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Research ArticleSpectral-Homotopy Perturbation Method forSolving Governing MHD Jeffery-Hamel Problem
Ahmed A Khidir
Faculty of Technology of Mathematical Sciences and Statistics Al-Neelain University Algamhoria StreetPO Box 12702 Khartoum Sudan
Correspondence should be addressed to Ahmed A Khidir ahmedkhidiryahoocom
Received 20 February 2014 Revised 3 July 2014 Accepted 7 July 2014 Published 14 July 2014
Academic Editor Xavier Ferrieres
Copyright copy 2014 Ahmed A Khidir This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We present a new modification of the homotopy perturbation method (HPM) for solving nonlinear boundary value problemsThe technique is based on the standard homotopy perturbation method and blending of the Chebyshev pseudospectral methodsThe implementation of the new approach is demonstrated by solving the MHD Jeffery-Hamel flow and the effect of MHD on theflow has been discussed Comparisons are made between the proposed technique the previous studies the standard homotopyperturbation method and the numerical solutions to demonstrate the applicability validity and high accuracy of the presentedapproach The results demonstrate that the new modification is more efficient and converges faster than the standard homotopyperturbation method at small orders The MATLAB software has been used to solve all the equations in this study
1 Introduction
The incompressible viscous fluid flow through convergent-divergent channels is one of the most applicable casesin fluid mechanics civil environmental mechanical andbiomechanical engineering The mathematical investigationsof this problem were pioneered by Jeffery [1] and Hamel [2]They presented an exact similarity solution of the Navier-Stokes equations in the special case of two-dimensional flowthrough a channel with inclined plane walls meeting at avertex and with a source or sink at the vertex and have beenextensively studied by several authors and discussed in manytextbooks for example [3 4] In the PhD thesis [5] wefind that Jeffery-Hamel flow used as asymptotic boundaryconditions to examine a steady of two-dimensional flow ofa viscous fluid in a channel But here certain symmetricsolutions of the flow has been considered by Sobey andDrazin [6] Although asymmetric solutions are both possibleand of physical interest
The classical Jeffery-Hamel problem was extended byAxford [7] to include the effects of an external magnetic fieldon an electrically conducting fluid in MHD Jeffery-Hamelproblems there are two additional nondimensional param-eters that determine the solutions namely the magnetic
Reynolds number and the Hartmann number Most scien-tific problems such as Jeffery-Hamel flows and other fluidmechanic problems are inherently in formof nonlinear differ-ential equations Except a limited number of these problemsmost of them do not have exact solution and some of thesolved by numerical methods Therefore these nonlinearequations should be solved using other methods Thereforemany different methods have recently introduced some waysto obtain analytical solution for these nonlinear problemssuch as the homotopy perturbation method (HPM) by He[8 9] the homotopy analysis method (HAM) by Liao [1011] the adomian decomposition method (ADM) [12ndash14] thevariational iterationmethod (VIM)byHe [15] the differentialtransformation method by Zhou [16] and recently spectralhomotopy analysis method (SHAM) by Motsa et al [17] Inthe numerical method stability and convergence should beconsidered so as to avoid divergence or inappropriate resultsSome of thesemethods used small parameter in the equationTherefore finding the small parameter and exerting it into theequation are deficiencies of these methods
Nonnumerical approaches include the classical power-series method and its variants for systems of nonlinear dif-ferential equations with small or large embedded parameterssuch as the homotopy perturbation method However it is
Hindawi Publishing CorporationJournal of Computational Methods in PhysicsVolume 2014 Article ID 512702 7 pageshttpdxdoiorg1011552014512702
2 Journal of Computational Methods in Physics
well-known that most of these perturbation solutions are notvalid in the whole physical region These methods do notguarantee the convergence of the series solution and theperturbation approximations may be only valid for weaklynonlinear problems Further disadvantages of perturbationmethods are that (i) they require the presence of a largeor small parameter in the problem while nonperturbationmethods require a careful selection of initial approximationsand linear operators and (ii) linearization usually leads todifficulties in the integration of higher order deformationequations
In this work we present an alternative and improved formof the HPM called spectral-homotopy perturbation method(SHPM) that blends the traditional homotopy perturbationmethodwith theChebyshev spectral collocationmethodTheadvantage of this approach is that it is more flexible thanHPM for choosing a linear operator and initial guess InHPM one is restricted to choosing a linear operator andinitial approximation that would make the integration ofthe higher-order differential equations possible whereas theSHPM allows us to have a wider range of selecting linearoperators and aninitial guessmay be used as long as it satisfiesthe boundary conditions
The aim of this study is to apply spectral homotopy per-turbationmethod (SHPM) to find an approximate solution tothe nonlinear differential equation governing MHD Jeffery-Hamel flowWe havemade a comparison between the currentresults and other methods with the numerical solution Theresults proves the applicability accuracy and efficiency of the(SHPM)
2 Mathematical Formulation
Consider the steady two-dimensional flow of an incompress-ible conducting viscous fluid from a source or sink at theintersection between two rigid plane walls that the angelbetween them is 2120572 The grid walls are considered to bedivergent if 120572 gt 0 and convergent if 120572 lt 0 We assumethat the velocity is only along radial direction and depend on119903 and 120579 where 119903 and 120579 are radial and angular coordinatesrespectively so that k = (119906(119903 120579) 0) only as shown in Figure 1Using continuity equation and Navier-Stokes equations inpolar coordinates one has
where119880max is the velocity at the center of the channel (119903 = 0)
and 1198672
= 1205901198612
0120588] is the square of the Hartmann number
Now we solve (6) by using (SHPM)
3 The Homotopy Perturbation Method
The homotopy perturbation method is a combination of theclassical perturbation technique and homotopy technique
Journal of Computational Methods in Physics 3
To illustrate the basic ideas of the HPM we consider thefollowing nonlinear differential equation
119860 (119906) minus 119891 (r) = 0 r isin Ω (9)
with the boundary conditions
119861(119906120597119906
120597119899) = 0 r isin Γ (10)
where 119860 is a general operator 119861 is a boundary operator 119891(r)is a known analytic function and Γ is the boundary of thedomain Ω
The operator 119860 can generally speaking be divided intotwo parts 119871 and 119873 where 119871 is linear while 119873 is nonlinear(9) therefore can be written as follows
where 119901 isin [0 1] is an embedding parameter and 1199060is
an initial approximation of (9) wich satisfies the boundaryconditions Obviously from (12) we have
119867(V 0) = 119871 (V) minus 119871 (1199060) = 0
