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JOURNAL OF APPLIED SCIENCES RESEARCH
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2013 December; 9(13): pages 6316-6327.
Published Online: 15 January 2014.
Research Article
Analytical Solution Of The Electroosmotic Flow In Nano-Channels With Different
Cross-Sections
Mehran Khaki Jamei and Morteza Abbasi
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran.
Received: 12 November 2013; Revised: 14 December, 2013; Accepted: 20 December 2013.
© 2013
AENSI PUBLISHER All rights reserved
ABSTRACT
This paper studies an analytical solution of an electroosmotic flow through nano-channels with different cross sections. The
governing equations are derived from Navier-Stokes, Boltzmann distributions and Poisson theory. The smallest physical dimension of the
channel is greater than the electric double layer (EDL). Coupled governing equations are transformed into a partial differential equation by
the Debye-Huckel approximation. This equation is converted to an ordinary differential equation by a cosine transform. Maple software is
used to solve this equation and the unknown coefficients are obtained by applying boundary values of the hyper-ellipse function. Solutions
are compared with the results available in the references for rectangular and circular sections. Results show very good agreement and high
accuracy in comparison with other method. Results reveal that if the ratio of the Debye length to the channel height (ε = λ/h) equals to
0.04, the velocity profiles and the electric potential follow the trend of plug flow distribution. In other words, potential and velocity change
rapidly near the walls are a little further from the walls uniform. This causes considerable shear stresses occur close the walls. On the other
hand, reducing the channel dimensions, change the velocity profile from- plug flow to parabolic form resulting in shear stress decrease
close the wall.
Key words: electro-osmotic flow, analytical solutions, Poisson - Boltzmann, Hypralyps, nano- channels.
INTRODUCTION
In the present century, the swift development of
science, has originated many changes in human life.
In this regard, one of the effective ways is studying
various problems in micro and nano-scale. Micro
industrial in the fluid field had been employed by the
Canon company in 1978 at first for fabricating ink jet
printers. With the extension of this industry in
mechanical and electrical mechanisms, using of the
terms micro-electro-mechanics and nano-electromechanics have become prevalent [1]. Using of
MEMEs is seen pervasively in many types of micro
valves, micro channels, micro pumps, small heating
systems [2]. All of these equipments are called
microfluidics. Micro pumps are one of the important
systems in this field that have lots of applications
[3,4,5]. Electric actuation is one of the important
actuations that because of low energy consumption
and simple control, has a lot of applications. In this
model, various mixtures like mixture of water and
sodium chloride are used for fluid flow and positive
and negative ions in the electrolyte with actuation of
the magnetic field
cause the fluid to flow.
Researchers have considered this sort of problems for
a long time. The first work has been done by
Burgreen and Nakache [6]. They assumed that the
density of ions follows the Boltzmann distribution.
Later, Rice and Whitehead [7] presented an analysis
of electro-osmotic flow in a capillary. Barcilon et al,
employed nonlinear perturbation and numerical
methods for solving Poisson-Nernst-Plank equations
in channels with ion flow problems. In this field
some authors have done other similar works [8,9].
Governing differential equations are partial and
coupled. So the majority of these investigations is
numerically or experimentally. Analytical solution of
the one-dimensional electro-osmotic flow by
employing Debye-Huckel approximation [11] is done
by Abhishek and Jensen [12]. They used onedimensional Poisson-Boltzmann equation in the
direction normal to the plane for potential
distribution. Analytical solution to the above
conditions has been done by other researchers
[13,14,15,16].
In this study, 3-dimensional Boltzmann
distribution is used for electric potential distribution.
For non-dimensionalizing the governing equations,
parameters were selected in such a way that the
dimensionless form of the electric charge and
momentum equations are similar. And also in a
special case, the boundary conditions are similar. In
this situation the solution of the electric charge
equation and the velocity are similar and just one
equation need to be solved. The above differential
equations are transformed into a partial differential
Corresponding Author: Mehran Khaki Jamei, Department of Mechanical Engineering, Sari Branch, Islamic Azad
University, Sari, Iran.
E-mail: [email protected]; Phone: +98-911-154-2593; fax: +98-151-222-4342)
6317
Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
equation by the Debye-Huckel approximation. And
by a cosine transform, an ordinary differential
equation is obtained. The general solution of this
equation can be achieved by Maple software. In this
solution the number of the unknown coefficients
depends on the number of the cosine transform
approximation terms. The functions for unknown
coefficients are obtained by applying boundary
conditions to the general solution in the specified
points. And the electric charge distribution and then
ion concentration and shear stress are calculated by
inverse cosine transform.
Mathematical models:
The aim of this paper is to study electro-osmotic
flow in channels with different cross sections
analytically. Studied channels are obtained by the
hyper-ellipse function as follows.
Hyper-ellipse function:
In the Cartesian coordinate system, the hyperellipse function is defined as follows:
x
a
n

