Journal of Applied Mechanics and Technical Physics, Vol. 54, No. 2, pp. 212–217, 2013. c N.M. Zubarev, E.A. Kochurin. Original Russian Text 

THREE-DIMENSIONAL NONLINEAR WAVES AT THE INTERFACE BETWEEN DIELECTRIC FLUID IN AN EXTERNAL HORIZONTAL ELECTRIC FIELD N. M. Zubarev and E. A. Kochurin

UDC 532.59

Abstract: The nonlinear dynamics of the interface between dielectric fluids in a strong horizontal electric field is studied. It is shown that three-dimensional waves of small but finite amplitude can propagate without shape distortions either in a direction coinciding with the direction of the external field vector or in the direction opposite to this vector, along the interface of the fluids, whose density ratio is directly proportional to the ratio of their dielectric constants. For this particular case, an analytical description of the interaction of counterpropagating weakly nonlinear waves of arbitrary shape is given. Keywords: nonlinear waves, interface, dielectric fluids, electrohydrodynamics, horizontal electric field. DOI: 10.1134/S0021894413020053 INTRODUCTION An external horizontal electric field has a stabilizing effect on the free surface of a dielectric fluid and on the interface between dielectric fluids [1–5]. Features of the behavior of nonlinear waves at the interface of fluids in the presence of a horizontal field were analyzed, e.g., in [6–9]. Equations describing the evolution of the interface usually do not have analytical solutions, which is why their solution requires numerical methods. Nevertheless, studies have shown that a significant progress in the analytical study of the interface dynamics is possible sometimes [10, 11]. In particular, it was found that plane nonlinear waves can propagate without distortion on the free surface of a nonconducting fluid with great dielectric permittivity in a direction coinciding with the direction of the strong external field vector. This result was extended to the case of three-dimensional waves of arbitrary shape [12, 13]. It is shown in this paper that weakly nonlinear waves possess similar properties at the interface of two ideal nonconductive liquids whose permittivity ratio is directly proportional to the ratio of their densities. In the case of a strong external field (where electrostatic forces dominate capillary and gravitational forces), threedimensional nonlinear waves can propagate without dispersion along the external horizontal field. The interaction of counterpropagating waves of relatively low amplitude is described within Hamiltonian formalism.

1. BASIC EQUATIONS We consider the dynamics of the interface of two immiscible nonconductive fluids in an external horizontal electric field. The unperturbed interface is a plane z = 0 [the x and y axes of the Cartesian coordinate system lie

Institute of Electrophysics, Ural Branch, Russian Academy of Sciences, Ekaterinburg, 620016 Russia; [email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 54, No. 2, pp. 52–58, March–April, 2013. Original article submitted October 2, 2012. 212

c 2013 by Pleiades Publishing, Ltd. 0021-8944/13/5402-0212 

e2, r 2

z

E

y

z=0 x z =n(x,y,t) e1, r 1 Fig. 1. Diagram of the problem.

in this plane, and the z axis is directed to it along the normal (Fig. 1)]. The external electric field is directed along the x axis, and it is equal to E in the absolute value. We assume that the deviation of the interface from the plane z = 0 is given by the function η(x, y, t), i.e., the equation z = η defines the profile of the interface. We assume that both fluids are inviscid and incompressible and that their flow is irrotational (potential). The velocity potential Φ1,2 and the electric field potential ϕ1,2 in the fluids satisfy the Laplace equations ∇2 Φ1,2 = 0,

∇2 ϕ1,2 = 0.

Here the subscripts 1 and 2 correspond to the fluids in the semispaces z < 0 and z > 0. The condition of equality of the normal velocity components of the upper and lower fluids should be satisfied at the interface: ∂n Φ1 = ∂n Φ2 ,

z = η(x, y, t).

(1)

Here ∂n is the derivative in the direction of the normal to the surface z = η. For the electric field potentials, the following boundary conditions are valid: ϕ1 = ϕ2 ,

ε1 ∂n ϕ1 = ε2 ∂n ϕ2 ,

z = η(x, y, t).

