research article spectral-homotopy perturbation method...

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

120588

119903

120597

120597119903(119903119906 (119903 120579)) = 0 (1)

119906 (119903 120579)120597119906 (119903 120579)

120597119903

= minus1

120588

120597119875

120597119903+ ]

times [1205972119906 (119903 120579)

1205971199032

+1

119903

120597119906 (119903 120579)

120597119903+

1

1199032

1205972119906 (119903 120579)

1205971205792

minus119906 (119903 120579)

1199032

]

minus1205901198612

0

1205881199032119906 (119903 120579)

(2)

minus1

120588119903

120597119875

120597120579+

2]1199032

120597119906 (119903 120579)

120597120579= 0 (3)

B0

120579u(r 120579)

120572

Figure 1 Geometry of the MHD Jeffery-Hamel flow

where 119875 is the fluid pressure 1198610is the electromagnetic

induction 120590 is the conductivity of the fluid 120588 is the fluiddensity and ] is the coefficient of kinematic viscosity From(1) one has

119906 (119903 120579) =119891 (120579)

119903 (4)

Using the dimensionless parameters

119865 (120578) =119891 (120579)

119891max 120578 =

120579

120572 (5)

and with eliminating 119875 from (2) and (3) we obtain thefollowing ordinary differential equation for the normalizedfunction profile 119865(120578)

119865101584010158401015840

(120578) + 2120572Re119865 (120578) 1198651015840(120578) + (4 minus 119867) 120572

21198651015840(120578) = 0 (6)

subject to the boundary conditions

119865 (0) = 1 1198651015840(0) = 0 119865 (1) = 0 (7)

where Re is the Reynolds number

Re =120572119891max]

=119880max119903120572

]= (

divergent channel 120572 gt 0 119880max gt 0

convergent channel 120572 lt 0 119880max lt 0)

(8)

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

119871 (119906) + 119873 (119906) minus 119891 (r) = 0 (11)

By the homotopy technique (see [10 11]) we construct ahomotopy V(119903 119901) Ω times [0 1] rarr R wich satisfies

119867(V 119901) = (1 minus 119901) [119871 (V) minus 119871 (1199060)]

+ 119901 [119860 (V) minus 119891 (r)] = 0 119901 isin [0 1] r isin Ω

(12)or

119867(V 119901) = 119871 (V) minus 119871 (1199060) + 119901119871 (119906

0) + 119901 [119873 (V) minus 119891 (r)] = 0

(13)

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

119891101584010158401015840

(119909) + 1198921(119909) 1198911015840(119909) + 119892

2(119909) 119891 (119909)

+1

2120572Re119891 (119909) 119891

1015840(119909) = 119866 (119909)

(19)

subject to

119891 (minus1) = 119891 (1) = 1198911015840(minus1) = 0 (20)

where

1198921(119909) =

1

8(3 minus 2119909 minus 119909

3) +

1

41205722(2 + 2119909 minus 119867)

1198922(119909) = minus

1

4120572Re (119909 + 1)

119866 (119909) =1

16120572Re (119909 + 1) (3 minus 2119909 minus 119909

2)

+1

81205722(119909 + 1) (4 minus 119867)

(21)

To apply the SHPM to the differential equation (19) we maychoose the following linear differential operator

L =1198893

1198891199093+ 1198921(119909)

119889

119889119909+ 1198922(119909) (22)

The initial approximation for the solution of (19) is obtainedfrom the solution to the linear part of (19) Consider

119891101584010158401015840

0(119909) + 119892

1(119909) 1198911015840

0+ 1198922(119909) 1198910(119909) = 119866 (119909) (23)


1198910(minus1) = 119891

0(1) = 119891

1015840

0(minus1) = 0 (24)

If an exact solution of (23) cannot be found we use theChebyshev pseudospectral method to solve the equation Wenow construct the homotopy

119867(119865 119901) = L [119865] minus L [1198910] + 119901L [119891

0]

+ 119901 (119873 (119865) minus 119866 (119909)) = 0

(25)


where 119865 is an approximate series solution of given by

119865 = 1198910+ 1199011198911+ 11990121198912+ sdot sdot sdot (26)