y
b
n
 1.
Iovanich et al. used this function for solving
Poisson's equation and creating different geometries
in 1997. Actually, they studied fully developed flow
in a channel and two dimensional conduction and
their non-dimensioned equations by definition of the
hydraulic diameter and lastly, they presented their
results by similarity solution. If we use equation 1 for
creating a quarter of the fluid flow section, we can
eliminate the absolute value sing and then hyperellipse function can be written as follows:
 x n  y n
      1.
 a 
 b 
(2)
b
, equation 2 and
By definition of ε =
a
substituting x = r.cos(θ ) and y = r.sin(θ ) ,
hyper-ellipse function is defined in a cylindrical
coordinate system as follows:
a
r 
.
1

n

n  sin()   n
 
  cos()   

   


(3)
(1)
a: n=0.5
b: n=1
c: n=2
d: n=4
d: n=4
f: n=400
Fig. 1: Created sections by the hyper-ellipse function with different values of n
6318
Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
In Eq. 3 a is half of the large diameter, b is half
of the small diameter, ε is the ratio of the small
diameter to the large diameter and n is a constant. If
quantities of n change, various sections will be
created. For creating a star, a rhombus, a ellipse,a
rounded edges square and a rectangle, n values must
be 0.5, 1, 2, 4 and infinity respectively. Figure 1
shows the different created sections by hyper-ellipse
function. If n is bigger than 2 in the above function,
the rounded edge squares will be created. By
increasing n to infinity, the rounded edges turn to the
sharp edges and then a rectangular section creates.
Equations in a cylindrical coordinate system are
used for non-rectangular sections that include:
Poisson, Navier-Stokes and ion conservation of mass
equations. In the third chapter the dimensionless
form of the electric potential distribution equation in
cylindrical coordinate system is defined as follows:
FCR2
2  
z X .
 i i i
e 0
ε1 =
(4)
R
 1 , Eq. 4 can be achieved:
L
1 2 
FCR2
 2  1 



z X .
 i i i
r 2 r r
r 2 2
e 0
(5)
have
2 
1 
1 2 



  z X .
i i
r r
r 2
r 2 2
2 i


0
 0

r


 0,
r 0

can be written with respect to the φ . According to
z
the Boltzmann ion distribution, X  X 0 .e i and
i
i
by substituting it in Eq. 6 we will have:
z 
1 2 
2 1 



  X 0e i
i
r r
r 2
r 2 2
2 i
(7)

4
or
2 1 
1  2



  (X 0 e  X 0 e )


2
r

r
2
2
r
r 
2
(8)
By using of Taylor expansion for small
quantities of ϕ and arrangement Eq. 8 turns to:
1 2
2 1 




(X 0  X 0 )  (X 0  X 0 ))




2
r

r
2
2
r
r 
2


(9)
C
λ
Substituting β =
and ε =
in Eq. 5 we
I
R
(r0 , )  0
xi
Eq. 7 is simplified for monovalent electrolytes
such as mixture of sodium chloride and water as
follows:
Governing Equations:
Since
By using the Debye-Huckel approximation,
(6)
By investigating the sections in Fig. 1, it is
shown that in all of sections, the section shape is
symmetric with respect to θ , at θ = 0 and
θ=
π
4
. It means that all variables do not vary with
θ , at these two angles and the derivative
with respect to θ is zero. On the other hand, because
respect to
of the symmetry in the center of the section, the
velocity values and electric potential are maximum.
So the boundary conditions can be written as
follows:
(0, )  max
0
(10)
Eq. 9 and the boundary conditions are similar in the potential distribution function and the velocity.
Analytical solution and Conclusion:
Eq.9 is partial in the r and θ directions and since
In the analytical solution of Eq. 9, the
the boundary conditions are derivative type so the
combination of cosine method and numeric
Eq. 9 can be converted to an ordinary differential
approximation in boundaries by using of Hiralips
equation by a cosine transform. Cosine transform in
function are employed.
the direction to the q, is as follows:
Cosine Transform Of Eq.9:
6319
Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
  2  1 
1 2 


