Application of homotopy analysis method to solve MHD Jeffery–
Hamel flows in non-parallel walls
S.M. Moghimi1, G. Domairry2, H.Bararnia2, E.Ghasemi2, S.Soleimani2
1
Department of Mechanical Engineering, Islamic Azad University,Qaemshahr , Iran
2
Babol University of Technology, Department of Mechanical Engineering, P. O. Box 484, Babol, Iran
Abstract
The MHD Jeffery–Hamel flows in non-parallel walls are investigated analytically for strongly
nonlinear ordinary differential equations using homotopy analysis method (HAM). Results for
velocity profiles in divergent and convergent channels are presented for various values of
Hartmann and Reynolds numbers. The convergence of the obtained series solutions is explicitly
studied and a proper discussion is given for the obtained results. Comparison between the HAM
and numerical solutions showed excellent agreement.
Keywords: MHD Jeffery–Hamel flows; HAM; Nonlinear ordinary differential equations.
1. Introduction
The incompressible viscous fluid flow through convergent and divergent channels is one
of the most applicable cases in fluid mechanics, electrical and bio-mechanical engineerings. The
mathematical investigations of this problem were discussed by [1,2]. Jeffery-Hamel flow are of
the Navier-Stokes equations in the particular case of two-dimensional flow through a channel
with inclined walls [3-12]. One of the most` significant examples of Jeffery-Hamel problems are
those subjected to an applied magnetic field. The equations of magnetohydrodynamics have been
solved exactly for the case of two-dimensional steady flow between non-parallel walls of a
viscous, incompressible, electrically conducting fluid; this is a straightforward extension of the
famous Jeffrey-Hamel problem in ordinary hydrodynamics [13]. It has been showed that for the
Jeffrey-Hamel problem, the equations of magnetohydrodynamics can be reduced to a set of three
ordinary differential equations, two of which are linear and of first order [14]. In addition, these

Corresponding Author: Tel/Fax: +98 111 3234205
E–mail address: amirganga111@yahoo.com
kinds of problems have been well studied in literature [5-11]. Liao [15,16] proposed a new
asymptotic technique for non-linear ordinary differential equations (ODES) and partial
differential equations (PDES) , named the homotopy analysis method (HAM)[17,18]. Based on
the homotopy in topology, the homotopy analysis method contains obvious merits over
perturbation techniques: its validity does not depend upon whether nonlinear ODES or PDES
under consideration contain small/larger parameters. Thus, the HAM method can be applied to
analyze more of non-linear problems in science and engineering. Another advantage of the
homotopy analysis method is that it provides us with larger freedom to select initial guess
approximations, auxiliary linear operators and some other auxiliary parameters. As discussed by
Liao [15, 16], this kind of freedom ensures that approximation series given by the HAM are
convergent and uniformly valid. In [15] Liao systematically discussed the basis of the HAM and
proved two theorems of convergences. Liao [15] has proved that as long as the approximate
solution series is convergent, it must converge to one solution of the non-linear ODES or PDES
under consideration. Most recent problems such as Jeffery-Hamel flow and other fluid mechanic
problems are inherently nonlinear. Except a limited number of these problems, most of them do
not have analytical solutions. So, these nonlinear equations should be solved utilizing other
methods. In this study, HAM has been successfully applied as an analytical methodology to
solve the nonlinear equations. The convergence of the series has been explicitly discussed.
Additionally, the governing equations are solved numerically using Maple software.
2. Problem formulation
Consider a system of cylindrical polar coordinates ( r , , z ) which steady two-dimensional
flow of an incompressible conducting viscous fluid from a source or sink at channel walls lie in
planes, and intersect in the axis of z. It is assumed that there are no changes with respect to z,
that the motion is purely radial direction and depends on r,  and that there is no magnetic field
in the z-direction. The governing equations are [15].

