Research Article On the Successive Linearisation Approach...
Transcript of Research Article On the Successive Linearisation Approach...
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 635392 7 pageshttpdxdoiorg1011552013635392
Research ArticleOn the Successive Linearisation Approach to the Flow ofReactive Third-Grade Liquid in a Channel with Isothermal Walls
S S Motsa1 O D Makinde2 and S Shateyi3
1 School of Mathematical Sciences University of KwaZulu-Natal Private Bag X01 Scottsville Pietermaritzburg 3209 South Africa2 Institute for Advanced Research in Mathematical Modelling and Computations Cape Peninsula University of TechnologyPO Box 1906 Bellville 7535 South Africa
3 Department of Mathematics amp Applied Mathematics University of Venda Private Bag X5050 Thohoyandou 0950 South Africa
Correspondence should be addressed to S S Motsa sandilemotsagmailcom
Received 27 March 2013 Accepted 30 April 2013
Academic Editor Anuar Ishak
Copyright copy 2013 S S Motsa et al 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
The nonlinear differential equations modeling flow of a reactive third-grade liquid between two parallel isothermal plates isinvestigated using a novel hybrid of numerical-analytical scheme known as the successive linearization method (SLM) Numericaland graphical results obtained show excellence in agreement with the earlier results reported in the literature A comparison withnumerical results generated using the inbuilt MATLAB boundary value solver bvp4c demonstrates that the new SLM approach isa very efficient technique for tackling highly nonlinear differential equations of the type discussed in this paper
1 Introduction
The rheological properties of many fluids used in indus-trial and engineering processes do exhibit non-Newtonianbehaviour [1 2] Meanwhile the study of heat transfer playsan important role during the handling and processing ofnon-Newtonian fluids [3ndash5] A complete thermodynamicsanalysis of the constitutive function for fluid of the differentialtype with the third-grade fluid being a special case has beenperformed by Fosdick and Rajagopal [6] Similar studies withrespect to non-Newtonian fluid are also reported byMakinde[7ndash9] Moreover the thermal boundary layer equationsfor non-Newtonian third-grade fluid constitute a nonlinearproblem and their solutions in space provide an insight intoan inherently complex physical process in the system Inmostcases the nonlinear nature of the model equations precludesits exact solution In recent time several approximationtechniques have been developed to tackle this problem [10ndash13] for example the Adomian decomposition method thevariation iteration method the improved finite differencesmethod the spectralmethod and so forthThe ideas of devel-oping new hybrids of numerical-analytical scheme to tacklenonlinear differential equations have experienced a revivalOne such trend is the spectral homotopy analysis method
that has recently been reported in [14 15] which is a hybridbetween the standard homotopy analysismethod [16] and theChebyshev spectral collocation method [17ndash19] Other novelstrategies involve using the Pade technique to improve theradius of convergence of the analytical methods of solutionRecent studies that make use of the Pade technique includethe Hermite-Pade [7] the Homotopy-Pade [16 20 21] andthe Hankel-Pade approaches
The purpose of the present work is to present a newmethod called the successive linearisation method (SLM) ofsolving nonlinear boundary value problemsWe demonstratethe applicability of SLM in tackling nonlinear differentiationequations modeling the flow of a reactive third-grade liquidbetween twoparallel isothermal platesThemathematical for-mulation of the problem is established in Section 2 In Section3 we introduce and apply some rudiments of SLM techniqueBoth numerical and graphical results are presented anddiscussed quantitatively with respect to various parametersembedded in the system in Section 4 A limited parametricstudy comparing numerical results generated using MAT-LABrsquos bvp4c boundary value solver is compared with theSLM results and good agreement is observed Using the SLMapproach multiple solutions which were theoretically provedto exist for such problems in [7] are also generated
2 Mathematical Problems in Engineering
2 Mathematical Formulation
Figure 1 depicts the problem geometry We consider thesteady flow of an incompressible third-grade reactive fluidplaced between two parallel isothermal plates It is assumedthat the flow is hydrodynamically and thermally fully devel-oped under the action of a constant axial pressure gradient
Following [1 4 5 7ndash9] the dimensionless governingequations for the momentum and energy balance can bewritten as
1198892119882
1198891199102+ 6120574
1198892119882
1198891199102(119889119882
119889119910)
2
= minus1 (1)
1198892120579
1198891199102+ 120582[119890
120579(1+120576120579)+ 119898(
119889119882
119889119910)
2
times(1 + 2120574(119889119882
119889119910)
2
)] = 0
(2)
where 119882 is the dimensionless velocity component 119910 isthe dimensionless normal coordinate 120579 is the dimension-less temperature and 120582 120576 120574 and 119898 represent the Frank-Kamenetskii parameter activation energy parameter thedimensionless non-Newtonian parameter and the viscousheating parameter respectively The additional Arrheniuskinetics term in energy balance equation (2) is due to [3]The appropriate boundary conditions in dimensionless formare given as follows the surface of the channel is fixedimpermeable and maintained at a given temperature
119882(1) = 0 120579 (1) = 0 at 119910 = 1 (3)
and the symmetry condition along the centerline that is
119889119882
119889119910=119889120579
119889119910= 0 at 119910 = 0 (4)
We have employed the following nondimensional quan-tities in (1)ndash(4)
120579 =119864 (119879 minus 119879
0)
1198771198792
0
119910 =119910
119886
120582 =119876119864119860119886
21198620119890minus1198641198771198790
1198792
0119877119896
119882 =119906
119880119866
119898 =120583119866211988021198901198641198771198790
11987611986011988621198620
120576 =1198771198790
119864
119866 = minus1198862
120583119880
119889119875
119889119909 120574 =
120573311988021198662
1198862120583
(5)
where 119879 is the absolute temperature 119880 is the fluid charac-teristic velocity 119879
0is the plate temperature 119896 is the thermal
conductivity of the material 119876 is the heat of reaction 119860 isthe rate constant 119864 is the activation energy 119877 is the universalgas constant 119862
0is the initial concentration of the reactant
species 119886 is the channel half width 1205733is the material coef-
ficient 119875 is the modified pressure and 120583 is the fluid dynamic
Third-grade reactive fluid
119906 = 0119910
119879 = 1198790 119910 = 119886
119906 = 0 119879 = 1198790 119910 = minus119886
Figure 1 Geometry of the problem
viscosity coefficient In the following sections (1)ndash(4) aresolved numerically using the new successive linearizationtechnique
3 Successive Linearisation Method (SLM)
In this section we apply the proposed linearisationmethod ofsolution hereinafter referred to as the successive linearisationmethod (SLM) to solve the governing equations (1) and (2)Before applying the SLM we first note that the problem canbe significantly simplified by finding the explicit analyticalsolution for the derivative 119889119882119889119910 First (1) is rewritten as
119889
119889119910[119889119882
119889119910+ 2120574(
119889119882
119889119910)
3
] = minus1 (6)
The above equation is then integrated on both sides and thesymmetry boundary condition 119889119882119889119910(0) = 0 is used toevaluate the resulting integrating constant to give
2120574(119889119882
119889119910)
3
+ (119889119882
119889119910) + 119910 = 0 (7)
We note that (7) is a cubic equation which can have eitherone or three real solutions If only positive values of 120574 areconsidered (7) will have a unique real solution which can becomputed using Maple and is given by
119889119882
119889119910=
1
6120574[120595(119910)]
13minus
1
[120595 (119910)]13
(8)
where
120595 (119910) = (minus54119910 + 6radic3radic2 + 27120574119910
2
120574)1205742 (9)
The analytical result is very important because when eva-luated at 119910 = 1 it gives an explicit analytical expression forthe skin friction coefficient (119862
119891) A close inspection of (8)
indicates that if 119862119891has a critical point then (120574
119888 119862119891) = (minus32
minus227) This critical point was reported as a bifurcation pointin [7] It is worth noting that 119889119882119889119910 and hence the solution119882(119910) is only valid when 120574 ge minus227
Since the momentum equation (1) is decoupled from theenergy equation (2) we solve for the velocity119882(119910) first then
Mathematical Problems in Engineering 3
substitute the result in the energy equation to obtain 120579(119910) Tosolve119882(119910) we write (1) as
1198892119882
1198891199102= 119891 (119910) (10)
where 119891(119910) is a known explicit function of 119910 given by
119891 (119910) = minus1
1 + 6120574(119889119882119889119910)2 (11)
with 119889119882119889119910 given by (8) Equation (10) can easily be inte-grated using any numerical methodThe SLM is based on theassumption that the unknown function 120579(119910) can be expandedas
120579 (119910) = 119879119894(119910) +
119894minus1
sum
119899=0
120579119899(119910) 119894 = 1 2 3 (12)
where 119879119894are unknown functions and 120579
119899are successive
approximations whose solutions are obtained recursivelyfrom solving the linear part of the equation that results fromsubstituting (12) in the governing equations (2) using 120579
0(119910) as
an initial approximationThe linearisation technique is basedon the assumption that 119879
119894becomes increasingly smaller as 119894
becomes larger that is
lim119894rarrinfin
119879119894= 0 (13)
Substituting (12) in (2) gives
11987910158401015840
119894+ 120582 exp[
119879119894+ sum119894minus1
119899=0120579119899
