Formation of Singularities on the Interface of Dielectric Liquids in - - PowerPoint PPT Presentation
Formation of Singularities on the Interface of Dielectric Liquids in a Strong Vertical Electric Field E. A. Kochurin & N. M. Zubarev Institute of Electrophysics, UD, RAS, Ekaterinburg, Russia Vertical electric field. Horizontal
Institute of Electrophysics, UD, RAS, Ekaterinburg, Russia
Vertical electric field. Horizontal electric field.
The vertical electric field has a destabilizing effect on the interface of dielectric liquids.
( , , ) z x y t η = ( , , ) z x y t η =
1,2 1,2
, ϕ Φ
The functions are the velocity and electric field potentials. We assume that both liquids are inviscid and incompressible, and the flow is irrotational (potential). As opposed to the vertical field, the horizontal electric field has a stabilizing effect on the interface.
Initial equations
1 1
= 0, = 0, ( , , ), z x y t ϕ η ΔΦ Δ <
1,2 1,2 1 2
0, , , , . z Ex z E E E ϕ Φ → → ∞ → − → ∞ = ≡ ∓ ∓
1 2 1 2 1 1 2 2
= , ( ) = ( ) , = ( , , ). z x y t z z ϕ ϕ ϕ ϕ ε η ϕ ε η ϕ η
⊥ ⊥ ⊥ ⊥
∂ ∂ ⎛ ⎞ ⎛ ⎞ − ∇ ⋅∇ − ∇ ⋅∇ ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠
( )
2 2 1 1 2 2 2 1 1 2 1 2 1 2
( ) ( ) ( ) = ( ) , ( , , ), 2 2 2 E E z x y t t t ε ε ε ρ ρ ϕ ϕ η ⎛ ⎞ ⎛ ⎞ ∂Φ ∇Φ ∂Φ ∇Φ − + − + − ∇ ⋅∇ = ⎜ ⎟ ⎜ ⎟ ∂ ∂ ⎝ ⎠ ⎝ ⎠
1 2 1 2
= ( ) = ( ), = ( , , ), z x y t t z z η η η η
⊥ ⊥ ⊥ ⊥
∂ ∂Φ ∂Φ − ∇ ⋅∇ Φ − ∇ ⋅∇ Φ ∂ ∂ ∂
1,2 1,2 1,2 1 1 2 2
0, , , , . z E z z E E ϕ ε ε Φ → → ∞ → − → ∞ = ∓ ∓
2 2
= 0, = 0, ( , , ), z x y t ϕ η ΔΦ Δ >
Conditions at infinity
Hamiltonian formalism
The equations of motion can be written in the Hamiltonian form [1,2]: where and
1 1 2 2
( , , ) = | |
z z
x y t
η η
ψ ρ φ ρ φ
= =
−
2 2 3 3 1 2 1 2 2 2 2 2 3 3 1 1 2 2 0 1 2
( ) ( ) = 2 2 ( ) ( ) . 2 2
z z z z
H d r d r E E d r d r
η η η η
ρ ρ ϕ ϕ ε ε ε ε
≤ ≥ ≤ ≥
∇Φ ∇Φ + ∇ − ∇ − − −
∫ ∫ ∫ ∫
[1]. V.E. Zakharov, Prikl. Mekh. Tekh. Fiz. 2, 86 (1968). [2]. E.A. Kuznetsov, M.D. Spector, JETP 71, 22 (1976).
= , = .
