 Hm . n Δtn−2 (Δtn−2 + Δtn−1 )Hm

In addition, the spatial grid is refined during the calculation to get better resolution of the solution. This is done by doubling the number of mesh points when the magnitude of more than 2/3 of the Fourier modes is larger than a set tolerance of 10−13 . (The fast Fourier transform is used as an accuracy diagnostic.) The numerical method has been described and implemented for solutions without any assumed symmetry. If f4 ≡ 0, however, as is the case here, and the initial condition is an even function, then H remains even for all time. In this case we can consider 2L-periodic even solutions and discretize the equation on the interval [0, L] alone, thus halving the number of unknowns. The appropriate boundary conditions are (5.7)

Hx (0, t) = Hxxx (0, t) = 0,

Hx (L, t) = Hxxx (L, t) = 0,

with periodicity used as needed in calculating difference formulas. 6. Numerical results. The code was validated by reproducing the results of Yiantsios and Higgins [30], who √ solved √ (3.12) √ with C = 1, G = −1, We = 0, on periodic intervals of lengths 2 2π, 4 2π, 6 2π, 5π. We also reproduced the results of Bertozzi and Pugh [3]. Part of their work involved the numerical solution of (3.12) on the interval [−1, 1], with C = 1, G = −80, We = 0. Our code has reproduced these results with indistinguishable differences at t = 100, which is the largest time that Bertozzi and Pugh [3] report. For the results presented here we take C = 1, G = −1 or G = 1, and increase We to enhance the instability. We take L = 10 and an initial condition (6.1)

H(x, 0) = 1 + 0.1 cos(πx/L).

Thus, without loss of generality the mean value of H(x, 0) over a period is taken to be 1. If it were d > 0, then the change of time scale, t → d−3 t, leaves the evolution equation (3.12) unchanged but normalizes the initial condition to have unit mean. 6.1. Films wetting the underside of the plate, G < 0. As explained previously, when G < 0 the flat film state is long-wave unstable even when We = 0. We present results for fixed G = −1 and C = 1 as the electric field parameter We increases. For these parameters with We = 0 and L = 10, the first two harmonics are linearly unstable modes of the flat state, as can be seen from the linear result (6.2) below. We aim to systematically quantify the dynamics in the nonlinear regime as We increases. The first set of results is presented in Figure 3 for We = 0. There are eight panels in the figure depicting the evolution over 1000 time units. The plots of the

1323


0

−2 −1 0 H, P

H

1

2

1 2

3

3

−10

−5

0 x

5

4 −10

10

−5

0 x

5

10

15

45

10 x 2

35

||H ||2

2

||H||2

40

30

5

25 0

20 0

200

400

600

800

1000

0

200

400

t

800

1000

600

800

1000

600

800

1000

120 integral of 1/H

xx 2

15

||H ||2

600 t

10

5

100 80 60 40 20

0 0

200

400

600

800

1000

0

200

400

t

t 0

10

3.5

min(H)

max(H)

3 2.5 2

−1

10

1.5 1 0

200

400

600 t

800

1000

0

200

400 t

Fig. 3. Evolution of the spatially periodic interface for C = 1, G = −1, We = 0. The equation was integrated for 0 ≤ t ≤ 1000. The upper-left panel shows the evolution of the profile H (the time interval between the plots is 10). The upper-right panel shows the profile H (thin line) and the pressure P given by (3.11) (solid line) at t = 1000. Also, the evolution of H22 , Hx 22 , Hxx 22 , 10 (1/H)dx and the maximum and minimum of H are shown. (For the as well as the evolution of −10 minimum we use a log-linear plot.)

interface are reflected about the x-axis to emphasize that we are dealing with hanging films. The interface evolution is shown in the top-left panel, and the top-right panel shows the solution at the last computed time t = 1000; the thin line curve represents the interface H(x, 1000), and the thick line curve the corresponding perturbation pressure distribution P (x, 1000) given by (3.11) (note that the subscript zero has been dropped). It can be seen that the pressure is essentially uniform and negative in the regions where large drops are forming (the pressure has different values in different-sized drops), and uniform and positive in the thinning regions between large drops. The resulting pressure gradient acts to push fluid out of the thinning regions and into the larger drops. This mechanism is at play for all computed results presented here and is responsible for the asymptotic thinning of the regions between the larger