 Cm
C m 
2
r r
r 2  2 
 r
 

   (X 0  X0 )  (X 0  X0 ))  
 2

(11)
Which
C
 2 
 
 C  (r ) ,
m  2 
m
 r 
C
 1  
1

  C  (r )
m  r r 
r m
(12)
And

2 
 4m 2
 
8 
 1  
C (r ) 
u (r , 0)  (1)n  1u (r , ) 




 
 4 
m 2 2
m
r 2 
r2
 r  
2
 1 2 
4m 

 C m 
C m (r )
2
2 
r2
 r  
  0
  0
 
(X   X0 ) m  0
0
Cm 
(X   X )    2
 2
 
m 0
 0
  0

 0
Cm 
(X   X0 )  
(X   X0 )C m (r )
 2

2
C
(12)
Substituting Eq. (12) in Eq. (11) two ordinary differential equations will be as follows:
1


C  (r )  C   (X 0  X 0 )C (r )  (X 0  X 0 )  0
0
0
0




r
2
2
(13)
1
1

C  (r )  C   (4m )2 C (r )  (X 0  X 0 )C (r )  0 m  1,2, 3,...
m
m

 m
r m
r2
2
(14)
The general solution of Eqs. (13) and (14) are obtained respectively as follows:
1
C 0 (r )  0.0415  A .Cosh(
62222r ) (15)
0
20
1
C (r )  A .Cosh(
62222 r )
m
m
20
(16)
The general solution of electric potential in polar coordinates is obtained by applying the inverse cosine
transform on Eqs. (15) and (16) so we have:
φ (r ,θ ) =
0.0415 + A 0Cosh (
K
1
1
6222r ) + 2∑ A m .Cosh (
6222r ).Cos (4 mθ )
20
20
m =1
(17)
The unknown coefficients, Am, can be achieved by applying boundary conditions at certain points between
θ=0 and θ=π/4.
Evaluation of flow in rectangular (n = 400):
6320
Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
If n = 400 is selected, a rectangular cross section will be created. By choosing K = 5, the coefficients Am is
obtained as follows:
A0  -1.371107624  10-7 ,
A1  -7.903530548  10-8 ,
A2  -1.157685701  10-8 ,
A3  6.741212138  10-11,
A4  1.482504049  10-10 ,
A5  -8.042757168  10-11.
(18)
Substituting equation (18) in equation (17) the electric potential in polar coordinates to rectangular crosssections will be as follows:
1

5
(r ,  )  u(r ,  )  0.0415  0.137110  10 cosh 
62222r 
 20

 1

 1



6
7
0.1580710 cosh
62222r  cos 4 0.23153710 cosh
62222r  cos 8 
 20

 20





1
1
62222r  cos 16 
0.134824109 cosh
62222r  cos 12  0.296501109 cosh
 20
 20




1

0.160855  10 9 cosh 
62222r  cos  20 
 20

.
Using Boltzmann ion distribution functions, X
X


(19)
and X can be obtained from the following relations:

5
0 (r , )
cosh
 X e
 0.000275  exp(( 0.0415  0.137110  10

1

 20
62222r







7
1
1
 20 62222r  cos  4   0.231537  10 cosh  20 62222r  cos  8  
1

1

9
9
cosh 
0.134824  10
 62222r  cos  12   0.296501  10 cosh  62222r  cos  16 
20
20


1
9
cosh 
0.160855  10
 62222r  cos  20 )),
20
(20)
1

 62222r 
 20

1

1

6
7
0.15807  10
cosh 
 62222r  cos  4   0.231537  10 cosh  62222r  cos  8 
20
20
1

1

9
9
0.134824  10
cosh 
 62222r  cos  12   0.296501  10 cosh  62222r  cos  16 
20
20