t t
H H δ δ ψ η δη δψ −
Vertical electric field; the small-angle approximation
Let us pass to dimensionless variables:
| | 1. η α
⊥
∇ << ∼
1 1 1 1 1 1
, , , . E t r t r k k E k k ρ η ψ ψ ε ε ρ η ε ε → → → →
( )
( )
( )
( )
2 2 2 2 2
1 ˆ ˆ ˆ ˆ = ( ) ( ) ( ) ( ) , 4 1
E E E
A A H k A k dxdy k A k dxdy A ψ ψ η ψ ψ η η η η η
⊥ ⊥
⎛ ⎞ + ⎛ ⎞ − − ∇ − + − ∇ ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠
∫ ∫
1 2 1 2
= ( ) ( ) A ρ ρ ρ ρ − +
kr kr 3/ 2 2
1 ( , ) ˆ ˆ , k . 2 ( ) ( )
i i
f x y kf dx dy ke e x x y y π ′ ′ ′ ′ = − = ′ ′ ⎡ ⎤ − − − ⎣ ⎦
∫∫
where is the Atwood number, and We consider that Expanding the integrand in the Hamiltonian in powers of the canonical variables up to the second- and third-order terms, we get:
1 2 1 2
= ( ) ( )
E
A ε ε ε ε − +
is its analog for the dielectric constants. Here
Vertical electric field; the small-angle approximation
= ( ) 2, = ( ) 2. f c g c ψ η ψ η + −
2 2 3 2 2 3
( ) ( ) ˆ ˆ ˆ ˆ = ( ) ( ) ( ) ( ) ( ), 4 2 ( ) ( ) ˆ ˆ ˆ ˆ = ( ) ( ) ( ) ( ) ( ). 4 2
E E t E E t
A A A A f kf kf f k f kf f f O A A A A g kg kf f k f kf f f O τ α τ α
⊥ ⊥ ⊥ ⊥ ⊥ ⊥
+ − ⎡ ⎤ ⎡ ⎤ − − ∇ + + ∇ ∇ + ⎣ ⎦ ⎣ ⎦ + + ⎡ ⎤ ⎡ ⎤ + − ∇ + + ∇ ∇ + ⎣ ⎦ ⎣ ⎦
(1 )(1 ) 1 1 = , = . 2 | | | | 1
E E E E
A A A c A A A τ − + − +
1 2 1 2 1 2 2 1
= / = / , = / = / .
E E
A A A A ρ ρ ε ε ρ ρ ε ε + ⇔ − ⇔
( ) ( ) ( ) ( )
2 3 3 2 2 2 2
2 (1 ) 2 ˆ ˆ ˆ ˆ ˆ = ( ) ( ) ( ) ( ) ( ) ( ) , 1 4 1 1 1 (1 ) ˆ ˆ ˆ = ( ) ( ) . 2 2
E E E t E E E t
A A A A A k k k k k A A A A A A k k k ψ η ψ ψ η η η η η η η ψ η ψ η ψ
⊥ ⊥ ⊥ ⊥ ⊥ ⊥
⎛ ⎞ ⎛ ⎞ + − − ∇ + − ∇ + + ∇ ∇ ⎜ ⎟ ⎜ ⎟ − − − ⎝ ⎠ ⎝ ⎠ + + ⎛ ⎞ − − + ∇ ∇ ⎜ ⎟ ⎝ ⎠
1. 2. Two special cases: The equations of motion: Let us introduce the new functions The equations take the form Here
Some pairs of immiscible dielectric liquids
3 2,
/ kg m ρ
3 1,
/ kg m ρ
A
1
1 vacuum 125 1.05 LH 1 0.98 1 1 air 1000 81 water 0.084
930 3.2 linseed oil 1100 2.7 PMPS 0.12 0.17 870 1.9 spindle oil 1100 2.7 PMPS
Upper fluid Lower fluid
The conditions or are satisfied with acceptable accuracy for these pairs. = ,
E
A A =
E
A A − Here PMPS is liquid organosilicon polymer, the polymethylphenylsiloxane; LH is liquid helium with the free surface charged by the electrons [3,4].
E
A
2
ε
1
ε
[3]. N.M. Zubarev, JETP Lett. 71, 367 (2000). [4]. N.M. Zubarev, JETP 94, 534 (2002).
