The Effect of an Electric Field on the Rotating Flows of a Thin Film Using Perturbation Technique

Abstract

The motion of a thin suspended film of an incompressible fluid under the effect of an external electric field is studied. The effect of the interfacial Maxwell stress, surface tension and inter-molecular forces is studied, in which the forces are included in the Navier-Stokes equations. The perturbation technique is used to solve for a given model. The obtained results show that, the fluid moving in a rotating patterns and the fluid particles move along the streamlines with different velocities. The free boundaries show good results in comparison with experiments data. In addition, the stability criteria are examined in the preset model.

Full text

1
The Effect of an Electric Field on the Rotating Flows of a Thin
Film Using Perturbation Technique
A. M. Morad a,∗, M. Abu-Shady a, G. I. Elsawy b
aDepartment of Mathematics and Computer Science, Faculty of Science, Menoufia
University, 32511 Egypt
bFaculty of Engineering, Modern Academy for Engineering and Technology, Cairo, Egypt
Corresponding Author: ∗dr adel morad@yahoo.com
Abstract
The motion of a thin suspended film of an incompressible fluid under the effect of an external
electric field is studied. The effect of the interfacial Maxwell stress, surface tension and inter-
molecular forces is studied, in which the forces are included in the Navier-Stokes equations. The
perturbation technique is used to solve for a given model. The obtained results show that, the
fluid moving in a rotating patterns and the fluid particles move along the streamlines with dif-
ferent velocities. The free boundaries show good results in comparison with experiments data.
In addition, the stability criteria are examined in the preset model.
1 Introduction
Many scientific and industrial problems connect to the flow of thin liquid
films (see, for instance [1–8]). Thin film technology is used extensively
in many applications including microelectronics, optics, magnetic, hard
and corrosion resistant coatings, biotechnology, micro-mechanics, laser,
medicine, etc. Also, a new mathematical formulation of the electrohydro-
dynamics (EHD) flows in a thin suspended liquid film has been introduced
in [9]. At larger scales the ascent of buoyant molten rocks (magma) below
solid ones and the spreading of lava on volcanoes are further problems of
geological researches [10–12].
In [9], authors calculated an averaged rotating flow in a thin film by
the edge effects in which the surface tension and the deviations of the free
surfaces of a film from the planes are not considered. Also, their work
showed that the jump of an electric field across a water dielectric interface
can produce (due to the electrokinetic effects) the tangential velocity of a
fluid that, in turn, can maintain the rotating flow to be steady in the film.

2
The effect of electric field generated stress (Maxwell stress), free surface
potential, surface tension, and inter-molecular van der Waals force have
been studied in many works [13–16].
The aim of this paper is to study the EHD of a thin suspended liquid
film where the flows are driven by constant external electric field applied
at the edges of the film in the free surface flow. In the present model, the
surface tension and the deviations of the free surfaces of a film from the
planes are included which are not considered in the recent works such as
Ref. [9] and references therein.
This paper is divided to five sections as follows: In the first section, the
characteristics of the physical system are presented. In the second section,
theoretical formulation of the rotating flows in a square film is studied by
the presenting of their velocity fields. In the third section, dimensionless
of given equations is obtained. The formulation of the preset model and
results are given in the fourth and fifth sections.
2 Theoretical formulation
The rotating electro-hydrodynamic flows in a suspended liquid film subject
to an applied electric filed at the edges of the film can be determined using
the Navier-Stokes conservation of mass and momentum equations. The
physical system under study consists of a rectangular thin liquid film with
a free surface. The nonlinear evolution equation for the deforming surface
will be derived by considering both the hydrodynamic stresses and the
Maxwell’s stresses with appropriate boundary conditions.
The location of the free surface is represented by
z=h(x, y, t).(2.1)
The external electric field leads to Maxwell stress σMon the model.
The total stress, σTis the combination of Maxwell’s stresses and the hy-
drodynamic stresses
σT=σM+σH(2.2)
In the absence of magnetic field, σMcan be written as

3
σM=ε(EE −0.5(E.E)I).(2.3)
So, the components of the total stress tensor are
σH
ij =µ∂vi
∂xj
+∂vj
∂xi−δijp, σM
ij =εEiEj−1
2εδijEkEk,
where, ε,p,δij,Iare the fluid permittivity, pressure, Kronecker delta and
unit matrix, respectively [17].
The total stress, σTcan therefore be expressed as
σT=µ∇v+∇vT−p+ε|E|2
2I+εEE.(2.4)
The corresponding electric field can be calculated as
E=−∇φ, (2.5)
where φis the electric potential. The first part in (2.4) is the viscid hy-
drodynamic contribution, the second and third parts arise from interfacial
electric field stresses given by the Maxwell stress tensor.
Along the shear line (z= 0), no-slip and no-penetration boundary con-
ditions (v= 0) are assumed. At the liquid free surface, z=h(x, y, t), the
normal and tangential stress balances are enforced and can be written as,
respectively,
n.σT.n=γκ. (2.6)
τ.σT.n= 0.(2.7)
with the tangent vectors
τx=(1,0, hx)
p1 + h2
x
,τy=(0,1, hy)
q1 + h2
y
,(2.8)
and the normal vector pointing outward
n=(−hx,−hy,1)
q1 + h2
x+h2
y
.(2.9)

