Contents lists available atScienceDirect
Computers and Mathematics with Applications
journal homepage:www.elsevier.com/locate/camwaThe application of homotopy perturbation method for MHD flows of
UCM fluids above porous stretching sheets
Behrouz Raftari, Ahmet Yildirim
∗Department of Mathematics, Islamic Azad University, Kermanshah branch, P.C. 6718997551, Kermanshah, Iran Department of Mathematics, Ege University, 35100, Bornova, Izmir, Turkey
a r t i c l e i n f o Article history:
Received 11 December 2009 Received in revised form 9 March 2010 Accepted 11 March 2010
Keywords:
Homotopy perturbation method MHD flow
UCM fluid
Porous stretching sheet
a b s t r a c t
In this study, by means of homotopy perturbation method (HPM) an approximate analytical solution of the magnetohydrodynamic (MHD) boundary layer flow of an upper-convected Maxwell (UCM) fluid over a porous stretching sheet is obtained. The main feature of the HPM is that it deforms a difficult problem into a set of problems which are easier to solve. HPM produces analytical expressions for the solution of nonlinear differential equations. The obtained analytic solution is in the form of an infinite power series. In this work, the analytical solution obtained by using only two terms from HPM solution. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear problems. Also it is shown that this method coincides with homotopy analysis method (HAM) for the studied problem.
©2010 Elsevier Ltd. All rights reserved.
1. Introduction
Sheet stretching is an important operation in polymer industry [1]. Production of plastic sheets and foils, for example,
involves extrusion of molten polymers through a slit die with the extrudate collected by a wind-up roll upon solidification. This process is normally accompanied with both heat and momentum transfer aspects. But there are also cases in which a plastic sheet may be stretched with no heat transfer involved. Cold drawing of plastic sheets, for example, is an important operation in which a plastic sheet is elongated in certain direction(s) in order to improve its mechanical properties in that
direction(s) [2–4]. In all such operations (with or without heat transfer involved) the force required to pull the sheet is one
of the most important design criteria. This force is known to depend to a large extent on the extensional viscosity of the sheet material. It also depends on the physical/rheological properties of the fluid surrounding the sheet. While traditionally Newtonian fluids are used for this purpose (e.g., water or air), but, in recent years it has been shown that there might be
some advantages (particularly in flows involving heat transfer [5]) if the fluid surrounding the sheet can be made viscoelastic,
say through the use of polymeric additives [6]. Alternatively, one may resort to injection or suction in order to modify flow
kinematics provided the sheet is porous at the first place [7]. And in cases where the fluid surrounding the sheet is electrically
conducting, one may equally well rely on applying a sufficiently strong magnetic field to modify flow kinematics [8,9].
Conceivably, one might also envisage cases in which a combination of all these methods might be involved simultaneously
to achieve the best result [10].
Among the techniques mentioned above to control flow kinematics, the idea of using magnetic fields appears to be the most attractive one both because of its ease of implementation and also because of its non-intrusive nature. The idea is not
new and has shown its effectiveness for both Newtonian and non-Newtonian fluids alike [11–16]. There have been extensive
efforts in the past, in both theoretical and experimental areas alike, to better understand such magnetohydrodynamic (MHD)
∗Corresponding author at: Department of Mathematics, Ege University, 35100, Bornova, Izmir, Turkey. Tel.: +90 232 388 4000; fax: +90 232 3881036.
E-mail address:ahmet.yildirim@ege.edu.tr(A. Yildirim).
0898-1221/$ – see front matter©2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.camwa.2010.03.018
B. Raftari, A. Yildirim / Computers and Mathematics with Applications 59 (2010) 3328–3337 3329 flows. For Newtonian fluids, theoretical results are in favor of the experimental observations. For non-Newtonian fluids, in contrast, theoretical results are not always in support of experimental findings. This is not surprising realizing the fact that most studies carried out in the past on MHD flows of non-Newtonian fluids were concerned primarily with simple
rheological models such as second-order model, or third-order model at best [5]. And this has been the case irrespective of
the fact that these widely-used rheological models are known to violate certain thermodynamical constraints [17,18]. To
this should be added the fact that these simple rheological models are known to be good only for fluids of low-elasticity in slow and/or slowly-varying flows. In practice, however, such restrictive conditions may not prevail. For reasons like these, the reliability of theoretical results obtained using such simple rheological models are in serious doubt.