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)


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)


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


Table 1 Comparison between the HPM SHPM and numerical results of F(120578) for different values of 119867 when 120572 = 75∘ and Re = 50

120578 01 02 03 04 05 06 07 08 09

119867 = 0

HPM 0977071 0911454 0810411 0685923 0549843 0413170 0284602 0170279 0074423SHPM 0977136 0911454 0810955 0686840 0551195 0415007 0286958 0173044 0076826Numeric 0977136 0911453 0810954 0686838 0551193 0415005 0286957 0173044 0076826

119867 = 250

HPM 0983734 0936343 0861689 0765340 0653396 0531462 0403813 0272871 0138943SHPM 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045Numeric 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045

119867 = 500

HPM 0988320 0953804 0897852 0822470 0729682 0620975 0496664 0355212 0192382SHPM 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418Numeric 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418

119867 = 1000

HPM 0993761 0974806 0942279 0894743 0829747 0743476 0629987 0479934 0278279SHPM 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418Numeric 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418

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

120578divergent-channel 120572 = 5

∘ convergent-channel 120572 = minus5∘

3rd order 4th order 6th order 8th order Numerical 3rd order 4th order 6th order 8th order Numerical01 0983499 0983500 0983500 0983500 0983500 0994672 0994672 0994672 0994672 099467202 0935285 0935287 0935286 0935286 0935286 0978184 0978184 0978184 0978184 097818403 0858989 0858991 0858991 0858991 0858991 0948987 0948986 0948986 0948986 094898604 0759953 0759957 0759957 0759957 0759957 0904400 0904399 0904399 0904399 090439905 0644378 0644384 0644384 0644384 0644384 0840477 0840475 0840475 0840475 084047506 0518442 0518448 0518448 0518448 0518448 0751876 0751874 0751874 0751874 075187407 0387576 0387581 0387581 0387581 0387581 0631800 0631798 0631798 0631798 063179808 0255978 0255981 0255981 0255981 0255981 0472168 0472166 0472166 0472166 047216609 0126379 0126380 0126380 0126380 0126380 0264232 0264231 0264231 0264231 0264231

Table 3 Comparison between DTM HPM HAM SHPM and numerical solution of F(120578) for various values of 120578 when Re = 80 120572 = minus5∘

and 119867 = 0

120578 ADM HPM HAM SHPM Numerical00 1 1 1 1 1

01 099596039 099606719 099596062 099596206 099596205

02 098327455 098369594 098327553 098328122 098328121

03 096017756 096107588 096017989 096019253 096019253

04 092351707 092492452 092352157 092354347 092354347

05 086845113 087019977 086845890 086849152 086849152

06 078807854 078983259 078809102 078813414 078813414

07 067312484 067453350 067314377 067319429 067319429

08 051196441 051283731 051199099 051204134 051204134

09 029152801 029189370 029155802 029159436 029159436

10 000000000 000000000 minus000000115 000000000 000000000

6 Conclusions

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


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


[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

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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

120588

119903

120597

120597119903(119903119906 (119903 120579)) = 0 (1)

119906 (119903 120579)120597119906 (119903 120579)

120597119903

= minus1

120588

120597119875

120597119903+ ]

times [1205972119906 (119903 120579)

1205971199032

+1

119903

120597119906 (119903 120579)

120597119903+

1

1199032

1205972119906 (119903 120579)

1205971205792

minus119906 (119903 120579)

1199032

]

minus1205901198612

0

1205881199032119906 (119903 120579)

(2)

minus1

120588119903

120597119875

120597120579+

2]1199032

120597119906 (119903 120579)

120597120579= 0 (3)

B0

120579u(r 120579)

120572

Figure 1 Geometry of the MHD Jeffery-Hamel flow

where 119875 is the fluid pressure 1198610is the electromagnetic

induction 120590 is the conductivity of the fluid 120588 is the fluiddensity and ] is the coefficient of kinematic viscosity From(1) one has

119906 (119903 120579) =119891 (120579)

119903 (4)