(2)

Here ε1,2 are the fluid permittivities. These requirements are consistent with the conditions of continuity of the tangential component of the electric field vector and the normal component of the electric induction vector at the interface (assuming that the free surface charges are absent at the interface). At an infinite distance from the interface, the following condition is valid: Φ1,2 → 0,

ϕ1,2 → −Ex,

z → ∓∞,

i.e., the fluid velocities tend to zero, and the electric field becomes homogeneous. We introduce the auxiliary function     − ρ2 Φ2  ψ(x, y, t) = ρ1 Φ1  z=η

z=η

(ρ1,2 are the fluid densities). The equations of motion of the interface z = η can be represented in the form of Hamilton’s equations, and the functions η and ψ are canonically conjugated quantities [14, 15]: δH δH , ηt = . (3) ψt = − δη δψ The Hamiltonian H, which coincides with the total energy of the system, is expressed as     (∇Φ1 )2 3 (∇Φ2 )2 3 (∇ϕ1 )2 − E 2 3 (∇ϕ2 )2 − E 2 3 d r + ρ2 d r − ε0 ε1 d r − ε0 ε2 d r, H = ρ1 (4) 2 2 2 2 zη

zη

zη

zη

where ε0 is the electric constant. These relations are a closed system of equations describing the motion of dielectric fluids with an interface influenced by the electrostatic forces due to the presence of an external horizontal electric field. The influence of cap illary and gravitational forces is not considered, which corresponds to the strong-field limit: ε0 E 2  gσ|ρ1 − ρ2 | (σ is the surface tension, and g is the acceleration of gravity). 213

2. EQUATION OF MOTION OF THE INTERFACE We now convert to dimensionless notation (with primes):  ρ1 Φ1,2 ϕ1,2 ψ  , ψ = , ϕ1,2 = , Φ1,2 = √ Eλ ε0 ε1 Eλ Eλ ε0 ε1 ρ1

η η = , λ 

r r = , λ

E t =t λ







ε0 ε1 ρ1

(λ is the characteristic wavelength). We introduce perturbations of the dimensionless electric field potentials φ1,2 = ϕ1,2 + x , such that φ1,2 → 0 for z  → ∓∞. Applying Green’s first theorem and using the boundary conditions (1) and (2), we can represent the Hamiltonian (4) as an integral over the interface (the primes are omitted below)    φ1 ηx 1  ρ2 ε2   Φ1 − dS, (5) H= Φ 2 ∂n Φ 1 − 1 − 2 ρ1 ε1 1 + (∇⊥ η)2 S

where dS is the surface differential and ∇⊥ is the gradient in the plane {x, y}. We consider the behavior of the system in the approximation of smallness of slope angles of the interface |∇⊥ η| ∼ α 1. The integrand in the Hamiltonian (5) is expanded in powers of the canonical variables η and ψ. For harmonic functions decaying at z → ∓∞, the following relations are valid:   f1,2 

z=η

=

∞ 

(±η)n kˆn  f1,2  , n! z=0 n=0

  ∂z f1,2 

z=0

 ˆ 1,2  = ±kf

z=0

.

Here kˆ is the dimensional integral operator with a kernel whose Fourier transform is equal to the modulus of the wave vector: kˆ eikr = |k| eikr . Using these relations for the potentials φ1,2 and Φ1,2 and their derivatives, with accuracy up to cubic terms, we find   1+A ˆ − A η((kψ) ˆ 2 − (∇⊥ ψ)2 ) dx dy ψ kψ H= 4 A2E + 1 + AE

  ˆ x + ηx kˆ−1 (∇⊥ η · ∇⊥ kˆ−1 ηx )) dx dy, ηx kˆ−1 ηx + AE (ηηx2 − ηx kˆ−1 η kη

where A = (ρ1 − ρ2 )/(ρ1 + ρ2 ) is the Atwood number (0 < A < 1 for ρ1 > ρ2 ); AE = (ε1 − ε2 )/(ε1 + ε2 ) is the analog of the Atwood number for permittivities. The equations of motion of the interface corresponding to this Hamiltonian are obtained by varying (3): ψt −