1 + 120576 (119879119894+ sum119894minus1
119899=0120579119899)
] = 119892 (119910) minus
119894minus1
sum
119899=0
12057910158401015840
119899 (14)
where 119892(119910) is a known function (from (8) and (9)) given by
119892 (119910) = minus120582119898(119889119882
119889119910)
2
[1 + 2120574(119889119882
119889119910)
2
] (15)
We choose 1205790= 0 as an initial approximation which is chosen
to satisfy the boundary conditions The subsequent solutionsfor 120579119899(119899 ge 1) are obtained by successively solving the line-
arised form of (14) which are given as
12057910158401015840
119894+ 119886119894minus1120579119894= 119903119894minus1 (16)
subject to the boundary conditions
120579119894(minus1) = 120579
119894(1) = 0 (17)
where
119903119894minus1
= 119892 (119910) minus [
119894minus1
sum
119899=0
12057910158401015840
119899+ 120582 exp(
sum119894minus1
119899=0120579119899
1 + 120576sum119894minus1
119899=0120579119899
)]
119886119894minus1
=120582
(1 + 120576sum119894minus1
119899=0120579119899)2exp[
sum119894minus1
119899=0120579119899
1 + 120576sum119894minus1
119899=0120579119899
]
(18)
Once each solution for 120579119894(119894 ge 1) has been found from
successively solving (11) and for each 119894 the approximatesolutions for 120579(119910) are obtained as
120579 (119910) asymp
119872
sum
119899=0
120579119899(119910) (19)
where119872 is the order of SLM approximationWe remark that the coefficient parameter 119886
119894minus1and the
right-hand side 119903119894minus1
of (16) for 119894 = 1 2 3 are known (fromprevious iterations) Thus (16) can easily be solved usinganalyticalmeans (whenever possible) or any numericalmeth-ods such as finite differences finite elements Runge-Kutta-based shooting methods or collocation methods In thiswork (10) and (16) are solved using the Chebyshev spectralcollocation method This method is based on approximatingthe unknown functions by the Chebyshev interpolating poly-nomials in such a way that they are collocated at the Gauss-Lobatto points defined as
119910119895= cos
120587119895
119873 119895 = 0 1 119873 (20)
where 119873 is the number of collocation points used (see eg[17 19]) The derivatives are approximated at the collocationpoints by
1198892119882
1198891199102=
119873
sum
119896=0
D2119896119895119882(119910119896)
1198892120579119894
1198891199102=
119873
sum
119896=0
D2119896119895120579119894(119910119896)
119895 = 0 1 119873
(21)
whereD is the Chebyshev spectral differentiationmatrix (seeeg [17 19]) Substituting (21) in (10) and (16) leads to matrixequations given by
D2W = F 119882 (1199100) = 119882(119910
119873) = 0 (22)
AT119894= R119894minus1 120579
119894(1199100) = 120579119894(119910119873) = 0 (23)
in which A is a (119873 + 1) times (119873 + 1) square matrix and TW Fand R are (119873 + 1) times 1 column vectors defined by
A = D2 + a119894minus1 (24)
F = [119891 (1199100) 119891 (119910
1) 119891 (119910
119873minus1) 119891 (119910
119873)]119879
(25)
T119894= [120579119894(1199100) 120579119894(1199101) 120579
119894(119910119873minus1
) 120579119894(119910119873)]119879
(26)
W = [119882 (1199100) 119882 (119910
1) 119882 (119910
119873minus1) 119882 (119910
119873)]119879
(27)
R119894minus1
= [119903119894minus1
(1199100) 119903119894minus1
(1199101) 119903
119894minus1(119910119873minus1
) 119903119894minus1
(119910119873)]119879
(28)
In the above definitions a119894minus1
is a diagonal matrix of size(119873+1)times(119873+1) and the superscript119879 denotes transpose After
4 Mathematical Problems in Engineering
Velo
city
120574 = 01
120574 = 1
120574 = 5
05
045
04
035
03
025
02
015
01
005
0minus1 minus05 0 05 1
119910
(a)
0
005
01
015
02
025
Tem
pera
ture
minus1 minus05 0 05 1119910
120574 = 01 1 5
(b)
Figure 2 Velocity and temperature profiles when 120574 = 01 1 5 when119898 = 1 120582 = 03 and 120576 = 01 SLM results (circles) bvp4c (solid line)
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1119910
05
04
03
02
01
0
120582 = 06
120582 = 05
120582 = 04
(a)
0
1
2
3
4
5
6
7Te
mpe
ratu
re
minus1 minus05 0 05 1119910
Lower branch
Upper branch120582 = 04 05 06
120582 = 04 05 06
(b)
Figure 3 Temperature profiles when 120582 = 04 05 06 when119898 = 1 120574 = 01 and 120576 = 01 SLM results (circles) bvp4c (solid line)
modifying the matrix system (22) and (23) to incorporateboundary conditions the solutions for 119882(119910) and 120579
119894(119910) are
obtained as
W = (D2)minus1F
T119894= Aminus1S
119894minus1
(29)
4 Results
In this sectionwe present the results showing the velocity dis-tribution and temperature distribution for different values ofthe governing parameters To check the accuracy of the pro-posed successive linearisation method (SLM) comparison ismade with numerical solutions obtained using the MATLABroutine bvp4c which is an adaptive Lobatto quadrature
scheme All the SLM results presented in this work weregenerated using119873 = 60 collocation points
Figure 2 depicts the effect of non-Newtonian parameter(120574) on both the velocity and temperature profiles Generallyboth fluid velocity and temperature profiles attained theirmaximum values along the channel centerline andminimumat the walls satisfying the boundary conditions Moreovera gradual decrease in the magnitude of fluid velocity andtemperature profiles is noticed with an increase in the valueof 120574 This can be attributed to the fact that as 120574 increasesthe fluid viscosity increases leading to a decrease in the flowrate In Figure 3 we observed that the fluid temperaturegenerally increases with an increase in the value of the Frank-Kamenetskii parameter (120582) due to the Arrhenius kineticsThe effect of viscous dissipation parameter (119898) on thefluid temperature is displayed in Figure 4 The internal heat
Mathematical Problems in Engineering 5
Tem
pera
ture
Lower branch119898 = 20
119898 = 15
119898 = 10
119898 = 5
119898 = 0
14
12
1
08
06
04
02
0minus1 minus05 0 05 1
119910
(a)
0
1
2
3
4
5
6
Tem
pera
ture
minus1 minus05 0 05 1119910
Upper branch119898 = 0 5 10 15 20
Lower branch119898 = 0 5 10 15 20
(b)
Figure 4 Upper branch and lower branch temperature profiles when 119898 = 0 5 10 15 20 when 120582 = 05 120574 = 1 and 120576 = 01 SLM results(circles) bvp4c (solid line)
Table 1 Comparison between the SLM results at different orders and the bvp4c numerical results for wall heat flux Nu for various values of119898 120574 120576 and 120582
119898 120574 120576 120582 1st order 2nd order 3rd order bvp4c
2 1 01 01 014948880 014958883 014958883 0149588834 1 01 01 019550530 019565401 019565401 0195654016 1 01 01 024152181 024172906 024172906 0241729068 1 01 01 028753831 028781405 028781405 0287814051 1 01 01 012648054 012655991 012655991 0126559911 1 01 02 026159825 026233446 026233447 0262334471 1 01 03 040652882 040944698 040944728 0409447281 1 01 04 056267511 057092737 057093089 0570930891 1 01 01 012648054 012655991 012655991 0126559911 3 01 01 012168410 012175980 012175980 0121759801 5 01 01 011956272 011963676 011963676 0119636761 10 01 01 011691095 011698289 011698289 0116982891 1 1 05 073172481 071360852 071359336 0713593361 1 5 05 073172481 066332507 066312547 0663125471 1 10 05 073172481 064501234 064484620 0644846201 1 20 05 073172481 063173135 063164065 063164065
generation due to viscous heating increases as the parametervalue of119898 increases leading to a general increase in the fluidtemperature Meanwhile the possibility of a lower and uppersolution branches is also highlighted in Figures 3 and 4 Thiscan be attributed to the nonlinear nature of the Arrheniuskinetics in the governing thermal boundary layer equation(2) It is noteworthy that the fluid temperature decreaseswith an increase in the activation energy parameter (120576) asillustrated in Figure 5 As 120576 increases the fluid becomes lessvolatile and its activation energy decreases
A slice of the bifurcation diagram for 120574 gt 0 in the(120582 minus120579
1015840(1)) plane is shown in Figure 6 It represents the varia-
tion of wall heat flux with the Frank-Kamenetskii parameter(120582) In particular for every 0 le 120576 le 01 there is a critical
value 120582119888(a turning point) such that for 0 le 120582 lt 120582
119888there are
two solutionsThis result is in perfect agreement with the onereported in Makinde [7]
In Table 1 we show the computations illustrating thecomparison between the SLM results at different orders andthe bvp4c numerical results for wall heat flux Nu for variousvalues of 119898 120574 120576 and 120582 It can be seen from the table thatthe SLM results are in very good agreement with the bvp4cnumerical results
5 Conclusion
In this work we employed a very powerful new linearisationtechnique known as the successive linearisation method
6 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1
120576 = 0 1 5 20
119910
Figure 5 Temperature profiles when 120576 = 0 1 5 20 when 120582 = 05120574 = 01 and119898 = 1 SLM results (circles) bvp4c (solid line)
0 02 04 06 08 10
1
2
3
4
5
6
7
120582119888 = 0936
120582
minus120579998400
(1)
Figure 6 Bifurcation diagram when 120574 = 01 120576 = 01 and119898 = 1
(SLM) to investigate the flow of reactive third-grade liquid ina channel with isothermal wallsThe SLM results for the gov-erning flow parameters were compared with results obtain-ed using MATLABrsquos bvp4c function and excellent agree-ment was observed From the results obtained in the studythe following was observed
(i) An increase in the non-Newtonian parameter (120574)leads to a gradual decrease in the magnitude of fluidvelocity and temperature profiles
(ii) The fluid temperature generally increases with an in-crease in the value of the Frank-Kamenetskii param-eter (120582)
(iii) The internal heat generation due to viscous heatingincreases as the parameter value 119898 increases leadingto a general increase in the fluid temperature
It was also shown that the governing nonlinear equationsadmitmultiple solutions Using the SLM approach lower andupper branch solutions were obtained and discussed
References