1324

DMITRI TSELUIKO AND DEMETRIOS T. PAPAGEORGIOU 0

0

0.5

0.5

1

1

H

H

1.5

1.5

2 2 2.5 2.5 3 3 3.5 −10

−8

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2

4

6

8

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−0.5 0

0

0.5 H, P

H, P

1 1 1.5

2

2 3 2.5 4

3 3.5 −10

−8

−6

−4

−2

0 x

2

4

6

8

10

−10

Fig. 4. Evolution for C = 1, G = −1. The top row shows the interface evolution for We = 0.5 (left) and 1 (right); the bottom row shows the corresponding profiles H (thin line) and the pressures P (solid line) at the final times.

quasi-static drops. The other six panels contain information on the evolution of different norms of H, namely H 22 , Hx 22 , and Hxx 22 (labeled in the figure); the L evolution of −L (1/H)dx; and the evolution of the maximum and minimum of H over the spatial domain, max(H) and min(H), respectively (the latter is plotted using log-linear scales). The boundedness of H 22 and Hx 22 (equivalently of the L2 - and H 1 -norms) is in line with the rigorous results of section 4, as is the at most linear L growth of the integral −L (1/H)dx. The evolution of Hxx 22 (equivalently the H 2 norm) indicates that it is bounded (moreover, from numerics it can be seen that max |Hxx | is bounded as well)—this is used as an assumption in producing a proof that the interface cannot touch down in finite time but can do so asymptotically in infinite time (see section 4.3). The log-linear evolution of min(H) also provides strong evidence of an asymptotic touchdown after infinite time in line with the conjecture in section 4.3. Finally, we note that the profile at large time contains two large drops (see the top-right panel), and this number coincides with the number of linearly unstable modes. Nonlinearity produces small daughter drops between the main mother drops that cannot be predicted by linear theory. This phenomenon is generic and holds when We is nonzero also (see Figures 4, 5, 6 also). Figure 4 contains results for nonzero electric fields with We = 0.5 and 1. The top row shows the evolution of the interface for We = 0.5 (left) and We = 1 (right), and the bottom row shows the corresponding final computed interfacial profiles and pressure distributions. The calculations were carried out to 100 and 30 time units for We = 0.5 and 1, respectively, and profiles in the top row are depicted at intervals of 4 time units. Once more we see main drops forming at large times with thinning regions between them containing smaller humps. The pressure distribution is uniform in the larger drops with maxima in the thinning regions, producing the draining mechanism discussed earlier. The main difference between the two cases is that for We = 0.5 we ultimately have four drops forming, while for We = 1 we have six. This can be explained using linear theory and (3.13). The wavenumber of the maximally unstable


1325

mode on 2π-periodic domains is kmax = (3CWe + 9(CWe )2 − 8CG)/4. Modifying this to 2L-periodic domains gives L 3CWe + 9(CWe )2 − 8CG . (6.2) kmax = π 4 The value of kmax provides a qualitative estimate of the main features of the interface at large times; for example, in the results of Figure 3 we have kmax = 2.25, which explains the two large drops that form. For the parameters of Figure 4 we have kmax = 3.74 and kmax = 5.67, which explain the four and six drops formed. The smaller drops forming in the thinning regions are due to nonlinearity and cannot be explained using a simple linear theory. We have also monitored norms and other diagnostics as in Figure 3 and have found similar behavior. Most notably, max |Hxx | remains bounded in time. 6.2. Films wetting the upper side of the plate, G > 0. Here we present results for G = 1, C = 1, and increasing values of We . As noted earlier, if We = 0, the flat state is stable—the solutions to the initial value problem are damped and produce the uniform trivial state at large times (this has been confirmed numerically also). Support for this is also provided by the linear result (3.13), since s(k) < 0 for all k = 0. If We exceeds the critical value Wec = (G/C)1/2 , instability sets in over a band of wavenumbers kL < k < kR , using the notation of section 3. For our parameters, Wec = 1, and in what follows we present results for increasing We > 1. The first set of results has We = 1.02, which is just above critical. According to (6.2), kmax = 3.37, and so we may expect three drops to form at large times. The results are shown in Figure 5; the integration was carried out to t = 5000. The format of the figure is the same as that of Figure 3, and the results are qualitatively similar. The top-right panel shows an enlargement of the solution and the corresponding pressure distribution in the vicinity of the first thinning region to the left of the origin. Again we see essentially uniform negative pressure in the main drops and a pressure maximum in the thinning region in between, so that the fluid draining mechanism described earlier is seen to operate. The other diagnostics are in agreement with the analytical results. In Figure 6 we present results for We = 1.1, 1.5, and 2.0, respectively. The top row shows the time evolution of H as We increases from the left to the right, and the bottom row shows the corresponding final computed profiles and pressure distributions (thin and thick solid lines, respectively)—for We = 1.1 this is enlarged accordingly. The computations were carried out to 1000, 75, and 20 time units for We = 1.1, 1.5, and 2, respectively, and profiles are plotted every 10, 1, and 1 time units, respectively. The pressure gradient draining mechanism is operational throughout, and uniform but different negative pressures are attained inside the large drops. The number of drops formed at large times is again in excellent agreement with linear theory (linear theory cannot provide the volumes of the drops or the formation of smaller daughter droplets in the thinning regions). For example, the values of kmax given by (6.2) are 3.98, 6.37, and 8.99 for We = 1.1, 1.5, and 2.0, respectively, while the numbers of the main computed drops are 4, 6, and 9, respectively. It can be concluded, therefore, that linear theory can be used in a very simple way to predict how many drops will form at large time. Details, including drop volumes, must be calculated numerically by solving the nonlinear problem. 7. Conclusions. We have derived and studied analytically and numerically a nonlinear nonlocal evolution equation that describes the evolution of thin films wetting

