Journal of

ELECTROSTATICS ELSEVIER Journal of Electrostatics 40&41 (1997) 55-60

Nonlinear waves in a viscous horizontal film in the presence of an electric field

A. Gonzglez 1,2 and A. Castellanos 2 1. Departamento de Electr6nica y glectromagnetismo. Facultad de Ffsica. Avenida Reina Met- cedes s/n. 410I~ Sevilla, Spain ~. Departamento de Ffsica Aplicada, E.S.L Avenida Reina Mercedes , /n . 41012 Sevilla, Spain

A b s t r a c t

In this work, we consider the influence of non-conservative forces in the propagation of weakly nonlinear waves on the surface of a shallow viscous perfectly conducting liquid layer stressed by a normal electric field. We introduce the viscous losses at the bottom solving the boundary layer equations for the velocity. We derive an integro-differential equation for the deflection of the interface. To prevent the decay of the waves, we discuss the possibility of sustaining them through an external input of electrical energy. For an ohmic liquid there appear tangential interfacial stresses that could pull the liquid. We perform an energy balance of the system and deduce the order of magnitude for the electric fi.elds needed to maintain these waves.

1 I n t r o d u c t i o n

The problem of damping in surface waves, due to viscous dissipation at the bottom, has been previously studied [1], [2]. Keulegan solved the boundary layer equations for the case of waves travelling at constant speed keeping their form unchanged. Here, we extend his analysis to arbitrary waves. The latter solution is then used in the horizontal component of the Navier-Stokes equations it~ the long wave limit, including weak nonlinearity and dispersion. Integrating this component across the layer, and using the

This work was carried out with financial support from DGICYT (Spanish Government Agency) under contract PB93-1182.

0304-3886/97/$17.00 © Elsevier Science B.V. All fights reserved. PII S0304-3886(97)00014-4

56 A. Gonz:ilez, A. Castellanos/Journal of Electrostatics 40&41 (1997) 55-60

T Z

,b Fig. 1. Liquid layer between two electrodes

boundary conditions at the free surface, we obtain an integro-differential equation for the mass flux. A standard procedure using the continuity equation allows us to determine the equation governing the deflection of the interface. The damping of waves may be obtained solving this equation.

In the last section, we discuss the possibility of sustaining stationary waves through an external input of electrical energy. To this end we need to consider the effect of a finite conductivity. In this case, a component of the electric field tangential to the surface appears. The action of this component over the surface charge produce a tangential stress that may be able in some circumstances to maintain the liquid motion. An order of magnitude analysis suffices to indicate the value of the conductivity needed to sustain these waves.

2 Governing equations

Our system consists of a liquid layer of thickness d resting on a grounded electrode at z = - d . Another electrode, with fixed potential V0, is placed at a distance z = h above the free surface (figure 1). The liquid is assumed to be incompressible and to have mass density p, kinematic viscosity v, electrical permittivity, e, and infinite electrical conductivity. The surface tension between the liquid and the air is 7. The system is subjected to a vertical gravitational acceleration g and to a constant external pressure Patm.

The system extends indefinitely in the horizontal directions but only disturbances depending on the x-coordinate will be considered, hence we restrict the s tudy to a 2D geometry (x, z). The surface deformation can

A. Gonzdlez, A. Castellanos/Journal of Electrostatics 40&41 (1997) 55-60 57

then be described by a function ri(x , t). Let u and w denote the horizontal and vertical velocity components , p

the pressure and ¢I, the electric potential in the air. In order to make the system dimensionless, we choose as scale for the

vertical distances d, while for the horizontal ones, we take a typical length, A, usually larger than d. For the horizontal component of the fluid velocity the uni t will be co = (gd) z12, the velocity of the linear waves in the ab- sence of electric field. However, in a thin layer the transversal velocities are smaller than the longitudinal ones. Thus we take dco/A as the scale for u. As t ime scale, we choose the mechanical t ime A/co. The pressure will be measured in terms of pgd, and the electric potential in the air will scale as Y0.

With this selection of units there appears a set of dimensionless param- eters, namely, the relative thickness D = h/d , the relative depth 5 = d/A, the Reynolds number R = cod~v, the Weber number W = 7/pgA2and a parameter tha t measures the ratio of the electrical energy density to the hydrostat ic pressure and tha t we call Electric Weber number, We = v02/pgh 3

The dimensionless Navier-Stokes equations together with the appropri- ate boundary conditions are s tated in [3]. We consider first this system of equations in the long wave limit for shallow liquid layers. In our terms, this means to take 5 -~ 0, nevertheless keeping (RS) 1/2 large compared to unity.