[1] K R Rajagopal ldquoOn boundary conditions for fluids of the dif-ferential typerdquo inNavier-Stokes Equations andRelatedNonlinearProblems (Funchal 1994) pp 273ndash278 Plenum New York NYUSA 1995
[2] A M Siddiqui M Ahmed and Q K Ghori ldquoCouette andpoiseuille flows for non-newtonian fluidsrdquo International Journalof Nonlinear Sciences and Numerical Simulation vol 7 no 1 pp15ndash26 2006
[3] D A Frank KamenetskiiDiffusion and Heat Transfer in Chem-ical Kinetics Plenum Press New York NY USA 1969
[4] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[5] M Yurusoy and M Pakdemirli ldquoApproximate analytical solu-tions for the flow of a third-grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[6] R L Fosdick and K R Rajagopal ldquoThermodynamics and sta-bility of fluids of third graderdquo Proceedings of the Royal Society Avol 369 no 1738 pp 351ndash377 1980
[7] O D Makinde ldquoHermite-Pade approximation approach tothermal criticality for a reactive third-grade liquid in a channelwith isothermal wallsrdquo International Communications in Heatand Mass Transfer vol 34 no 7 pp 870ndash877 2007
[8] O DMakinde ldquoThermal criticality for a reactive gravity driventhin film flow of a third-grade fluid with adiabatic free surfacedown an inclined planerdquo Applied Mathematics and Mechanicsvol 30 no 3 pp 373ndash380 2009
[9] O D Makinde ldquoAnalysis of non-Newtonian reactive flow in acylindrical piperdquo Journal of Applied Mechanics vol 76 no 3Article ID 034502 pp 1ndash5 2009
[10] 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
[11] O D Makinde and R J Moitsheki ldquoOn nonperturbative tech-niques for thermal radiation effect on natural convection past avertical plate embedded in a saturated porous mediumrdquoMath-ematical Problems in Engineering vol 2008 Article ID 68907411 pages 2008
[12] O D Makinde ldquoOn the chebyshev collocation spectral ap-proach to stability of fluid in a porous mediumrdquo InternationalJournal for Numerical Methods in Fluids vol 59 no 7 pp 791ndash799 2009
[13] A Shidfar M Djalalvand and M Garshasbi ldquoA numeri-cal scheme for solving special class of nonlinear diffusion-convection equationrdquo Applied Mathematics and Computationvol 167 no 2 pp 1080ndash1089 2005
[14] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homo-topy analysismethod for solving a nonlinear second order BVPrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 9 pp 2293ndash2302 2010
[15] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA new spe-ctral-homotopy analysis method for the MHD Jeffery-Hamelproblemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash12252010
Mathematical Problems in Engineering 7
[16] S LiaoBeyondPerturbation Introduction to theHomotopyAna-lysis Method Chapman amp HallCRC Boca Raton Fla USA2003
[17] C Canuto M Y Hussaini A Quarteroni and T A Zang Spe-ctral Methods in Fluid Dynamics Springer Series in Computa-tional Physics Springer New York NY USA 1988
[18] 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
[19] L N Trefethen Spectral Methods in MATLAB vol 10 of Soft-ware Environments and Tools SIAM Philadelphia Pa USA2000
[20] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009
[21] C Yang and S Liao ldquoOn the explicit purely analytic solutionof Von Karman swirling viscous flowrdquoCommunications in Non-linear Science andNumerical Simulation vol 11 no 1 pp 83ndash932006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
2 Mathematical Formulation
Figure 1 depicts the problem geometry We consider thesteady flow of an incompressible third-grade reactive fluidplaced between two parallel isothermal plates It is assumedthat the flow is hydrodynamically and thermally fully devel-oped under the action of a constant axial pressure gradient
Following [1 4 5 7ndash9] the dimensionless governingequations for the momentum and energy balance can bewritten as
1198892119882
1198891199102+ 6120574
1198892119882
1198891199102(119889119882
119889119910)
2
= minus1 (1)
1198892120579
1198891199102+ 120582[119890
120579(1+120576120579)+ 119898(
119889119882
119889119910)
2
times(1 + 2120574(119889119882
119889119910)
2
)] = 0
(2)
where 119882 is the dimensionless velocity component 119910 isthe dimensionless normal coordinate 120579 is the dimension-less temperature and 120582 120576 120574 and 119898 represent the Frank-Kamenetskii parameter activation energy parameter thedimensionless non-Newtonian parameter and the viscousheating parameter respectively The additional Arrheniuskinetics term in energy balance equation (2) is due to [3]The appropriate boundary conditions in dimensionless formare given as follows the surface of the channel is fixedimpermeable and maintained at a given temperature
119882(1) = 0 120579 (1) = 0 at 119910 = 1 (3)
and the symmetry condition along the centerline that is
119889119882
119889119910=119889120579
119889119910= 0 at 119910 = 0 (4)
We have employed the following nondimensional quan-tities in (1)ndash(4)
120579 =119864 (119879 minus 119879
0)
1198771198792
0
119910 =119910
119886
120582 =119876119864119860119886
21198620119890minus1198641198771198790
1198792
0119877119896
119882 =119906
119880119866
119898 =120583119866211988021198901198641198771198790
11987611986011988621198620
120576 =1198771198790
119864
119866 = minus1198862
120583119880
119889119875
119889119909 120574 =
120573311988021198662
1198862120583
(5)
where 119879 is the absolute temperature 119880 is the fluid charac-teristic velocity 119879
0is the plate temperature 119896 is the thermal
conductivity of the material 119876 is the heat of reaction 119860 isthe rate constant 119864 is the activation energy 119877 is the universalgas constant 119862
0is the initial concentration of the reactant
species 119886 is the channel half width 1205733is the material coef-
ficient 119875 is the modified pressure and 120583 is the fluid dynamic
Third-grade reactive fluid
119906 = 0119910
119879 = 1198790 119910 = 119886
119906 = 0 119879 = 1198790 119910 = minus119886
Figure 1 Geometry of the problem
viscosity coefficient In the following sections (1)ndash(4) aresolved numerically using the new successive linearizationtechnique
3 Successive Linearisation Method (SLM)
In this section we apply the proposed linearisationmethod ofsolution hereinafter referred to as the successive linearisationmethod (SLM) to solve the governing equations (1) and (2)Before applying the SLM we first note that the problem canbe significantly simplified by finding the explicit analyticalsolution for the derivative 119889119882119889119910 First (1) is rewritten as
119889
119889119910[119889119882
119889119910+ 2120574(
119889119882
119889119910)
3
] = minus1 (6)
The above equation is then integrated on both sides and thesymmetry boundary condition 119889119882119889119910(0) = 0 is used toevaluate the resulting integrating constant to give
2120574(119889119882
119889119910)
3
+ (119889119882
119889119910) + 119910 = 0 (7)
We note that (7) is a cubic equation which can have eitherone or three real solutions If only positive values of 120574 areconsidered (7) will have a unique real solution which can becomputed using Maple and is given by
119889119882
119889119910=
1
6120574[120595(119910)]
13minus
1
[120595 (119910)]13
(8)
where
120595 (119910) = (minus54119910 + 6radic3radic2 + 27120574119910
2
120574)1205742 (9)
The analytical result is very important because when eva-luated at 119910 = 1 it gives an explicit analytical expression forthe skin friction coefficient (119862
119891) A close inspection of (8)
indicates that if 119862119891has a critical point then (120574
119888 119862119891) = (minus32
minus227) This critical point was reported as a bifurcation pointin [7] It is worth noting that 119889119882119889119910 and hence the solution119882(119910) is only valid when 120574 ge minus227
Since the momentum equation (1) is decoupled from theenergy equation (2) we solve for the velocity119882(119910) first then
Mathematical Problems in Engineering 3
substitute the result in the energy equation to obtain 120579(119910) Tosolve119882(119910) we write (1) as
1198892119882
1198891199102= 119891 (119910) (10)
where 119891(119910) is a known explicit function of 119910 given by
119891 (119910) = minus1
1 + 6120574(119889119882119889119910)2 (11)
with 119889119882119889119910 given by (8) Equation (10) can easily be inte-grated using any numerical methodThe SLM is based on theassumption that the unknown function 120579(119910) can be expandedas
120579 (119910) = 119879119894(119910) +
119894minus1
sum
119899=0
120579119899(119910) 119894 = 1 2 3 (12)
where 119879119894are unknown functions and 120579
119899are successive
approximations whose solutions are obtained recursivelyfrom solving the linear part of the equation that results fromsubstituting (12) in the governing equations (2) using 120579
0(119910) as
an initial approximationThe linearisation technique is basedon the assumption that 119879
119894becomes increasingly smaller as 119894
becomes larger that is
lim119894rarrinfin
119879119894= 0 (13)
Substituting (12) in (2) gives
11987910158401015840
119894+ 120582 exp[
119879119894+ sum119894minus1
119899=0120579119899
1 + 120576 (119879119894+ sum119894minus1
119899=0120579119899)
] = 119892 (119910) minus
119894minus1
sum
119899=0
12057910158401015840
119899 (14)
where 119892(119910) is a known function (from (8) and (9)) given by
119892 (119910) = minus120582119898(119889119882
119889119910)
2
[1 + 2120574(119889119882
119889119910)
2
] (15)
We choose 1205790= 0 as an initial approximation which is chosen
to satisfy the boundary conditions The subsequent solutionsfor 120579119899(119899 ge 1) are obtained by successively solving the line-
arised form of (14) which are given as
12057910158401015840
119894+ 119886119894minus1120579119894= 119903119894minus1 (16)
subject to the boundary conditions
120579119894(minus1) = 120579
119894(1) = 0 (17)
where
119903119894minus1
= 119892 (119910) minus [
119894minus1
sum
119899=0
12057910158401015840
119899+ 120582 exp(
sum119894minus1
119899=0120579119899
1 + 120576sum119894minus1
119899=0120579119899
)]
119886119894minus1
=120582