4
where γκ is the capillary force with γis the surface tension and κ= div n
is the local interfacial curvature of the film interface.
The location of the liquid-air free surface, z=h(x, y, t) is defined by
the following condition
−1
|∇F|∂F
∂t =v.n,|∇F|="∂F
∂xi2#1/2
.(2.10)
where, F(x, y, z, t) = z−h(x, y, t) is a a single-valued function of xand
ythat vanishes on the surface and vis the fluid velocity in Cartesian
coordinates
v=v(x, t),v= (u, v, w),x≡xi= (x, y, z) (2.11)
The dimensional equations governing the EHD incompressible flow of
a Newtonian fluid are the mass and momentum conservation with the
Maxwell’s and hydrodynamic stresses contributions
ρ∂v
∂t + (v.∇)v=−∇p+νρ∆v+ div σM
=−∇p+νρ∆v+ε∆φ∇φ. (2.12)
The continuity equation
div v= 0,(2.13)
with the velocities satisfying the no-slip and no-penetration boundary con-
dition
v= 0.
The charge density is determined from summing the concentration dis-
tributions ck
ρe=F a X
k
ekck.(2.14)
Here, F a denotes Faraday’s constant and ekthe valence number of the
kth species. This charge distribution along with the external applied elec-
tric potentials generate an electric field within the liquid that can be de-
termined from the Poisson-Boltzmann relation

5
div(εE) = q. (2.15)
Here, vis the flow velocity (m/s), pis the pressure (N/m2), ρis the
liquid density (kg/m3), qis the charge density (C/m3), φis the electric
potential (V), Eis the electric field strength (V/m), ckis the molar concen-
tration for the kth component of mixture (mol/m3), ikis the density fluxes
for concentrations (mol/(m2s),νis the kinematic viscosity (m2/s), ek
are the electric charges of components (in the units of electron charge), ε
is the permittivity of the liquid ε=εrε0, where εris the relative permit-
tivity and ε0is the absolute permittivity, C/(V m)], and F a is the Faraday
constant (C/mol). In addition, we define Dkand γkas the molecular dif-
fusivity (m2/s) and electric mobilities m2/(V s)for the components of
a mixture, respectively. Using differential geometry, the surface curvature
can be written as
κ=(hxx +hyy) + hxxh2
y+hyyh2
x−2hxhyhxy
1 + h2
x+h2
y3/2
The second boundary condition at the free surface (called ”kinematic”)
is formulated by considering that a fluid particle on the free surface remains
on it. Therefore, executing the dot product v.n, the condition (2.10) can
be rewritten as
w=ht+uhx+vhyon z=h(x, y, t) (2.16)
The normal and tangential components of the viscous stress vector at the
interface, in which ni,τ(k)
iare the i–direction components of unit vectors,
the outward normal to the free surface and the two tangential to the free
surface in the (x, z) and (y, z) planes, respectively,
σn≡niσT
jinj=γκ, σ(k)
τ≡τ(k)
iσT
jinj= 0, k =x, y. (2.17)
Equation (2.16) represents a non-linear boundary condition; the free
surface his an unknown function of time and space and must be deter-
mined as part of the solution. Using the relations (2.1)–(2.9), the dynamic

6
boundary conditions (2.6), (2.7) are written as
σ(x)
τ=(σ13 +σ33hx)−hx(σ11 +σ31hx)−hy(σ12 +σ32hx)
p1 + h2
xq1 + h2
x+h2
y
(2.18)
σ(y)
τ=(σ23 +σ33hy)−hx(σ21 +σ31hy)−hy(σ22 +σ32hy)
q1 + h2
yq1 + h2
x+h2
y
(2.19)
µ[(1 −h2
x)(uz+wx)+2hx(wz−ux)−hy(uy+vx)−hxhy(vz+wy)]+
+εφxφz(1 −h2
x)−hxhyφyφz−hyφxφy= 0 (2.20)
µ[(1 −h2
y)(vz+wy)+2hy(wz−vy)−hx(uy+vx)−hxhy(uz+wx)]+
+εφyφz(1 −h2
y)−hxhyφxφz−hxφxφy= 0 (2.21)
for the tangential directions
−p+2νρ
1 + h2
x+h2
y
(wz−hx(uz+wx)−hy(vz+wy)+h2
xux+hxhy(vx+uy)+h2
yvy)+
+ε
1 + h2
x+h2
y
(h2
xφ2
x+h2
yφ2
y+h2
zφ2
z−2hxhyφxφy−2hyφyφz−2hxφxφz) =
=γhxx 1 + h2
y−2hxhyhxy +hyy 1 + h2
x
1 + h2
x+h2
y3/2(2.22)
for the normal direction
For small slopes, the tangent and normal vectors and the curvature can
be written in the following form
τx'(1,0, hx),τy'(0,1, hy),
n'(−hx,−hy,1),and κ'hxx +hyy (2.23)