(ru (r , ))  0
rr
  2 u (r ,  ) 1 u (r ,  )
u (r ,  )
1 P
1  2 u (r ,  ) u (r ,  )  B02
u (r ,  )





u (r ,  )

2
r
 r
r
r
r2
 2
r 2  r 2
 r
1 P 2 u (r , )

0
r  r 2 
1
2
3
Where B0 is the electromagnetic induction,  the conductivity of the fluid, u ( r , ) is the velocity
along radial direction, P is the fluid pressure,  the coefficient of kinematic viscosity and  the
fluid density .From Eq.1.
f ( )  ru (r , )
4
Using dimensionless parameters
f ( ) 

f ( ) ,


f max
5
Substituting equation 5 and into equation 2, 3 and eliminating P, we obtain an ordinary
differential equation for the normalized function profile F ( ) [20]:
f ( )  2 Re f ( ) f ( )  (4  H ) 2 f ( )  0
6
The boundary conductions are :
f (0)  1 , f (0)  0 , f (1)  0
The Reynolds number is :
f  U r  divergent  channel :   0, f max  0 

Re  max  max 

  convergent  channel :   0, f max  0 
The Hartmann number is :
H
B02

7
8
9
3. Homotopy analysis solution
Consider the equation 6. We define a nonlinear operator as follows:
 ( ; q) 
 3 ( , q)
 ( , q)
 ( , q)
 2 Re  ( , q)
 (4  H ) 2
3



10
where q  [0,1] the embedding parameter ,   0 is a nonzero auxiliary parameter. As the
embedding parameter increases from 0 to 1,  ( , q) varies from the initial guess f 0 ( ) to the
exact solution f ( ) :
 ( ,0)  f 0 ( ) ,  ( ,1)  f ( )
Expanding  ( , q) in Taylor series with respect to q , we have:
11

 ( , q )  f 0 ( )   f m ( )q m
12
m 1
Where:
f m ( ) 
1  m ( , q)
| q 0
m! q m
13
Homotopy analysis method can be expressed by many different base functions [21], according to
the governing equation; it is straightforward to use a set of base function:
{ 2 n  n  0,1,2,3,...}
In the form :
14

f ( )   bn  2 n
15
n 1
Where bn is a coefficient to be determined. Besides, a set of base function, the auxiliary
function H ( ) , initial approximation f 0 ( ) and the auxiliary linear operator L must be chosen in
such a way that all solutions of the corresponding high-order deformation equations exist and can
be express by this set of the base function and the other expressions such as  n Sin (m ) must be
avoided. This provides us with the rule of solution expression [21]. We choose the linear
operator, as follow :
 3 ( , q)
L[( ( , q)] 
 3
That the L is :
16
L[0.5c1 2  c2  c3 ]  0
17
Where c1 , c2 and c3 are constants. We must guess the initial value of f ( ) so that it to satisfy
the boundary conditions. According to the discussed limitation and under the rule of solution
expression and initial conditions, the initial guess is :
f 0 ( )  0.5c1 2  c 2  c3
18
c1  2, c 2  0, c3  1
The zero order deformation equation is :
(1  q) L[ ( , q)  F0 ( )]  qH ( ) N[ ( , q)]
 (0, q)  1,  (1, q)  0,
19
 (0, q)
0

20
According to the rule of solution expression denoted by Eq.10, and from Eq.19, the auxiliary
function H(η) can be chosen as follows:
H ( )   p
Considering [21,22] we choose P  0 , then auxiliary function is
21
H ( )  1 .
Differentiating eq.19, m times with respect to the embedding parameter q and then setting q=0
and finally dividing them by m! and From Eqs.10,17. We have the mth-order deformation
equation for m≥1:

x
y
0
0
0
f m ( )   m f m1 ( )    dx  dy  H ( ) Rm ( f m 1 )d  c1 2  c 2  c 3
22
f m (0)  0, f m (1)  0, f m (0)  0
23
Where:
m 1
Rm ( f m 1 )  f m1 ( )  2 Re  f n ( ) f m 1 n ( )  (4  H ) 2 f m 1 ( )
24
n 0
And :
0, m  1
1, m  1
m 
25
We now successively obtain:
f 0 ( )  1   2
1
1
1
1
2
1
1
f 1 ( )  
 Re  6  (  Re   2   2 H ) 4  (  2 H   2   2 H ) 2
300
6
3
12
15
3
12
1
1
1 3
1
1 4 2 1 3
1
f 2 ( )  
 2 Re 2  10  (  2 Re 2 
 Re H   3 Re) 8  (
 H   Re H   3 Re
1350
140
280
70
360
60
15
2
1
1
1
1
1 4 2 1 4
1
1
  4   4 H   2 Re 2 ) 6  (  4   3 Re 
 H   H   2 Re 2   3 Re H ) 4
45
45
50
9
10
144
18
45
40
1 4
1 4 2 1 4 1 3
1 2 2 1 3
2
(
 H
 H     Re   Re   Re H )
300
240
15
21
23
86
26
27
28
We have solved the mth-order approximation of f(η) can be expressed by :
f ( ) 
2 m 1

n0
m, n
() 2 n
Series 29, converges by choosing of the auxiliary parameter
dependent to