119867 (V 1) = 119860 (V) minus 119891 (r) = 0
(14)
The changing process of 119901 from 0 to 1 is just that of V(r 119901)
from 1199060(r) to 119906(r) In topology this is called deformation and
119871(V) minus 119871(1199060) 119860(V) minus 119891(r) are called homotopic
According to HPM we can first use the embeddingparameter 119901 as small parameter and assume that the solutionof (12) can be written as a power series ip 1199014
V = V0+ 119901V1+ 1199012V2+ sdot sdot sdot (15)
setting 119901 = 1 results in the approximation to the solution of(9)
119906 = lim119901rarr1
V = V0+ V1+ V2+ sdot sdot sdot (16)
The series (16) is convergent for most cases however theconvergent depends upon the nonlinear operator119860(V) Somecriteria suggested for convergence of the series (16) and thefollowing opinions are suggested by He [18 19]
(1) The second derivative of 119873(V) with respect to V mustbe small because the parameter 119901 may be relativelylarge that is 119901 rarr 1
(2) The norm of 119871minus1
(120597119873120597V)must be smaller than one sothat the series converges
4 The Spectral HomotopyPerturbation Method
To solve the nonlinear ordinary differential equation (6)using the SHPM we start by transfom the domain of theproblem from [0 1] to the domain [minus1 1] on which theChebyshev spectral method can be implemented using thealgebraic mapping
119909 = 2120578 minus 1 119909 isin [minus1 1] (17)
It is also convenient to make the boundary conditionshomogeneous by making use of the transformation
119865 (120578) = 119891 (119909) + 1 minus 1205782 (18)
Substituting (18) and (17) in (6) and the boundary conditions(7) gives
And119873(119865) is the nonlinear part of (19) Substitute (22) in (25)to get
1198893119865
1198891199093
+ 1198921(119909)
119889119865
119889119909+ 1198922(119909) 119865 + (1 minus 119901)
times (11988931198910
1198891199093
+ 1198921(119909)
1198891198910
119889119909+ 1198922(119909) 1198910)
+ 119901(1
2120572Re119865
119889119865
119889119909minus 119866 (119909))
= 0
(27)
Wemake a comparison between the power of1199011 in both sidesof (27) to obtain the following system of equations
A1198911= B1 (28)
subject to the boundary conditions
1198911(minus1) = 119891
1(1) = 119891
1(minus1) = 0 (29)
where
A = [D3+ diag [119892
1(119909119894)]D + diag [119892
2(119909119894)]]
B1 = minus[A1198910+
1
2120572Re119891
0(D21198910) minus 119866 (119909
119894)]
119879
(30)
where 119879 denotes transpose diag [] is a diagonal matrix ofsize (119873 + 1) times (119873 + 1) and D is the Chebyshev spectraldifferentiation matrix whose entries (see [20 21]) are givenby
D119895119896
=
119888119895
119888119896
(minus1)119895+119896
119909119895minus 119909119896
119895 = 119896 119895 119896 = 0 1 119873
D119896119896
= minus119909119896
2 (1 minus 1199092
119896)
119896 = 1 2 119873 minus 1
D00
=21198732+ 1
6= minusD
119873119873
(31)
Here 1198880
= 119888119873
= 2 and 119888119895= 1 with 1 le 119895 le 119873 minus 1 119909
119895are the
Chebyshev collocation points (see [20]) defined by
119909119895= cos
119895120587
119873 119895 = 0 1 2 119873 (32)
the solution of (28) can be given by
1198911= Aminus1B1 (33)
To get more higher order approximations for (19) we com-pare between the coefficients of 119901119894 (119894 = 2 3 4 ) in (25) toobtain the following approximations
119891119894= Aminus1Bi (34)
subject to the boundary conditions
119891119894(minus1) = 119891
119894(1) = 119891
1015840
119894(minus1) = 0 (35)
where
Bi = minus1
2120572Re[
119894minus1
sum
119899=0
119891119899(D119891119894minus1minus119899
)] 119894 ge 2 (36)
Thematrix119860 has dimensions (119873+1)times(119873+1)while matrices119861119894and 119891
119894have a dimensions (119873 + 1) times 1
To implement the boundary conditions (29) and (35) tothe systems (28) and (34) respectively we delete the first andthe last rows and columns of 119860 and delete the first and lastrows of and 119891
1and 119891
119894 also we replace the results of last row
of the modified matrix119860 and setting the results of last row ofthe modified matrices 119861
1and 119861
119894to be zeroThen the solution
of (19) is given by substituting the series119891119894in (26) after setting
119901 = 1
5 Results and Discussion
In this section we present the obtained results of the solutionsfor MHD Jeffery-Hame flow using the HPM SHPM anda numerical solution Here we used the inbuilt MATLABboundary value problems solver bvp4c for the numericalsolution approach In generating the presented results it wasdetermined through numerical experimentation that 119873 =
60 Table 1 shows a comparison between the HPM SHPMand numerical approximate solutions of 119865(120578) The tableshows that the results of the present method are in excellentagreement with those of the numerical ones Also in Table 2we give a comparison of the SHPM results for divergent andconvergent channels and fixed values of 119867 and Re when120578 is varied at different orders of approximation against thenumerical results It can be seen from Table 2 that SHPMresults converge rapidly to the numerical solution Table 3gives a comparison of the differential transformationmethod(DTM) HPM homotopy analysis method (HAM) given byJoneidi et al [22] and SHPM results for 119865(120578) against thenumerical results when 120578 is varied It can be seen from thistable that the approximate solution of MHD Jeffery-Hamelflows obtained by SHPM is very accurate and it is convergesmuch more rapidly to the numerical result compared to theDTM HPM and HAM
Figures 2(a) and 2(b) show firstly the influence of themagnetic field parameter on the velocity profile for diver-gent and convergent channels and secondly a comparisonbetween the present results and numerical results to give asense of the accuracy and convergence rate of the SHPMThe figures indicate that there is very good match betweenthe two sets of results even at very low orders of SHPMapproximations series compared with the numerical resultsThese findings firmly establish the SHPM as an accurateand alternative to the HPM Also it can seen that the fluidvelocity increases with increasingHartman numbers for bothconvergent and divergent channels
Journal of Computational Methods in Physics 5
Table 1 Comparison between the HPM SHPM and numerical results of F(120578) for different values of 119867 when 120572 = 75∘ and Re = 50
Table 2 Comparison of the values of the SHPM approximate solutions for F(120578) with the numerical solution for various values of 120578 when119867 = 100 and Re = 50 for divergent and convergent channels
In this study the SHPM were applied successfully to findan approximate solution of a nonlinear MHD Jeffery-HamelflowsThe effect of themagnetic field parameter on the veloc-ity profile for convergent and divergent channels has beendetermined It could be noticed that by increasing magneticfield parameter the velocity profile increases resulting in
a rise in the flow rate for both convergent and divergentchannels The obtained results compared with the numericalsolution of the governing nonlinear equation and with DTMHPM and HAM Also the tables and figures clearly showhigh accuracy of the method and the convergent is very fastto solve MHD Jeffery-Hamel problem An important aspectof this work has been the need to prove the computationalefficiency and accuracy of the SHPM in solving nonlinear
6 Journal of Computational Methods in Physics
0 02 04 06 08 10
02
04
06
08
1
H = 0H = 250H = 500H = 1000
120572 = 5