7
so that the balance of the normal and the two tangential stress components
at the free interface z=h(x, y, t) may be described conveniently by
νρ (uz+wx) + ε(φxφz−hyφxφy) = 0 (2.24)
νρ (vz+wy) + ε(φyφz−hxφxφy) = 0 (2.25)
−p+ 2νρwz−2ε(hyφyφz+hxφxφz) = γ(hxx +hyy).(2.26)
3 Dimensionless equations
For the dimensionless variables, we utilize the following transformations to
render the problem dimensionless
[x, y] = a∗,[z, h] = h∗,and [t] = T∗.(3.1)
The velocities, on the other hand, scale as,
[u, v] = a∗
T∗
,and [w] = h∗
T∗
.(3.2)
where, a∗is the characteristic length in the plane of the film, h∗is the
dimensional half thicknesses of the film and T∗is the characteristic time.
The molar concentration, electric field, electric potential, charge density
and pressure are dimensionalized as, respectively,
[ck] = C∗,[E] = E∗,[φ] = E∗a∗,(3.3)
[q] = F∗C∗,and [p] = F∗C∗E∗a∗δ2,
with the quantities
Υ = F∗E∗a∗
R∗T∗
,T2
∗=ρ∗a∗
F∗C∗E∗δ2, δ2=h2
∗
a2
∗
and γ∗=γF∗C∗E∗a2
∗δ2(3.4)
Here, C∗is the molar concentration, F∗C∗is the characteristic charge
density, E∗a∗is the characteristic difference of electric potentials in the x
direction, R∗is the universal gas constant, and T∗is the absolute temper-
ature of a solution.

8
The dimensional values of the kinematic viscosity ν∗, diffusion coeffi-
cients D∗
k, and dielectric permittivity ε∗are linked to their dimensionless
counterparts as
ν=ν∗T∗
a2
∗
, Dk=D∗
kT∗
a2
∗
,and ε=ε∗E∗
a∗F∗C∗
.(3.5)
The governing equations describing the electrohydrodynamics flows of
multicomponent fluid are converted to the following dimensionless forms
by choosing the references scales listed above
δ2Du
Dt =−δ2∇0p+δ2ν∆0u+ν∂zzu−q∇0φ, (3.6)
δ4Dw
Dt =−δ2∂zp+δ4ν∆0w+δ2ν∂zzw−q∂zφ, (3.7)
div0u+∂zw= 0,(3.8)
εδ2∆0φ+∂zz φ=−δ2q, q =X
k
ekck,(3.9)
The nondimensional form of the conservation equations of mixture is
δ2Dck
Dt +δ2div0ik+∂zIk= 0,(3.10)
ik=−Dk(∇0ck+ekΥck∇0φ),(3.11)
Ik=−Dk(∂zck+ekΥck∂zφ),(3.12)
The material derivative, the gradient and horizontal Laplacian operators
in the xy plane can be expressed as, respectively,
D
Dt =∂t+u.∇0+w∂z,∇0= (∂x, ∂y),and
∆0= (∂xx, ∂yy).(3.13)
Here, v= (u, w) is the velocity field and u= (u, v) are their xand
yprojections, ikand Ikare the planar and transversal density fluxes for

9
concentrations, the parameter Υ characterizes the ratio between the trans-
ports of concentrations by an electric field and by diffusion.
The corresponding dimensionless version of the boundary conditions:
On the film boundaries:
u=v=w= 0.(3.14)
The no-leak conditions for concentrations:
Ik|z=−1= 0,Ik|z=+h= 0.(3.15)
The vanishing of the normal electric current:
∂zφ|z=−1= 0, ∂zφ|z=+h= 0.(3.16)
The kinematic condition (2.16) at the free surface z=h(x, y, t) is the
following:
w=∂th+u∂xh+v∂yh. (3.17)
Using the condition (3.16), the continuity of shear and normal stresses
(2.24)–(2.26) at the free surface z=h(x, y, t) can be written as, respec-
tively,
δ2ν(uz+δ2wx) + (φxφz−δ2hyφxφy) = 0,(3.18)
δ2ν(vz+δ2wy) + (φyφz−δ2hxφxφy) = 0,(3.19)
−δ2p+ 2δ2νwz−2(hyφzφy+hxφzφx) = γδ3(hyy +hxx).(3.20)
The dimensionless form of the Navier-Stokes equation (2.12) with the
Maxwell’s and hydrodynamic stresses contributions can be written as
δ2Du
Dt =−δ2∇0p+δ2ν∆0u+ν∂zzu+ε(∆0φ+1
δ2∂zz φ)∇0φ, (3.21)
δ4Dw
Dt =−δ2∂zp+δ4ν∆0w+δ2ν∂zzw+ε(∆0φ+1
δ2∂zz φ)∂zφ, (3.22)

10
4 Formulation of the model
4.1 The total electric potential and velocity
The electric potential field in the system is developed as a superposition
of three potential field the first one is due to the formation of the internal
potential and is represent by the potential φ1(z), the second and the third
are due to the external electric field E0which are represented as a gradient
of the potential φ2(x), φ3(y) then the total electric potential can be written
as
φ(x, y, z) = φ1(z) + φ2(x) + φ3(y).(4.23)
Upon using the classical Poisson-Boltzmann relation, the potential dis-
tribution is obtained as
φzz =−
−
−δ2
q, q
=ρ, (4.24)
where, ρis the charge density and is the dielectric permittivity, such
that the charge density follows the Boltzmann distribution. So, the total
electric potential can be obtained as
φ(x, y, z) = −x
e1−y
e2
+( 1
Sinh( hδ
Dn))(ZPSinh( zδ
Dn
)+Sinh( δ
Dn
(h−z))) (4.25)
where the Debye number is written as an equation of Debye length and the
film thickness as
Dn=λd
H
with Debye length which depends on the ionic concentration as
λd∝1
√Ck
where Ckis the ionic concentration, e1and e2are the external electric field
in xdirection and ydirection.
Now, we can define the zeta potential ratio, ZPone of the most impor-
tant functions of our study as
ZP=ηinterface
ηfilm

