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MHD Flow of A Third Grade Fluid with Heat Transfer And
Slip Boundary Condition Down An Inclined Plane
1
Aiyesimi, Y. M., 2Okedayo, G. T. and 1*Lawal O. W.
1. Department of Mathematics, Federal University of Technology Minna, Niger State, Nigeria
2. Department of Mathematics, Ondo State University of Science and Technology Okitipupa, Ondo State, Nigeria
* E-mail of the corresponding author: waheedlawal207@yahoo.com
Abstract
In this work, we consider the combine effects of slip boundary ohmic heating on MHD flow of a third grade fluid down an
inclined plane. The couple non-linear ordinary differential equations arising from the model were solved using both the
regular and homotopy perturbation. Effects of the various thermo physical parameters are studied and depicted graphically.
Keywords: Slip boundary, MHD, Third grade fluid, Ohmic heating, inclined plane.
1. Introduction
Non-Newtonian fluid flow play an important role’s in several industrial manufacturing processes. An example of such
non-Newtonian fluids includes drilling mud, polymer solutions or melts, certain oils and greases and many other emulsions.
Some of the typical applications of non-Newtonian fluid flow are noticed in the drilling of oil and gas wells, polymer sheet
extrusion from a dye, glass fibber and paper production, drawing of plastic films, waste fluids e.t.c.
Because of the complexity of non-Newtonian fluids, it is very difficulty to analyse a single model that exhibits all its
properties. The normal stress differences is describe in second grade of non-Newtonian fluid, but it can not predict shear
thinning or thickening properties due to its constant apparent viscosity. The third grade fluid model attempt to include such
characteristics of visco-elastics fluids.
Several researchers discussed the slip effect on fluid flow. Asghar et. al. (2006) examines the effects of partial slip on the
rotating flow of an incompressible third grade fluid past a uniformly porous plate. Miccal and James (2008) discuss the
effect of replacing the standard no slip boundary condition of fluid mechanics applying for the so called Falkner-Skan
solutions, with a boundary condition that allows some degree of tangential fluid slip. Ellahi (2009) discuss the slip condition
of an Oldroyd 8 – constant fluid and Sajid (2008) investigate the effect of slip condition on thin film flow.
In this paper, we examine the effects of slip boundary condition on a thin film flow of an MHD third grade fluid down an
inclined plane. The heat transfer analysis is also carried out.
Many analytical and numerical techniques have been proposed by various authors for the solution of governing nonlinear
differential equations of non-Newtonian fluids. We have solved the governing nonlinear equation of present problem using
the traditional perturbation method (Nayfeh,1979) and homotopy perturbation method (He 2003 & 2009). We noticed that
the solution obtain from the two methods are in a complete agreement. It is also observed that homotopy perturbation
method is powerful analytical technique that is simple and straight forward and does not require the existence of any small
or large parameter as does in traditional perturbation method. The plan of the paper is as follow: Section 2 contains the basic
equations. Section 3 deals with the formulation of the problem with slip condition and also include solutions of the problem
by traditional perturbation method and homotopy perturbation method. Results and discussion are presented in section 4
while section 5 concludes the paper.
2. Basic Equation
The fundamental equations governing the MHD flow of an incompressible electrically conducting fluid are the field
equation:
∇⋅v = 0 (1)
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Dv
ρ = −∇p + divτ + J × B + ρf (2)
Dt
where ρ is the density f the fluid, v is the fluid velocity, B is the magnetic induction so that
B = B0 + b (3)
and
J = σ ( E + v × B) (4)
is the current density, σ is the electrical conductivity, E is the electrical filed which is not considered (i.e E = 0 ),
B0 and b are applied and induced magnetic filed respectively, D Dt denote the material derivative, p is the pressure,
f is the external body force and τ is the Cauchy stress tensor which for a third grade fluid satisfies the constitutive
equation
τ = − pI + µA1 + α 1 A2 + α 2 A12 + β1 A3 + β 2 ( A1 A2 + A2 A1 ) + β 3 (trA12 ) A1 (5)
DAn −1
An = + An−1∇v + (∇v) ⊥ An −1 , n ≥1 (6)
Dt
where pI is the isotropic stress due to constraint incompressibility, µ is the dynamics viscosity, α1 , α 2 , β1 , β 2 , β 3 are
the material constants; ⊥ indicate the matrix transpose, A1 , A2 , A3 are the first three Rivlin-Ericken tensors and A0 = I
is the identity tensor.
3. Problem Formulation
We consider a thin film of an incompressible MHD fluid of a third grade flowing in an inclined plane. The ambient air is
assumed stationary so that the flow is due to gravity alone.
By neglect the surface tension of the fluid and the film is of uniform thickness δ , we seek a velocity field of the form
v = [u ( y ),0,0, ] (7)
In the absence of modified presence gradient, equation (1)-(4) along with equation (5)-(7) yields
2
d 2u  du  d 2 u
+ 6β  
 dy  dy 2 + K − Mu = 0 (8)
dy  
Subject to the boundary condition
  du   d 2u  
3
u − Λ  µ   + 2 β  2   = 0 at y = 0
   dy  (9)
  dy 
   