F(120578)
120578
(a)
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
H = 0H = 250H = 500H = 1000120572 = minus5
F(120578)
120578
(b)
Figure 2 Comparison between the numerical solution and SHPM of 119865(120578) for divergent and convergent channels when Re = 50
differential equations and this method is in general moreaccurate than theHPMThe results in this paper confirm thatthe SHPM is a powerful and efficient technique for findingsolutions for nonlinear differential equations in differentfields of science and engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] G B Jeffery ldquoThe two-dimensional steady motion of a viscousfluidrdquo Philosophical Magazine vol 6 pp 455ndash465 1915
[2] G Hamel ldquoSpiralformige Bewegungen zaher FlussigkeitenrdquoJahresbericht der Deutschen Mathematiker-Vereinigung vol 25pp 34ndash60 1916
[3] L Rosenhead ldquoThe steady two-dimensional radial flow ofviscous fluid between two inclined plane wallsrdquo Proceedings ofthe Royal Society A vol 175 pp 436ndash467 1940
[4] A McAlpine and P G Drazin ldquoOn the spatio-temporal devel-opment of small perturbations of Jeffery-Hamel flowsrdquo FluidDynamics Research vol 22 no 3 pp 123ndash138 1998
[5] R M Sadri Channel entrance flow [PhD thesis] Departmentof Mechanical Engineering the University of Western Ontario1997
[6] I J Sobey and P G Drazin ldquoBifurcations of two-dimensionalchannel flowsrdquo Journal of Fluid Mechanics vol 171 pp 263ndash2871986
[7] W I Axford ldquoThe magnetohydrodynamic Jeffrey-Hamel prob-lem for a weakly conducting fluidrdquo The Quarterly Journal ofMechanics and Applied Mathematics vol 14 pp 335ndash351 1961
[8] J He ldquoA review on some new recently developed nonlinear ana-lytical techniquesrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 1 no 1 pp 51ndash70 2000
[9] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 2 pp 207ndash208 2005
[10] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[11] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[12] Q Esmaili A Ramiar E Alizadeh and D D Ganji ldquoAnapproximation of the analytical solution of the Jeffery-Hamelflow by decomposition methodrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 372 no 19 pp 3434ndash34392008
[13] O D Makinde and P Y Mhone ldquoHermite-Pade approximationapproach to MHD Jeffery-Hamel flowsrdquo Applied Mathematicsand Computation vol 181 no 2 pp 966ndash972 2006
[14] O D Makinde ldquoEffect of arbitrary magnetic Reynolds numberonMHDflows in convergent-divergent channelsrdquo InternationalJournal of Numerical Methods for Heat amp Fluid Flow vol 18 no5-6 pp 697ndash707 2008
[15] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[17] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[18] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] J H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000
Journal of Computational Methods in Physics 7
[21] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[22] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
well-known that most of these perturbation solutions are notvalid in the whole physical region These methods do notguarantee the convergence of the series solution and theperturbation approximations may be only valid for weaklynonlinear problems Further disadvantages of perturbationmethods are that (i) they require the presence of a largeor small parameter in the problem while nonperturbationmethods require a careful selection of initial approximationsand linear operators and (ii) linearization usually leads todifficulties in the integration of higher order deformationequations
In this work we present an alternative and improved formof the HPM called spectral-homotopy perturbation method(SHPM) that blends the traditional homotopy perturbationmethodwith theChebyshev spectral collocationmethodTheadvantage of this approach is that it is more flexible thanHPM for choosing a linear operator and initial guess InHPM one is restricted to choosing a linear operator andinitial approximation that would make the integration ofthe higher-order differential equations possible whereas theSHPM allows us to have a wider range of selecting linearoperators and aninitial guessmay be used as long as it satisfiesthe boundary conditions
The aim of this study is to apply spectral homotopy per-turbationmethod (SHPM) to find an approximate solution tothe nonlinear differential equation governing MHD Jeffery-Hamel flowWe havemade a comparison between the currentresults and other methods with the numerical solution Theresults proves the applicability accuracy and efficiency of the(SHPM)
2 Mathematical Formulation
Consider the steady two-dimensional flow of an incompress-ible conducting viscous fluid from a source or sink at theintersection between two rigid plane walls that the angelbetween them is 2120572 The grid walls are considered to bedivergent if 120572 gt 0 and convergent if 120572 lt 0 We assumethat the velocity is only along radial direction and depend on119903 and 120579 where 119903 and 120579 are radial and angular coordinatesrespectively so that k = (119906(119903 120579) 0) only as shown in Figure 1Using continuity equation and Navier-Stokes equations inpolar coordinates one has
where119880max is the velocity at the center of the channel (119903 = 0)
and 1198672
= 1205901198612
0120588] is the square of the Hartmann number
Now we solve (6) by using (SHPM)
3 The Homotopy Perturbation Method
The homotopy perturbation method is a combination of theclassical perturbation technique and homotopy technique
Journal of Computational Methods in Physics 3
To illustrate the basic ideas of the HPM we consider thefollowing nonlinear differential equation
119860 (119906) minus 119891 (r) = 0 r isin Ω (9)
with the boundary conditions
119861(119906120597119906
120597119899) = 0 r isin Γ (10)
where 119860 is a general operator 119861 is a boundary operator 119891(r)is a known analytic function and Γ is the boundary of thedomain Ω
The operator 119860 can generally speaking be divided intotwo parts 119871 and 119873 where 119871 is linear while 119873 is nonlinear(9) therefore can be written as follows
where 119901 isin [0 1] is an embedding parameter and 1199060is
an initial approximation of (9) wich satisfies the boundaryconditions Obviously from (12) we have
119867(V 0) = 119871 (V) minus 119871 (1199060) = 0
119867 (V 1) = 119860 (V) minus 119891 (r) = 0
(14)
The changing process of 119901 from 0 to 1 is just that of V(r 119901)
from 1199060(r) to 119906(r) In topology this is called deformation and
119871(V) minus 119871(1199060) 119860(V) minus 119891(r) are called homotopic
According to HPM we can first use the embeddingparameter 119901 as small parameter and assume that the solutionof (12) can be written as a power series ip 1199014
V = V0+ 119901V1+ 1199012V2+ sdot sdot sdot (15)
setting 119901 = 1 results in the approximation to the solution of(9)
119906 = lim119901rarr1
V = V0+ V1+ V2+ sdot sdot sdot (16)
The series (16) is convergent for most cases however theconvergent depends upon the nonlinear operator119860(V) Somecriteria suggested for convergence of the series (16) and thefollowing opinions are suggested by He [18 19]
(1) The second derivative of 119873(V) with respect to V mustbe small because the parameter 119901 may be relativelylarge that is 119901 rarr 1
(2) The norm of 119871minus1
(120597119873120597V)must be smaller than one sothat the series converges
4 The Spectral HomotopyPerturbation Method