Recently, Hayat et al. [19] tried to investigate MHD flow of a more realistic viscoelastic fluid model, i.e., the so-called
upper-convected Maxwell model, above a porous stretching sheet. Also a study on MHD flows of UCM fluids above porous
stretching sheets using HAM purposed by Sadeghy et al. [20]
One of the newest analytical methods to solve the mathematical problems is using both homotopy and perturbation methods. In recent years, the application of the homotopy perturbation method in nonlinear problems has been devoted by scientists and engineers, because this is to continuously deform a simple problem easy to solve into the difficult problem
under study. Homotopy techniques were applied to find all roots of nonlinear equations first in [21–28]. Recently, the
applications of homotopy theory becomes a powerful mathematical tool, when it is successfully coupled with perturbation
theory [29–39]. The interested reader can see [40–44] for last development and interpretation of HPM.
Recently a great deal of interest has been focused on the application of homotopy perturbation method to solve a wide variety of problems. In this work, we use the homotopy perturbation method to solve the highly nonlinear ODE to derive an approximate analytical solution. HPM is an analytical procedure for finding the solutions of problems which is based on the
constructing a homotopy with an embedding parameterpthat is considered as a small parameter.
The purpose of this paper is to apply homotopy perturbation method which is useful for finding the approximate analytical solution of MHD flows of UCM fluids above porous stretching sheets.
2. Equations of motion
The constitutive equation for a Maxwell fluid is
T
= −
pI+
S,
(1)whereTis the Cauchy stress tensor and the extra stress tensorSsatisfies
S
+
λ
dS dt−
LS−
SL >=
µ
A1, (2)in which
µ
is the viscosity,λ
is the relaxation time and the Rivlin–Ericksen tensorA1is defined throughA1
= ∇
V+
(
∇
V)
>.
(3)For the steady two-dimensional flow, the equations of continuity and momentum for the magnetohydrodynamic flow are
∂
u∂
x+
∂v
∂
y=
0,
(4)ρ
u∂
u∂
x+
v
∂
u∂
y= −
∂
p∂
x+
∂
Sxx∂
x+
∂
Sxy∂
y−
σ
B 2 0u,
(5)ρ
u∂v
∂
x+
v
∂v
∂
y= −
∂
p∂
x+
∂
Syx∂
x+
∂
Syy∂
y,
(6)where
ρ
is the fluid density,σ
is the electrical conductivity,B0is the constant applied magnetic field in they-direction andSxx
,
Sxy,
SyxandSyyare the components of the extra stress tensor. Using the following boundary layer approximations [45,46] u
=
O(
1),
v
=
O(δ),
x=
O(
1),
y=
O(δ),
(7) Txxρ
=
O(
1),
Txyρ
=
O(δ),
Tyyρ
=
O(δ
2),
(8)the flow in the absence of the pressure gradient is governed by Eq.(4)and
u
∂
u∂
x+
v
∂
u∂
y+
λ
u2∂
2u∂
x2+
v
2∂
2u∂
y2+
2uv
∂
2u∂
x∂
y=
v
∂
2u∂
y2−
σ
B2 0ρ
u,
(9)where
δ
being the boundary layer thickness. The appropriate boundary conditions on the flow areu
=
Bx,
v
=
V0 aty=
0,
Fig. 1. Effects of suction velocityRon HPM approximation off0
(β=0.3 andM=0.5).
whereV0
>
0 is the suction velocity andV0<
0 gives the injection velocity. Introducingη
=
r
Bν
y,
u=
Bxf 0(η),
v
= −
√
ν
B (11)the governing problem is transformed to
f000
−
M2f0−
f02+
ff00+
β(
2ff0f00−
f2f000)
=
0,
(12) f=
R,
f0=
1 atη
=
0,
f0
→
0 asη
→ ∞
.
