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 electric field. = η = η ( , , ) z x y t ( , , ) z x y t The vertical electric field has As opposed to the vertical field, a destabilizing effect on the the horizontal electric field has a interface of dielectric liquids. stabilizing effect on the interface. We assume that both liquids are inviscid and incompressible, and the flow is irrotational (potential). Φ , ϕ The functions are the velocity and electric field potentials. 1,2 1,2

Initial equations ΔΦ Δ ϕ < η = 0, = 0, ( , , ), z x y t 1 1 ΔΦ Δ ϕ > η = 0, = 0, ( , , ), z x y t 2 2 ⎛ ⎞ ⎛ ⎞ ∂Φ ∇Φ ∂Φ ∇Φ ε ε − ε 2 2 ( ) ( ) ( ) ( ) ρ + − ρ + − ∇ ϕ ⋅∇ ϕ = η 1 1 2 2 0 2 1 = ( ) , ( , , ), ⎜ ⎟ ⎜ ⎟ E E z x y t ∂ ∂ 1 2 1 2 1 2 ⎝ 2 ⎠ ⎝ 2 ⎠ 2 t t ∂ η ∂Φ ∂Φ − ∇ η ⋅∇ Φ − ∇ η ⋅∇ Φ η 1 2 = ( ) = ( ), = ( , , ), z x y t ⊥ ⊥ ⊥ ⊥ ∂ ∂ ∂ 1 2 t z z ∂ ϕ ∂ ϕ ⎛ ⎞ ⎛ ⎞ ϕ ϕ ε − ∇ η ⋅∇ ϕ ε − ∇ η ⋅∇ ϕ η 1 2 = , ( ) = ( ) , = ( , , ). ⎜ ⎟ ⎜ ⎟ z x y t ⊥ ⊥ ⊥ ⊥ ∂ ∂ 1 2 1 1 2 2 ⎝ ⎠ ⎝ ⎠ z z Conditions at infinity 1. Vertical electric field: 2. Horizontal electric field: Φ → → ∞ ∓ 0, , Φ → → ∞ z ∓ 0, , z 1,2 1,2 ϕ → − → ∞ ∓ , , E z z ϕ → − → ∞ ∓ , , Ex z 1,2 1,2 1,2 ε = ε . = ≡ E E . E E E 1 1 2 2 1 2

Hamiltonian formalism The equations of motion can be written in the Hamiltonian form [1,2]: δ δ H H ψ − η = , = . δη δψ t t ψ ρ φ − ρ φ ( , , ) = | | x y t where and = η = η 1 1 z 2 2 z ∇Φ ∇Φ 2 2 ( ) ( ) ∫ ∫ ρ + ρ 3 3 1 2 = H d r d r 1 2 2 2 ≤ η ≥ η z z ∇ ϕ − ∇ ϕ − 2 2 2 2 ( ) ( ) E E ∫ ∫ − ε ε − ε ε 3 3 1 1 2 2 . d r d r 0 1 0 2 2 2 ≤ η ≥ η z z [1]. V.E. Zakharov, Prikl. Mekh. Tekh. Fiz. 2, 86 (1968). [2]. E.A. Kuznetsov, M.D. Spector, JETP 71, 22 (1976).

Vertical electric field; the small-angle approximation Let us pass to dimensionless variables: η ρ E t r ψ → ψ ε ε ρ η → → → 1 1 , , , . t r 0 1 1 ε ε k k E k k 1 0 1 ∇ η α << ∼ | | 1. 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: ) ) ( ⎛ ⎞ ( + ( ) ( ) ⎛ ⎞ 2 1 A A ∫ ∫ ˆ ˆ ˆ ˆ ψ ψ − η ψ − ∇ ψ − η η + η η − ∇ η 2 2 2 2 ⎜ ⎟ = ⎜ ⎟ ( ) ( ) E ( ) ( ) , H k A k dxdy k A k dxdy ⊥ ⊥ − ⎝ ⎠ E 4 ⎝ 1 ⎠ A E ρ − ρ ρ + ρ ε − ε ε + ε where is the Atwood number, and = ( ) ( ) = ( ) ( ) A A E 1 2 1 2 1 2 1 2 is its analog for the dielectric constants. ′ ′ 1 ( , ) f x y ∫∫ Here ˆ ′ ′ ˆ = − = i kr i kr , k . kf dx dy ke e π 3/ 2 ′ ′ 2 ⎡ − − − ⎤ 2 ( ) ( ) x x y y ⎣ ⎦