(1 + 120576sum119894minus1
119899=0120579119899)2exp[
sum119894minus1
119899=0120579119899
1 + 120576sum119894minus1
119899=0120579119899
]
(18)
Once each solution for 120579119894(119894 ge 1) has been found from
successively solving (11) and for each 119894 the approximatesolutions for 120579(119910) are obtained as
120579 (119910) asymp
119872
sum
119899=0
120579119899(119910) (19)
where119872 is the order of SLM approximationWe remark that the coefficient parameter 119886
119894minus1and the
right-hand side 119903119894minus1
of (16) for 119894 = 1 2 3 are known (fromprevious iterations) Thus (16) can easily be solved usinganalyticalmeans (whenever possible) or any numericalmeth-ods such as finite differences finite elements Runge-Kutta-based shooting methods or collocation methods In thiswork (10) and (16) are solved using the Chebyshev spectralcollocation method This method is based on approximatingthe unknown functions by the Chebyshev interpolating poly-nomials in such a way that they are collocated at the Gauss-Lobatto points defined as
119910119895= cos
120587119895
119873 119895 = 0 1 119873 (20)
where 119873 is the number of collocation points used (see eg[17 19]) The derivatives are approximated at the collocationpoints by
1198892119882
1198891199102=
119873
sum
119896=0
D2119896119895119882(119910119896)
1198892120579119894
1198891199102=
119873
sum
119896=0
D2119896119895120579119894(119910119896)
119895 = 0 1 119873
(21)
whereD is the Chebyshev spectral differentiationmatrix (seeeg [17 19]) Substituting (21) in (10) and (16) leads to matrixequations given by
D2W = F 119882 (1199100) = 119882(119910
119873) = 0 (22)
AT119894= R119894minus1 120579
119894(1199100) = 120579119894(119910119873) = 0 (23)
in which A is a (119873 + 1) times (119873 + 1) square matrix and TW Fand R are (119873 + 1) times 1 column vectors defined by
A = D2 + a119894minus1 (24)
F = [119891 (1199100) 119891 (119910
1) 119891 (119910
119873minus1) 119891 (119910
119873)]119879
(25)
T119894= [120579119894(1199100) 120579119894(1199101) 120579
119894(119910119873minus1
) 120579119894(119910119873)]119879
(26)
W = [119882 (1199100) 119882 (119910
1) 119882 (119910
119873minus1) 119882 (119910
119873)]119879
(27)
R119894minus1
= [119903119894minus1
(1199100) 119903119894minus1
(1199101) 119903
119894minus1(119910119873minus1
) 119903119894minus1
(119910119873)]119879
(28)
In the above definitions a119894minus1
is a diagonal matrix of size(119873+1)times(119873+1) and the superscript119879 denotes transpose After
4 Mathematical Problems in Engineering
Velo
city
120574 = 01
120574 = 1
120574 = 5
05
045
04
035
03
025
02
015
01
005
0minus1 minus05 0 05 1
119910
(a)
0
005
01
015
02
025
Tem
pera
ture
minus1 minus05 0 05 1119910
120574 = 01 1 5
(b)
Figure 2 Velocity and temperature profiles when 120574 = 01 1 5 when119898 = 1 120582 = 03 and 120576 = 01 SLM results (circles) bvp4c (solid line)
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1119910
05
04
03
02
01
0
120582 = 06
120582 = 05
120582 = 04
(a)
0
1
2
3
4
5
6
7Te
mpe
ratu
re
minus1 minus05 0 05 1119910
Lower branch
Upper branch120582 = 04 05 06
120582 = 04 05 06
(b)
Figure 3 Temperature profiles when 120582 = 04 05 06 when119898 = 1 120574 = 01 and 120576 = 01 SLM results (circles) bvp4c (solid line)
modifying the matrix system (22) and (23) to incorporateboundary conditions the solutions for 119882(119910) and 120579
119894(119910) are
obtained as
W = (D2)minus1F
T119894= Aminus1S
119894minus1
(29)
4 Results
In this sectionwe present the results showing the velocity dis-tribution and temperature distribution for different values ofthe governing parameters To check the accuracy of the pro-posed successive linearisation method (SLM) comparison ismade with numerical solutions obtained using the MATLABroutine bvp4c which is an adaptive Lobatto quadrature
scheme All the SLM results presented in this work weregenerated using119873 = 60 collocation points
Figure 2 depicts the effect of non-Newtonian parameter(120574) on both the velocity and temperature profiles Generallyboth fluid velocity and temperature profiles attained theirmaximum values along the channel centerline andminimumat the walls satisfying the boundary conditions Moreovera gradual decrease in the magnitude of fluid velocity andtemperature profiles is noticed with an increase in the valueof 120574 This can be attributed to the fact that as 120574 increasesthe fluid viscosity increases leading to a decrease in the flowrate In Figure 3 we observed that the fluid temperaturegenerally increases with an increase in the value of the Frank-Kamenetskii parameter (120582) due to the Arrhenius kineticsThe effect of viscous dissipation parameter (119898) on thefluid temperature is displayed in Figure 4 The internal heat
Mathematical Problems in Engineering 5
Tem
pera
ture
Lower branch119898 = 20
119898 = 15
119898 = 10
119898 = 5
119898 = 0
14
12
1
08
06
04
02
0minus1 minus05 0 05 1
119910
(a)
0
1
2
3
4
5
6
Tem
pera
ture
minus1 minus05 0 05 1119910
Upper branch119898 = 0 5 10 15 20
Lower branch119898 = 0 5 10 15 20
(b)
Figure 4 Upper branch and lower branch temperature profiles when 119898 = 0 5 10 15 20 when 120582 = 05 120574 = 1 and 120576 = 01 SLM results(circles) bvp4c (solid line)
Table 1 Comparison between the SLM results at different orders and the bvp4c numerical results for wall heat flux Nu for various values of119898 120574 120576 and 120582
119898 120574 120576 120582 1st order 2nd order 3rd order bvp4c
2 1 01 01 014948880 014958883 014958883 0149588834 1 01 01 019550530 019565401 019565401 0195654016 1 01 01 024152181 024172906 024172906 0241729068 1 01 01 028753831 028781405 028781405 0287814051 1 01 01 012648054 012655991 012655991 0126559911 1 01 02 026159825 026233446 026233447 0262334471 1 01 03 040652882 040944698 040944728 0409447281 1 01 04 056267511 057092737 057093089 0570930891 1 01 01 012648054 012655991 012655991 0126559911 3 01 01 012168410 012175980 012175980 0121759801 5 01 01 011956272 011963676 011963676 0119636761 10 01 01 011691095 011698289 011698289 0116982891 1 1 05 073172481 071360852 071359336 0713593361 1 5 05 073172481 066332507 066312547 0663125471 1 10 05 073172481 064501234 064484620 0644846201 1 20 05 073172481 063173135 063164065 063164065
generation due to viscous heating increases as the parametervalue of119898 increases leading to a general increase in the fluidtemperature Meanwhile the possibility of a lower and uppersolution branches is also highlighted in Figures 3 and 4 Thiscan be attributed to the nonlinear nature of the Arrheniuskinetics in the governing thermal boundary layer equation(2) It is noteworthy that the fluid temperature decreaseswith an increase in the activation energy parameter (120576) asillustrated in Figure 5 As 120576 increases the fluid becomes lessvolatile and its activation energy decreases
A slice of the bifurcation diagram for 120574 gt 0 in the(120582 minus120579
1015840(1)) plane is shown in Figure 6 It represents the varia-
tion of wall heat flux with the Frank-Kamenetskii parameter(120582) In particular for every 0 le 120576 le 01 there is a critical
value 120582119888(a turning point) such that for 0 le 120582 lt 120582
119888there are
two solutionsThis result is in perfect agreement with the onereported in Makinde [7]
In Table 1 we show the computations illustrating thecomparison between the SLM results at different orders andthe bvp4c numerical results for wall heat flux Nu for variousvalues of 119898 120574 120576 and 120582 It can be seen from the table thatthe SLM results are in very good agreement with the bvp4cnumerical results
5 Conclusion
In this work we employed a very powerful new linearisationtechnique known as the successive linearisation method
6 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1
120576 = 0 1 5 20
119910
Figure 5 Temperature profiles when 120576 = 0 1 5 20 when 120582 = 05120574 = 01 and119898 = 1 SLM results (circles) bvp4c (solid line)
0 02 04 06 08 10
1
2
3
4
5
6
7
120582119888 = 0936
120582
minus120579998400
(1)
Figure 6 Bifurcation diagram when 120574 = 01 120576 = 01 and119898 = 1
(SLM) to investigate the flow of reactive third-grade liquid ina channel with isothermal wallsThe SLM results for the gov-erning flow parameters were compared with results obtain-ed using MATLABrsquos bvp4c function and excellent agree-ment was observed From the results obtained in the studythe following was observed
(i) An increase in the non-Newtonian parameter (120574)leads to a gradual decrease in the magnitude of fluidvelocity and temperature profiles
(ii) The fluid temperature generally increases with an in-crease in the value of the Frank-Kamenetskii param-eter (120582)
(iii) The internal heat generation due to viscous heatingincreases as the parameter value 119898 increases leadingto a general increase in the fluid temperature
It was also shown that the governing nonlinear equationsadmitmultiple solutions Using the SLM approach lower andupper branch solutions were obtained and discussed
References
[1] K R Rajagopal ldquoOn boundary conditions for fluids of the dif-ferential typerdquo inNavier-Stokes Equations andRelatedNonlinearProblems (Funchal 1994) pp 273ndash278 Plenum New York NYUSA 1995
[2] A M Siddiqui M Ahmed and Q K Ghori ldquoCouette andpoiseuille flows for non-newtonian fluidsrdquo International Journalof Nonlinear Sciences and Numerical Simulation vol 7 no 1 pp15ndash26 2006
[3] D A Frank KamenetskiiDiffusion and Heat Transfer in Chem-ical Kinetics Plenum Press New York NY USA 1969
[4] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[5] M Yurusoy and M Pakdemirli ldquoApproximate analytical solu-tions for the flow of a third-grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[6] R L Fosdick and K R Rajagopal ldquoThermodynamics and sta-bility of fluids of third graderdquo Proceedings of the Royal Society Avol 369 no 1738 pp 351ndash377 1980
[7] O D Makinde ldquoHermite-Pade approximation approach tothermal criticality for a reactive third-grade liquid in a channelwith isothermal wallsrdquo International Communications in Heatand Mass Transfer vol 34 no 7 pp 870ndash877 2007