11
where ηinterface is the potential of the interface and ηfilm is the potential of
the film. The zeta potential ratio (ZP) at the free surface is a function of a
variety of parameters involving fluid and interface properties which intro-
duced in Ref. [17]. The solution of the model equation at the boundaries
can be obtained by using the uniform property of the thin film thickness.
Therefore, the basic equations can be formulated for initial the velocity
U(z) as follows
ν∂zz U=qφx,(4.26)
with boundary condition
ν∂zU+φxφz= 0,(4.27)
then the fluid velocity at the free surface is
U= ( 
δ2νe1
)Csch( hδ
Dn
)(−Sinh( hδ
Dn
) + Sinh((h−z)δ
Dn
) + Sinh( zδ
Dn
)ZP)
Fig. 1: The initial velocity is plotted as a function of the wave number at different values
of the surface zeta potential ratio, (ZP)
.
Figure 1shows that the initial velocity follows the same behavior under
the surface at different values of the surface zeta potential ratio (ZP) then it

12
changes on the surface, this behavior produced what is called the interfacial
stress. During the study of system stability, the interfacial stress was
detected by the interfacial polarity. The interfacial polarity with respect
to zeta potential control the system stability as it reduces the interfacial
stress, such that the system is stable when (ZP) is positive (due to the
same polarity to the surface), but when (ZP) is negative, this makes the
system more unstable (due to the opposite polarity to the surface).
4.2 Normal mode analysis
The perturbation of the variables as follows
u(x, y, z) = U(z) + ¯u(x, z, y),(4.28)
v(x, y, z) = ¯v(x, z, y),(4.29)
w(x, y, z) = ¯w(x, z, y),(4.30)
h(x, y, t) = h+¯
h(x, y).(4.31)
where the variables bare correspond to the perturbation. Using the normal
mode analysis with perturbation parameters, the stream function is given
as ˜
ψ(x, y, z, t) = ψ(z)ei k(x+y−ct),(4.32)
¯
h(x, y) = H0ei k(x+y−ct).(4.33)
where kis the wave number and cis the wave velocity.
4.3 Derivation of the model
The model equation is obtained by using derivation of the Navier-Stokes
equations, then the final equation becomes
δ2(uuxz +wuzz −δ2wtx +utz −δ2uwxx) + qxφz−qzφx+
ν(δ2uxxz +δ2uyyz +uzzz −δ4wxxx −δ4wyyx −δ2wzzx) = 0 (4.34)

13
By eliminating pressure from the Navier-Stokes equations, we obtain
δ2(ut+uux+wuz)−2(hxxφxφz+hyxφyφz)+
qφx−ν(δ2uxx +δ2uyy +uzz −2δ2wxz )
=γδ3(hxxx +hyyx) (4.35)
4.4 The Orr-Sommerfeld equations
The Orr-Sommerfeld equations are obtained by using the perturbation
technique as shown in [14], and then the model equation describing the
thin film moving with the initial velocity under the effect of an electric
field is given by
¯u=∂˜
ψ
∂z = Dψ ei k(x+y−ct)
¯w=∂˜
ψ
∂x = i k ψ(z)ei k(x+y−ct)
After substitution with ¯uand ¯w, we get the mathematical model which is
given by
(ν((D2−3
2k2δ2)2−δ4k4
4)−ikδ2((U−c)(D2−k2δ2)) −D2U)ψ(z)+
qxφz−qzφx= 0,(4.36)
the boundary conditions for the normal direction
νD(D2−4k2δ2)−ikδ2((U−c)Dψ(h)−ψ(h)DU)
= qφx+ 2k2H0(iγδ3k+(φx+φy)φz),(4.37)
for the tangential direction
νδ2(D2+k2δ2)ψ(h) + (φxφz−ikδ2H0φxφy) = 0,(4.38)
and the kinematic condition
ψ(h) + H0(U−c) = 0 (4.39)
where Dis the derivative of ψ(z).

14
The solution of the eigenvalue problem is obtained by considering an
expansion of the eigenvalue and the eigenfunction around their solutions
for
c=c0+ i kc1
ψ=ψ0+ i kψ1
The solution of the Orr-Sommerfeld equation is obtained using the per-
turbation expansion method.
The set of equations with zero order are given by
ν(D4ψ) = qzφx−qxφz(4.40)
with the boundary conditions
νD3ψ(h) = qφx,(4.41)
νδ2D2ψ(h) = −φxφz,(4.42)
νDψ(0) = 0,(4.43)
νψ(0) = 0.(4.44)
By solving the above set of equations, we get
ψ0=−z
νδ2e1
+Csch( hδ
Dn)(Cosh( hδ
Dn)−Cosh((h−z)δ
Dn)−ZP+ Cosh( zδ
Dn)ZP)Dn
δ3νe1(4.45)
c0=−h
νδ2e1H0
+Csch( hδ
Dn)(−1 + Cosh( hδ
Dn)−ZP+ Cosh( hδ
Dn)ZP)Dn
δ3νe1H0
+
νe1δ2(ZP−1) (4.46)
The set of equations with first order
ν(D4ψ) = δ2((U−c0)D2ψ0(z)−ψ0(z)D2U) (4.47)
with the boundary conditions
νD3ψ(h) = −δ2(ψ0(h)DU −(U−c0)Dψ0(h)−2γH0δk2),(4.48)