To solve the nonlinear ordinary differential equation (6)using the SHPM we start by transfom the domain of theproblem from [0 1] to the domain [minus1 1] on which theChebyshev spectral method can be implemented using thealgebraic mapping
119909 = 2120578 minus 1 119909 isin [minus1 1] (17)
It is also convenient to make the boundary conditionshomogeneous by making use of the transformation
119865 (120578) = 119891 (119909) + 1 minus 1205782 (18)
Substituting (18) and (17) in (6) and the boundary conditions(7) gives
And119873(119865) is the nonlinear part of (19) Substitute (22) in (25)to get
1198893119865
1198891199093
+ 1198921(119909)
119889119865
119889119909+ 1198922(119909) 119865 + (1 minus 119901)
times (11988931198910
1198891199093
+ 1198921(119909)
1198891198910
119889119909+ 1198922(119909) 1198910)
+ 119901(1
2120572Re119865
119889119865
119889119909minus 119866 (119909))
= 0
(27)
Wemake a comparison between the power of1199011 in both sidesof (27) to obtain the following system of equations
A1198911= B1 (28)
subject to the boundary conditions
1198911(minus1) = 119891
1(1) = 119891
1(minus1) = 0 (29)
where
A = [D3+ diag [119892
1(119909119894)]D + diag [119892
2(119909119894)]]
B1 = minus[A1198910+
1
2120572Re119891
0(D21198910) minus 119866 (119909
119894)]
119879
(30)
where 119879 denotes transpose diag [] is a diagonal matrix ofsize (119873 + 1) times (119873 + 1) and D is the Chebyshev spectraldifferentiation matrix whose entries (see [20 21]) are givenby
D119895119896
=
119888119895
119888119896
(minus1)119895+119896
119909119895minus 119909119896
119895 = 119896 119895 119896 = 0 1 119873
D119896119896
= minus119909119896
2 (1 minus 1199092
119896)
119896 = 1 2 119873 minus 1
D00
=21198732+ 1
6= minusD
119873119873
(31)
Here 1198880
= 119888119873
= 2 and 119888119895= 1 with 1 le 119895 le 119873 minus 1 119909
119895are the
Chebyshev collocation points (see [20]) defined by
119909119895= cos
119895120587
119873 119895 = 0 1 2 119873 (32)
the solution of (28) can be given by
1198911= Aminus1B1 (33)
To get more higher order approximations for (19) we com-pare between the coefficients of 119901119894 (119894 = 2 3 4 ) in (25) toobtain the following approximations
119891119894= Aminus1Bi (34)
subject to the boundary conditions
119891119894(minus1) = 119891
119894(1) = 119891
1015840
119894(minus1) = 0 (35)
where
Bi = minus1
2120572Re[
119894minus1
sum
119899=0
119891119899(D119891119894minus1minus119899
)] 119894 ge 2 (36)
Thematrix119860 has dimensions (119873+1)times(119873+1)while matrices119861119894and 119891
119894have a dimensions (119873 + 1) times 1
To implement the boundary conditions (29) and (35) tothe systems (28) and (34) respectively we delete the first andthe last rows and columns of 119860 and delete the first and lastrows of and 119891
1and 119891
119894 also we replace the results of last row
of the modified matrix119860 and setting the results of last row ofthe modified matrices 119861
1and 119861
119894to be zeroThen the solution
of (19) is given by substituting the series119891119894in (26) after setting
119901 = 1
5 Results and Discussion
In this section we present the obtained results of the solutionsfor MHD Jeffery-Hame flow using the HPM SHPM anda numerical solution Here we used the inbuilt MATLABboundary value problems solver bvp4c for the numericalsolution approach In generating the presented results it wasdetermined through numerical experimentation that 119873 =
60 Table 1 shows a comparison between the HPM SHPMand numerical approximate solutions of 119865(120578) The tableshows that the results of the present method are in excellentagreement with those of the numerical ones Also in Table 2we give a comparison of the SHPM results for divergent andconvergent channels and fixed values of 119867 and Re when120578 is varied at different orders of approximation against thenumerical results It can be seen from Table 2 that SHPMresults converge rapidly to the numerical solution Table 3gives a comparison of the differential transformationmethod(DTM) HPM homotopy analysis method (HAM) given byJoneidi et al [22] and SHPM results for 119865(120578) against thenumerical results when 120578 is varied It can be seen from thistable that the approximate solution of MHD Jeffery-Hamelflows obtained by SHPM is very accurate and it is convergesmuch more rapidly to the numerical result compared to theDTM HPM and HAM
Figures 2(a) and 2(b) show firstly the influence of themagnetic field parameter on the velocity profile for diver-gent and convergent channels and secondly a comparisonbetween the present results and numerical results to give asense of the accuracy and convergence rate of the SHPMThe figures indicate that there is very good match betweenthe two sets of results even at very low orders of SHPMapproximations series compared with the numerical resultsThese findings firmly establish the SHPM as an accurateand alternative to the HPM Also it can seen that the fluidvelocity increases with increasingHartman numbers for bothconvergent and divergent channels
Journal of Computational Methods in Physics 5
Table 1 Comparison between the HPM SHPM and numerical results of F(120578) for different values of 119867 when 120572 = 75∘ and Re = 50
Table 2 Comparison of the values of the SHPM approximate solutions for F(120578) with the numerical solution for various values of 120578 when119867 = 100 and Re = 50 for divergent and convergent channels
In this study the SHPM were applied successfully to findan approximate solution of a nonlinear MHD Jeffery-HamelflowsThe effect of themagnetic field parameter on the veloc-ity profile for convergent and divergent channels has beendetermined It could be noticed that by increasing magneticfield parameter the velocity profile increases resulting in
a rise in the flow rate for both convergent and divergentchannels The obtained results compared with the numericalsolution of the governing nonlinear equation and with DTMHPM and HAM Also the tables and figures clearly showhigh accuracy of the method and the convergent is very fastto solve MHD Jeffery-Hamel problem An important aspectof this work has been the need to prove the computationalefficiency and accuracy of the SHPM in solving nonlinear
6 Journal of Computational Methods in Physics
0 02 04 06 08 10
02
04
06
08
1
H = 0H = 250H = 500H = 1000
120572 = 5
F(120578)
120578
(a)
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
H = 0H = 250H = 500H = 1000120572 = minus5
F(120578)
120578
(b)
Figure 2 Comparison between the numerical solution and SHPM of 119865(120578) for divergent and convergent channels when Re = 50
differential equations and this method is in general moreaccurate than theHPMThe results in this paper confirm thatthe SHPM is a powerful and efficient technique for findingsolutions for nonlinear differential equations in differentfields of science and engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] G B Jeffery ldquoThe two-dimensional steady motion of a viscousfluidrdquo Philosophical Magazine vol 6 pp 455ndash465 1915
[2] G Hamel ldquoSpiralformige Bewegungen zaher FlussigkeitenrdquoJahresbericht der Deutschen Mathematiker-Vereinigung vol 25pp 34ndash60 1916
[3] L Rosenhead ldquoThe steady two-dimensional radial flow ofviscous fluid between two inclined plane wallsrdquo Proceedings ofthe Royal Society A vol 175 pp 436ndash467 1940
[4] A McAlpine and P G Drazin ldquoOn the spatio-temporal devel-opment of small perturbations of Jeffery-Hamel flowsrdquo FluidDynamics Research vol 22 no 3 pp 123ndash138 1998