[8] O DMakinde ldquoThermal criticality for a reactive gravity driventhin film flow of a third-grade fluid with adiabatic free surfacedown an inclined planerdquo Applied Mathematics and Mechanicsvol 30 no 3 pp 373ndash380 2009
[9] O D Makinde ldquoAnalysis of non-Newtonian reactive flow in acylindrical piperdquo Journal of Applied Mechanics vol 76 no 3Article ID 034502 pp 1ndash5 2009
[10] 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
[11] O D Makinde and R J Moitsheki ldquoOn nonperturbative tech-niques for thermal radiation effect on natural convection past avertical plate embedded in a saturated porous mediumrdquoMath-ematical Problems in Engineering vol 2008 Article ID 68907411 pages 2008
[12] O D Makinde ldquoOn the chebyshev collocation spectral ap-proach to stability of fluid in a porous mediumrdquo InternationalJournal for Numerical Methods in Fluids vol 59 no 7 pp 791ndash799 2009
[13] A Shidfar M Djalalvand and M Garshasbi ldquoA numeri-cal scheme for solving special class of nonlinear diffusion-convection equationrdquo Applied Mathematics and Computationvol 167 no 2 pp 1080ndash1089 2005
[14] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homo-topy analysismethod for solving a nonlinear second order BVPrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 9 pp 2293ndash2302 2010
[15] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA new spe-ctral-homotopy analysis method for the MHD Jeffery-Hamelproblemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash12252010
Mathematical Problems in Engineering 7
[16] S LiaoBeyondPerturbation Introduction to theHomotopyAna-lysis Method Chapman amp HallCRC Boca Raton Fla USA2003
[17] C Canuto M Y Hussaini A Quarteroni and T A Zang Spe-ctral Methods in Fluid Dynamics Springer Series in Computa-tional Physics Springer New York NY USA 1988
[18] 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
[19] L N Trefethen Spectral Methods in MATLAB vol 10 of Soft-ware Environments and Tools SIAM Philadelphia Pa USA2000
[20] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009
[21] C Yang and S Liao ldquoOn the explicit purely analytic solutionof Von Karman swirling viscous flowrdquoCommunications in Non-linear Science andNumerical Simulation vol 11 no 1 pp 83ndash932006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
substitute the result in the energy equation to obtain 120579(119910) Tosolve119882(119910) we write (1) as
1198892119882
1198891199102= 119891 (119910) (10)
where 119891(119910) is a known explicit function of 119910 given by
119891 (119910) = minus1
1 + 6120574(119889119882119889119910)2 (11)
with 119889119882119889119910 given by (8) Equation (10) can easily be inte-grated using any numerical methodThe SLM is based on theassumption that the unknown function 120579(119910) can be expandedas
120579 (119910) = 119879119894(119910) +
119894minus1
sum
119899=0
120579119899(119910) 119894 = 1 2 3 (12)
where 119879119894are unknown functions and 120579
119899are successive
approximations whose solutions are obtained recursivelyfrom solving the linear part of the equation that results fromsubstituting (12) in the governing equations (2) using 120579
0(119910) as
an initial approximationThe linearisation technique is basedon the assumption that 119879
119894becomes increasingly smaller as 119894
becomes larger that is
lim119894rarrinfin
119879119894= 0 (13)
Substituting (12) in (2) gives
11987910158401015840
119894+ 120582 exp[
119879119894+ sum119894minus1
119899=0120579119899
1 + 120576 (119879119894+ sum119894minus1
119899=0120579119899)
] = 119892 (119910) minus
119894minus1
sum
119899=0
12057910158401015840
119899 (14)
where 119892(119910) is a known function (from (8) and (9)) given by
119892 (119910) = minus120582119898(119889119882
119889119910)
2
[1 + 2120574(119889119882
119889119910)
2
] (15)
We choose 1205790= 0 as an initial approximation which is chosen
to satisfy the boundary conditions The subsequent solutionsfor 120579119899(119899 ge 1) are obtained by successively solving the line-
arised form of (14) which are given as
12057910158401015840
119894+ 119886119894minus1120579119894= 119903119894minus1 (16)
subject to the boundary conditions
120579119894(minus1) = 120579
119894(1) = 0 (17)
where
119903119894minus1
= 119892 (119910) minus [
119894minus1
sum
119899=0
12057910158401015840
119899+ 120582 exp(
sum119894minus1
119899=0120579119899
1 + 120576sum119894minus1
119899=0120579119899
)]
119886119894minus1
=120582
(1 + 120576sum119894minus1
119899=0120579119899)2exp[
sum119894minus1
119899=0120579119899
1 + 120576sum119894minus1
119899=0120579119899
]
(18)
Once each solution for 120579119894(119894 ge 1) has been found from
successively solving (11) and for each 119894 the approximatesolutions for 120579(119910) are obtained as
120579 (119910) asymp
119872
sum
119899=0
120579119899(119910) (19)
where119872 is the order of SLM approximationWe remark that the coefficient parameter 119886
119894minus1and the
right-hand side 119903119894minus1
of (16) for 119894 = 1 2 3 are known (fromprevious iterations) Thus (16) can easily be solved usinganalyticalmeans (whenever possible) or any numericalmeth-ods such as finite differences finite elements Runge-Kutta-based shooting methods or collocation methods In thiswork (10) and (16) are solved using the Chebyshev spectralcollocation method This method is based on approximatingthe unknown functions by the Chebyshev interpolating poly-nomials in such a way that they are collocated at the Gauss-Lobatto points defined as
119910119895= cos
120587119895
119873 119895 = 0 1 119873 (20)
where 119873 is the number of collocation points used (see eg[17 19]) The derivatives are approximated at the collocationpoints by
1198892119882
1198891199102=
119873
sum
119896=0
D2119896119895119882(119910119896)
1198892120579119894
1198891199102=
119873
sum
119896=0
D2119896119895120579119894(119910119896)
119895 = 0 1 119873
(21)
whereD is the Chebyshev spectral differentiationmatrix (seeeg [17 19]) Substituting (21) in (10) and (16) leads to matrixequations given by
D2W = F 119882 (1199100) = 119882(119910
119873) = 0 (22)
AT119894= R119894minus1 120579
119894(1199100) = 120579119894(119910119873) = 0 (23)
in which A is a (119873 + 1) times (119873 + 1) square matrix and TW Fand R are (119873 + 1) times 1 column vectors defined by
A = D2 + a119894minus1 (24)
F = [119891 (1199100) 119891 (119910
1) 119891 (119910
119873minus1) 119891 (119910
119873)]119879
(25)
T119894= [120579119894(1199100) 120579119894(1199101) 120579
119894(119910119873minus1
) 120579119894(119910119873)]119879
(26)
W = [119882 (1199100) 119882 (119910
1) 119882 (119910
119873minus1) 119882 (119910
119873)]119879
(27)
R119894minus1
= [119903119894minus1
(1199100) 119903119894minus1
(1199101) 119903
119894minus1(119910119873minus1
) 119903119894minus1
(119910119873)]119879
(28)
In the above definitions a119894minus1
is a diagonal matrix of size(119873+1)times(119873+1) and the superscript119879 denotes transpose After
4 Mathematical Problems in Engineering
Velo
city
120574 = 01
120574 = 1
120574 = 5
05
045
04
035
03
025
02
015
01
005
0minus1 minus05 0 05 1
119910
(a)
0
005
01
015
02
025
Tem
pera
ture
minus1 minus05 0 05 1119910
120574 = 01 1 5
(b)
Figure 2 Velocity and temperature profiles when 120574 = 01 1 5 when119898 = 1 120582 = 03 and 120576 = 01 SLM results (circles) bvp4c (solid line)
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1119910
05
04
03
02
01
0
120582 = 06
120582 = 05
120582 = 04
(a)
0
1
2
3
4
5
6
7Te
mpe
ratu
re
minus1 minus05 0 05 1119910
Lower branch
Upper branch120582 = 04 05 06
120582 = 04 05 06
(b)
Figure 3 Temperature profiles when 120582 = 04 05 06 when119898 = 1 120574 = 01 and 120576 = 01 SLM results (circles) bvp4c (solid line)
modifying the matrix system (22) and (23) to incorporateboundary conditions the solutions for 119882(119910) and 120579
119894(119910) are
obtained as
W = (D2)minus1F
T119894= Aminus1S
119894minus1
(29)
4 Results
In this sectionwe present the results showing the velocity dis-tribution and temperature distribution for different values ofthe governing parameters To check the accuracy of the pro-posed successive linearisation method (SLM) comparison ismade with numerical solutions obtained using the MATLABroutine bvp4c which is an adaptive Lobatto quadrature
scheme All the SLM results presented in this work weregenerated using119873 = 60 collocation points
Figure 2 depicts the effect of non-Newtonian parameter(120574) on both the velocity and temperature profiles Generallyboth fluid velocity and temperature profiles attained theirmaximum values along the channel centerline andminimumat the walls satisfying the boundary conditions Moreovera gradual decrease in the magnitude of fluid velocity andtemperature profiles is noticed with an increase in the valueof 120574 This can be attributed to the fact that as 120574 increasesthe fluid viscosity increases leading to a decrease in the flowrate In Figure 3 we observed that the fluid temperaturegenerally increases with an increase in the value of the Frank-Kamenetskii parameter (120582) due to the Arrhenius kineticsThe effect of viscous dissipation parameter (119898) on thefluid temperature is displayed in Figure 4 The internal heat
Mathematical Problems in Engineering 5
Tem
pera
ture
Lower branch119898 = 20
119898 = 15
119898 = 10
119898 = 5
119898 = 0
14
12
1
08
06
04
02
0minus1 minus05 0 05 1
119910
(a)
0
1
2
3
4
5
6
Tem
pera
ture
minus1 minus05 0 05 1119910
Upper branch119898 = 0 5 10 15 20
Lower branch119898 = 0 5 10 15 20
(b)
Figure 4 Upper branch and lower branch temperature profiles when 119898 = 0 5 10 15 20 when 120582 = 05 120574 = 1 and 120576 = 01 SLM results(circles) bvp4c (solid line)
Table 1 Comparison between the SLM results at different orders and the bvp4c numerical results for wall heat flux Nu for various values of119898 120574 120576 and 120582
119898 120574 120576 120582 1st order 2nd order 3rd order bvp4c