With this hypothesis, the electric potential and the pressure are r e , l i l y obtained

O - z - y WeD 3 p = p~t,,~ + ,j - z w r l ~ ; ( 1 )

D - r/ ' 2(D - r/) 2

and the system reduces to the pair of equations

u z + w z = O ,

ut + uux + wuz = --Tl~ (1

with the boundary conditions

U ~ W ~ 0

(2)

W!D 3 .'~ 1 ( O - ~/)31 + Wr/z . . + ~--~uzz. (3)

( z = - l ) (4)

z=0 (z=O) (5) w = r/t + ur/~

To simplify this system" we assume that the viscosity is very low, so tha t

58 A. Gonzdlez, A. Castellanos/Journal of Electrostatics 40&41 (1997) 55-60

we can approximate the velocity profile by its inviscid limit except in a thin boundary layer near the bottom. In this layer the velocity profile can be written formally as

207 ( R ~ ( z + l ) 2 ) e - ~ d ~ u=u0--~ uo x,t ~ (6)

being u0 the inviscid solution at the bottom (in our case this value is the same for all z).

If we integrate the horizontal Navier-Stokes equation over a cross section we can reduce the original system to a pair of equations for the surface deflection and the mass flux.

,7, + = o (7)

( q2 ) ( WLD_a ~ ( l + y ) +

+ + ' ) + g ot, , , = , _ . a s (8)

In the linear limit and neglecting dispersion and viscous effects eqs. (7) and (8) reduce to a nondispersive wave equation with wave velocity c = (1 - We) 1/2. To obtain the corrections to this linear equation we can in- clude weakly nonlinear terms, dispersion in the electric potential and in the velocity, capillarity (see [4] for technical details), and viscous dissipation. Assuming that the zero order solution is a right-moving wave we get

OO

~h + Cqx + Aq~lx + BTlzxx + C f ~h(x,t - (~2)d(~ =- O (9) 0

Where

3( W~(D + 1).) (10) c = ~ / 1 - W ~ A = ~ 1 D

1 B = 6c (c2 + W*D2 - 3W)

2 C = V/~o~ (11)

Now 7/is of order a/d, small compared to unity but not negligible. The equation (9) is quite similar to the famous KdV equation, except

for the new term due to the viscous dissipation. This viscous-modified KdV

A. Gonzdlez, A. Castellanos/Journal of Electrostatics 40&41 (1997) 55-60 59

equation predicts that, if We < 1, all waves must disappear for long times, as the only nonconservative term is a dissipative one.

In [3] only viscous losses in the bulk were considered and a Korteweg- de Vries-Burgers equation was deduced. A combination of both effects would result in the addition of the term proportional to r/x~ in (9).

3 Energy balance

To sustain the waves in a viscous layer an external input of energy is necessary. The question is whether the electric field could be this agent or not. For a perfectly conductor liquid the answer is negative. The forces are always normal to the surface and are always conservative. The only way is to include the effects of a finite conductivity. In this case, a component of the electric field tangential to the surface may appear. The action of this component over the surface charge can produce a tangential stress, able to push the surface and maintain the liquid motion. The reason lies in an external factor, the generator. To sustain the fixed potential it must do work, which accounts for the injected and dissipated energies.

The order of magnitude of the rate of energy losses per unit time is given by

Ou I Pdi~ " P f UO~z ~=_d dx "~ Pvll2cS/2)~l/2a2d-2 - - 0 0

(12)

as in the weakly nonlinear limit, u ,.., acid. Here viscous losses in the bulk and in the boundary layer at the free surface are neglected. This is justified for thin layers.

On the other hand the input of electrical energy can be obtained from the work done for the electric field.

Oo

Pinj ~ / q~Etuo dx (13) - - O 0

Here, the surface charge, q,, is approximately the same surface charge as in the perfect conductor case q, ~_ -EoV/h.

From the charge conservation equation in this limit

Oqx Ouo 0"-~ + q~-~x - aSz,iq ~- 0 (14)

60 A. Gonz/,lez, A. Castellanos /Journal of Electrostatics 40&41 (1997) 55-60

and from the estimations

Oqx eoVoa Ouo ,., eoVoa Ot - Th 2 qs Ox Thd

we have

E~i q ~" ~T(D -- 1)ahVO

(15)

(16)

Taking into account that V A E -- 0 we get

~T . , da Vo Et -~ E , ,,, (D - 1) ~ (17)

Finally we have

Pinj "~ z°V°2(D - 1)dadC h3A (is)

Comparing these two powers we arrive at the following criterion

coco 1 W ( D - 1) aA N (19)

For example, for a layer with d = 2cm, h = l cm, a = 0.1 cm, A = 10cm, u = 10 -6 kg /m s, and E0 = 2.3 MV/m, we have We = 1/2 (still far from the linear criterion of instability, We = 1), we need a liquid of conductivity of the order of a ..~ 10 -9 to inject enough electrical energy to sustain the waves .

R e f e r e n c e s

[1] G.H. Keulegan. Journal of Research of the National Bureau of Standards 40 p. 487 (1948)

[2] A.J. Bernoff, L.P. Kwok and S. Lichter, Phys. of fluids A 1 p. 678 (1989)

[3] A. Gonz£1ez and A. Castellanos, Phys. Rev. E 49 p. 2935 (1994)

[4] G.B. Witham Linear and Nonlinear waves, John Wyley, New York (1974)