The document analyzes the stability of a liquid film between two fluids in an electric field. It presents the governing equations for electric potential, surface charge density, and Stokes flow. Dimensionless parameters are defined through non-dimensionalization. The base state and linear stability analysis via normal mode decomposition are described. Results show the growth rate of instabilities peaks at a particular wavenumber and streamlines/pressure contours for different systems and wavenumbers.

1. Stability Analysis of Liquid Film in
an Electric Field
Yixuan Jia
Nov. 30th 2016

2. Problem Definition
h
h
✏, , µ
✏, , µ
¯✏, ¯, ¯µ
S1, q1
S2, q2
fluid #1
fluid #1
', E, u, pH
fluid
#2
', E, u, pH
¯', ¯E, ¯u, ¯pH
E0 = E0 · ˆz
Governing equations:
a) Electric potential
r2
' = 0
r2
¯' = 0
'(x) = ¯'(x) for x 2 S
'(x) ! 'e(x) = E0 · x as | x |! 1
@tq + n ·
⇥
[ E]
⇤
+ rs · (qu) = 0
BCs:
q(x) = n ·
⇥
[✏E]
⇤
= n · (¯✏¯E ✏E), E = r'
⇥
[ E]
⇤
= ¯ ¯E E, rs ⌘ (I nn) · r
where
r · u = 0 , µr2
u + rpH
= 0
r · ¯u = 0 , ¯µr2
¯u + r¯pH
= 0
b) Stokes flow
u = ¯u for x 2 S
u ! 0 as | x |! 1
⇥
[fE
]
⇤
+
⇥
[fH
]
⇤
= (rs · n)n for x 2 S
BCs:
⇥
[fE
]
⇤
= n ·
⇥
[✏(EE
1
2
E2
I)]
⇤
⇥
[fH
]
⇤
= n ·
⇥
[ pH
I + µ(ru + ruT
)]
⇤
where

3. Non-dimensionalization
h
h
✏, , µ
✏, , µ
¯✏, ¯, ¯µ
S1, q1
S2, q2
fluid #1
fluid #1
', E, u, pH
fluid
#2
', E, u, pH
¯', ¯E, ¯u, ¯pH
E0 = E0 · ˆz
a) Characteristic scale:
• length: h
• time: ⌧MW = ¯✏+2✏
¯+2
• pressure: ✏E2
0
• velocity: h
⌧MW
• electric potential: E0h
b) Dimensionless numbers:
• Material properties
Q =
¯✏
✏
, R =
¯
, =
¯µ
µ
• Electric capillary number CaE, Mason number Ma:
CaE =
h✏E2
0
, Ma =
2µ
✏⌧MW E2
0
• Electric Reynolds number ReE
ReE =
1
Ma
2(1 + 2R)
R(Q + 2)

4. Non-dimensionalization
h
h
✏, , µ
✏, , µ
¯✏, ¯, ¯µ
S1, q1
S2, q2
fluid #1
fluid #1
', E, u, pH
fluid
#2
', E, u, pH
¯', ¯E, ¯u, ¯pH
E0 = E0 · ˆz
• Surface charge density BCs
@t
⇥
n · (Q¯E E)
⇤
+
2
MaReE
n · (
1
R
¯E E)
+ rs ·
⇥
n · (Q¯E E)u
⇤
= 0
• Dynamic BCs
n ·
⇥
Q(¯E¯E
1
2
¯E2
I) (EE
1
2
E2
I)
⇤
+ n ·
⇥
¯pH
I
+
Ma
2
(r¯u + r¯uT
) + pH
I
Ma
2
(ru + ruT
)
⇤
=
1
CaE
(rs · n)n
c) Dimensionless governing equations:
• Electric potential
r2
' = 0
r2
¯' = 0
• Stokes flow
r · u = 0
r2
u +
2
Ma
rpH
= 0
r · ¯u = 0
r2
¯u +
2
Ma
r¯pH
= 0

5. Base Flow
h
h
✏, , µ
✏, , µ
¯✏, ¯, ¯µ
S1, q1
S2, q2
fluid #1
fluid #1
', E, u, pH
fluid
#2
', E, u, pH
¯', ¯E, ¯u, ¯pH
E0 = E0 · ˆz
1. Velocity
¯U = U = 0
2. Hydrodynamic pressure
¯PH
+ PH
=
1
2
(QR2
1)
3. Electric potential
=
8
><
>:
z + (1 R) z > 1
Rz 1 < z < 1
z (1 R) z < 1
Electric field
E = (0 0 1), ¯E = (0 0 R)
Surface charge density
Q1 = 1 RQ, Q2 = 1 + RQ
n1 = (0 0 1)
n2 = (0 0 1)