2 1 01 01 014948880 014958883 014958883 0149588834 1 01 01 019550530 019565401 019565401 0195654016 1 01 01 024152181 024172906 024172906 0241729068 1 01 01 028753831 028781405 028781405 0287814051 1 01 01 012648054 012655991 012655991 0126559911 1 01 02 026159825 026233446 026233447 0262334471 1 01 03 040652882 040944698 040944728 0409447281 1 01 04 056267511 057092737 057093089 0570930891 1 01 01 012648054 012655991 012655991 0126559911 3 01 01 012168410 012175980 012175980 0121759801 5 01 01 011956272 011963676 011963676 0119636761 10 01 01 011691095 011698289 011698289 0116982891 1 1 05 073172481 071360852 071359336 0713593361 1 5 05 073172481 066332507 066312547 0663125471 1 10 05 073172481 064501234 064484620 0644846201 1 20 05 073172481 063173135 063164065 063164065
generation due to viscous heating increases as the parametervalue of119898 increases leading to a general increase in the fluidtemperature Meanwhile the possibility of a lower and uppersolution branches is also highlighted in Figures 3 and 4 Thiscan be attributed to the nonlinear nature of the Arrheniuskinetics in the governing thermal boundary layer equation(2) It is noteworthy that the fluid temperature decreaseswith an increase in the activation energy parameter (120576) asillustrated in Figure 5 As 120576 increases the fluid becomes lessvolatile and its activation energy decreases
A slice of the bifurcation diagram for 120574 gt 0 in the(120582 minus120579
1015840(1)) plane is shown in Figure 6 It represents the varia-
tion of wall heat flux with the Frank-Kamenetskii parameter(120582) In particular for every 0 le 120576 le 01 there is a critical
value 120582119888(a turning point) such that for 0 le 120582 lt 120582
119888there are
two solutionsThis result is in perfect agreement with the onereported in Makinde [7]
In Table 1 we show the computations illustrating thecomparison between the SLM results at different orders andthe bvp4c numerical results for wall heat flux Nu for variousvalues of 119898 120574 120576 and 120582 It can be seen from the table thatthe SLM results are in very good agreement with the bvp4cnumerical results
5 Conclusion
In this work we employed a very powerful new linearisationtechnique known as the successive linearisation method
6 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1
120576 = 0 1 5 20
119910
Figure 5 Temperature profiles when 120576 = 0 1 5 20 when 120582 = 05120574 = 01 and119898 = 1 SLM results (circles) bvp4c (solid line)
0 02 04 06 08 10
1
2
3
4
5
6
7
120582119888 = 0936
120582
minus120579998400
(1)
Figure 6 Bifurcation diagram when 120574 = 01 120576 = 01 and119898 = 1
(SLM) to investigate the flow of reactive third-grade liquid ina channel with isothermal wallsThe SLM results for the gov-erning flow parameters were compared with results obtain-ed using MATLABrsquos bvp4c function and excellent agree-ment was observed From the results obtained in the studythe following was observed
(i) An increase in the non-Newtonian parameter (120574)leads to a gradual decrease in the magnitude of fluidvelocity and temperature profiles
(ii) The fluid temperature generally increases with an in-crease in the value of the Frank-Kamenetskii param-eter (120582)
(iii) The internal heat generation due to viscous heatingincreases as the parameter value 119898 increases leadingto a general increase in the fluid temperature
It was also shown that the governing nonlinear equationsadmitmultiple solutions Using the SLM approach lower andupper branch solutions were obtained and discussed
References
[1] K R Rajagopal ldquoOn boundary conditions for fluids of the dif-ferential typerdquo inNavier-Stokes Equations andRelatedNonlinearProblems (Funchal 1994) pp 273ndash278 Plenum New York NYUSA 1995
[2] A M Siddiqui M Ahmed and Q K Ghori ldquoCouette andpoiseuille flows for non-newtonian fluidsrdquo International Journalof Nonlinear Sciences and Numerical Simulation vol 7 no 1 pp15ndash26 2006
[3] D A Frank KamenetskiiDiffusion and Heat Transfer in Chem-ical Kinetics Plenum Press New York NY USA 1969
[4] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[5] M Yurusoy and M Pakdemirli ldquoApproximate analytical solu-tions for the flow of a third-grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[6] R L Fosdick and K R Rajagopal ldquoThermodynamics and sta-bility of fluids of third graderdquo Proceedings of the Royal Society Avol 369 no 1738 pp 351ndash377 1980
[7] O D Makinde ldquoHermite-Pade approximation approach tothermal criticality for a reactive third-grade liquid in a channelwith isothermal wallsrdquo International Communications in Heatand Mass Transfer vol 34 no 7 pp 870ndash877 2007
[8] O DMakinde ldquoThermal criticality for a reactive gravity driventhin film flow of a third-grade fluid with adiabatic free surfacedown an inclined planerdquo Applied Mathematics and Mechanicsvol 30 no 3 pp 373ndash380 2009
[9] O D Makinde ldquoAnalysis of non-Newtonian reactive flow in acylindrical piperdquo Journal of Applied Mechanics vol 76 no 3Article ID 034502 pp 1ndash5 2009
[10] 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
[11] O D Makinde and R J Moitsheki ldquoOn nonperturbative tech-niques for thermal radiation effect on natural convection past avertical plate embedded in a saturated porous mediumrdquoMath-ematical Problems in Engineering vol 2008 Article ID 68907411 pages 2008
[12] O D Makinde ldquoOn the chebyshev collocation spectral ap-proach to stability of fluid in a porous mediumrdquo InternationalJournal for Numerical Methods in Fluids vol 59 no 7 pp 791ndash799 2009
[13] A Shidfar M Djalalvand and M Garshasbi ldquoA numeri-cal scheme for solving special class of nonlinear diffusion-convection equationrdquo Applied Mathematics and Computationvol 167 no 2 pp 1080ndash1089 2005
[14] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homo-topy analysismethod for solving a nonlinear second order BVPrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 9 pp 2293ndash2302 2010
[15] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA new spe-ctral-homotopy analysis method for the MHD Jeffery-Hamelproblemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash12252010
Mathematical Problems in Engineering 7
[16] S LiaoBeyondPerturbation Introduction to theHomotopyAna-lysis Method Chapman amp HallCRC Boca Raton Fla USA2003
[17] C Canuto M Y Hussaini A Quarteroni and T A Zang Spe-ctral Methods in Fluid Dynamics Springer Series in Computa-tional Physics Springer New York NY USA 1988
[18] 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
[19] L N Trefethen Spectral Methods in MATLAB vol 10 of Soft-ware Environments and Tools SIAM Philadelphia Pa USA2000
[20] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009
[21] C Yang and S Liao ldquoOn the explicit purely analytic solutionof Von Karman swirling viscous flowrdquoCommunications in Non-linear Science andNumerical Simulation vol 11 no 1 pp 83ndash932006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
Velo
city
120574 = 01
120574 = 1
120574 = 5
05
045
04
035
03
025
02
015
01
005
0minus1 minus05 0 05 1
119910
(a)
0
005
01
015
02
025
Tem
pera
ture
minus1 minus05 0 05 1119910
120574 = 01 1 5
(b)
Figure 2 Velocity and temperature profiles when 120574 = 01 1 5 when119898 = 1 120582 = 03 and 120576 = 01 SLM results (circles) bvp4c (solid line)
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1119910
05
04
03
02
01
0
120582 = 06
120582 = 05
120582 = 04
(a)
0
1
2
3
4
5
6
7Te
mpe
ratu
re
minus1 minus05 0 05 1119910
Lower branch
Upper branch120582 = 04 05 06
120582 = 04 05 06
(b)
Figure 3 Temperature profiles when 120582 = 04 05 06 when119898 = 1 120574 = 01 and 120576 = 01 SLM results (circles) bvp4c (solid line)
modifying the matrix system (22) and (23) to incorporateboundary conditions the solutions for 119882(119910) and 120579
119894(119910) are
obtained as
W = (D2)minus1F
T119894= Aminus1S
119894minus1
(29)
4 Results
In this sectionwe present the results showing the velocity dis-tribution and temperature distribution for different values ofthe governing parameters To check the accuracy of the pro-posed successive linearisation method (SLM) comparison ismade with numerical solutions obtained using the MATLABroutine bvp4c which is an adaptive Lobatto quadrature
scheme All the SLM results presented in this work weregenerated using119873 = 60 collocation points
Figure 2 depicts the effect of non-Newtonian parameter(120574) on both the velocity and temperature profiles Generallyboth fluid velocity and temperature profiles attained theirmaximum values along the channel centerline andminimumat the walls satisfying the boundary conditions Moreovera gradual decrease in the magnitude of fluid velocity andtemperature profiles is noticed with an increase in the valueof 120574 This can be attributed to the fact that as 120574 increasesthe fluid viscosity increases leading to a decrease in the flowrate In Figure 3 we observed that the fluid temperaturegenerally increases with an increase in the value of the Frank-Kamenetskii parameter (120582) due to the Arrhenius kineticsThe effect of viscous dissipation parameter (119898) on thefluid temperature is displayed in Figure 4 The internal heat
Mathematical Problems in Engineering 5
Tem
pera
ture
Lower branch119898 = 20
119898 = 15
119898 = 10
119898 = 5
119898 = 0
14
12
1
08
06
04
02
0minus1 minus05 0 05 1
119910
(a)
0
1
2
3
4
5
6
Tem
pera
ture
minus1 minus05 0 05 1119910
Upper branch119898 = 0 5 10 15 20
Lower branch119898 = 0 5 10 15 20
(b)
Figure 4 Upper branch and lower branch temperature profiles when 119898 = 0 5 10 15 20 when 120582 = 05 120574 = 1 and 120576 = 01 SLM results(circles) bvp4c (solid line)
Table 1 Comparison between the SLM results at different orders and the bvp4c numerical results for wall heat flux Nu for various values of119898 120574 120576 and 120582
119898 120574 120576 120582 1st order 2nd order 3rd order bvp4c