1
9
0.160855  10
cosh 
 62222r  cos  20 ).
20
(21)
0.15807  10
6
cosh
0  (r ,  )
X   X e
 0.000254  exp( 0.0415  0.137110  10
The accuracy achieved will depend on the
number of terms n. In order to obtain the appropriate
number of sentences, the relative error of the electric
potential and velocity has been drawn versus the
number of sentences. Figure 2, shows that when k =
3, the relative error of five percent level. And taking
5
cosh
six of the first sentence (k = 5), the error is less than
one percent. To Evaluation of validity of the above
solution, the change of electric potential and ion
fraction of anion and cation, in a two-dimensional
has been drawn and compared with numerical
method.
6321
Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
Fig. 2: The variation of the relative error in terms of expanding the cosine function of velocity and electric
potential in the channel with dimensions of 20 × 20 nm (n = 400).
For closer examination, these deviation variables are compared to the numerical procedure that is outlined
below.
Fig. 3: Comparison of velocity profiles and electric potential obtained from the combination of in a cylindrical
coordinate system with finite difference method [16], in the channel with dimensions of 20 × 20 nm.
Fig. 4: Comparison of cation mole fraction obtained from the combination of in a cylindrical coordinate system
with finite difference method [16], in the channel with dimensions of 20 × 20 nm.
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Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
A figure 3, 4 and 5, represents the comparison of a combined approach with the present method and a
numerical solution at the reference [16]. Results show very good agreement and high accuracy in comparison
with other techniques.
Fig. 5: Comparison of anion mole fraction obtained from the combination techniquein a cylindrical coordinate
system with finite difference method [16], in the channel with dimensions of 20 × 20 nm.
Fig. 6: Distribution of velocity and electric potential obtained from the proposed combined method in the
channel with dimensions of 20 × 20 using a cylindrical coordinate system, the boundary Hypralyps,
cosine transform.
Fig. 7: Distribution of cation and anion mole fractions of electric potential obtained from the proposed hybrid
method using a cylindrical coordinate system, the boundary Hypralyps, cosine transform channel
dimensions 20 x 20.
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Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
To Evaluation of more closely the behaviour of
fluid, three-dimensional form of velocity profiles and
mole fraction of cation and anion, respectively, is
depicted in Figures 6 and 7. A value obtained with
the reference [16] is also quite consistent.
θ, is constant, the coefficients Am are calculated as
follows.
A0  3.18064617  10-7 ,
A1  0.
Evaluation of the circular flow (n = 2):
In order to create a circular cross section using
the function hyper-ellipse, It is enough to take n = 2
and the calculation of r0, but in this case all values of
(r ,  )  u(r ,  ) 
0.041503  0.3180646  106 cosh
 201
Substituting Eq. (22) in Eq. (17), the following
equations of change of electric potential and velocity
will be obtained:
62222r
.
(23)
The functions of X + and X- can be written as follows:
0 e(r , )  0.000275  exp  
6
1
X  X
 0.0415030.318064610 cosh 20 62222r


,

 20
.
X  X0e (r , ) 
(24)


0.00027254  exp  0.0415030.3180646106 cosh 1 62222r 

Equations 24, 25 and 26, all coefficients except for A0 and A1 due to the symmetry with respect to
(25)
θ
are
zero
Fig. 8: Comparison of velocity and electric potential contours in a circular channel with radius 10 nm radius in
two models, a: one-dimensional equations in cylindrical coordinates[17] b: two-dimensional cylindrical
coordinate system and using the boundary function hyper-ellipse
Fig. (8) represents the dimensionless velocity and electric potential contour with the present method and
reference [17]. Fig. (8) reveals that the present method has good accuracy in comparison with the numerical
results.
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Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
Fig. 9: Distribution of velocity and electric potential obtained from the proposed combined method using a
cylindrical coordinate system, the boundary hyper-ellipse, cosine transforms the circular channel (n = 2).
Fig. 10: Distribution of cation and anion mole fraction of electric potential obtained from the proposed
combined method using a cylindrical coordinate system, the boundary hyper-ellipse, cosine
transforms the circular channel
The present method for two points, n = 400 and
n = 2, and comparison with other methods confirms
the validity of this method. Now, the other sections
according to different values of n, by using this
method will be investigated.
A0  1.498319  10-7 ,
Flow rectangular with rounded edges (n = 4):
A4  3.142326  10-11,
Again, by using hyper-ellipse boundary function
for n = 4, the values of r0 on the basis of θ are
obtained and by substituting it into equation (17), the
coefficients Am are obtained as follows:
A1  7.152169  10-8 ,
A2  1.214084  10-8 ,
A3  4.841598  10-10 ,
A5  9.619619  10-13.
(26)
Substituting of in equation (18), the fluid
velocity and potential function is obtained as follows:
6325
Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
1