15
νδ2D2ψ(H) = δ2φxφyH0,(4.49)
νDψ(0) = 0, νψ(0) = 0.(4.50)
Then we get νψ1as an expansion such that
νψ1=A0+A1Dn+A2Dn2+A3Dn3+A4Dn4.(4.51)
We can get c1from the kinematic boundary condition as
c1=ψ1
H0
,
therefore, it is expanded as follows
c1=B0+B1Dn+B2Dn2+B3Dn3+B4Dn4.(4.52)
The solutions of c1and ψ1are calculated in appendix A.
5 Results and discussion
5.1 The stream function
After the solution of the zero and first order of the Orr-Sommerfeld equa-
tions, the stream function becomes
˜
ψ(x, y, z, t)=(ψ0+ i k ψ1)eik(x+y−c0t)+k2c1t(5.53)
Fig. 2: The stream function is plotted as a function of the wave number, kand the surface
tension, γ.

16
Figure 2shows the three dimensional behavior of the stream function
with the surface tension and the wave number. This behavior can be
appeared through the effect of the local curvature. Thus, the external
electric field appeared in which the thin suspended film rotates.
Fig. 3: The stream lines of the rotating film at t= 5 seconds (the left graph) and at
t= 15 second (the right graph).
Figure 3shows that the angular velocity of the fluid increases toward
the center and the fluid is stable.
Fig. 4: The stream lines of the rotating film at t= 10 seconds (the left graph) and at
t= 20 second (the right graph).

17
Figure 4shows that the angular velocity of the fluid increases toward
the edges of the box.
When the external electric field is connected, the angular velocity of the
fluid moves up to reach the maximum at the center as shown in Fig. 3,
then the angular velocity of the fluid increases again until it reaches the
maximum at the edges as seen in Fig. 4. At this time, the fluid is in a state
of stability and is identical to what happens in the laboratory experiment.
In Ref. [9], authors are devoted to the electrohydrodynamic of a thin
suspended liquid film where the flows are given by constant electric field at
the edges of the film using averaging method. Comparing the results of our
model and the results they reached, it found a great match in the graphics
and their physical interpretation but our model is superior in studying the
thin film motion in three dimensions and outstanding in studying of the
impact of many variables on the system stability.
5.2 The real part of the growth rate function
The real part of the growth rate function is obtained from the solution of
the Orr-Sommerfeld equations
G=k2g(ZP, Dn, , ν, δ, e1, e2) + f(γ)k4(5.54)
where k2c1is equivalent to the real part of the growth function. It has
many variables through which the study of the effect of external electric
field and natural interpretation can be achieved. The stability can be
studied at different values of variables though the wave number. In Ref.
[14], authors use Orr-Sommerfeld equation as a method of solution in 2d.
In the present model we have generalized the problem to three dimensional
spaces which led to the growth function rich in variables.
G=F0+F1Dn+F2Dn2+F3Dn3+F4Dn4
The solution of Gis calculated in the appendix B.

18
5.3 The growth rate function at different values of interface zeta
potential ratio
Fig. 5: The growth rate function is plotted as a function of the wave number at different
values of ZP,γ= 10.
Fig. 6: The growth rate function is plotted as a function of the wave number at different
values of ZP,γ= 30.
Figure 5shows that the growth rate function follows the same behavior
under the surface at different values of the interface zeta potential ratio(ZP)
then changes on the surface this behavior produced what is called the
interfacial stress. The interfacial polarity reveals the interfacial stress in
the system stability in the equation of the real part of the growth rate. The
interfacial polarity with respect to the interface zeta potential control the

19
system stability as it reduces the interfacial stress when (ZP) is negative (of
the opposite polarity to the surface) make the system unstable, but when
(ZP) is positive (of the same polarity to the surface) make the system more
stable. This gives good agreement with the figure of the initial velocity with
(ZP) as shown in Fig. 1. Figure 6shows that the growth rate function
becomes unstable at large values of the surface tension whatever taking
the values of (ZP).
5.4 The growth rate function at different values of the Debye
number
Fig. 7: The growth rate function is plotted as a function of the wave number at different
values of Dn.
Figure 7shows the effect of the Debye number, Dnon the stability of the
system, this can be explained by the thin film thickness and the Debye
length, λd. The fluid can be more stable by increasing the value of Dn
which depends on decreasing the thin film thickness and increasing λd.
Subsequently, increasing ionic concentration due to external electric field
leads the system to be unstable. The values of the Dncan be considered
as an indirect variable to study the effect of the external electric field. It
should be noted that both of thin film thickness and Debye length have
not any effect when the film thickness in the same rang of λd. We extend
the model which introduced in Ref. [14] to treat the general case at the
boundary conditions are Maxwell and hydrodynamic stresses, the local

20
curvature and the surface tension. When we solve the open problem we
find that the thin film goes to be more stable.
5.5 The growth rate function at different values of the surface
tension
Fig. 8: The growth rate function is plotted as a function of the wave number at different
values of surface tension.
Fig. 9: The growth rate function is plotted as a function of the wave number and surface
tension.
The effect of the surface tension on the stability of the system, the surface
tension increases the interfacial stability so the surface tension has a big