2 1 01 01 014948880 014958883 014958883 0149588834 1 01 01 019550530 019565401 019565401 0195654016 1 01 01 024152181 024172906 024172906 0241729068 1 01 01 028753831 028781405 028781405 0287814051 1 01 01 012648054 012655991 012655991 0126559911 1 01 02 026159825 026233446 026233447 0262334471 1 01 03 040652882 040944698 040944728 0409447281 1 01 04 056267511 057092737 057093089 0570930891 1 01 01 012648054 012655991 012655991 0126559911 3 01 01 012168410 012175980 012175980 0121759801 5 01 01 011956272 011963676 011963676 0119636761 10 01 01 011691095 011698289 011698289 0116982891 1 1 05 073172481 071360852 071359336 0713593361 1 5 05 073172481 066332507 066312547 0663125471 1 10 05 073172481 064501234 064484620 0644846201 1 20 05 073172481 063173135 063164065 063164065
generation due to viscous heating increases as the parametervalue of119898 increases leading to a general increase in the fluidtemperature Meanwhile the possibility of a lower and uppersolution branches is also highlighted in Figures 3 and 4 Thiscan be attributed to the nonlinear nature of the Arrheniuskinetics in the governing thermal boundary layer equation(2) It is noteworthy that the fluid temperature decreaseswith an increase in the activation energy parameter (120576) asillustrated in Figure 5 As 120576 increases the fluid becomes lessvolatile and its activation energy decreases
A slice of the bifurcation diagram for 120574 gt 0 in the(120582 minus120579
1015840(1)) plane is shown in Figure 6 It represents the varia-
tion of wall heat flux with the Frank-Kamenetskii parameter(120582) In particular for every 0 le 120576 le 01 there is a critical
value 120582119888(a turning point) such that for 0 le 120582 lt 120582
119888there are
two solutionsThis result is in perfect agreement with the onereported in Makinde [7]
In Table 1 we show the computations illustrating thecomparison between the SLM results at different orders andthe bvp4c numerical results for wall heat flux Nu for variousvalues of 119898 120574 120576 and 120582 It can be seen from the table thatthe SLM results are in very good agreement with the bvp4cnumerical results
5 Conclusion
In this work we employed a very powerful new linearisationtechnique known as the successive linearisation method
6 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1
120576 = 0 1 5 20
119910
Figure 5 Temperature profiles when 120576 = 0 1 5 20 when 120582 = 05120574 = 01 and119898 = 1 SLM results (circles) bvp4c (solid line)
0 02 04 06 08 10
1
2
3
4
5
6
7
120582119888 = 0936
120582
minus120579998400
(1)
Figure 6 Bifurcation diagram when 120574 = 01 120576 = 01 and119898 = 1
(SLM) to investigate the flow of reactive third-grade liquid ina channel with isothermal wallsThe SLM results for the gov-erning flow parameters were compared with results obtain-ed using MATLABrsquos bvp4c function and excellent agree-ment was observed From the results obtained in the studythe following was observed
(i) An increase in the non-Newtonian parameter (120574)leads to a gradual decrease in the magnitude of fluidvelocity and temperature profiles
(ii) The fluid temperature generally increases with an in-crease in the value of the Frank-Kamenetskii param-eter (120582)
(iii) The internal heat generation due to viscous heatingincreases as the parameter value 119898 increases leadingto a general increase in the fluid temperature
It was also shown that the governing nonlinear equationsadmitmultiple solutions Using the SLM approach lower andupper branch solutions were obtained and discussed
References
[1] K R Rajagopal ldquoOn boundary conditions for fluids of the dif-ferential typerdquo inNavier-Stokes Equations andRelatedNonlinearProblems (Funchal 1994) pp 273ndash278 Plenum New York NYUSA 1995
[2] A M Siddiqui M Ahmed and Q K Ghori ldquoCouette andpoiseuille flows for non-newtonian fluidsrdquo International Journalof Nonlinear Sciences and Numerical Simulation vol 7 no 1 pp15ndash26 2006
[3] D A Frank KamenetskiiDiffusion and Heat Transfer in Chem-ical Kinetics Plenum Press New York NY USA 1969
[4] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[5] M Yurusoy and M Pakdemirli ldquoApproximate analytical solu-tions for the flow of a third-grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[6] R L Fosdick and K R Rajagopal ldquoThermodynamics and sta-bility of fluids of third graderdquo Proceedings of the Royal Society Avol 369 no 1738 pp 351ndash377 1980
[7] O D Makinde ldquoHermite-Pade approximation approach tothermal criticality for a reactive third-grade liquid in a channelwith isothermal wallsrdquo International Communications in Heatand Mass Transfer vol 34 no 7 pp 870ndash877 2007
[8] O DMakinde ldquoThermal criticality for a reactive gravity driventhin film flow of a third-grade fluid with adiabatic free surfacedown an inclined planerdquo Applied Mathematics and Mechanicsvol 30 no 3 pp 373ndash380 2009
[9] O D Makinde ldquoAnalysis of non-Newtonian reactive flow in acylindrical piperdquo Journal of Applied Mechanics vol 76 no 3Article ID 034502 pp 1ndash5 2009
[10] 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
[11] O D Makinde and R J Moitsheki ldquoOn nonperturbative tech-niques for thermal radiation effect on natural convection past avertical plate embedded in a saturated porous mediumrdquoMath-ematical Problems in Engineering vol 2008 Article ID 68907411 pages 2008
[12] O D Makinde ldquoOn the chebyshev collocation spectral ap-proach to stability of fluid in a porous mediumrdquo InternationalJournal for Numerical Methods in Fluids vol 59 no 7 pp 791ndash799 2009
[13] A Shidfar M Djalalvand and M Garshasbi ldquoA numeri-cal scheme for solving special class of nonlinear diffusion-convection equationrdquo Applied Mathematics and Computationvol 167 no 2 pp 1080ndash1089 2005
[14] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homo-topy analysismethod for solving a nonlinear second order BVPrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 9 pp 2293ndash2302 2010
[15] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA new spe-ctral-homotopy analysis method for the MHD Jeffery-Hamelproblemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash12252010
Mathematical Problems in Engineering 7
[16] S LiaoBeyondPerturbation Introduction to theHomotopyAna-lysis Method Chapman amp HallCRC Boca Raton Fla USA2003
[17] C Canuto M Y Hussaini A Quarteroni and T A Zang Spe-ctral Methods in Fluid Dynamics Springer Series in Computa-tional Physics Springer New York NY USA 1988
[18] 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
[19] L N Trefethen Spectral Methods in MATLAB vol 10 of Soft-ware Environments and Tools SIAM Philadelphia Pa USA2000
[20] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009
[21] C Yang and S Liao ldquoOn the explicit purely analytic solutionof Von Karman swirling viscous flowrdquoCommunications in Non-linear Science andNumerical Simulation vol 11 no 1 pp 83ndash932006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Tem
pera
ture
Lower branch119898 = 20
119898 = 15
119898 = 10
119898 = 5
119898 = 0
14
12
1
08
06
04
02
0minus1 minus05 0 05 1
119910
(a)
0
1
2
3
4
5
6
Tem
pera
ture
minus1 minus05 0 05 1119910
Upper branch119898 = 0 5 10 15 20
Lower branch119898 = 0 5 10 15 20
(b)
Figure 4 Upper branch and lower branch temperature profiles when 119898 = 0 5 10 15 20 when 120582 = 05 120574 = 1 and 120576 = 01 SLM results(circles) bvp4c (solid line)
Table 1 Comparison between the SLM results at different orders and the bvp4c numerical results for wall heat flux Nu for various values of119898 120574 120576 and 120582
119898 120574 120576 120582 1st order 2nd order 3rd order bvp4c
2 1 01 01 014948880 014958883 014958883 0149588834 1 01 01 019550530 019565401 019565401 0195654016 1 01 01 024152181 024172906 024172906 0241729068 1 01 01 028753831 028781405 028781405 0287814051 1 01 01 012648054 012655991 012655991 0126559911 1 01 02 026159825 026233446 026233447 0262334471 1 01 03 040652882 040944698 040944728 0409447281 1 01 04 056267511 057092737 057093089 0570930891 1 01 01 012648054 012655991 012655991 0126559911 3 01 01 012168410 012175980 012175980 0121759801 5 01 01 011956272 011963676 011963676 0119636761 10 01 01 011691095 011698289 011698289 0116982891 1 1 05 073172481 071360852 071359336 0713593361 1 5 05 073172481 066332507 066312547 0663125471 1 10 05 073172481 064501234 064484620 0644846201 1 20 05 073172481 063173135 063164065 063164065
generation due to viscous heating increases as the parametervalue of119898 increases leading to a general increase in the fluidtemperature Meanwhile the possibility of a lower and uppersolution branches is also highlighted in Figures 3 and 4 Thiscan be attributed to the nonlinear nature of the Arrheniuskinetics in the governing thermal boundary layer equation(2) It is noteworthy that the fluid temperature decreaseswith an increase in the activation energy parameter (120576) asillustrated in Figure 5 As 120576 increases the fluid becomes lessvolatile and its activation energy decreases
A slice of the bifurcation diagram for 120574 gt 0 in the(120582 minus120579
1015840(1)) plane is shown in Figure 6 It represents the varia-
tion of wall heat flux with the Frank-Kamenetskii parameter(120582) In particular for every 0 le 120576 le 01 there is a critical
value 120582119888(a turning point) such that for 0 le 120582 lt 120582
119888there are
two solutionsThis result is in perfect agreement with the onereported in Makinde [7]
In Table 1 we show the computations illustrating thecomparison between the SLM results at different orders andthe bvp4c numerical results for wall heat flux Nu for variousvalues of 119898 120574 120576 and 120582 It can be seen from the table thatthe SLM results are in very good agreement with the bvp4cnumerical results
5 Conclusion
In this work we employed a very powerful new linearisationtechnique known as the successive linearisation method
6 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1
120576 = 0 1 5 20
119910
Figure 5 Temperature profiles when 120576 = 0 1 5 20 when 120582 = 05120574 = 01 and119898 = 1 SLM results (circles) bvp4c (solid line)
0 02 04 06 08 10
1
2
3
4
5
6
7
120582119888 = 0936
120582
minus120579998400
(1)
Figure 6 Bifurcation diagram when 120574 = 01 120576 = 01 and119898 = 1