[5] R M Sadri Channel entrance flow [PhD thesis] Departmentof Mechanical Engineering the University of Western Ontario1997
[6] I J Sobey and P G Drazin ldquoBifurcations of two-dimensionalchannel flowsrdquo Journal of Fluid Mechanics vol 171 pp 263ndash2871986
[7] W I Axford ldquoThe magnetohydrodynamic Jeffrey-Hamel prob-lem for a weakly conducting fluidrdquo The Quarterly Journal ofMechanics and Applied Mathematics vol 14 pp 335ndash351 1961
[8] J He ldquoA review on some new recently developed nonlinear ana-lytical techniquesrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 1 no 1 pp 51ndash70 2000
[9] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 2 pp 207ndash208 2005
[10] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[11] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[12] Q Esmaili A Ramiar E Alizadeh and D D Ganji ldquoAnapproximation of the analytical solution of the Jeffery-Hamelflow by decomposition methodrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 372 no 19 pp 3434ndash34392008
[13] O D Makinde and P Y Mhone ldquoHermite-Pade approximationapproach to MHD Jeffery-Hamel flowsrdquo Applied Mathematicsand Computation vol 181 no 2 pp 966ndash972 2006
[14] O D Makinde ldquoEffect of arbitrary magnetic Reynolds numberonMHDflows in convergent-divergent channelsrdquo InternationalJournal of Numerical Methods for Heat amp Fluid Flow vol 18 no5-6 pp 697ndash707 2008
[15] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[17] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[18] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] J H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000
Journal of Computational Methods in Physics 7
[21] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[22] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
To illustrate the basic ideas of the HPM we consider thefollowing nonlinear differential equation
119860 (119906) minus 119891 (r) = 0 r isin Ω (9)
with the boundary conditions
119861(119906120597119906
120597119899) = 0 r isin Γ (10)
where 119860 is a general operator 119861 is a boundary operator 119891(r)is a known analytic function and Γ is the boundary of thedomain Ω
The operator 119860 can generally speaking be divided intotwo parts 119871 and 119873 where 119871 is linear while 119873 is nonlinear(9) therefore can be written as follows
where 119901 isin [0 1] is an embedding parameter and 1199060is
an initial approximation of (9) wich satisfies the boundaryconditions Obviously from (12) we have
119867(V 0) = 119871 (V) minus 119871 (1199060) = 0
119867 (V 1) = 119860 (V) minus 119891 (r) = 0
(14)
The changing process of 119901 from 0 to 1 is just that of V(r 119901)
from 1199060(r) to 119906(r) In topology this is called deformation and
119871(V) minus 119871(1199060) 119860(V) minus 119891(r) are called homotopic
According to HPM we can first use the embeddingparameter 119901 as small parameter and assume that the solutionof (12) can be written as a power series ip 1199014
V = V0+ 119901V1+ 1199012V2+ sdot sdot sdot (15)
setting 119901 = 1 results in the approximation to the solution of(9)
119906 = lim119901rarr1
V = V0+ V1+ V2+ sdot sdot sdot (16)
The series (16) is convergent for most cases however theconvergent depends upon the nonlinear operator119860(V) Somecriteria suggested for convergence of the series (16) and thefollowing opinions are suggested by He [18 19]
(1) The second derivative of 119873(V) with respect to V mustbe small because the parameter 119901 may be relativelylarge that is 119901 rarr 1
(2) The norm of 119871minus1
(120597119873120597V)must be smaller than one sothat the series converges
4 The Spectral HomotopyPerturbation Method
To solve the nonlinear ordinary differential equation (6)using the SHPM we start by transfom the domain of theproblem from [0 1] to the domain [minus1 1] on which theChebyshev spectral method can be implemented using thealgebraic mapping
119909 = 2120578 minus 1 119909 isin [minus1 1] (17)
It is also convenient to make the boundary conditionshomogeneous by making use of the transformation
119865 (120578) = 119891 (119909) + 1 minus 1205782 (18)
Substituting (18) and (17) in (6) and the boundary conditions(7) gives
And119873(119865) is the nonlinear part of (19) Substitute (22) in (25)to get
1198893119865
1198891199093
+ 1198921(119909)
119889119865
119889119909+ 1198922(119909) 119865 + (1 minus 119901)
times (11988931198910
1198891199093
+ 1198921(119909)
1198891198910
119889119909+ 1198922(119909) 1198910)
+ 119901(1
2120572Re119865
119889119865
119889119909minus 119866 (119909))
= 0
(27)
Wemake a comparison between the power of1199011 in both sidesof (27) to obtain the following system of equations
A1198911= B1 (28)
subject to the boundary conditions
1198911(minus1) = 119891
1(1) = 119891
1(minus1) = 0 (29)
where
A = [D3+ diag [119892
1(119909119894)]D + diag [119892
2(119909119894)]]
B1 = minus[A1198910+
1
2120572Re119891
0(D21198910) minus 119866 (119909
119894)]
119879
(30)
where 119879 denotes transpose diag [] is a diagonal matrix ofsize (119873 + 1) times (119873 + 1) and D is the Chebyshev spectraldifferentiation matrix whose entries (see [20 21]) are givenby
D119895119896
=
119888119895
119888119896
(minus1)119895+119896
119909119895minus 119909119896
119895 = 119896 119895 119896 = 0 1 119873
D119896119896
= minus119909119896
2 (1 minus 1199092
119896)
119896 = 1 2 119873 minus 1
D00
=21198732+ 1
6= minusD
119873119873
(31)
Here 1198880
= 119888119873
= 2 and 119888119895= 1 with 1 le 119895 le 119873 minus 1 119909
119895are the
Chebyshev collocation points (see [20]) defined by
119909119895= cos
119895120587
119873 119895 = 0 1 2 119873 (32)
the solution of (28) can be given by
1198911= Aminus1B1 (33)
To get more higher order approximations for (19) we com-pare between the coefficients of 119901119894 (119894 = 2 3 4 ) in (25) toobtain the following approximations
119891119894= Aminus1Bi (34)
subject to the boundary conditions
119891119894(minus1) = 119891
119894(1) = 119891
1015840
119894(minus1) = 0 (35)
where
Bi = minus1
2120572Re[
119894minus1
sum
119899=0
119891119899(D119891119894minus1minus119899
)] 119894 ge 2 (36)
Thematrix119860 has dimensions (119873+1)times(119873+1)while matrices119861119894and 119891
119894have a dimensions (119873 + 1) times 1
To implement the boundary conditions (29) and (35) tothe systems (28) and (34) respectively we delete the first andthe last rows and columns of 119860 and delete the first and lastrows of and 119891
1and 119891
119894 also we replace the results of last row
of the modified matrix119860 and setting the results of last row ofthe modified matrices 119861
1and 119861
119894to be zeroThen the solution
of (19) is given by substituting the series119891119894in (26) after setting
119901 = 1
5 Results and Discussion
In this section we present the obtained results of the solutionsfor MHD Jeffery-Hame flow using the HPM SHPM anda numerical solution Here we used the inbuilt MATLABboundary value problems solver bvp4c for the numericalsolution approach In generating the presented results it wasdetermined through numerical experimentation that 119873 =