+

2 A2E ˆ−1 A (1 + A)  ˆ 2 (kψ) − (∇⊥ ψ)2 k ηxx = 1 + AE 4

A3E  2 ηx + 2ηηxx + (∇⊥ kˆ−1 ηx )2 − 2kˆ−1 ∂x (η kˆ ηˆx − ∇⊥ η · ∇⊥ kˆ−1 ηx ) ; 1 + AE ηt −

A (1 + A)  ˆ ˆ 1+A ˆ kψ = − k(η kψ) + ∇⊥ (η∇⊥ ψ) . 2 2

(6)

(7)

Eliminating the function ψ from Eqs. (6) and (7), we get a (2 + 1)-dimensional equation for the evolution of the interface, in which quadratic nonlinearities are taken into account: 2 2 2 ˆ ˆ 2 ηtt − v02 ηxx = (k/2)(v 0 AE ηx − A ηt ) + k(v0 AE ηηxx − A ηηtt ) 2 ˆ−1 ηx )2 − A (∇⊥ kˆ−1 ηt )2 ] ˆ + (k/2)[v 0 AE (∇⊥ k

+ ∇⊥ [v02 AE ∂x (η∇⊥ kˆ−1 ηx ) − A ∂t (η∇⊥ kˆ−1 ηt )].

(8)  Here v0 = AE (1 + A)/(1 + AE ) is the constant having the meaning of the propagation velocity of linear waves. 214

3. INTERACTION OF OPPOSING WAVES In the linear approximation, Eq. (8) is the (1 + 1)-dimensional wave equation ηtt = v02 ηxx , whose general solution can be represented as a superposition of waves propagating along the x axis in opposite directions: η(x, y, t) = f (x − v0 t, y) + g(x + v0 t, y) (f and g are arbitrary functions of two variables). Note that the variable y is a parameter in this expression. This is due to the fact that it is not part of the wave equation in explicit form. In general, the nonlinear equation (8) cannot be solved analytically. However, it is not hard to see that a significant progress in its solution is possible with ρ1 /ρ2 = ε1 /ε2 ,

(9)

i.e., if the ratio of the fluid densities is directly proportional to the ratio of their permittivities (hence A = AE = v0 ). Indeed, in this case, Eq. (8) takes the simpler form 3 2 3 ˆ ˆ ηx − A ηt2 ) + (k/2)[A (∇⊥ kˆ−1 ηx )2 − A (∇⊥ kˆ−1 ηt )2 ] ηtt − A2 ηxx = (k/2)(A

+ ∇⊥ [A3 ∂x (η∇⊥ kˆ−1 ηx ) − A ∂t (η∇⊥ kˆ−1 ηt )] + O(α3 ).

(10)

It is easy to show that Eq. (10) has a pair of exact solutions: η(x, y, t) = f (x − A t, y),

η(x, y, t) = g(x + A t, y).

(11)

In accordance with these solutions, three-dimensional nonlinear waves of arbitrary form can propagate without distortion either in a direction coinciding with the direction of the external electric field vector, or in the direction opposite to this vector, i.e., similarly to the propagation of the waves described by the simplest linear equations ηt = ∓ A ηx . Thus, the nonlinearity of the problem does not affect the wave propagation velocity and does not lead to wave breaking. The influence of nonlinearity should be considered only in the analysis of interaction of counterpropagating waves. Consider the interaction of counterpropagating waves in the case where condition (9) is satisfied. Equation (10), up to a scale transformation, coincides with the equation derived in [12] to describe the evolution of the free surface of a fluid with high permittivity in a horizontal electric field. In the notation used in the present paper, the case considered in [12] corresponds to the limits ρ2 /ρ1 → 0 and ε2 /ε1 → 0 (i.e., A = AE = 1). Similarly to [12], we obtain the general solution of Eq. (10) in the form: ˆ g + ∇⊥ kˆ−1 f · ∇⊥ kˆ−1 g) η(x, y, t) = f (x − A t, y) + g(x + A t, y) − (A /2)k(f − (A /2)∇⊥ (f ∇⊥ kˆ−1 g + g∇⊥ kˆ−1 f ) + O(α3 ).