Using the dimensionless parameters

119865 (120578) =119891 (120579)

119891max 120578 =

120579

120572 (5)

and with eliminating 119875 from (2) and (3) we obtain thefollowing ordinary differential equation for the normalizedfunction profile 119865(120578)

119865101584010158401015840

(120578) + 2120572Re119865 (120578) 1198651015840(120578) + (4 minus 119867) 120572

21198651015840(120578) = 0 (6)


119865 (0) = 1 1198651015840(0) = 0 119865 (1) = 0 (7)

where Re is the Reynolds number

Re =120572119891max]

=119880max119903120572

]= (

divergent channel 120572 gt 0 119880max gt 0

convergent channel 120572 lt 0 119880max lt 0)

(8)

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



119860 (119906) minus 119891 (r) = 0 r isin Ω (9)


119861(119906120597119906

120597119899) = 0 r isin Γ (10)



119871 (119906) + 119873 (119906) minus 119891 (r) = 0 (11)


119867(V 119901) = (1 minus 119901) [119871 (V) minus 119871 (1199060)]


(12)or

119867(V 119901) = 119871 (V) minus 119871 (1199060) + 119901119871 (119906

0) + 119901 [119873 (V) minus 119891 (r)] = 0

(13)



119867(V 0) = 119871 (V) minus 119871 (1199060) = 0

119867 (V 1) = 119860 (V) minus 119891 (r) = 0

(14)







119906 = lim119901rarr1








119909 = 2120578 minus 1 119909 isin [minus1 1] (17)


119865 (120578) = 119891 (119909) + 1 minus 1205782 (18)


119891101584010158401015840

(119909) + 1198921(119909) 1198911015840(119909) + 119892

2(119909) 119891 (119909)

+1

2120572Re119891 (119909) 119891

1015840(119909) = 119866 (119909)

(19)

subject to

119891 (minus1) = 119891 (1) = 1198911015840(minus1) = 0 (20)

where

1198921(119909) =

1

8(3 minus 2119909 minus 119909

3) +

1

41205722(2 + 2119909 minus 119867)

1198922(119909) = minus

1

4120572Re (119909 + 1)

119866 (119909) =1

16120572Re (119909 + 1) (3 minus 2119909 minus 119909

2)

+1

81205722(119909 + 1) (4 minus 119867)

(21)


L =1198893

1198891199093+ 1198921(119909)

119889

119889119909+ 1198922(119909) (22)


119891101584010158401015840

0(119909) + 119892

1(119909) 1198911015840

0+ 1198922(119909) 1198910(119909) = 119866 (119909) (23)


1198910(minus1) = 119891

0(1) = 119891

1015840

0(minus1) = 0 (24)


119867(119865 119901) = L [119865] minus L [1198910] + 119901L [119891

0]

+ 119901 (119873 (119865) minus 119866 (119909)) = 0

(25)



119865 = 1198910+ 1199011198911+ 11990121198912+ sdot sdot sdot (26)


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)


A1198911= B1 (28)


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)


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


119895are the


119909119895= cos

119895120587

119873 119895 = 0 1 2 119873 (32)


1198911= Aminus1B1 (33)


119891119894= Aminus1Bi (34)


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)




1and 119891



1and 119861



119901 = 1







120578 01 02 03 04 05 06 07 08 09

119867 = 0

HPM 0977071 0911454 0810411 0685923 0549843 0413170 0284602 0170279 0074423SHPM 0977136 0911454 0810955 0686840 0551195 0415007 0286958 0173044 0076826Numeric 0977136 0911453 0810954 0686838 0551193 0415005 0286957 0173044 0076826

119867 = 250

HPM 0983734 0936343 0861689 0765340 0653396 0531462 0403813 0272871 0138943SHPM 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045Numeric 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045

119867 = 500

HPM 0988320 0953804 0897852 0822470 0729682 0620975 0496664 0355212 0192382SHPM 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418Numeric 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418