du
=0 at y =1 (10)
dy
where
β=
(β 2 + β 3 )µ is third grade fluid parameter
δ4
δ 3 ρg
K = f1 sin α while f1 = is the gravitational parameter.
µ
δ 2σB02
M= is the magnetic parameter.
µ
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Equation (9) is the slip conditions where Λ is coefficient of slip and equation (10) comes from τ yx = 0 at y = δ .
In the sequence, we take ε = β and solve the system of equation (8)-(10) by the traditional perturbation method and also by
the homotopy perturbation.
Heat Transfer Analysis
The thermal boundary layer equation for the thermodynamically compatible third grade fluid with viscous dissipation, work
done due to deformation and joule heating is given as
2 4
d 2T  du   du 
k 2 + µ   + 2( β 2 + β 3 )  + σB02 u 2 = 0
   dy  (11)
dy  dy   
with boundary condition
T ( y ) = Tw at y=0 (12)
T ( y ) = Tδ at y= δ (13)
where k is the thermal conductivity, T is the temperature, and Tδ is the temperature of the ambient fluid.
Introducing the following dimensionless variable
u T − T1
u= , T = (14)
u′ T2 − T1
where T1 = Tw and T2 = Tδ
The system of equation (11)-(13) and (14) after dropping the caps take the following form:
2 4
d 2T  du   du 
2
+ Br   + 2 Br β   + Br Mu 2 = 0
   dy  (15)
dy  dy   
T ( y ) = 0 at y=0 (16)
T ( y ) = 1 at y =1 (17)
µ3
where Br = is the Brinkman number
kδ 2 (T2 − T1 )
σB02δ 2
M= is the magnetic parameter
µ
β=
(β 2 + β 3 )µ is third grade fluid parameter
δ4
Again, in the sequence, we take ε = β and solve the system of equation (15)-(17) by the traditional perturbation method
and also by the homotopy perturbation.
(A) Solution of the problem by Regular perturbation method
Let us assume ε as a small parameter in order to solve equation (8) by this method, we expand
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u ( y, ε ) = u0 ( y ) + εu1 ( y ) + ε 2u2 ( y ) + ⋅ ⋅ ⋅ ⋅ ⋅ (18)
Substituting equation (18) into equation (8) and rearranging based on powers of ε − terms.
We obtain the following problems of different order with their boundary conditions:
Zeroth – order problem and its solution
d 2u 0
− Mu + K = 0 (19)
dy 2
 du 
u 0 − Λµ  0
 dy  = 0 at y = 0
 (20)
 
du0
=0 at y= 1 (21)
dy
The solution of system of equation (19)-(21) is given by
K
u 0 = c1e y M
+ c2 e − y M
+ (22)
M
First order problem and its solution
2
d 2u1 d 2u  du 
+ 6 20  0  − Mu1 = 0 (23)
dy 2 dy  dy 
 
3
 du   du 
u1 − Λµ  1  − 2Λµ  0
 dy   dy  =0
 at y=0 (24)
   
du1
=0 at y= 1 (25)
dy
The solution of system of equation (23)-(25) is given by
u1 = (c7 + c8 y )e y M
+ (c9 + c10 y )e − y M
+ c11e 3 y M
+ c12 e −3 y M
(26)
Second order problem and its solution
2
d 2u 2 d 2u  du  du du d 2u0
+ 6 21  0  + 12 0 1 − Mu2 = 0 (27)
dy 2 dy  dy 
  dy dy dy 2
2
 du   du   du1 
u 2 − Λµ  2
 dy  − 6 Λµ  0
  dy 
 
 dy  = 0
 at y=0 (28)
     
du 2
=0 at y= 1 (29)
dy
The solution of system of equation (27)-(29) is given by
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(
u 2 = c 25 + c 26 y + c 27 y 2 e y ) M
(
+ c28 + c 29 y + c30 y 2 e − y ) M
+ (c31 + c32 y )e 3 y M
(30)
+ (c33 + c34 y )e −3 y M
+ c35 e 5 y M
+ c36 e −5 y M
Next, we find the approximate solution for temperature distribution, for which we write
T ( y, ε ) = T0 ( y ) + εT1 ( y ) + ε 2T2 ( y ) + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ (31)
Substituting equation (31) into equation (15) –(17) and collecting the same power of ε , yields different order problems.
Zeroth –order problem with boundary conditions:
d 2T0  du  2 
+ Br  0  + Mu 0  = 0
  (32)
dy 2  dy  
 