(13)HereM2
=
σ
B20/ρB, β
=
λ
B,
R=
V0/√
ν
B, whereR>
0 corresponds to suction andR<
0 for injection. Eq.(12)isnonlinear differential equation which can be solved analytically by HPM. 3. Homotopy perturbation method
Now we will use the method in order to obtain the solution of Eq.(12). Assumingu
=
f, Eq.(12)can be written in thefollowing form:
u000
+
F(
u)
=
0,
(14)in which
F
(
u)
= −
M2u0−
u02+
uu00+
β(
2uu0u00−
u2u000).
(15)According to the homotopy perturbation method [23], we construct a homotopy in the form
u000
−
α
2u0+
p[
F(
u)
+
α
2u0] =
0,
(16)with the initial conditions
u
(
0)
=
R,
u0(
0)
=
1,
u0(
∞
)
=
0.
(17)Whenp
=
0, Eq.(16)becomes a linearized equation,u000−
α
2u0=
0, whereα
is an unknown parameter to be furtherdetermined; whenp
=
1, it turns out to be the original one. The embedded parameterpmonotonically increases from zeroto unit as the trivial problem,u000
−
α
2u0=
0, is continuously deformed to the original problem, Eq.(14). According to thehomotopy perturbation method, we assume that the solution to Eq.(16)may be written as a power series inp:
B. Raftari, A. Yildirim / Computers and Mathematics with Applications 59 (2010) 3328–3337 3331
Fig. 2. Effects of injection velocityRon HPM approximation off0
(β=0.3 andM=0.5).
Fig. 3. Effects of suction velocityRon HPM approximation off(β=0.3 andM=0.5).
Substituting Eq.(18)into Eq.(16)and equating the terms with the identical powers ofp, we have
p0
:
u0000−
α
2u00=
0,
u0(0)
=
R,
u00(0)
=
1,
u00(∞
)
=
0,
(19) p1:
u0001−
α
2u01−
(
u0)0 2+
u0u000−
β(
2u0u00u 00 0−
u 2 0u 000)
+
(α
2−
M2)
u0 0=
0,
u1(0)
=
0,
u01(0)
=
0,
u01(∞
)
=
0.
(20)The solution of Eq.(19)can be readily obtained, which reads
u0(η)
=
R+
1Fig. 4. Effects of injection velocityRon HPM approximation off(β=0.3 andM=0.5).
Fig. 5. Effects of relaxation timeβon HPM approximation off0
is drawn in case of suction (R=0.3 andM=0.5).
Substituting Eq.(21)into Eq.(20)results in
u0001
−
α
2u01=
(
M2+
Rα
+
β
R2α
2+
2β
Rα
+
β
−
α
2+
1)
e(
−
αη)
−
β
exp(
−
3αη).
(22)In caseR2
(
2β
+
1)
2−
(β
R2−
1)(
4M2+
3β
+
4) >
0 we can solveu1from Eq.(22)under the initial/boundary conditionsu1(0
)
=
0,
u01(0)
=
0 andu01(∞
)
=
0 and obtain the following solution with ease:u1(η)
= −
β
24α
3+
M2+
Rα
+
β
R2α
2+
2β
Rα
+
β
−
α
2+
1 2α
2η
exp(
−
αη)
+
β
24α
3exp(
−
3αη),
(23)B. Raftari, A. Yildirim / Computers and Mathematics with Applications 59 (2010) 3328–3337 3333
Fig. 6. Effects of relaxation timeβon HPM approximation off0
is drawn in case of injection (R= −0.3 andM=0.5).
Fig. 7. Effects of relaxation timeβon HPM approximation offis drawn in case of suction (R=0.3 andM=0.5).
which
α
=
−
R(
2β
+
1)
−
p
R2
(
2β
+
1)
2−
(β
R2−
1)(
4M2+
3β
+
4)
2
(β
R2−
1)
.
(24)Therefore, we obtain the first-order approximate solution forR2
(
2β
+
1)
2−
(β
R2−
1)(
4M2+
3β
+
4) >
0, which readsFig. 8. Effects of relaxation timeβon HPM approximation offis drawn in case of injection (R= −0.3 andM=0.5).