1326


0.6

2

0.5 1.5

0.4

H, P

H 1

0.3 0.2 0.1

0.5

0 0 −10

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5

10

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−3

−2.5

−2

3000

4000

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x

32

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30 8 ||Hx||22

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||H||2

28 26 24

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t 500

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2

||Hxx||2

8 6 4 2

300 200 100

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4000

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2000

t

t 0

2.2

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max(H)

2

1.6 1.4

−1

10

−2

10

1.2 1 0

1000

2000

3000 t

4000

5000

0

1000

2000 t

Fig. 5. Evolution of the spatially periodic interface for C = 1, G = 1, We = 1.02. The equation was integrated for 0 ≤ t ≤ 5000. The upper-left panel shows the evolution of the profile H (the time interval between the plots is 100). The upper-right panel shows the profile H (thin line) and the pressure P given by (3.11) (solid line) at the final time. Also, the evolution of H22 , Hx 22 , 10 (1/H)dx and the maximum and minimum of H, are shown Hxx 22 , as well as the evolution of −10 (for the minimum we use a log-linear plot).

a horizontal plate in the presence of a vertical electric field. The field introduces the instability when G > 0 (films wetting the upper side of the plate) and enhances the instability when G < 0 (films wetting the underside side of the plate). By extending previous analytical studies to incorporate nonlocal terms, we have proved a no-blow-up theorem of positive smooth solutions of the evolution equation. Using an estimate of the integral of the reciprocal of the solution and assuming that max |Hxx | is uniformly bounded (this is suggested by extensive numerical work), we have also presented a conjecture that the film cannot touch down in finite time but can do so only asymptotically in infinite time. This also holds in the absence of the field. All rigorous results are seen in the numerical solutions, thus providing additional accuracy checks for the latter. Extensive numerical experiments have been carried out to describe the salient features of thin electrified film dynamics. Initially, the evolution follows the predictions of linear theory, and the solution grows exponentially. As time increases, higher

1327


2.5 3

2.5 2

2.5

2 1.5 H

H

H

2 1.5

1.5 1 1 1 0.5

0.5

0 −10

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1

1

4 2

0.4 0.5

0

0.2

0

0

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−0.5 −4

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−2.5 x

−2

−1.5

−1

−1 −10

−8

−6

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2

4

6

8

10

−4 −10

Fig. 6. Evolution for C = 1, G = 1. The top row shows the interface evolution for We = 1.1, 1.5, and 2, from left to right; the bottom row shows the corresponding profiles H (thin line) and the pressures P (solid line) at the final times.