21
effect at large value but its effect is not noticeable at small value as shown
in Fig. 8. By using the values of variables in the mathematical model
which introduced Ref. [14], we find that the growth function follows the
same behavior in the absence of surface tension in their model. So, there is
no significant effect on the behavior of the growth function with the surface
tension. The effect of the surface tension at large values makes the thin
film unstable as seen in Fig. 9.
6 Conclusion
In this paper, the motion of thin suspended film of an incompressible
fluid under the effect of an electric field has been studied. Based on the
depth-averaged and multi-scale asymptotic expansion methods the Orr-
Sommerfeld model for this electrohydrodynamic system has been deduced.
The perturbation technique is used to solve the present model, in which
the model is generalized to 3D space. Also, we have investigated periodic
solutions of the obtained model in different cases of instability to charac-
terize the behavior of the thin suspended film. Many variables have an
effect on the stability of the system such that the Debye number, Dn, the
zeta potential, ZPand the surface tension, S. The fluid can be more stable
by increasing the value of Dnwhich depends on decreasing the thin film
thickness and increasing the Debye length, λd. The interfacial polarity
with respect to zeta potential control the system stability as it reduces the
interfacial stress when ZPis negative (due to the opposite polarity to the
surface) make the system more unstable, but when ZPis positive (due to
the same polarity to the surface) make the system more stable as seen in
[20,21]. The surface tension has a big effect at large value but its effect is
not noticeable at small values. The results indicate that the system reveals
instable behavior characterized by the perturbations on the linear region
as seen in [22–24]. Good agreement of the theory produces in the presence
of free boundaries was founded with the experiments. In addition, the
stability properties are presented graphically in which stability figures are
illustrated.

22
7 Appendices
A Appendix A
Coefficients of the terms in the expansion of (νψ1) are
A0=hz22Csch( hδ
Dn)2
2δ2ν2e2
1−z32Csch( hδ
Dn)2
6δ2ν2e2
1
+h2z22
2δ2ν2e2
1H0−hz32
6δ2ν2e2
1H0
−hk2z2γδ3H0+1
3k2z3γδ3H0+z2H0
2e1e2−hz22ZP
δ2ν2e2
1
+z32ZP
6δ2ν2e2
1−hz22ZP2
2δ2ν2e2
1
−hz22Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ2ν2e2
1
+z32Coth( hδ
Dn)Csch( hδ
Dn)ZP
3δ2ν2e2
1
+z32ZP2
6δ2ν2e2
1
+hz22Coth( hδ
Dn)2ZP2
2δ2ν2e2
1−z32Coth( hδ
Dn)2ZP2
6δ2ν2e2
1
(1.55)
A1=−z22Csch( hδ
Dn)
δ3ν2e2
1−hz22Coth( hδ
Dn)
2δ3ν2e2
1H0
+z32Coth( hδ
Dn)
6δ3ν2e2
1H0
+hz22Csch( hδ
Dn)
δ3ν2e2
1H0
−z32Csch( hδ
Dn)
6δ3ν2e2
1H0
+3z22Coth( hδ
Dn)ZP
2δ3ν2e2
1−z22Csch( hδ
Dn)ZP
2δ3ν2e2
1−hz22Coth( hδ
Dn)ZP
δ3ν2e2
1H0
+z32Coth( hδ
Dn)ZP
6δ3ν2e2
1H0
+hz22Csch( hδ
Dn)ZP
2δ3ν2e2
1H0−z32Csch( hδ
Dn)ZP
6δ3ν2e2
1H0
+z22Coth( hδ
Dn)ZP2
2δ3ν2e2
1−z22Csch( hδ
Dn)ZP2
2δ3ν2e2
1
(1.56)
A2=5z2
δ4ν2e2
1
+3z2Coth( hδ
Dn)2
δ4ν2e2
1−hz2
δ4ν2e2
1H0−z22Coth( hδ
Dn)Csch( hδ
Dn)
2δ4ν2e2
1H0
+z22Csch( hδ
Dn)2
2δ4ν2e2
1H0
+z2ZP
δ4ν2e2
1−6z2Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ4ν2e2
1
+z22Coth( hδ
Dn)2ZP
2δ4ν2e2
1H0−z22Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ4ν2e2
1H0
+z22Csch( hδ
Dn)2ZP
2δ4ν2e2
1H0
+3z2Csch( hδ
Dn)2ZP2
δ4ν2e2
1
+z22Coth( hδ
Dn)2ZP2
2δ4ν2e2
1H0−z22Coth( hδ
Dn)Csch( hδ
Dn)ZP2
2δ4ν2e2
1H0
(1.57)