Vertical electric field; the small-angle approximation The equations of motion: ⎛ ⎞ ⎛ ⎞ + ( ) ( ) ( ) 2 3 3 2 (1 ) 2 A A A A A ψ − ˆ η ˆ ψ − ∇ ψ + ˆ η − ∇ η + ˆ η η ˆ + ∇ η ∇ η 2 2 2 2 ⎜ E ⎟ = ( ) ( ) E ( ) ( ) ⎜ E ⎟ ( ) ( ) , k k k k k ⊥ ⊥ ⊥ ⊥ − − − t ⎝ 1 ⎠ 4 1 ⎝ 1 ⎠ A A A E E E + + ( ) ⎛ ⎞ 1 (1 ) A A A ˆ ˆ ˆ η − ψ − η ψ + ∇ η ∇ ψ ⎜ ⎟ = ( ) ( ) . k k k ⊥ ⊥ t ⎝ ⎠ 2 2 ψ + η ψ − η Let us introduce the new functions = ( ) 2, = ( ) 2. f c g c The equations take the form + − ( ) ( ) A A A A ⎡ ⎤ ⎡ ⎤ ˆ ˆ ˆ ˆ τ − − ∇ + + ∇ ∇ + α 2 2 3 = E ( ) ( ) E ( ) ( ) ( ), f kf kf f k f kf f f O ⎣ ⎦ ⎣ ⎦ ⊥ ⊥ ⊥ t 4 2 + + ( ) ( ) A A A A ⎡ ⎤ ⎡ ⎤ ˆ ˆ ˆ ˆ τ + − ∇ + + ∇ ∇ + α 2 2 3 = E ( ) ( ) E ( ) ( ) ( ). g kg kf f k f kf f f O ⎣ ⎦ ⎣ ⎦ ⊥ ⊥ ⊥ t 4 2 − + − (1 )(1 ) A A 1 1 A Here τ E = , = E . c + 2 | | | | 1 A A A E E + ⇔ ρ ρ ε ε = / = / , A A 1. 1 2 1 2 E Two special cases: − ⇔ ρ ρ ε ε = / = / . A A 2. 1 2 2 1 E

Some pairs of immiscible dielectric liquids Lower Upper ε ρ ρ ε 3 3 A 1 , / 2 , / kg m kg m A fluid fluid 1 E 2 PMPS 2.7 1100 spindle oil 1.9 870 0.17 0.12 PMPS 2.7 1100 linseed oil 3.2 930 -0.084 0.084 LH 1.05 125 vacuum 1 0 -1 (formally) 1 water 81 1000 air 1 1 0.98 1 Here PMPS is liquid organosilicon polymer, the polymethylphenylsiloxane; LH is liquid helium with the free surface charged by the electrons [3,4]. − = , = The conditions or are satisfied with acceptable accuracy A A A A E E for these pairs. [3]. N.M. Zubarev, JETP Lett. 71 , 367 (2000). [4]. N.M. Zubarev, JETP 94 , 534 (2002).