119867 = 1000

HPM 0993761 0974806 0942279 0894743 0829747 0743476 0629987 0479934 0278279SHPM 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418Numeric 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418






and 119867 = 0


01 099596039 099606719 099596062 099596206 099596205

02 098327455 098369594 098327553 098328122 098328121

03 096017756 096107588 096017989 096019253 096019253

04 092351707 092492452 092352157 092354347 092354347

05 086845113 087019977 086845890 086849152 086849152

06 078807854 078983259 078809102 078813414 078813414

07 067312484 067453350 067314377 067319429 067319429

08 051196441 051283731 051199099 051204134 051204134

09 029152801 029189370 029155802 029159436 029159436

10 000000000 000000000 minus000000115 000000000 000000000

6 Conclusions




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)





References





























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119860 (119906) minus 119891 (r) = 0 r isin Ω (9)


119861(119906120597119906

120597119899) = 0 r isin Γ (10)



119871 (119906) + 119873 (119906) minus 119891 (r) = 0 (11)


119867(V 119901) = (1 minus 119901) [119871 (V) minus 119871 (1199060)]


(12)or

119867(V 119901) = 119871 (V) minus 119871 (1199060) + 119901119871 (119906

0) + 119901 [119873 (V) minus 119891 (r)] = 0

(13)



119867(V 0) = 119871 (V) minus 119871 (1199060) = 0

119867 (V 1) = 119860 (V) minus 119891 (r) = 0

(14)







119906 = lim119901rarr1








119909 = 2120578 minus 1 119909 isin [minus1 1] (17)


119865 (120578) = 119891 (119909) + 1 minus 1205782 (18)


119891101584010158401015840

(119909) + 1198921(119909) 1198911015840(119909) + 119892

2(119909) 119891 (119909)

+1

2120572Re119891 (119909) 119891

1015840(119909) = 119866 (119909)

(19)

subject to

119891 (minus1) = 119891 (1) = 1198911015840(minus1) = 0 (20)

where

1198921(119909) =

1

8(3 minus 2119909 minus 119909

3) +

1

41205722(2 + 2119909 minus 119867)

1198922(119909) = minus

1

4120572Re (119909 + 1)

119866 (119909) =1

16120572Re (119909 + 1) (3 minus 2119909 minus 119909

2)

+1

81205722(119909 + 1) (4 minus 119867)

(21)


L =1198893

1198891199093+ 1198921(119909)

119889

119889119909+ 1198922(119909) (22)


119891101584010158401015840

0(119909) + 119892

1(119909) 1198911015840

0+ 1198922(119909) 1198910(119909) = 119866 (119909) (23)


1198910(minus1) = 119891

0(1) = 119891

1015840

0(minus1) = 0 (24)


119867(119865 119901) = L [119865] minus L [1198910] + 119901L [119891

0]

+ 119901 (119873 (119865) minus 119866 (119909)) = 0

(25)



119865 = 1198910+ 1199011198911+ 11990121198912+ sdot sdot sdot (26)


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)


A1198911= B1 (28)


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)


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


119895are the


119909119895= cos

119895120587

119873 119895 = 0 1 2 119873 (32)


1198911= Aminus1B1 (33)


119891119894= Aminus1Bi (34)


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)




1and 119891



1and 119861



119901 = 1







120578 01 02 03 04 05 06 07 08 09

119867 = 0

HPM 0977071 0911454 0810411 0685923 0549843 0413170 0284602 0170279 0074423SHPM 0977136 0911454 0810955 0686840 0551195 0415007 0286958 0173044 0076826Numeric 0977136 0911453 0810954 0686838 0551193 0415005 0286957 0173044 0076826

119867 = 250

HPM 0983734 0936343 0861689 0765340 0653396 0531462 0403813 0272871 0138943SHPM 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045Numeric 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045

119867 = 500

HPM 0988320 0953804 0897852 0822470 0729682 0620975 0496664 0355212 0192382SHPM 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418Numeric 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418