Fig. 9. Effects of MHD parameterMon HPM approximation off0
is drawn in case of suction (R=0.3 andβ=0).
4. Results and discussion
Figs. 1–12have been made in order to see the effects of the relaxation time
β
, suction/injection parameterRand the MHDparameterMon the velocity field.Figs. 1–4are made to see the effect of the suction and injection velocityRon the velocity
componentsf0andfrespectively for the magnetohydrodynamic flow. Similar to the results obtained by HAM [19], it is found
that for case of suction inFigs. 1and3, the velocity componentf0decreases butfincreases initially and then for the value of
η
between 1 and 2 it decreases with an increase inR. The boundary layer thickness decreases in both cases. For the case ofinjection inFigs. 2and4, we have the opposite effect.Figs. 5–8are drawn for the effects of non-dimensional relaxation time
B. Raftari, A. Yildirim / Computers and Mathematics with Applications 59 (2010) 3328–3337 3335
Fig. 10. Effects of MHD parameterMon HPM approximation off0
is drawn in case of injection (R= −0.3 andβ=0).
Fig. 11. Effects of MHD parameterMon HPM approximation offis drawn in case of suction (R=0.3 andβ=0).
f0andfdecrease and the boundary layer thickness decreases. But in the case of injection both decreases with an increase of.
Figs. 9–12are sketched in order to see the effects of MHD parameterMon the velocity componentsf0andf. These figures
show thatf0andf decreases by increasingMfor the case of suction and injection. InFigs. 1–12we find a good agreement
between the results obtained by Hayat et al. [19] and present method.
We, therefore, can easily obtain the wall shear stress which reads
f00
(
0)
=
u00(
0).
(26)For different
β
,MandRthe values of the wall shear stress are illustrated inTables 1and2. In these Tables, we present aFig. 12. Effects of MHD parameterMon HPM approximation offis drawn in case of injection (R= −0.3 andβ=0).
Table 1
Comparison of the values off00(
0)obtained by HPM and HAM for the case of suction (R=0.3).
β M HPM (present method) HAM [20]
0 0 −1.161187 −1.1589 5 −5.251225 −5.25119 10 −10.200995 −10.201 5 0 −7.633621 −7.0737 5 −11.228550 −10.8399 10 −17.305342 −17.0558 10 0 −64.418655 −64.1885 5 −68.023447 −67.7827 10 −77.157946 −76.9326 Table 2
Comparison of the values off00(
0)obtained by HPM and HAM for the case of injection (R= −0.3).
β M HPM (present method) HAM [20]
0 0 −0.861187 −0.861506 5 −4.951225 −4.95122 10 −9.900995 −9.901 5 0 −3.804673 −0.700263 5 −5.575188 −4.74886 10 −11.403791 −11.0275 10 0 −6.051008 −0.269701 5 −6.198820 −4.15231 10 −14.520849 −13.5784
Results show that the methods are in a good agreement with each other. In the case injection (forM
=
0 andβ >
0),wesee a remarkable difference between the results obtained by HPM and HAM. But our results are more acceptable than results
obtained by Sadeghy et al. [20] because in the all of cases wall shear stress decreases when the value of
β
increases.5. Conclusions
In this study, we have applied homotopy perturbation method in solving the MHD flows of UCM fluids above porous stretching sheets. Comparison between the HPM and HAM methods for the studied problem show a remarkable agreement and reveal that the HPM need less work. It is important that we applied HPM for the equation in unbounded domain without
B. Raftari, A. Yildirim / Computers and Mathematics with Applications 59 (2010) 3328–3337 3337 using Padé approximants. It is of utter simplicity and effectiveness; the first-order approximate solution leads to a very highly accurate solution. It is a promising method and might find wide applications.
Acknowledgements
The authors are grateful to the referees for their invaluable suggestions and comments for the improvement of the paper. The authors are grateful to Islamic Azad University, Kermanshah branch for financial support through Research Project (The optimal homotopy perturbation method for solving the nonlinear ODE arising in magnetohydrodynamics), for completing this work.
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