T0 (0) = 0 at y=0 (33)
T0 (1) = 1 at y =1 (34)
with the solution
T0 = c 41 e y M
+ c 42 e − y M
+ c 43 e 2 y M
+ c 44 e −2 y M
+ c 45 e 4 y M
+ c 46 e −4 y M
(35)
+ c 47 y 2 + c39 y + c 40
First –order problem with boundary conditions:
d 2T1  du du  du  4 
+ 2 Br  0 1 +  0  + Mu 0 u1  = 0
  (36)
dy 2  dy dy  dy  
 
T1 (0) = 0 at y=0 (37)
T1 (1) = 1 at y =1 (38)
with the solution
T1 = (c50 + c51 y )e y M
+ (c 52 + c 53 y )e − y M
+ (c 54 + c 55 y )e 2 y M
+ (c56 + c57 y )e − 2 y M
+ c58 e 3 y M
+ c59 e −3 y M
+ c 60 e 4 y M
(39)
+ c 61 e − 4 y M
+ c 62 y 2 + c 48 y + c 49
Second–order problem with boundary conditions:
d 2T2  du du  du  2  du  3 du 
2
+ Br 2 0 2 +  1  + 8 0 
 dy   dy  dy + 2 Mu 0 u 2 + Mu1  = 0
1 2
(40)
dy  dy dy 
    

T2 (0) = 0 at y=0 (41)
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T2 (1) = 1 at y =1 (42)
with the solution
(
T 2 = c 65 + c 66 y + c 67 y 2 e y ) M
(
+ c 68 + c 69 y + c 70 y 2 e − y ) M
(
+ c 71 + c 72 y + c 73 y 2 e 2 y ) M
(
+ c 74 + c 75 y + c 76 y 2 e − 2 y) M
+ (c 77 + c 78 y )e 3 y M
+ (c 79 + c 80 y )e − 3 y M
+ (c 81 + c 82 y )e 4 y M
(43)
+ (c 83 + c 84 y )e − 4 y M
+ c 85 e 5 y M
+ c 86 e − 5 y M
+ c 87 e 6 y M
+ c 88 e − 6 y M
+ c 89 y 4 + c 90 y 3 + c 90 y 2 + c 63 y + c 64
(B) Solution by Homotopy perturbation method
The problem under consideration i.e equations (8)-(10) can be written as
 β  dv  2 d 2 v 
L(v) − L(u 0 ) + qL(u 0 ) + q 6  
  − Mv + K  = 0 (44)
 µ  dy  dy
 

d2
where L= and equation (28) is the initial guess approximation.
dy 2
Let v = v 0 + qv1 + q v 2 + ⋅ ⋅ ⋅ ⋅
2
(45)
Substitute equation (45) into (44) and equating the coefficient of like powers of q , we have
Zeroth –order problem with boundary conditions:
d 2 v0 d 2 u 0
− =0 (46)
dy 2 dy 2
 dv 
v 0 − Λµ  0
 dy =0
 at y=0 (47)
 
dv 0
=0 at y =1 (48)
dy
with the solution
v 0 = c1e y M
+ c2 e − y M
+ c3 y + c 4
* *
(49)
First –order problem with boundary conditions:
2
d 2 v1 d 2 u 0 β d 2 v0  dv0 
+ +6 
 dy  − Mv0 + K = 0
 (50)
dy 2 dy 2 µ dy 2  
3
 dv   dv 
v1 − Λµ  0
 dy  − 2 Λµ  0
  dy  =0
 at y=0 (51)
   
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dv1
=0 at y =1 (52)
dy
with the solution
v1 = c8 e y
* M
+ c9 e − y
* M
+ c10 e 2 y
* M
+ c11 e −2 y
* M
+ c12 e 3 y
* M
+ c13 e −3 y
* M
(53)
+ c14 y 2 + c 6 y + c 7
* * *
Second –order problem with boundary conditions:
β  d 2 v dv dv  dv  d 2 v1 
2
d 2 v1
+ 6 2 20 0 1 +  0 
 dy 2  − Mv1 = 0 (54)
dy 2 µ  dy dy dy  dy
   