23
A3=−32Coth( hδ
Dn)
δ5ν2e2
1
+22Cosh((h−z)δ
Dn)Csch( hδ
Dn)
δ5ν2e2
1
+2Csch( hδ
Dn)Sinh((h−z)δ
Dn)
δ5ν2e2
1
+2Coth( hδ
DN)Csch( hδ
Dn)Sinh((h−z)δ
Dn)
δ5ν2e2
1
+h2Coth( hδ
Dn)
δ5ν2e2
1H0
+z2Coth( hδ
Dn)
δ5ν2e2
1H0
−z2Csch( hδ
Dn)
δ5ν2e2
1H0−h2Cosh((h−z)δ
Dn)Csch( hδ
Dn)
δ5ν2e2
1H0−2Coth( hδ
Dn)ZP
δ5ν2e2
1
+32Csch( hδ
Dn)ZP
δ5ν2e2
1
+2Cosh((h−z)δ
Dn)Csch( hδ
Dn)ZP
δ5ν2e2
1−22Cosh( zδ
Dn)Csch( hδ
Dn)ZP
δ5ν2e2
1
−2Csch( hδ
Dn)2Sinh((h−z)δ
Dn)ZP
δ5ν2e2
1
+2Csch( hδ
Dn)Sinh( zδ
Dn)ZP
δ5ν2e2
1
+
+2Coth( hδ
Dn)Csch( hδ
Dn)Sinh( zδ
Dn)ZP
δ5ν2e2
1
+z2Coth( hδ
Dn)ZP
δ5ν2e2
1H0−h2Csch( hδ
Dn)ZP
δ5ν2e2
1H0
−z2Csch( hδ
Dn)ZP
δ5ν2e2
1H0
+h2Cosh( zδ
Dn)Csch( hδ
Dn)ZP
δ5ν2e2
1H0
+2Csch( hδ
Dn)ZP2
δ5ν2e2
1
−2Cosh( zδ
Dn)Csch( hδ
Dn)ZP2
δ5ν2e2
1−2Csch( hδ
Dn)2Sinh( zδ
Dn)ZP2
δ5ν2e2
1−2
δ5ν2e2
1
(1.58)
A4=−2Coth( hδ
Dn)2
δ6ν2e2
1H0
+2Coth( hδ
Dn)Csch( hδ
Dn)
δ6ν2e2
1H0
+2Cosh((h−z)δ
Dn)Coth( hδ
Dn)Csch( hδ
Dn)
δ6ν2e2
1H0
−2Cosh((h−z)δ
Dn)Csch( hδ
Dn)2
δ6ν2e2
1H0−2Coth( hδ
Dn)2ZP
δ6ν2e2
1H0
+22Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ6ν2e2
1H0
+2Cosh((h−z)δ
Dn)Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ6ν2e2
1H0−2Cosh( zδ
Dn)Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ6ν2e2
1H0
−2Csch( hδ
Dn)2ZP
δ6ν2e2
1H0−2Cosh((h−z)δ
Dn)Csch( hδ
Dn)2ZP
δ6ν2e2
1H0
+2Cosh( zδ
Dn)Csch( hδ
Dn)2ZP
δ6ν2e2
1H0
+2Coth( hδ
Dn)Csch( hδ
Dn)ZP2
δ6ν2e2
1H0−2Cosh( zδ
Dn)Coth( hδ
Dn)Csch( hδ
Dn)ZP2
δ6ν2e2
1H0

24
−2Csch( hδ
Dn)2ZP2
δ6ν2e2
1H0
+2Cosh( zδ
Dn)Csch( hδ
Dn)2ZP2
δ6ν2e2
1H0
(1.59)
and coefficients of the terms in the expansion of c1are
B0=h32Csch( hδ
Dn)2
3δ2ν3H0e2
1
+h42
3δ2ν3e2
1H02−2
3νh3k2γδ3+h2
2e1νe2−5h32ZP
6δ2ν3H0e2
1
−2h32Coth( hδ
Dn)Csch( hδ
Dn)ZP
3δ2ν3H0e2
1−h32ZP2
3δ2ν3H0e2
1
+h32Coth( hδ
Dn)2ZP2
3δ2ν3H0e2
1
(1.60)
B1=−h22Csch( hδ
Dn)
δ3ν3H0e2
1−h32Coth( hδ
Dn)
3δ3ν3e2
1H02+5h32Csch( hδ
Dn)
6δ3ν3e2
1H02+3h22Coth( hδ
Dn)ZP
2δ3ν3H0e2
1
−h22Csch( hδ
Dn)ZP
2δ3ν3H0e2
1−5h32Coth( hδ
Dn)ZP
6δ3ν3e2
1H02+h32Csch( hδ
Dn)ZP
3δ3ν3e2
1H02
+h22Coth( hδ
Dn)ZP2
2δ3ν3H0e2
1−h22Csch( hδ
Dn)ZP2
2δ3ν3H0e2
1
(1.61)
B2=5h2
δ4ν3H0e2
1
+3h2Coth( hδ
Dn)2
δ4ν3H0e2
1−h22
δ4ν3e2
1H02−h22Coth( hδ
Dn)Csch( hδ
Dn)
2δ4ν3e2
1H02
+h22Csch( hδ
Dn)2
2δ4ν3e2
1H02+h2ZP
δ4ν3H0e2
1−6h2Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ4ν3H0e2
1
+h22Coth( hδ
Dn)2ZP
2δ4ν3e2
1H02−h22Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ4ν3e2
1H02+h22Csch( hδ
Dn)2ZP
2δ4ν3e2
1H02
+3h2Csch( hδ
Dn)2ZP2
δ4ν3H0e2
1
+h22Coth( hδ
Dn)2ZP2
2δ4ν3e2
1H02−h22Coth( hδ
Dn)Csch( hδ
Dn)ZP2
2δ4ν3e2
1H02
(1.62)
B3=−2
δ5ν3H0e2
1−32Coth( hδ
Dn)
δ5ν3H0e2
1
+22Csch( hδ
Dn)
δ5ν3H0e2
1
+2h2Coth( hδ
Dn)
δ5ν3e2
1H02
−2h2Csch( hδ
Dn)
δ5ν3e2
1H02+2ZP
δ5ν3H0e2
1−22Coth( hδ
Dn)ZP
δ5ν3H0e2
1
+42Csch( hδ
Dn)ZP
δ5ν3H0e2
1
+2h2Coth( hδ
Dn)ZP
δ5ν3e2
1H02−2h2Csch( hδ
Dn)ZP
δ5ν3e2
1H02−2Coth( hδ
Dn)ZP2
δ5ν3H0e2
1
(1.63)