60 Table 1 shows a comparison between the HPM SHPMand numerical approximate solutions of 119865(120578) The tableshows that the results of the present method are in excellentagreement with those of the numerical ones Also in Table 2we give a comparison of the SHPM results for divergent andconvergent channels and fixed values of 119867 and Re when120578 is varied at different orders of approximation against thenumerical results It can be seen from Table 2 that SHPMresults converge rapidly to the numerical solution Table 3gives a comparison of the differential transformationmethod(DTM) HPM homotopy analysis method (HAM) given byJoneidi et al [22] and SHPM results for 119865(120578) against thenumerical results when 120578 is varied It can be seen from thistable that the approximate solution of MHD Jeffery-Hamelflows obtained by SHPM is very accurate and it is convergesmuch more rapidly to the numerical result compared to theDTM HPM and HAM
Figures 2(a) and 2(b) show firstly the influence of themagnetic field parameter on the velocity profile for diver-gent and convergent channels and secondly a comparisonbetween the present results and numerical results to give asense of the accuracy and convergence rate of the SHPMThe figures indicate that there is very good match betweenthe two sets of results even at very low orders of SHPMapproximations series compared with the numerical resultsThese findings firmly establish the SHPM as an accurateand alternative to the HPM Also it can seen that the fluidvelocity increases with increasingHartman numbers for bothconvergent and divergent channels
Journal of Computational Methods in Physics 5
Table 1 Comparison between the HPM SHPM and numerical results of F(120578) for different values of 119867 when 120572 = 75∘ and Re = 50
Table 2 Comparison of the values of the SHPM approximate solutions for F(120578) with the numerical solution for various values of 120578 when119867 = 100 and Re = 50 for divergent and convergent channels
In this study the SHPM were applied successfully to findan approximate solution of a nonlinear MHD Jeffery-HamelflowsThe effect of themagnetic field parameter on the veloc-ity profile for convergent and divergent channels has beendetermined It could be noticed that by increasing magneticfield parameter the velocity profile increases resulting in
a rise in the flow rate for both convergent and divergentchannels The obtained results compared with the numericalsolution of the governing nonlinear equation and with DTMHPM and HAM Also the tables and figures clearly showhigh accuracy of the method and the convergent is very fastto solve MHD Jeffery-Hamel problem An important aspectof this work has been the need to prove the computationalefficiency and accuracy of the SHPM in solving nonlinear
6 Journal of Computational Methods in Physics
0 02 04 06 08 10
02
04
06
08
1
H = 0H = 250H = 500H = 1000
120572 = 5
F(120578)
120578
(a)
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
H = 0H = 250H = 500H = 1000120572 = minus5
F(120578)
120578
(b)
Figure 2 Comparison between the numerical solution and SHPM of 119865(120578) for divergent and convergent channels when Re = 50
differential equations and this method is in general moreaccurate than theHPMThe results in this paper confirm thatthe SHPM is a powerful and efficient technique for findingsolutions for nonlinear differential equations in differentfields of science and engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] G B Jeffery ldquoThe two-dimensional steady motion of a viscousfluidrdquo Philosophical Magazine vol 6 pp 455ndash465 1915
[2] G Hamel ldquoSpiralformige Bewegungen zaher FlussigkeitenrdquoJahresbericht der Deutschen Mathematiker-Vereinigung vol 25pp 34ndash60 1916
[3] L Rosenhead ldquoThe steady two-dimensional radial flow ofviscous fluid between two inclined plane wallsrdquo Proceedings ofthe Royal Society A vol 175 pp 436ndash467 1940
[4] A McAlpine and P G Drazin ldquoOn the spatio-temporal devel-opment of small perturbations of Jeffery-Hamel flowsrdquo FluidDynamics Research vol 22 no 3 pp 123ndash138 1998
[5] R M Sadri Channel entrance flow [PhD thesis] Departmentof Mechanical Engineering the University of Western Ontario1997
[6] I J Sobey and P G Drazin ldquoBifurcations of two-dimensionalchannel flowsrdquo Journal of Fluid Mechanics vol 171 pp 263ndash2871986
[7] W I Axford ldquoThe magnetohydrodynamic Jeffrey-Hamel prob-lem for a weakly conducting fluidrdquo The Quarterly Journal ofMechanics and Applied Mathematics vol 14 pp 335ndash351 1961
[8] J He ldquoA review on some new recently developed nonlinear ana-lytical techniquesrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 1 no 1 pp 51ndash70 2000
[9] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 2 pp 207ndash208 2005
[10] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[11] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[12] Q Esmaili A Ramiar E Alizadeh and D D Ganji ldquoAnapproximation of the analytical solution of the Jeffery-Hamelflow by decomposition methodrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 372 no 19 pp 3434ndash34392008
[13] O D Makinde and P Y Mhone ldquoHermite-Pade approximationapproach to MHD Jeffery-Hamel flowsrdquo Applied Mathematicsand Computation vol 181 no 2 pp 966ndash972 2006
[14] O D Makinde ldquoEffect of arbitrary magnetic Reynolds numberonMHDflows in convergent-divergent channelsrdquo InternationalJournal of Numerical Methods for Heat amp Fluid Flow vol 18 no5-6 pp 697ndash707 2008
[15] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[17] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[18] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] J H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000
Journal of Computational Methods in Physics 7
[21] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[22] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
And119873(119865) is the nonlinear part of (19) Substitute (22) in (25)to get
1198893119865
1198891199093
+ 1198921(119909)
119889119865
119889119909+ 1198922(119909) 119865 + (1 minus 119901)
times (11988931198910
1198891199093
+ 1198921(119909)
1198891198910
119889119909+ 1198922(119909) 1198910)
+ 119901(1
2120572Re119865
119889119865
119889119909minus 119866 (119909))
= 0
(27)
Wemake a comparison between the power of1199011 in both sidesof (27) to obtain the following system of equations
A1198911= B1 (28)
subject to the boundary conditions
1198911(minus1) = 119891
1(1) = 119891
1(minus1) = 0 (29)
where
A = [D3+ diag [119892
1(119909119894)]D + diag [119892
2(119909119894)]]
B1 = minus[A1198910+
1
2120572Re119891
0(D21198910) minus 119866 (119909
119894)]
119879
(30)
where 119879 denotes transpose diag [] is a diagonal matrix ofsize (119873 + 1) times (119873 + 1) and D is the Chebyshev spectraldifferentiation matrix whose entries (see [20 21]) are givenby
D119895119896
=
119888119895
119888119896
(minus1)119895+119896
119909119895minus 119909119896
119895 = 119896 119895 119896 = 0 1 119873
D119896119896
= minus119909119896
2 (1 minus 1199092
119896)
119896 = 1 2 119873 minus 1
D00
=21198732+ 1
6= minusD
119873119873
(31)
Here 1198880
= 119888119873
= 2 and 119888119895= 1 with 1 le 119895 le 119873 minus 1 119909
119895are the
Chebyshev collocation points (see [20]) defined by
119909119895= cos
119895120587
119873 119895 = 0 1 2 119873 (32)
the solution of (28) can be given by
1198911= Aminus1B1 (33)
To get more higher order approximations for (19) we com-pare between the coefficients of 119901119894 (119894 = 2 3 4 ) in (25) toobtain the following approximations
119891119894= Aminus1Bi (34)
subject to the boundary conditions
119891119894(minus1) = 119891
119894(1) = 119891
1015840
119894(minus1) = 0 (35)
where
Bi = minus1
2120572Re[
119894minus1