2
 dv   dv   dv1 
v 2 − Λµ  1  − 6Λµ  0
 dy   dy 
 
 dy  = 0
 at y=0 (55)
     
dv 2
=0 at y =1 (56)
dy
with the solution
(
v2 = c17 + c18 y e y
* *
) M *
(
+ c19 + c20 y e − y
*
) M
(
+ c21 + c22 y e 2 y
* *
) M *
( *
)
+ c23 + c24 y e −2 y M
+ c25 e 3 y
* M
+ c26 e −3 y
* M
+ c27 e 4 y
* M
+ c28 e −4 y
* M
+ c29 e 5 y
* M
(57)
+ c30 e −5 y
* M
+ c31 y 4 + c32 y 3 + c33 y 2 + c15 y + c16
* * * *
Next, we find approximate solution of temperature profile using homotopy perturbation by written equation (15) as
  dv  2  dv 
4

L(T ) − L(θ 0 ) + qL(θ 0 ) + q  Br   + 2 Br β   + Br Mv 2  = 0
 dy    (58)

    dy  

Let T = T0 + qT1 + q T2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
2
(59)
Substitute equation (55) into (54) and equating the coefficient of like powers of q , we have
Zeroth –order problem with boundary conditions:
d 2T0 d 2θ 0
− =0 (60)
dy 2 dy 2
T0 (0) = 0 at y=0 (61)
T0 (1) = 1 at y =1 (62)
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Using equation (35) as θ0 to serve as initial guess approximation, we have
T0 = c34e y
* M
+ c35e− y
* M
+ c36e2 y
* M
+ c37e−2 y
* M
+ c38e4 y
* M
+ c39e−4 y
* M
(63)
+ c40 y2 + c41 y + c42
* * *
First-order problem with boundary conditions:
2
d 2T1 d 2θ 0  dv 
2
− 2
+ Br  0
 dy  + Br Mv0 = 0