(SLM) to investigate the flow of reactive third-grade liquid ina channel with isothermal wallsThe SLM results for the gov-erning flow parameters were compared with results obtain-ed using MATLABrsquos bvp4c function and excellent agree-ment was observed From the results obtained in the studythe following was observed
(i) An increase in the non-Newtonian parameter (120574)leads to a gradual decrease in the magnitude of fluidvelocity and temperature profiles
(ii) The fluid temperature generally increases with an in-crease in the value of the Frank-Kamenetskii param-eter (120582)
(iii) The internal heat generation due to viscous heatingincreases as the parameter value 119898 increases leadingto a general increase in the fluid temperature
It was also shown that the governing nonlinear equationsadmitmultiple solutions Using the SLM approach lower andupper branch solutions were obtained and discussed
References
[1] K R Rajagopal ldquoOn boundary conditions for fluids of the dif-ferential typerdquo inNavier-Stokes Equations andRelatedNonlinearProblems (Funchal 1994) pp 273ndash278 Plenum New York NYUSA 1995
[2] A M Siddiqui M Ahmed and Q K Ghori ldquoCouette andpoiseuille flows for non-newtonian fluidsrdquo International Journalof Nonlinear Sciences and Numerical Simulation vol 7 no 1 pp15ndash26 2006
[3] D A Frank KamenetskiiDiffusion and Heat Transfer in Chem-ical Kinetics Plenum Press New York NY USA 1969
[4] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[5] M Yurusoy and M Pakdemirli ldquoApproximate analytical solu-tions for the flow of a third-grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[6] R L Fosdick and K R Rajagopal ldquoThermodynamics and sta-bility of fluids of third graderdquo Proceedings of the Royal Society Avol 369 no 1738 pp 351ndash377 1980
[7] O D Makinde ldquoHermite-Pade approximation approach tothermal criticality for a reactive third-grade liquid in a channelwith isothermal wallsrdquo International Communications in Heatand Mass Transfer vol 34 no 7 pp 870ndash877 2007
[8] O DMakinde ldquoThermal criticality for a reactive gravity driventhin film flow of a third-grade fluid with adiabatic free surfacedown an inclined planerdquo Applied Mathematics and Mechanicsvol 30 no 3 pp 373ndash380 2009
[9] O D Makinde ldquoAnalysis of non-Newtonian reactive flow in acylindrical piperdquo Journal of Applied Mechanics vol 76 no 3Article ID 034502 pp 1ndash5 2009
[10] 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
[11] O D Makinde and R J Moitsheki ldquoOn nonperturbative tech-niques for thermal radiation effect on natural convection past avertical plate embedded in a saturated porous mediumrdquoMath-ematical Problems in Engineering vol 2008 Article ID 68907411 pages 2008
[12] O D Makinde ldquoOn the chebyshev collocation spectral ap-proach to stability of fluid in a porous mediumrdquo InternationalJournal for Numerical Methods in Fluids vol 59 no 7 pp 791ndash799 2009
[13] A Shidfar M Djalalvand and M Garshasbi ldquoA numeri-cal scheme for solving special class of nonlinear diffusion-convection equationrdquo Applied Mathematics and Computationvol 167 no 2 pp 1080ndash1089 2005
[14] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homo-topy analysismethod for solving a nonlinear second order BVPrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 9 pp 2293ndash2302 2010
[15] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA new spe-ctral-homotopy analysis method for the MHD Jeffery-Hamelproblemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash12252010
Mathematical Problems in Engineering 7
[16] S LiaoBeyondPerturbation Introduction to theHomotopyAna-lysis Method Chapman amp HallCRC Boca Raton Fla USA2003
[17] C Canuto M Y Hussaini A Quarteroni and T A Zang Spe-ctral Methods in Fluid Dynamics Springer Series in Computa-tional Physics Springer New York NY USA 1988
[18] 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
[19] L N Trefethen Spectral Methods in MATLAB vol 10 of Soft-ware Environments and Tools SIAM Philadelphia Pa USA2000
[20] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009
[21] C Yang and S Liao ldquoOn the explicit purely analytic solutionof Von Karman swirling viscous flowrdquoCommunications in Non-linear Science andNumerical Simulation vol 11 no 1 pp 83ndash932006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
0
005
01
015
02
025
03
035
04
Tem
pera
ture
Lower branch
minus1 minus05 0 05 1
120576 = 0 1 5 20
119910
Figure 5 Temperature profiles when 120576 = 0 1 5 20 when 120582 = 05120574 = 01 and119898 = 1 SLM results (circles) bvp4c (solid line)
0 02 04 06 08 10
1
2
3
4
5
6
7
120582119888 = 0936
120582
minus120579998400
(1)
Figure 6 Bifurcation diagram when 120574 = 01 120576 = 01 and119898 = 1
(SLM) to investigate the flow of reactive third-grade liquid ina channel with isothermal wallsThe SLM results for the gov-erning flow parameters were compared with results obtain-ed using MATLABrsquos bvp4c function and excellent agree-ment was observed From the results obtained in the studythe following was observed
(i) An increase in the non-Newtonian parameter (120574)leads to a gradual decrease in the magnitude of fluidvelocity and temperature profiles
(ii) The fluid temperature generally increases with an in-crease in the value of the Frank-Kamenetskii param-eter (120582)
(iii) The internal heat generation due to viscous heatingincreases as the parameter value 119898 increases leadingto a general increase in the fluid temperature
It was also shown that the governing nonlinear equationsadmitmultiple solutions Using the SLM approach lower andupper branch solutions were obtained and discussed
References
[1] K R Rajagopal ldquoOn boundary conditions for fluids of the dif-ferential typerdquo inNavier-Stokes Equations andRelatedNonlinearProblems (Funchal 1994) pp 273ndash278 Plenum New York NYUSA 1995
[2] A M Siddiqui M Ahmed and Q K Ghori ldquoCouette andpoiseuille flows for non-newtonian fluidsrdquo International Journalof Nonlinear Sciences and Numerical Simulation vol 7 no 1 pp15ndash26 2006
[3] D A Frank KamenetskiiDiffusion and Heat Transfer in Chem-ical Kinetics Plenum Press New York NY USA 1969
[4] M Massoudi and I Christie ldquoEffects of variable viscosity andviscous dissipation on the flow of a third grade fluid in a piperdquoInternational Journal of Non-LinearMechanics vol 30 no 5 pp687ndash699 1995
[5] M Yurusoy and M Pakdemirli ldquoApproximate analytical solu-tions for the flow of a third-grade fluid in a piperdquo InternationalJournal ofNon-LinearMechanics vol 37 no 2 pp 187ndash195 2002
[6] R L Fosdick and K R Rajagopal ldquoThermodynamics and sta-bility of fluids of third graderdquo Proceedings of the Royal Society Avol 369 no 1738 pp 351ndash377 1980
[7] O D Makinde ldquoHermite-Pade approximation approach tothermal criticality for a reactive third-grade liquid in a channelwith isothermal wallsrdquo International Communications in Heatand Mass Transfer vol 34 no 7 pp 870ndash877 2007
[8] O DMakinde ldquoThermal criticality for a reactive gravity driventhin film flow of a third-grade fluid with adiabatic free surfacedown an inclined planerdquo Applied Mathematics and Mechanicsvol 30 no 3 pp 373ndash380 2009
[9] O D Makinde ldquoAnalysis of non-Newtonian reactive flow in acylindrical piperdquo Journal of Applied Mechanics vol 76 no 3Article ID 034502 pp 1ndash5 2009
[10] 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
[11] O D Makinde and R J Moitsheki ldquoOn nonperturbative tech-niques for thermal radiation effect on natural convection past avertical plate embedded in a saturated porous mediumrdquoMath-ematical Problems in Engineering vol 2008 Article ID 68907411 pages 2008
[12] O D Makinde ldquoOn the chebyshev collocation spectral ap-proach to stability of fluid in a porous mediumrdquo InternationalJournal for Numerical Methods in Fluids vol 59 no 7 pp 791ndash799 2009
[13] A Shidfar M Djalalvand and M Garshasbi ldquoA numeri-cal scheme for solving special class of nonlinear diffusion-convection equationrdquo Applied Mathematics and Computationvol 167 no 2 pp 1080ndash1089 2005
[14] S S Motsa P Sibanda and S Shateyi ldquoA new spectral-homo-topy analysismethod for solving a nonlinear second order BVPrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 15 no 9 pp 2293ndash2302 2010
[15] S S Motsa P Sibanda F G Awad and S Shateyi ldquoA new spe-ctral-homotopy analysis method for the MHD Jeffery-Hamelproblemrdquo Computers and Fluids vol 39 no 7 pp 1219ndash12252010
Mathematical Problems in Engineering 7
[16] S LiaoBeyondPerturbation Introduction to theHomotopyAna-lysis Method Chapman amp HallCRC Boca Raton Fla USA2003
[17] C Canuto M Y Hussaini A Quarteroni and T A Zang Spe-ctral Methods in Fluid Dynamics Springer Series in Computa-tional Physics Springer New York NY USA 1988
[18] 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
[19] L N Trefethen Spectral Methods in MATLAB vol 10 of Soft-ware Environments and Tools SIAM Philadelphia Pa USA2000
[20] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009
[21] C Yang and S Liao ldquoOn the explicit purely analytic solutionof Von Karman swirling viscous flowrdquoCommunications in Non-linear Science andNumerical Simulation vol 11 no 1 pp 83ndash932006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
[16] S LiaoBeyondPerturbation Introduction to theHomotopyAna-lysis Method Chapman amp HallCRC Boca Raton Fla USA2003
[17] C Canuto M Y Hussaini A Quarteroni and T A Zang Spe-ctral Methods in Fluid Dynamics Springer Series in Computa-tional Physics Springer New York NY USA 1988
[18] 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
[19] L N Trefethen Spectral Methods in MATLAB vol 10 of Soft-ware Environments and Tools SIAM Philadelphia Pa USA2000
[20] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009
[21] C Yang and S Liao ldquoOn the explicit purely analytic solutionof Von Karman swirling viscous flowrdquoCommunications in Non-linear Science andNumerical Simulation vol 11 no 1 pp 83ndash932006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of