sum
119899=0
119891119899(D119891119894minus1minus119899
)] 119894 ge 2 (36)
Thematrix119860 has dimensions (119873+1)times(119873+1)while matrices119861119894and 119891
119894have a dimensions (119873 + 1) times 1
To implement the boundary conditions (29) and (35) tothe systems (28) and (34) respectively we delete the first andthe last rows and columns of 119860 and delete the first and lastrows of and 119891
1and 119891
119894 also we replace the results of last row
of the modified matrix119860 and setting the results of last row ofthe modified matrices 119861
1and 119861
119894to be zeroThen the solution
of (19) is given by substituting the series119891119894in (26) after setting
119901 = 1
5 Results and Discussion
In this section we present the obtained results of the solutionsfor MHD Jeffery-Hame flow using the HPM SHPM anda numerical solution Here we used the inbuilt MATLABboundary value problems solver bvp4c for the numericalsolution approach In generating the presented results it wasdetermined through numerical experimentation that 119873 =
60 Table 1 shows a comparison between the HPM SHPMand numerical approximate solutions of 119865(120578) The tableshows that the results of the present method are in excellentagreement with those of the numerical ones Also in Table 2we give a comparison of the SHPM results for divergent andconvergent channels and fixed values of 119867 and Re when120578 is varied at different orders of approximation against thenumerical results It can be seen from Table 2 that SHPMresults converge rapidly to the numerical solution Table 3gives a comparison of the differential transformationmethod(DTM) HPM homotopy analysis method (HAM) given byJoneidi et al [22] and SHPM results for 119865(120578) against thenumerical results when 120578 is varied It can be seen from thistable that the approximate solution of MHD Jeffery-Hamelflows obtained by SHPM is very accurate and it is convergesmuch more rapidly to the numerical result compared to theDTM HPM and HAM
Figures 2(a) and 2(b) show firstly the influence of themagnetic field parameter on the velocity profile for diver-gent and convergent channels and secondly a comparisonbetween the present results and numerical results to give asense of the accuracy and convergence rate of the SHPMThe figures indicate that there is very good match betweenthe two sets of results even at very low orders of SHPMapproximations series compared with the numerical resultsThese findings firmly establish the SHPM as an accurateand alternative to the HPM Also it can seen that the fluidvelocity increases with increasingHartman numbers for bothconvergent and divergent channels
Journal of Computational Methods in Physics 5
Table 1 Comparison between the HPM SHPM and numerical results of F(120578) for different values of 119867 when 120572 = 75∘ and Re = 50
Table 2 Comparison of the values of the SHPM approximate solutions for F(120578) with the numerical solution for various values of 120578 when119867 = 100 and Re = 50 for divergent and convergent channels
In this study the SHPM were applied successfully to findan approximate solution of a nonlinear MHD Jeffery-HamelflowsThe effect of themagnetic field parameter on the veloc-ity profile for convergent and divergent channels has beendetermined It could be noticed that by increasing magneticfield parameter the velocity profile increases resulting in
a rise in the flow rate for both convergent and divergentchannels The obtained results compared with the numericalsolution of the governing nonlinear equation and with DTMHPM and HAM Also the tables and figures clearly showhigh accuracy of the method and the convergent is very fastto solve MHD Jeffery-Hamel problem An important aspectof this work has been the need to prove the computationalefficiency and accuracy of the SHPM in solving nonlinear
6 Journal of Computational Methods in Physics
0 02 04 06 08 10
02
04
06
08
1
H = 0H = 250H = 500H = 1000
120572 = 5
F(120578)
120578
(a)
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
H = 0H = 250H = 500H = 1000120572 = minus5
F(120578)
120578
(b)
Figure 2 Comparison between the numerical solution and SHPM of 119865(120578) for divergent and convergent channels when Re = 50
differential equations and this method is in general moreaccurate than theHPMThe results in this paper confirm thatthe SHPM is a powerful and efficient technique for findingsolutions for nonlinear differential equations in differentfields of science and engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] G B Jeffery ldquoThe two-dimensional steady motion of a viscousfluidrdquo Philosophical Magazine vol 6 pp 455ndash465 1915
[2] G Hamel ldquoSpiralformige Bewegungen zaher FlussigkeitenrdquoJahresbericht der Deutschen Mathematiker-Vereinigung vol 25pp 34ndash60 1916
[3] L Rosenhead ldquoThe steady two-dimensional radial flow ofviscous fluid between two inclined plane wallsrdquo Proceedings ofthe Royal Society A vol 175 pp 436ndash467 1940
[4] A McAlpine and P G Drazin ldquoOn the spatio-temporal devel-opment of small perturbations of Jeffery-Hamel flowsrdquo FluidDynamics Research vol 22 no 3 pp 123ndash138 1998
[5] R M Sadri Channel entrance flow [PhD thesis] Departmentof Mechanical Engineering the University of Western Ontario1997
[6] I J Sobey and P G Drazin ldquoBifurcations of two-dimensionalchannel flowsrdquo Journal of Fluid Mechanics vol 171 pp 263ndash2871986
[7] W I Axford ldquoThe magnetohydrodynamic Jeffrey-Hamel prob-lem for a weakly conducting fluidrdquo The Quarterly Journal ofMechanics and Applied Mathematics vol 14 pp 335ndash351 1961
[8] J He ldquoA review on some new recently developed nonlinear ana-lytical techniquesrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 1 no 1 pp 51ndash70 2000
[9] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 2 pp 207ndash208 2005
[10] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[11] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[12] Q Esmaili A Ramiar E Alizadeh and D D Ganji ldquoAnapproximation of the analytical solution of the Jeffery-Hamelflow by decomposition methodrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 372 no 19 pp 3434ndash34392008
[13] O D Makinde and P Y Mhone ldquoHermite-Pade approximationapproach to MHD Jeffery-Hamel flowsrdquo Applied Mathematicsand Computation vol 181 no 2 pp 966ndash972 2006
[14] O D Makinde ldquoEffect of arbitrary magnetic Reynolds numberonMHDflows in convergent-divergent channelsrdquo InternationalJournal of Numerical Methods for Heat amp Fluid Flow vol 18 no5-6 pp 697ndash707 2008
[15] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[17] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[18] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] J H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000
Journal of Computational Methods in Physics 7
[21] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[22] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
Table 2 Comparison of the values of the SHPM approximate solutions for F(120578) with the numerical solution for various values of 120578 when119867 = 100 and Re = 50 for divergent and convergent channels
In this study the SHPM were applied successfully to findan approximate solution of a nonlinear MHD Jeffery-HamelflowsThe effect of themagnetic field parameter on the veloc-ity profile for convergent and divergent channels has beendetermined It could be noticed that by increasing magneticfield parameter the velocity profile increases resulting in