(12)

This expression describes the nonlinear superposition of counterpropagating three-dimensional waves (11). It follows from (12) that during interaction the waves remain structurally stable. Figures 2 and 3 show the shape of the interface at the time of the meeting of the three-dimensional spatial localized waves described by the equations f (x, y) = exp (−x2 − y 2 ),

g(x, y) = − exp (−x2 − y 2 ).

(13)

It follows from Fig. 2 that as solitary waves approach each other, the influence of nonlinearity increases, and at the time of their meeting (t = 0), it determines the shape of the interface (in the linear approximation, the interface is flat at this moment). The influence of nonlinearity is also responsible for the lack of symmetry in the line x = 0 in Fig. 3. 215

z 1.0

y 1

1.0

0.5

0.5

2 3

0

0

_0.5

_0.5

_0.3 _0.2

_0.4 0.4

_0.1

0.3 0.2

_1.0

_1.0

0

_4

_3

_2

_1

0

1

2

3

4 x

_1.5 _2

Fig. 2.

_1

0.1

0

1

2 x

Fig. 3.

Fig. 2. Evolution of the three-dimensional counterpropagating waves described by Eqs. (12) and (13) for A = AE = 0.5, y = 0 and t = −6 (1), −3 (2), and 0 (3); dashed and solid curves refer to the linear and nonlinear analysis results, respectively. Fig. 3. Isolines of the function η(x, y, t) for the counterpropagating waves described by Eqs. (12) and (13) at time t = 0.4 for A = AE = 0.5.

CONCLUSIONS In this work, it is shown, using the model of three-dimensional electrohydrodynamics, that waves of small but finite amplitude can propagate without shape distortion along a direction coinciding with the direction of an external horizontal electric field vector at the interface of two ideal nonconductive fluids. This situation occurs for fluids whose density ratio is directly proportional to the ratio of the permittivities in the case of a strong external field, where the effect of electrostatic force is predominant. The solution of the equations of motion describing the interaction of nonlinear counterpropagating waves is obtained. In accordance with this solution, the interaction of counterpropagating solitary waves of arbitrary geometry does not influence their shape and phase. It should be noted that condition (9) is satisfied with reasonable accuracy for pairs of immiscible fluids such as polymethylphenylsiloxane, an organosilicon polymer (ρ ≈ 1100 kg/m3 and ε ≈ 2.7), mineral oil (ρ ≈ 880 kg/m3 and ε ≈ 2.2), linseed oil (ρ ≈ 930 kg/m3 and ε ≈ 3.2), and Freon (ρ ≈ 570 kg/m3 and ε ≈ 2.0). For the first of these pairs, A ≈ 0.10 and AE ≈ 0.11, and for the second pair, A ≈ 0.24 and AE ≈ 0.23, i.e., A ≈ AE in both cases. It should be noted that the above results are suitable (up to the replacement of the electric field E by the magnetic field H and the permittivity ε1,2 by the magnetic permeability μ1,2 ) to describing the propagation of waves at the interface between ferromagnetic fluids in a horizontal magnetic field. In this case, condition (9) takes the form ρ1 /ρ2 = μ1 /μ2 .

(14)

As is known, ferromagnetic fluids are colloidal systems consisting of magnetic nanoparticles suspended in carrier fluids. As a rule, the relative concentration of the particles is a few percent. When the required number of particles is added, condition (14) can be implemented for nanoparticles of virtually any material. This work was supported by the Russian Foundation for Basic Research (Grant No. 13-08-96010-r ural), Ural Branch of the Russian Academy of Sciences within the framework of the program of the Presidium of Russian Academy of Sciences “Fundamental Problems of Nonlinear Dynamics in Mathematical and Physical Sciences” (Project No. P-12-2-1023) and the program for support of young scientists and graduate students (Project No. 132-CB-380), and the Dinastiya non-profit foundation. 216

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