>K9m 2015, TOM 147, bmii. 1, cTp. 174 180
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EXACT SOLUTIONS FOR THE EVOLUTION OF A BUBBLE IN AN IDEAL LIQUID IN A UNIFORM EXTERNAL ELECTRIC FIELD
N. M. Zubareva'h t O. V. Zubarevaa
" Institute of Electrophysi.es, Russian Academy of Sciences, Ural Branch 620016, Ekaterinburg, Russia
hLebedev Physical Institute, Russian Academy of Sciences 119991, Moscow, Russia
Received May 23, 2014
The dynamics of a bubble in a dielectric liquid under the influence of a uniform external electric field are considered. It is shown that in the situation where the boundary motion is determined only by electrostatic forces, the special regime of fluid motion can be realized for which the velocity and electric field potentials are linearly related. In the two-dimensional case, the corresponding equations are reduced to an equation similar in structure to the well-known Laplacian growth equation, which, in turn, can be reduced to a finite number of ordinary differential equations. This allows us to obtain exact solutions for asymmetric bubble deformations resulting in the formation of a finite-time singularity (cusp).
DOI: 10.7868/S0044451015010162
1. INTRODUCTION
It is known that a liquid drop suspended in another liquid deforms when an external electric field is applied. In the case of two perfect dielectric fluids with no free charges at the interface, an initially spherical drop (or a gas bubble) is stretched by the electrostatic forces in the direction of the electric field fl 4]. For leaky-dielectric fluids, the drop behavior becomes more complicated; its deformation also depends on the ratio of the conductivities of the fluids (see [5,6] and the references therein).
Considerable interest is focused on the behavior of a conducting drop surrounded by an insulating fluid in an electric field [7 9]). In this situation, the electric field also stretches the drop. If the drop moves through the ambient fluid, the dynamic pressure of the flow should be taken into account. For irrotational flow, in the absence of an electric field, the drop is flattened along the direction of its motion (see, e.g., Ref. [10]). Considering a bubble instead of a drop corresponds to passing to the limit of zero density of the internal fluid. If the surface of the bubble is assumed to be perfectly conducting, then the electric field does not penetrate
E-mail: nick'fliep.uran.ru
into the interior of the bubble as well as it does not penetrate into the conducting drop.
The problem of bubble motion, as well as any other problem concerning the dynamics of a free surface or interface, is extremely difficult to solve. Therefore, it is important to find ways to simplify the corresponding equations of motion. One known approach is to consider the Stokes flow of a viscous incompressible fluid, where the stream function satisfies the bihar-monic equation (see, e.g., Refs. [8, 9, 11]). It is clear that the analysis essentially simplifies for a two-dimensional bubble [12]. The effect of an electric field on the motion of a two-dimensional bubble or drop surrounded by a viscous fluid was studied numerically in Refs. [13, 14]. In the case of two spatial dimensions, the conformal mapping technique can be effectively used for studying the bubble behavior. It allows one to reduce the original moving boundary problem to a fixed boundary problem (see the papers by Crowdy [11] and by Tan veer and Vasconcelos [15]).
In this paper, we show that if the boundary motion is determined only by electrostatic forces (capillary-forces being ignored), it is possible to use a completely-different method to simplify the equations of motion, which is applicable to studying the potential flow of an incompressible, inviscid fluid. The method is based on the consideration of a special regime of liquid 1110-
tion for which the velocity and electric field potentials are linearly dependent functions. Duo to this dependence, the number of equations required for describing the motion of the bubble boundary can be reduced by half. The reduction can be carried out in the general three-dimensional case. In the particular case of a two-dimensional bubble, where it is possible to use the conformal mapping technique, the problem reduces to an equation similar to the Laplacian growth equation (LGE), whose time-dependent exact solutions can be found analytically. Its simplest (quasistationary) solution corresponds to an elliptical bubble moving with a constant velocity along the direction of the external electric field. Other (nontrivial) solutions describe the development of instability of the steady flow. Initially small deviations from the elliptical shape of the bubble grow rapidly; the bubble boundary is deformed asymmetrically, resulting in the formation of a singularity (a cusp) in a finite time.
We note that a similar approach was previously-used in the analysis of the electrohydrodynamic instability of a charged free surface of liquid helium [10, 17] and also of an interface between two dielectric fluids [18]. A condition for instability of the plane boundary of liquid helium (the threshold value of the surface charge density) was found in Ref. [19]; it is a generalization of the instability criterion for the surface of a conducting liquid in an external electric field [20]. A functional relation between the electric and velocity potentials that underlies the analysis of boundary dynamics in Refs. [16 18] arises in the situation where electrostatic forces dominate over gravitational and capillary forces, i. e., if the system is far above the stability threshold [19].