25
B4=−2Coth( hδ
Dn)2
δ6ν3e2
1H02+22Coth( hδ
Dn)Csch( hδ
Dn)
δ6ν3e2
1H02−2Csch( hδ
Dn)2
δ6ν3e2
1H02−22Coth( hδ
Dn)2ZP
δ6ν3e2
1H02
+42Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ6ν3e2
1H02−22Csch( hδ
Dn)2ZP
δ6ν3e2
1H02−2Coth( hδ
Dn)2ZP2
δ6ν3e2
1H02
+22Coth( hδ
Dn)Csch( hδ
Dn)ZP2
δ6ν3e2
1H02−2Csch( hδ
Dn)2ZP2
δ6ν3e2
1H02(1.64)
B Appendix B
The solution of the growth rate function gives the following coefficients in
the expansion of G
F0=h3k22Csch( hδ
Dn)2
3δ2ν3H0e2
1
+h4k22
3δ2ν3e2
1H02−2
3νh3k4γδ3+h2k2
2e1νe2−5h3k22ZP
6δ2ν3H0e2
1
−2h3k22Coth( hδ
Dn)Csch( hδ
Dn)ZP
3δ2ν3H0e2
1−h3k22ZP2
3δ2ν3H0e2
1
+h3k22Coth( hδ
Dn)2ZP2
3δ2ν3H0e2
1
(2.65)
F1=−h2k22Csch( hδ
Dn)
δ3ν3H0e2
1−h3k22Coth( hδ
Dn)
3δ3ν3e2
1H02+5h3k22Csch( hδ
Dn)
6δ3ν3e2
1H02
−h2k22Csch( hδ
Dn)ZP
2δ3ν3H0e2
1−5h3k22Coth( hδ
Dn)ZP
6δ3ν3e2
1H02+h3k22Csch( hδ
Dn)ZP
3δ3ν3e2
1H02
+h2k22Coth( hδ
Dn)ZP2
2δ3ν3H0e2
1−h2k22Csch( hδ
Dn)ZP2
2δ3ν3H0e2
1
+3h2k22Coth( hδ
Dn)ZP
2δ3ν3H0e2
1
(2.66)
F2=5hk22
δ4ν3H0e2
1
+3hk22Coth( hδ
Dn)2
δ4ν3H0e2
1−h2k22
δ4ν3e2
1H02−h2k22Coth( hδ
Dn)Csch( hδ
Dn)
2δ4ν3e2
1H02
+h2k22Csch( hδ
Dn)2
2δ4ν3e2
1H02+hk22ZP
δ4ν3H0e2
1−6hk22Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ4ν3H0e2
1
+h2k22Coth( hδ
Dn)2ZP
2δ4ν3e2
1H02−h2k22Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ4ν3e2
1H02+h2k22Csch( hδ
Dn)2ZP
2δ4ν3e2
1H02

26
+3hk22Csch( hδ
Dn)2ZP2
δ4ν3H0e2
1
+h2k22Coth( hδ
Dn)2ZP2
2δ4ν3e2
1H02−h2k22Coth( hδ
Dn)Csch( hδ
Dn)ZP2
2δ4ν3e2
1H02
(2.67)
F3=−k22
δ5ν3H0e2
1−3k22Coth( hδ
Dn)
δ5ν3H0e2
1
+2k22Csch( hδ
Dn)
δ5ν3H0e2
1
+2hk22Coth( hδ
Dn)
δ5ν3e2
1H02
−2hk22Csch( hδ
Dn)
δ5ν3e2
1H02+k22ZP
δ5ν3H0e2
1−2k22Coth( hδ
Dn)ZP
δ5ν3H0e2
1
+4k22Csch( hδ
Dn)ZP
δ5ν3H0e2
1
+2hk22Coth( hδ
Dn)ZP
δ5ν3e2
1H02−2hk22Csch( hδ
Dn)ZP
δ5ν3e2
1H02−2k2Coth( hδ
Dn)ZP2
δ5ν3H0e2
1
(2.68)
F4=−k22Coth( hδ
Dn)2
δ6ν3e2
1H02+2k22Coth( hδ
Dn)Csch( hδ
Dn)
δ6ν3e2
1H02−k22Csch( hδ
Dn)2
δ6ν3e2
1H02
+4k22Coth( hδ
Dn)Csch( hδ
Dn)ZP
δ6ν3e2
1H02−2k22Csch( hδ
Dn)2ZP
δ6ν3e2
1H02−2k2Coth( hδ
Dn)2ZP2
δ6ν3e2
1H02
+2k22Coth( hδ
Dn)Csch( hδ
Dn)ZP2
δ6ν3e2
1H02−k22Csch( hδ
Dn)2ZP2
δ6ν3e2
1H02−2k22Coth( hδ
Dn)2ZP
δ6ν3e2
1H02
(2.69)
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