harmonics are generated due to the nonlinearities, and the most dominant mode appears to be the most unstable mode predicted by linear theory; note that this mode corresponds to the number of the drops which appear at large times. The qualitative features of the solutions for G < 0 and We zero or nonzero are similar to those with G > 0 and We > 0 (in the latter case a nonzero electric field is required to destabilize the flow and to produce nontrivial dynamics). An increase in We for fixed G and C (or equivalently a decrease of a negative G with fixed We and C), produces increasingly more drops at large times, whose number is predicted by linear theory. This drop-formation behavior is one of the main features of the dynamics as additional unstable modes enter. In all computed cases, as the time increases the evolution slows down (this can be seen in any of the different computational panels in the figures, but is most clearly evidenced by the evolution of min(H)). The spatial features at large times are also quite intricate: First, as the interface reaches the vicinity of the wall it tends to flatten, and after that the solution tends to bulge near the ends of the flat regions forming a secondary hump in between—see Figure 3, for example. All the results indicate that the solution remains positive—the film does not touch down within finite time. Also, the solution is bounded for all time (this has been proven rigorously), despite the fact that the electric field increases the instability and promotes the process of the formation of the increasingly larger numbers of drops. Finally, we comment on the possibility of coarsening dynamics, as seen in other thin film studies by Glasner and Witelski [12], for example. Even though it is not clear a priori whether neighboring drops communicate with each other in order to trigger merging and coarsening, the numerical solutions with electric fields present suggest that such dynamics is not seen. Our results suggest that the thinning of the interdrop regions feeds fluid into main drops and that the large time dynamics of the latter remain independent from each other, drops remaining fixed and not moving. It would be interesting to add a disjoining pressure in the manner of Glasner and Witelski [12] that prevents asymptotic thinning and allows communication between main drops. It is also interesting to revisit the calculations of Yiantsios and Higgins [30] for We = 0, where they do not impose symmetry and see that drops move (slowly) with the larger drop moving more. The calculations failed to see merging, however, due to a critical

1328 0

0

0.5

0.5

1

1

1.5

1.5

H

H


2

2

2.5

2.5

3

3 −6

−4

−2

0 x

2

4

6

−6

−4

−2

0 x

2

4

6

Fig. 7. Nonsymmetric initial conditions for hanging drops, C = 1, G = −1. Left panel, We = 0, and right panel, We = 0.1. The domain length is 5π, and the initial condition in both runs is H(x, 0) = 1 + 0.1 sin(2x/5) + 0.1 sin(4x/5). As time increases, the minimum film thickness decreases.

slow-down of drop motion as thinning of interdrop regions takes place. Following [30], we take C = 1, G = −1, and We = 0 and 0.1 with the length L = 2.5π and the initial condition H(x, 0) = 1 + 0.1 sin(2x/5) + 0.1 sin(4x/5). The results are shown in Figure 7, with We = 0 for the left panel and We = 0.1 for the right panel. For We = 0 two drops are formed and are approaching each other, with the larger drop being more mobile and moving from the right to the left, while the smaller drop moves from the left to the right. The rate of approach slows down critically (the integration is carried out to 2000 time units, and the profiles are shown every 400 units). For We = 0.1, two drops are formed but move in the same direction (from the right to the left) with the electric field present. The electric field significantly increases the mobility of the drops. The velocity of the larger drop is slowing down, while the velocity of the smaller drop is increasing with time, and the distance between the drops is decreasing with increasing time. We did not see merging again due to a critical slow-down of drop motion (the integration is carried out to 500 time units, and the profiles are shown every 100 units). REFERENCES [1] A. L. Bertozzi, The mathematics of moving contact lines in thin liquid films, Notices Amer. Math. Soc., 45 (1998), pp. 689–697. [2] A. L. Bertozzi and M. C. Pugh, The lubrication approximation for thin viscous films: The moving contact line with ‘porous media’ cut off of Van der Waals interactions, Nonlinearity, 7 (1994), pp. 1535–1564. [3] A. L. Bertozzi and M. C. Pugh, Long-wave instabilities and saturation in thin film equations, Comm. Pure Appl. Math., 51 (1998), pp. 625–661. [4] A. L. Bertozzi and M. C. Pugh, Finite-time blow-up of solutions of some long-wave unstable thin film equations, Indiana Univ. Math. J., 49 (2000), pp. 1323–1366. [5] R. V. Craster and O. Matar, Electrically induced pattern formation in thin leaky dielectric films, Phys. Fluids, 17 (2005), paper 032104. [6] J. A. Diez, L. Kondic, and A. L. Bertozzi, Global models for moving contact lines, Phys. Rev. E, 63 (2000), paper 011208. [7] E. B. V. Dussan, On the spreading of liquids on solid surfaces, static and dynamic contact angles, Ann. Rev. Fluid Mech., 11 (1979), pp. 371–400. [8] P. Ehrhard, The spreading of hanging drops, J. Colloid Interface Sci., 168 (1994), pp. 242–246. [9] P. Ehrhard and S. H. Davis, Nonisothermal spreading of liquid drops on horizontal plates, J. Fluid. Mech., 229 (1991), pp. 365–388. [10] E. D. Eidelman, Parabolic Systems, North-Holland, Amsterdam, 1969. [11] A. Friedman, Interior estimates for parabolic systems of partial differential equations, J. Math. Mech., 7 (1958), pp. 393–418. [12] K. B. Glasner and T. P. Witelski, Coarsening dynamics of dewetting films, Phys. Rev. E, 67 (2003), paper 016302.


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