29

and  m, n () which is a coefficient
.
4. Convergence of HAM solutions
In HAM it is necessary to find the values of auxiliary parameter  that converges the series
in 29 according to the various values of  , Re, H Liao[21]. This region of  can be found by
plotting the f (0) versus  (  -curve) and choosing  where f (0) is constant. For example in
Fig.2 , it is clear that the acceptable region of  is  0.6    0.2 for H  0 ,  0.4    1.8 for
H  500 ,  0.2    1.6 for H  1000 , and  0.3    1.1 for H  2000 , and also in Figs.3 and 4 ,
these ranges are  0.5    1.2 and  0.5    0.9 , respectively.
5. Results and discussion
The obtained results of HAM solution and numerical ones using Maple software are
shown through the following table and figures .Figs 5 and 6 show the values of velocity in
divergent and convergent channels, respectively. As seen, the value of velocity increases with
growth of H  0 (Hartmann number); the maximum value of velocity takes place at the center of
the channel whereas the values on the walls are zero. Fig.7 presents that as  increases the effect
of walls on fluid flow decreases when we move away from them which leads to increase of
velocity, while there is a reverse behavior in velocity profile in Fig.8 when the values of 
decreases .The effect of H  0 number is demonstrated in Fig.9. As shown, with increase of Re
number boundary layer thickness increases which results in decrease of velocity and also makes
the flow move reversely (for Re  250 ). Fig.10 represents that increase of H prevent from
producing of vorticity in fluid flow and therefore removes the changing direction of the flow.
Table 1 gives the comparison between the analytic and numerical results which are in excellent
agreement.
References
[1]
G.B. Jeffery, The two-dimensional steady motion of a viscous fluid, Phil. Mag. 6 (1915) 455-465.
[2]
G. Hamel, Spiralförmige Bewgungen Zäher Flüssigkeiten, Jahresber. Deutsch. Math.-Verein. 25
(1916) 34-60.
[3]
L. Rosenhead, The steady two-dimensional radial flow of viscous fluid between two inclined plane
walls, Proc. R. Soc. A 175 (1940) 436-467.
[4]
K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967.
[5]
R.M. Sadri, Channel entrance flow, Ph.D. Thesis, Department of Mechanical Engineering, The
University of Western Ontario, 1997.
[6]
I.J. Sobey, PG. Drazin, Bifurcations of two-dimensional channel flows, J. Fluid Mech. 171 (1986)
263-287.
[7]
M. Hamadiche, J. Scott, D. Jeandel, Temporal stability of Jeffery  Hamel flow, J. Fluid Mech. 268
(1994) 71-88.
[8]
L.E. Fraenkel, Laminar flow in symmetrical channels with slightly curved walls. I: On the
Jeffery  Hamel solutions for flow between plane walls,Proc. R. Soc. Lond. A 267 (1962) 119-138.
[9]
O.D. Makinde, P.Y. Mhone, Hermite_Padé approximation approach to MHD Jeffery  Hamel
flows, Appl. Math. Comput. 181 (2006) 966-972.
[10] H. Schlichting, Boundary-layer Theory, McGraw-Hill Press, New York, 2000.
[11] R.K. Rathy, An Introduction to Fluid Dynamics, Oxford and IBH Pl, New Delhi, 1976.
[12] A. McAlpine, P.G. Drazin, On the spatio-temporal development of small perturbations of
Jeffery  Hamel flows, Fluid Dyn. Res. 22 (1998) 123-138.
[13] S. Goldstein (ed.), Modem Developments in Fluid Dynamics, Vol. 1(Oxford,1938).
[14] W.I. Axford, The Magnetohydrodynamic Jeffrey  Hamel Problem For a Weakly Conducting Fluid,
14(1961) 335 – 351.
[15] Liao, S. J., An explicit, totally analytic approximate solution for Blasius’ viscous flow problems,
Journal of Non-Linear Mechanics, 34 (1999) 759-778.
[16] Liao, S. J., A uniformly valid analytic solution of two- dimensional viscous flow over a semiinfinite
flat plate, Journal of Fluid Mechanics, 385(1999)101-128.