2
(64)
dy dy  
T1 (0) = 0 at y=0 (65)
T1 (1) = 1 at y =1 (66)
with the solution
T1 = (c *
45 + c 46 y e y
*
) M
(
+ c 47 + c 48 y e − y
* *
) M
+ c 49 e 2 y
* M
+ c50 e −2 y
* M
(67)
+ c 51 e 4 y
* M
+ c 52 e − 4 y
* M
+ c53 y 4 + c 54 y 3 + c 55 y 2 + c56 y + c57
* * * * *
Second order problem with boundary conditions:
3
d 2T dv dv  dv  dv1
2
+ 2 Br 0 1 + 4 B r β  0
 dy 
 dy + 2 Br Mv0 v1 = 0 (68)
dy dy dy  
T2 (0) = 0 at y=0 (69)
T2 (1) = 1 at y =1 (70)
with solution
T2 = (c *
58 + c59 y + c60 y 2 e y
* *
) M *
(
+ c61 + c62 y + c63 y 2 e − y
* *
) M *
(
+ c64 + c65 y e 2 y
*
) M
(
+ c66 + c67 y e −2 y
* *
) M *
(
+ c68 + c69 y e 3 y
*
) M *
(
+ c70 + c71 y e −3 y
*
) M
+ c72 e 4 y
* M
(71)
−4 y M
+ c73
*
+ c74 e 5 y
* M
+ c75 e −5 y
* M
+ c76 e 6 y
* M
+ c77 e −6 y
* M
+ c78 y 5 + c79 y 4
* *
+ c80 y 3 + c81 y 2 + c82 y + c83
* * * *
4. Results and Discussion
Approximate analytical solutions for velocity and temperature distribution of a thin film flow of an MHD third grade fluid
down an inclined plane have been found. The governing nonlinear ordinary differential equations are solved using
traditional regular perturbation method as well as the recently introduced homotopy perturbation technique and the results
are compared.
Figure 1-6 shows that the solution obtained by two method are the same for identical value of ε and β . Therefore, we
discuss only solution obtained by homotopy perturbation method. Figure 1 and 2 illustrates the effect of magnetic parameter
M on the velocity and temperature distribution of the fluid down an inclined plane. Increase in the value of M reduces
the velocity distribution and increases the temperature distribution as slip parameter Λ is constant. Figure 3 and 4 shows
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the effect of parameter K on the velocity and temperature distribution. Clearly, increase in value of parameter K
increases the velocity and temperature distribution due to fact that increase in parameter K correspond to the increasing in
angle of inclination. In figure 5 and 6, we find out that increase in slip parameter Λ , the velocity and temperature
distribution increases.
5. Conclusion
A thin film flow of an MHD third grade fluid down an inclined plane with slip boundary condition has been discussed. It
was found that the solutions obtained by traditional perturbation and homotopy perturbation technique are identical at the
same values of ε and β , this is illustrated in Figures 1-6. The effect of slip parameter, Magnetic parameter and other
parameters involved in the problem are discussed and results are displayed in graph to see their effects.
References
Nayak I., Nayak A. K. & Padhy S., (2012), “Numerical solution for the flow and heat transfer of a third grade fluid past a
porous vertical plate”, Adv. Studies Theory Phys., Vol.6, No.13,615-624.
Asghar S, Gulzar M.M, & Ayub M., (2006), “Effects of partial slip on flow of a third grade fluid”, Acta
Mech Sinica , 22:393-396.
Miceel T.M. & Jamess M.H, (2008), “A note on the boundary layer equations with linear slip boundary condition”, Applied
Mathematics letters 21, 810-813.
Ellehi R., (2009), “Effects of the slip boundary condition on non-Newtonian flows in a channel”, Commu Nonlinear Science
Numerical Simul, 14,1377-1384.
Sajid M, Awals M, Nadeem S, & Hagat T., (2008), “The influence of slip condition on thin film flow of a fourth grade fluid
by the homotopy analysis method”, Comput Math Appl, 56, 2019-2026.
Nayfeh A.A, (1979), “Intrudction to perturbation techniques”, Wiley.
He J.A., (1999), “Homotopy perturbation technique”, Comput methods Appl Mech Eng ,178, 257-262.
He J.A, (2009), “Elementary introduction to the homotopy perturbation method”, Comput Math Appl., 57,
410-412.
Hayat. T, Saif S. & Abbas Z., (2008), “The influence of heart transfer in an MHD second grade fluid film
over an unsteady stretching sheet”, Physics Letters A372, 50037-5045.
Yongqi W. & Wu. W, (2007), “Unsteady flow of a fourth grade fluid to an oscillating plate”, International
Journal of non-linear Mechanics 42, 432-441.
Siddiqui A. M., Mahmood R. & Ghorn Q. K., (2008), “Homotopy Perturbation Method for thin film flow
of a third grade fluid down an inclined plane”, Chaos solitons and fractals 35,140-147
Nargis K. & Mahmood T. (2012), “The influence of slip condition on the thin film flow of a third order
fluid”, International Journal of Non-linear Science. Vol. 13:No. 1. pp. 105-116.
Makinde O. D. (2009), “Thermal critically for a reactive gravity driven thin film flow of a third grade fluid
with adiabatic free surface down an inclined plane”, Appl. Math. Mech. Ed. 30(3), 373-380.
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(a) Traditional Perturbation (b) Homotopy Perturbation
Figure 1: Velocity profile for different values of magnetic parameter M when δ = 1 , K = +1 , β = 0.001 ,
Λ =1
(a) Traditional Perturbation (b) Homotopy Perturbation
Figure 2: Temperature profile for different values of magnetic parameter M when δ = 1 , K = +1 ,
β = 0.001 , Λ = 1
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(a) Traditional Perturbation (b) Homotopy Perturbation
Figure 3: Velocity profile for different values of parameter K when δ = 1 , M = 5 , β = 0.001 , Λ = 1
(a) Traditional Perturbation (b) Homotopy Perturbation
Figure 4: Temperature profile for different values of parameter K when δ = 1 , M = 5 , β = 0.001 ,
Λ =1
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(a) Traditional Perturbation (b) Homotopy Perturbation
Figure 5: Velocity profile for different values of slip parameter Λ when δ = 1 , M = 5 , β = 0.001 ,
K = +1
(a) Traditional Perturbation (b) Homotopy Perturbation
Figure 6: Temperature profile for different values of slip parameter Λ when δ = 1 , M = 5 , β = 0.001 ,
K = +1
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IISTE journals can find the submission instruction on the following page:
http://www.iiste.org/Journals/
The IISTE editorial team promises to the review and publish all the qualified
submissions in a fast manner. All the journals articles are available online to the
readers all over the world without financial, legal, or technical barriers other than
those inseparable from gaining access to the internet itself. Printed version of the
journals is also available upon request of readers and authors.
IISTE Knowledge Sharing Partners
EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open
Archives Harvester, Bielefeld Academic Search Engine, Elektronische
Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial
Library , NewJour, Google Scholar