Dynamics of the interface for the case = A A E A ⎡ ⎤ τ − ˆ ˆ − ∇ 2 2 = ( ) ( ) , f kf kf f ⎣ ⎦ ⊥ The equations of motion t 2 (compare with Refs. [5,6]): A ⎡ ⎤ ⎡ ⎤ ˆ ˆ ˆ ˆ τ + − ∇ + +∇ ∇ 2 2 = ( ) ( ) ( ) ( ) . g kg kf f A k f kf f f ⎣ ⎦ ⎣ ⎦ ⊥ ⊥ ⊥ t 2 +∞ ′ ∂ φ ˆ 1 ( ) x ∫ ˆ ′ H − ˆ ˆ φ where is Hilbert transform. 2D geometry: = , ( ) = . . , k H H x p v dx ′ ∂ π − x x x −∞ τ + − 2 = , F iF AF t x x The equations take the following form: ( ) ˆ τ − − + 2 = 2 . G iG AF AP FF t x x x x ˆ ˆ ˆ − ˆ Here where is the projection operator. = , = , = (1 )/2 F Pf G Pg P iH These functions are analytical in the upper half-plane of the complex variable x. [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).

Dynamics of the interface for the case = A A E F The equation on transforms to the complex Hopf equation: τ + − = 2 , = . V iV AVV V F t x x x Its solution has the form: τ τ + + ' ' ' = ( ), = 2 ( ) , ( ) = | , V V x x x it AV x t V x V 0 0 0 t =0 The equation on can be also solved: G t 1 ( ) ∫ + τ − τ − + ˆ ' ' ' 2 = ( / / , ) , ( , ) = 2 . G Q x it it t dt Q x t AF AP FF τ x x x 0 − − − ∼ 3/2 | | . Weak root singularities are formed at the interface: z z x x c c For these singularities the curvature becomes infinite in a finite time, and the boundary remains smooth: η − − ⋅ − − ∼ 1/2 ( ) | | , x x x x x c c η − − − ∼ 1/2 ( , ) | | , x t x x xx c c − η − − ∼ 1/2 ( , ) ( ) . x t t t xx c c

− Dynamics of the interface for the case = A A E ⎡ ⎤ τ − ˆ ˆ ˆ + ∇ ∇ = ( ) ( ) , f kf A k fkf f f ⎣ ⎦ ⊥ ⊥ The equations of motion: t ˆ = 0. τ + g kg t g → 0. According to the second equation, + (1 ) A η ψ As a consequence, we can put = . 2 A The equation of interface motion in 2D geometry: ( ) ⎡ ( ) ⎤ η + ˆ η ˆ η ˆ η + ηη 2 = . AH A H H ⎣ ⎦ t x x x x x ( ) 2 ˆ + P η ˆ It can be rewritten as where = . = 2 , F iAF A P FF F t x x x This integro-differential equation can be reduced to the set of ordinary differential equations by the substitution: N N S /2 iS dp ∑ ∑ − + 2 j … ( , ) = n , n = , =1,2, , . F x t iA iA n N + − 2 ( ) x p t dt ( ) p p n =1 j =1 n n j

− Dynamics of the interface for the case = A A E N = 1: Exact particular solution for ⎛ ⎞ − 2 ( ) AS a t AS + − ⎜ ⎟ ( ) ln = ( ), > 0, a t A t t S ⎜ ⎟ + 0 ( ) Sa t 4 ⎝ 2 ( ) ⎠ a t AS η ( , ) = , where x t + 2 2 ⎛ ⎞ ( ) x a t | | 2 ( ) A S a t + − ⎜ ⎟ ( ) arctan = ( ), < 0. a t A t t S ⎜ ⎟ c 2 | | ⎝ A S ⎠ = : The boundary shape becomes singular at some moment t t c ⎛ ⎞ Sa η π δ < ( , ) = = ( ), 0. ⎜ ⎟ x t S x S lim + c ⎝ 2 2 ⎠ → x a 0 a → ρ ρ → 1, / 0 In the formal limit A the equation of motion is reduced to 2 1 the Laplace Growth E quation [7] . It describes the formation of cusps at the interface in a finite time, or the formation of so-called “fingers” (see figures). [7]. N.M. Zubarev, Phys. Fluids 18 , art. no. 028103 (2006).