6. Perturbation Linearization
h
h
✏, , µ
✏, , µ
¯✏, ¯, ¯µ
S1, q1
S2, q2
fluid #1
fluid #1
', E, u, pH
fluid
#2
', E, u, pH
¯', ¯E, ¯u, ¯pH
E0 = E0 · ˆz
• Hydrodynamic pressure
pH
= PH
+ p0H
, ¯pH
= ¯PH
+ ¯p0H
• Electric potential
' = + '0
(x), ¯' = ¯ + ¯'0
(x)
• Electric field
E = (
@'
@x
@'
@y
1
@'
@z
), ¯E = (
@ ¯'
@x
@ ¯'
@y
R
@ ¯'
@z
)
Define the perturbation ⇣1 atz = 1 and ⇣2 at z = 1.
• Surface normal vector
n1 = (
@⇣1
@x
@⇣1
@y
1)
n2 = (
@⇣2
@x
@⇣2
@y
1)
• Velocity
u = (v0
x v0
y v0
z)
¯u = (¯v0
x ¯v0
y ¯v0
z)
n1
n2
⇣2
⇣1

7. Perturbation Linearization
• Surface charge density BCs
@t( Q
@ ¯'0
@z
+
@'0
@z
) +
2
MaReE
(
1
R
@ ¯'0
@z
+
@'0
@z
)+
(QR 1)(
@v0
x
@x
+
@v0
y
@y
) = 0, at z = ±1
• Dynamic BCs
z = 1
(
QR@ ¯'0
@x + @'0
@x + Ma
2 (
@¯v0
z
@x +
@¯v0
x
@z ) Ma
2 (
@v0
z
@x +
@v0
x
@z ) + (QR2
1)@⇣1
@x = 0
QR@ ¯'0
@z + @'0
@z ¯p0H
+ p0H
+ Ma
@¯v0
z
@z Ma
@v0
z
@z + 1
CaE
(@2
⇣1
@x2 + @2
⇣1
@y2 ) = 0
z = 1
(
QR@ ¯'0
@x + @'0
@x + Ma
2 (
@¯v0
z
@x +
@¯v0
x
@z ) Ma
2 (
@v0
z
@x +
@v0
x
@z ) + (QR2
1)@⇣2
@x = 0
QR@ ¯'0
@z + @'0
@z ¯p0H
+ p0H
+ Ma
@¯v0
z
@z Ma
@v0
z
@z
1
CaE
(@2
⇣2
@x2 + @2
⇣2
@y2 ) = 0
• Kinematic BCs
@⇣1
@t
v0
z = 0, at z = 1,
@⇣2
@t
v0
z = 0, at z = 1
• No jump condition at the interfaces – electric potential
¯'0
'0
+ (1 R)⇣1 = 0, at z = 1, ¯'0
'0
(1 R)⇣2 = 0, at z = 1
• No jump condition at the interfaces – velocity
¯u0
= u0
, at z = ±1

8. Normal Mode Analysis
'0
= ˆ'(z)eikx+st
, ¯'0
= ˆ¯'(z)eikx+st
u0
=
⇥
ˆvx(z) ˆvy(z) ˆvz(z)
⇤
eikx+st
, ¯u0
=
⇥
ˆ¯vx(z) ˆ¯vy(z) ˆ¯vz(z)
⇤
eikx+st
p0H
= ˆpH
(z)eikx+st
, ¯p0H
= ˆ¯pH
(z)eikx+st
⇣1 = ˆ⇣1eikx+st
, ⇣2 = ˆ⇣2eikx+st
Electric potential profile
ˆ' =
8
><
>:
A1e kz
z > 1
A2e kz
+ B2ekz
1 < z < 1
B3ekz
z < 1
Velocity profile
ˆvz =
8
><
>:
(C1 + D1z)e kz
z > 1
(C2 + D2z)e kz
+ (E2 + F2z)ekz
1 < z < 1
(E3 + F3z)ekz
z < 1
Hydraulic pressure profile
ˆpH
=
8
><
>:
MaD1e kz
z > 1
MaD2e kz
+ MaF2ekz
1 < z < 1
MaF3ekz
z < 1
k2
ˆ' +
d2
ˆ'
dz2
= 0
d4
ˆvz
dz4
2k2 d2
ˆvz
dz2
+ k4
ˆvz = 0

9. Results — Growth Rate
k
0 1 2 3 4 5 6
sr
0
500
1000
1500
2000
2500
3000
3500
4000
Biggest growth rate sr
(k)
system 1a
system 1b
system 1c
system 2

10. Results — Streamlines
x
-25 -20 -15 -10 -5 0 5 10 15 20 25
z
-6
-4
-2
0
2
4
6
Streamlines z = [-5,5], System 1a and k = 0.3

11. Results — Streamlines
x
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
z
-6
-4
-2
0
2
4
6
Streamlines z = [-5,5], System 1a and k = 3

12. Results — Streamlines
x
-25 -20 -15 -10 -5 0 5 10 15 20 25
z
-6
-4
-2
0
2
4
6
Streamlines z = [-5,5], System 2 and k = 0.3

13. Results — Streamlines
x
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
z
-6
-4
-2
0
2
4
6
Streamlines z = [-5,5], System 2 and k = 3

14. Results — Hydraulic Pressure Contour

15. Thanks!