a rise in the flow rate for both convergent and divergentchannels The obtained results compared with the numericalsolution of the governing nonlinear equation and with DTMHPM and HAM Also the tables and figures clearly showhigh accuracy of the method and the convergent is very fastto solve MHD Jeffery-Hamel problem An important aspectof this work has been the need to prove the computationalefficiency and accuracy of the SHPM in solving nonlinear
6 Journal of Computational Methods in Physics
0 02 04 06 08 10
02
04
06
08
1
H = 0H = 250H = 500H = 1000
120572 = 5
F(120578)
120578
(a)
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
H = 0H = 250H = 500H = 1000120572 = minus5
F(120578)
120578
(b)
Figure 2 Comparison between the numerical solution and SHPM of 119865(120578) for divergent and convergent channels when Re = 50
differential equations and this method is in general moreaccurate than theHPMThe results in this paper confirm thatthe SHPM is a powerful and efficient technique for findingsolutions for nonlinear differential equations in differentfields of science and engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] G B Jeffery ldquoThe two-dimensional steady motion of a viscousfluidrdquo Philosophical Magazine vol 6 pp 455ndash465 1915
[2] G Hamel ldquoSpiralformige Bewegungen zaher FlussigkeitenrdquoJahresbericht der Deutschen Mathematiker-Vereinigung vol 25pp 34ndash60 1916
[3] L Rosenhead ldquoThe steady two-dimensional radial flow ofviscous fluid between two inclined plane wallsrdquo Proceedings ofthe Royal Society A vol 175 pp 436ndash467 1940
[4] A McAlpine and P G Drazin ldquoOn the spatio-temporal devel-opment of small perturbations of Jeffery-Hamel flowsrdquo FluidDynamics Research vol 22 no 3 pp 123ndash138 1998
[5] R M Sadri Channel entrance flow [PhD thesis] Departmentof Mechanical Engineering the University of Western Ontario1997
[6] I J Sobey and P G Drazin ldquoBifurcations of two-dimensionalchannel flowsrdquo Journal of Fluid Mechanics vol 171 pp 263ndash2871986
[7] W I Axford ldquoThe magnetohydrodynamic Jeffrey-Hamel prob-lem for a weakly conducting fluidrdquo The Quarterly Journal ofMechanics and Applied Mathematics vol 14 pp 335ndash351 1961
[8] J He ldquoA review on some new recently developed nonlinear ana-lytical techniquesrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 1 no 1 pp 51ndash70 2000
[9] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 2 pp 207ndash208 2005
[10] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[11] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[12] Q Esmaili A Ramiar E Alizadeh and D D Ganji ldquoAnapproximation of the analytical solution of the Jeffery-Hamelflow by decomposition methodrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 372 no 19 pp 3434ndash34392008
[13] O D Makinde and P Y Mhone ldquoHermite-Pade approximationapproach to MHD Jeffery-Hamel flowsrdquo Applied Mathematicsand Computation vol 181 no 2 pp 966ndash972 2006
[14] O D Makinde ldquoEffect of arbitrary magnetic Reynolds numberonMHDflows in convergent-divergent channelsrdquo InternationalJournal of Numerical Methods for Heat amp Fluid Flow vol 18 no5-6 pp 697ndash707 2008
[15] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[17] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[18] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] J H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000
Journal of Computational Methods in Physics 7
[21] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[22] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
Figure 2 Comparison between the numerical solution and SHPM of 119865(120578) for divergent and convergent channels when Re = 50
differential equations and this method is in general moreaccurate than theHPMThe results in this paper confirm thatthe SHPM is a powerful and efficient technique for findingsolutions for nonlinear differential equations in differentfields of science and engineering
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
References
[1] G B Jeffery ldquoThe two-dimensional steady motion of a viscousfluidrdquo Philosophical Magazine vol 6 pp 455ndash465 1915
[2] G Hamel ldquoSpiralformige Bewegungen zaher FlussigkeitenrdquoJahresbericht der Deutschen Mathematiker-Vereinigung vol 25pp 34ndash60 1916
[3] L Rosenhead ldquoThe steady two-dimensional radial flow ofviscous fluid between two inclined plane wallsrdquo Proceedings ofthe Royal Society A vol 175 pp 436ndash467 1940
[4] A McAlpine and P G Drazin ldquoOn the spatio-temporal devel-opment of small perturbations of Jeffery-Hamel flowsrdquo FluidDynamics Research vol 22 no 3 pp 123ndash138 1998
[5] R M Sadri Channel entrance flow [PhD thesis] Departmentof Mechanical Engineering the University of Western Ontario1997
[6] I J Sobey and P G Drazin ldquoBifurcations of two-dimensionalchannel flowsrdquo Journal of Fluid Mechanics vol 171 pp 263ndash2871986
[7] W I Axford ldquoThe magnetohydrodynamic Jeffrey-Hamel prob-lem for a weakly conducting fluidrdquo The Quarterly Journal ofMechanics and Applied Mathematics vol 14 pp 335ndash351 1961
[8] J He ldquoA review on some new recently developed nonlinear ana-lytical techniquesrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 1 no 1 pp 51ndash70 2000
[9] J H He ldquoHomotopy perturbation method for bifurcation ofnonlinear problemsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 8 no 2 pp 207ndash208 2005
[10] S J Liao The proposed homotopy analysis technique for thesolution of nonlinear problems [PhD thesis] Shanghai Jiao TongUniversity 1992
[11] S Liao ldquoOn the homotopy analysis method for nonlinearproblemsrdquo Applied Mathematics and Computation vol 147 no2 pp 499ndash513 2004
[12] Q Esmaili A Ramiar E Alizadeh and D D Ganji ldquoAnapproximation of the analytical solution of the Jeffery-Hamelflow by decomposition methodrdquo Physics Letters A GeneralAtomic and Solid State Physics vol 372 no 19 pp 3434ndash34392008
[13] O D Makinde and P Y Mhone ldquoHermite-Pade approximationapproach to MHD Jeffery-Hamel flowsrdquo Applied Mathematicsand Computation vol 181 no 2 pp 966ndash972 2006
[14] O D Makinde ldquoEffect of arbitrary magnetic Reynolds numberonMHDflows in convergent-divergent channelsrdquo InternationalJournal of Numerical Methods for Heat amp Fluid Flow vol 18 no5-6 pp 697ndash707 2008
[15] J H He ldquoVariational iteration methodmdasha kind of non-linearanalytical technique Some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999
[16] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[17] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA newspectral-homotopy analysis method for the MHD Jeffery-Hamel problemrdquo Computers amp Fluids vol 39 no 7 pp 1219ndash1225 2010
[18] J H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[19] J H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000
[20] L N Trefethen Spectral Methods in MATLAB SIAM 2000
Journal of Computational Methods in Physics 7
[21] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[22] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010
[21] W S Don and A Solomonoff ldquoAccuracy and speed in com-puting the Chebyshev collocation derivativerdquo SIAM Journal onScientific Computing vol 16 no 6 pp 1253ndash1268 1995
[22] A A Joneidi G Domairry andM Babaelahi ldquoThree analyticalmethods applied to Jeffery-Hamel flowrdquo Communications inNonlinear Science and Numerical Simulation vol 15 no 11 pp3423ndash3434 2010