Dynamics of the interface for the case
=
E
A A
2 2 2 2
ˆ ˆ = ( ) ( ) , 2 ˆ ˆ ˆ ˆ = ( ) ( ) ( ) ( ) . 2
t t
A f kf kf f A g kg kf f A k f kf f f τ τ
⊥ ⊥ ⊥ ⊥
⎡ ⎤ − − ∇ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ + − ∇ + +∇ ∇ ⎣ ⎦ ⎣ ⎦
2D geometry:
1 ( ) ˆ ˆ ˆ = , ( ) = . . , x k H H x p v dx x x x φ φ π
+∞ −∞
′ ∂ ′ − ′ ∂ −
∫
ˆ H
where is Hilbert transform. The equations take the following form:
( )
2 2
= , ˆ = 2 .
t x x t x x x x
F iF AF G iG AF AP FF τ τ + − − − +
The equations of motion (compare with Refs. [5,6]): Here where is the projection operator. These functions are analytical in the upper half-plane of the complex variable x. ˆ ˆ = , = , F Pf G Pg ˆ ˆ = (1 )/2 P iH −
[5]. E.A. Kuznetsov, M.D. Spector, and V.E. Zakharov, Phys. Rev. E 49, 1283 (1994). [6]. N.M. Zubarev, JETP 114, 2043 (1998).
=0
= ( ), = 2 ( ) , ( ) = | ,
' ' ' t
V V x x x it AV x t V x V τ τ + +
1/2 1/2 1/2
( ) | | , ( , ) | | , ( , ) ( ) .
x c c xx c c xx c c
x x x x x t x x x t t t η η η
− − −
− − ⋅ − − − − − ∼ ∼ ∼
3/2
| | .
c c
z z x x − − − ∼
( )
2
1 ˆ = ( / / , ) , ( , ) = 2 .
t ' ' ' x x x
G Q x it it t dt Q x t AF AP FF τ τ τ + − − +
∫
Dynamics of the interface for the case
=
E
A A
Weak root singularities are formed at the interface: For these singularities the curvature becomes infinite in a finite time, and the boundary remains smooth: Its solution has the form: The equation on can be also solved:
G
The equation on transforms to the complex Hopf equation:
= 2 , = .
t x x x
V iV AVV V F τ + −
F
ˆ ˆ ˆ = ( ) ( ) , ˆ = 0.
t t
f kf A k fkf f f g kg τ τ
⊥ ⊥
⎡ ⎤ − + ∇ ∇ ⎣ ⎦ +
0. g →
(1 ) = . 2 A A η ψ +
According to the second equation,
( )
( )
2
ˆ ˆ ˆ = .
t x x x x x
AH A H H η η η η ηη ⎡ ⎤ + + ⎣ ⎦
( )
2 ˆ
= 2 ,
t x x x
F iAF A P FF +
2 2 =1 =1
/2 ( , ) = , = , =1,2, , . ( ) ( )
N N j n n n j n n j
S iS dp F x t iA iA n N x p t dt p p − + + −
∑ ∑
…
Dynamics of the interface for the case
=
E
A A −
The equations of motion: The equation of interface motion in 2D geometry: As a consequence, we can put ˆ = . F Pη It can be rewritten as where This integro-differential equation can be reduced to the set of ordinary differential equations by the substitution:
In the formal limit the equation of motion is reduced to the Laplace Growth Equation [7]. It describes the formation of cusps at the interface in a finite time, or the formation of so-called “fingers” (see figures).
2 2
( ) ( , ) = , ( ) Sa t x t x a t η +
2 ( ) ( ) ln = ( ), > 0, 4 2 ( ) | | 2 ( ) ( ) arctan = ( ), < 0. 2 | |
c
AS a t AS a t A t t S a t AS A S a t a t A t t S A S ⎛ ⎞ − + − ⎜ ⎟ ⎜ ⎟ + ⎝ ⎠ ⎛ ⎞ + − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
2 2
( , ) = = ( ), 0. lim
c a
Sa x t S x S x a η π δ
→
⎛ ⎞ < ⎜ ⎟ + ⎝ ⎠
Dynamics of the interface for the case
=
E
A A −
1: N =
2 1
1, / A ρ ρ → →
Exact particular solution for where The boundary shape becomes singular at some moment
:
c
t t =
[7]. N.M. Zubarev, Phys. Fluids 18, art. no. 028103 (2006).