119867 = 1000

HPM 0993761 0974806 0942279 0894743 0829747 0743476 0629987 0479934 0278279SHPM 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418Numeric 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418






and 119867 = 0


01 099596039 099606719 099596062 099596206 099596205

02 098327455 098369594 098327553 098328122 098328121

03 096017756 096107588 096017989 096019253 096019253

04 092351707 092492452 092352157 092354347 092354347

05 086845113 087019977 086845890 086849152 086849152

06 078807854 078983259 078809102 078813414 078813414

07 067312484 067453350 067314377 067319429 067319429

08 051196441 051283731 051199099 051204134 051204134

09 029152801 029189370 029155802 029159436 029159436

10 000000000 000000000 minus000000115 000000000 000000000

6 Conclusions




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)





References





























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119865 = 1198910+ 1199011198911+ 11990121198912+ sdot sdot sdot (26)


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)


A1198911= B1 (28)


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)


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


119895are the


119909119895= cos

119895120587

119873 119895 = 0 1 2 119873 (32)


1198911= Aminus1B1 (33)


119891119894= Aminus1Bi (34)


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)




1and 119891



1and 119861



119901 = 1







120578 01 02 03 04 05 06 07 08 09

119867 = 0

HPM 0977071 0911454 0810411 0685923 0549843 0413170 0284602 0170279 0074423SHPM 0977136 0911454 0810955 0686840 0551195 0415007 0286958 0173044 0076826Numeric 0977136 0911453 0810954 0686838 0551193 0415005 0286957 0173044 0076826

119867 = 250

HPM 0983734 0936343 0861689 0765340 0653396 0531462 0403813 0272871 0138943SHPM 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045Numeric 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045

119867 = 500

HPM 0988320 0953804 0897852 0822470 0729682 0620975 0496664 0355212 0192382SHPM 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418Numeric 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418

119867 = 1000

HPM 0993761 0974806 0942279 0894743 0829747 0743476 0629987 0479934 0278279SHPM 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418Numeric 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418






and 119867 = 0


01 099596039 099606719 099596062 099596206 099596205

02 098327455 098369594 098327553 098328122 098328121

03 096017756 096107588 096017989 096019253 096019253

04 092351707 092492452 092352157 092354347 092354347

05 086845113 087019977 086845890 086849152 086849152

06 078807854 078983259 078809102 078813414 078813414

07 067312484 067453350 067314377 067319429 067319429

08 051196441 051283731 051199099 051204134 051204134

09 029152801 029189370 029155802 029159436 029159436

10 000000000 000000000 minus000000115 000000000 000000000

6 Conclusions




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)





References





























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120578 01 02 03 04 05 06 07 08 09

119867 = 0

HPM 0977071 0911454 0810411 0685923 0549843 0413170 0284602 0170279 0074423SHPM 0977136 0911454 0810955 0686840 0551195 0415007 0286958 0173044 0076826Numeric 0977136 0911453 0810954 0686838 0551193 0415005 0286957 0173044 0076826

119867 = 250

HPM 0983734 0936343 0861689 0765340 0653396 0531462 0403813 0272871 0138943SHPM 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045Numeric 0983736 0936343 0861708 0765375 0653451 0531542 0403921 0273000 0139045

119867 = 500

HPM 0988320 0953804 0897852 0822470 0729682 0620975 0496664 0355212 0192382SHPM 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418Numeric 0988322 0953804 0897870 0822501 0729725 0621028 0496723 0355266 0192418

119867 = 1000

HPM 0993761 0974806 0942279 0894743 0829747 0743476 0629987 0479934 0278279SHPM 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418Numeric 0993765 0974806 0942318 0894809 0829843 0743602 0630138 0480096 0278418






and 119867 = 0


01 099596039 099606719 099596062 099596206 099596205

02 098327455 098369594 098327553 098328122 098328121

03 096017756 096107588 096017989 096019253 096019253

04 092351707 092492452 092352157 092354347 092354347

05 086845113 087019977 086845890 086849152 086849152

06 078807854 078983259 078809102 078813414 078813414

07 067312484 067453350 067314377 067319429 067319429

08 051196441 051283731 051199099 051204134 051204134

09 029152801 029189370 029155802 029159436 029159436

10 000000000 000000000 minus000000115 000000000 000000000

6 Conclusions




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)





References





























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




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)





References





























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics











FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article spectral-homotopy perturbation method...

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