2. INITIAL EQUATIONS
We consider the dynamics of the free boundary of a bubble in a perfect dielectric (nonconducting) fluid in the presence of an external uniform electric field. We assume that the electric field is directed along the x axis of the Cartesian coordinate system, and E is the external electric field strength. Let D(t) be the region occupied by the fluid, Dh(t) be the region corresponding to the bubble, and dD(t) be the bubble boundary. We suppose that the surface of the bubble is conductive and the charge relaxation time is small, and hence the surface can be considered equipotential in the characteristic times of electrohydrodynamic phenomena. This situation can correspond to the bubble filled with a dis-
charge plasma formed during electrical breakdown in a liquid dielectric.
We assume that the fluid is inviscid and incompressible and that the flow is irrotational (potential). The velocity and electric field potentials, <f> and p, satisfy the Laplace equations
V24> = 0, V'V = 0 in D(t). (1)
The velocity potential <f> obeys the dynamic boundary-condition (the nonstationary Bernoulli equation on a free surface),
P4>, + ^V4>)2 = Ap-£-f{V^)2 on 0D(t). (2)
Here, e0 is the vacuum permittivity, e is the dielectric constant of the fluid, p is its density, and Ap is the difference between the fluid pressure at infinity and the pressure in the bubble, Ap = pXt — pt, (the bubble is regarded as a constant-pressure region). We suppose that Ap does not vary with time and is defined by
which corresponds to volume-preserving deformations of the bubble. The last term in the right-hand side of Eq. (2) is responsible for the electrostatic pressure at the bubble boundary resulting from the interaction between free surface charges and the external electric field. We note that the surface tension effects are not taken into account in (2); this corresponds to the formal limit of a strong external electric field.
Without loss of generality, the electric field potential can be assumed to be zero at the bubble boundary:
Lp = {) on 0D(t). (3)
Formally, the equation
<p{x,y,z,t) = 0
is the equation of a free surface. Then the condition that the velocity of the bubble surface coincides with the normal velocity of the ambient liquid (the kinematic boundary condition) can be written as
<f,. + V<f-V<i) = 0 on dD(t). (4)
The system is closed by the conditions
0 —^ 0, <p -Ex, |r| oc, (5)
stating that the liquid is at rest and the electric field is uniform at an infinite distance from the bubble.
Multiplying kinematic boundary condition (4) by sjpeof, then adding and subtracting dynamic (2) and kinematic (4) boundary conditions, we find
= ^E2 on dD(t).
It follows from these expressions that it is convenient to introduce a pair of auxiliary potentials,
fl±) = <t>±f s/eoe/p.
Then the initial equations (1) (5) take the symmetric form
VV(±) = o in D(t), (6)
+ \ (V</'(±))2 = 011 dD(th
i(/,(±) =FEx \/e0e/f>, |r| oc. (8)
Equipotentiality condition (3) is then rewritten as
= on 8D(t). (9)
This form of the equations of motion turns out to be very convenient for the analysis of the bubble dynamics.
3. REDUCED EQUATIONS OF MOTION
An important feature of the system of equations (6) (9) is that they are compatible with the conditions
,J,i-) =+£.,■ /I°£ or 0<+) = _£J: /£2£.
VP VP
This proves the possibility of realizing the special regime of fluid motion for which the potentials are related by the linear expressions
<t> = ±s/eo£/p (f + Ex). (10)
As follows from them, there exists a moving coordinate system in which the liquid moves along the electric field lines. Relations (10) allow eliminating one of the potentials from the initial equations of motion, which significantly simplifies their form.
For convenience, we switch to dimensionless variables,
if fER, t t s/pR2/e0eE2, r rR,
where R. is the characteristic size of the bubble. The reduced equations of motion, written in terms of the electric field potential, have the form
VV=0 in D(t), (11)
f,. ± fr ± (Vip)2 = 0 on 8D(t), (12) f = 0 on 8D(t), (13)
f —¥ —X, |r| —¥ OG. (14)
These two systems differ only by the time direction (they are related by the replacement t —¥ —t). Without loss of generality, we can consider only the system with the upper signs in Eq. (12).
Thus, analyzing the initial equations (1) (5), we have shown that a special flow regi
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