Horizontal electric field; the small-angle approximation
( ) ( )
2 2 2 1 2 1 1
1 ˆ ˆ = ( ) ( ) 4 ˆ ˆ ˆ ˆ ( ) . 1
E x x E x x x x x E
A H k A k dxdy A k A k k k dxdy A ψ ψ η ψ ψ η η ηη η η η η η η
⊥ − − − ⊥ ⊥
+ ⎛ ⎞ ⎡ ⎤ − − ∇ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ⎛ ⎞ ⎡ ⎤ + + − + ∇ ⋅∇ ⎜ ⎟ ⎣ ⎦ + ⎝ ⎠
∫∫ ∫∫
The equations of motion:
( )
2 1 2 2 2 2 1 2 1 1
2 (1 ) ˆ ˆ = ( ) ( ) 1 4 ˆ ˆ ˆ ˆ 2 ( ) , 1 1 (1 ) ˆ ˆ ˆ = ( ) ( ) . 2 2
E t xx E E x xx x x x x E t
A A A k k A A k k k A A A A k k k ψ η ψ ψ η ηη η ηη η η η ψ η ψ η η
− ⊥ − − − ⊥ ⊥ ⊥ ⊥ ⊥
⎛ ⎞ + ⎡ ⎤ − − ∇ ⎜ ⎟ ⎣ ⎦ + ⎝ ⎠ ⎛ ⎞ ⎡ ⎤ + + + ∇ − ∂ − ∇ ⋅∇ ⎜ ⎟ ⎣ ⎦ + ⎝ ⎠ + + ⎛ ⎞ ⎡ ⎤ − − + ∇ ∇ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠
| | 1. η α
⊥
∇ << ∼
We consider that The Hamiltonian takes the following form:
The equation for the interface evolution:
( ) ( )
2 2 2 2 2 2 2 2 2 1 1
ˆ ˆ ˆ = ( ) ( ) 2 2 ˆ ˆ ( ) ( ) .
tt xx E x t E xx tt E x t E x x t t
k k v v A A k v A A v A A v A k A k η η η η ηη ηη η η η η η η
⊥ ⊥ − − ⊥ ⊥ ⊥
⎡ ⎤ − − + − + ∇ − ∇ ⎣ ⎦ ⎡ ⎤ +∇ ∂ ∇ − ∂ ∇ ⎣ ⎦
0 =
(1 )/(1 )
E E
v A A A + + Linear approximation:
2
= .
tt xx
v η η
( , , ) = ( , ) ( , ). x y t f x v t y g x v t y η − + +
= .
E
A A
( , ) ( , ), ( , ) ( , ). x y t f x At y x y t g x At y η η , = − , = +
The equation of motion admits two exact particular solutions: Here is the velocity of linear waves. General solution of the linear wave equation: Lets us consider the special case
Dynamics of the interface
Interaction of counter-propagating waves
The approximate solution of the equations of motion has the form:
1 1 1 1 3
ˆ ˆ ˆ ( , ) ( , ) ( , ) ( ) 2 ˆ ˆ ( ) ( ). 2 A x y t f x At y g x At y k fg k f k g A f k g g k f O η α
− − ⊥ ⊥ − − ⊥ ⊥ ⊥
, = − + + − + ∇ ⋅∇ − ∇ ∇ + ∇ +
2
0.5 ( ) ( ) (1 ) f x g x x = = +
1
E
A A = =
This formula describes the nonlinear superposition of the oppositely directed waves. Interaction of the plane solitary waves:
Interaction of two 3D solitary waves
0.4 t =
2 2 2 2
( , ) exp( ), ( , ) exp( ). f x y x y g x y x y = − − = − − −
0.5
E
A A = =
Interaction of two 3D solitary waves
Interaction of two 3D solitary waves
t=-1.5 t=-0.5 t=-0.25 t=-0.1
=
E
A A =
E
A A − =
E
A A
Vertical electric field (the interface is unstable) Horizontal electric field (the interface is linearly stable)
Formation of weak (root) singularities at the interface Formation of strong singularities (cusps) at the interface Nondispersive propagation of weakly nonlinear waves The problem becomes intergrable if upper fluid moves relative to the lower one The small angle approximation is valid The small angle approximation is violated The small angle approximation is valid The equations admit nonsingular exact solutions =
E
A A −
The nonlinear dynamics of the interface between two ideal dielectric liquids in an external electric field was considered. A number of particular cases, where the evolution of the interface can be effectively studied analytically, were revealed.
Conclusion