62222r 
 20

-7
(r ,  )  u (r ,  )  0.0415  1.498319 10 cosh 
-7 cosh  1 62222r  cos 4 - 2.428168  10 -8 cosh  1 62222r  cos 8
 
 



 20
 20

1

1

-9.68016610-10 cosh
62222r  cos 16 
62222r  cos 12  6.27584710-11 cosh

 20
 20



1
2.2267501012 cosh
62222r  cos 20 

 20
- 1.430435 10
.
(27)
Positive function of the mole fraction will be as follows:
0 e (r , )  0.000275  exp((0.0415  1.498319 10 -7 cosh  1 62222r 
X  X

 20

1

1

-1.43043510-7 cosh
62222r  cos 4 -2.42816810-8 cosh
62222r  cos 8 

 20
 20

 1

 1

-11
-10

62222r  cos 16 
-9.68016610
cosh
62222r  cos 12  6.27584710 cosh
 20

 20

 1

12


cosh
62222r  cos 20 ))
2.22675010

 20
,
(28)
And a negative function of the mole fraction is calculated as follows
1

62222r 
 20

 1

 1

-7
-8
-1.43043510 cosh
62222r  cos 4 -2.42816810 cosh
62222r  cos 8 


 20
 20




1
1
62222r  cos 16 
62222r  cos 12  6.27584710-11 cosh
-9.68016610-10 cosh

 20
 20

1

62222r  cos 20 )
2.2267501012 cosh
 20

X  X0e (r , ) 
0.000254  exp(0.0415  1.498319 10 -7 cosh 
.
(29)
Where, equations (27) to (29), representatives of electric potential, the mole fraction of positive and
negative mole fraction for channel dimensions 20 x 20 nm with rounded edges respectively.
Fig. 11: Contour velocity and electric potential distribution in the channel with dimensions of 20 × 20 nm
rounded edges of the cylindrical coordinate system and applying boundary conditions hyper-ellipse (n
= 4).
Figures 11 and 12, shows two-dimensional contour velocity and electric potential distribution in channel
with dimensions of 20 × 20 nm square with rounded edges respectively. In this case, rounded edges make
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Mehran Khaki Jamei and Morteza Abbasi, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6316-6327
forming flow at the edges toward the channel has a sharp edge. This causes the fluid shear stress at the corners is
reduced. Figure 13 shows Cation and anion mole fraction distribution in the channel.
Fig. 12: velocity and electric potential distribution in the channel with dimensions of 20 × 20 nm rounded edges
of the cylindrical coordinate system and applying boundary conditions hyper-ellipse (n = 4).
Fig. 13: Distribution of mole fraction of cation and anion channels with dimensions of 20 × 20 nm with rounded
edges using a cylindrical coordinate system and applying boundary conditions hyper-ellipse (n = 4).
As can be seen, the rounded edges, no significant effect on the distribution of ion concentration.
Conclusion:
In this paper, two-dimensional flows in nonrectangular channels were analysed. The equations in
cylindrical coordinates are introduced is used to
solve
this
problem.
The
Debye-Huckel
approximation is used to convert the differential
equations to one partial differential equation and
their general solution is obtained by using a cosine
transform. The validity of results is verified by
comparing the present method for rectangular and
circular sections with another technique. The result
shows that the new combined approaches have good
precision. In addition, the effect of rounded edge
rectangular channel flow was studied. As it's clear,
rounded edge, the local stress decreases and so, fluid
in most parts of the channels has maximum. The
results indicate that the velocity profiles and electric
potential distribution of the flow are following the
cap. In other words, the velocity and the potential
distribution near the wall is sharp and with a little
distance from the border the values will be uniform.
This phenomenon may cause significant shear
stresses in the vicinity of the wall. Because of drastic
changes in the velocity of the walls, can be the result
of the accumulation of ions with opposite charge and
make a considerable difference in the concentrations
of positive and negative ions, resulting in a
significant stimulation fluid.
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