[17] H. Bararnia , Abdoul R. Ghotbi , G. Domairry. On the analytical solution for MHD natural
convection flow and heat generation fluid in porous medium. Commun Nonlinear Sci Numer
Simulat 14 (2009) 2689–2701.
[18] Abdoul R. Ghotbi, H. Bararnia, G. Domairry , A. Barari, Investigation of a powerful analytical
method into natural convection boundary layer flow. Commun Nonlinear Sci Numer Simulat 14
(2009) 2222–2228.
[19] H.Bararnia, E.Ghasemi, G. Domairry , S.Soleimani. Behavior of micropolar flow due to linear
stretching of porous sheet with injection and suction. Advances in Engineering Software, In Press.
[20] H. Schlichting. Boundary-layer theory. New York: MCGraw-Hill Press; 2000.
[21] Liao, S. J. Beyond perturbation: Introduction to homotopy analysis method. Boca Raton: Chapman
and Hall/CRC Press; 2003.
[22] Liao, S. J. On the homotopy analysis method for nonlinear problems. Appl Math Comput
47(2004)499–513.
q
u(r, )
Fig. 1. Geometry of the problem
Fig.2. The ħ curve of f (0) for Re  50 ,   5  at 8th-order of approximation
Fig.3. The ħ of f (0) for Re  50 ,   5  , H  1000 for different orders of approximation
Fig.4. The ħ of f (0) for Re  50 ,   5  , H  1000 for different orders of approximation
Fig.5. The comparison of numerical results and HAM solution for velocity in divergent
channel for Re  50 and   5  .
Fig.6. The comparison of numerical results and HAM solution for velocity in convergent
channel for Re  50 and   5 .
Fig.7. The comparison of numerical results and HAM solution for velocity in divergent
channel for Re  50 and H  1000 .
Fig.8. The comparison of numerical results and HAM solution for velocity in convergent
channel for Re  50 and H  1000 .
Fig.9. The solution by HAM for velocity in divergence channel, H=1000 and   5  .
Fig.10. The solution by HAM for velocity in divergence channel, Re  250 and   5  .
Table1. The comparison between the Numerical results and HAM solution for velocity when
Re  50 and H  1000 .
HAM
  5
Numerical
Error
1.0000000000
0.9976051267
0.9904272156
0.9784856266
0.9618100746
0.9404368643
0.9144036501
0.8837428568
0.8484737066
0.8085929619
0.7640642412
0.7148059133
0.6606772666
0.6014624674
0.5368520875
0.4664210783
0.3896019056
0.3056518011
0.2136111723
0.1122503245
0.0000000000
1.0000000000
0.9976051266
0.9904272154
0.9784856255
0.9618100752
0.9404368631
0.9144036551
0.8837428679
0.8484737175
0.8085929634
0.7640642437
0.7148059216
0.6606772549
0.6014624512
0.5368520846
0.4664210525
0.3896018881
0.3056518077
0.2136112049
0.1122503834
0.0000000000
0.0000000000
0.0000000001
0.0000000002
0.0000000011
0.0000000006
0.0000000012
0.0000000050
0.0000000011
0.0000000110
0.0000000015
0.0000000025
0.0000000083
0.0000000117
0.0000000162
0.0000000029
0.0000000258
0.0000000175
0.0000000066
0.0000000326
0.0000000589
0.0000000000

0
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
HAM
  5
Numerical
Error
1.0000000000
0.9991974677
0.9967570409
0.9925789754
0.9864928100
0.9782509276
0.9675193145
0.9538653197
0.9367421766
0.9154700636
0.8892135408
0.8569552920
0.8174664644
0.7692740948
0.7106275597
0.6394657739
0.5533900638
0.4496485964
0.3251423732
0.1764658313
0.0000000000
1.0000000000
0.9991973819
0.9967567004
0.9925782199
0.9864914948
0.9782489428
0.9675165808
0.9538617983
0.9367379265
0.9154652381
0.8892083429
0.8569499740
0.8174612678
0.7692693193
0.7106233991
0.6394623698
0.5533874550
0.4496468019
0.3251413493
0.1764654178
0.0000000000
0.0000000000
0.0000000858
0.0000003405
0.0000007555
0.0000013152
0.0000019848
0.0000027337
0.0000035214
0.0000042501
0.0000048255
0.0000051979
0.0000053180
0.0000051966
0.0000047755
0.0000041606
0.0000034041
0.0000026088
0.0000017945
0.0000010239
0.0000004135
0.0000000000