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


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


Tem

pera

ture

Lower branch119898 = 20

119898 = 15

119898 = 10

119898 = 5

119898 = 0

14

12

1

08

06

04

02


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


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


[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

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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



1198892119882

1198891199102= 119891 (119910) (10)


119891 (119910) = minus1

1 + 6120574(119889119882119889119910)2 (11)


120579 (119910) = 119879119894(119910) +

119894minus1

sum

119899=0

120579119899(119910) 119894 = 1 2 3 (12)




0(119910) as




lim119894rarrinfin

119879119894= 0 (13)


11987910158401015840

119894+ 120582 exp[

119879119894+ sum119894minus1

119899=0120579119899

1 + 120576 (119879119894+ sum119894minus1

119899=0120579119899)

] = 119892 (119910) minus

119894minus1

sum

119899=0

12057910158401015840

119899 (14)


119892 (119910) = minus120582119898(119889119882

119889119910)

2

[1 + 2120574(119889119882

119889119910)

2

] (15)




12057910158401015840

119894+ 119886119894minus1120579119894= 119903119894minus1 (16)


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)



120579 (119910) asymp

119872

sum

119899=0

120579119899(119910) (19)


119894minus1and the



119910119895= cos

120587119895

119873 119895 = 0 1 119873 (20)


1198892119882

1198891199102=

119873

sum

119896=0

D2119896119895119882(119910119896)

1198892120579119894

1198891199102=

119873

sum

119896=0

D2119896119895120579119894(119910119896)

119895 = 0 1 119873

(21)


D2W = F 119882 (1199100) = 119882(119910

119873) = 0 (22)

AT119894= R119894minus1 120579

119894(1199100) = 120579119894(119910119873) = 0 (23)


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)




Velo

city

120574 = 01

120574 = 1

120574 = 5

05

045

04

035

03

025

02

015

01

005


119910

(a)

0

005

01

015

02

025

Tem

pera

ture

minus1 minus05 0 05 1119910

120574 = 01 1 5

(b)


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)



119894(119910) are

obtained as

W = (D2)minus1F

T119894= Aminus1S

119894minus1

(29)

4 Results





Tem

pera

ture


119898 = 15

119898 = 10

119898 = 5

119898 = 0

14

12

1

08

06

04

02


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)




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






119888there are



5 Conclusion



0

005

01

015

02

025

03

035

04

Tem

pera

ture

Lower branch


120576 = 0 1 5 20

119910


0 02 04 06 08 10

1

2

3

4

5

6

7

120582119888 = 0936

120582

minus120579998400

(1)







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198892119882

1198891199102= 119891 (119910) (10)


119891 (119910) = minus1

1 + 6120574(119889119882119889119910)2 (11)


120579 (119910) = 119879119894(119910) +

119894minus1

sum

119899=0

120579119899(119910) 119894 = 1 2 3 (12)




0(119910) as




lim119894rarrinfin

119879119894= 0 (13)


11987910158401015840

119894+ 120582 exp[

119879119894+ sum119894minus1

119899=0120579119899

1 + 120576 (119879119894+ sum119894minus1

119899=0120579119899)

] = 119892 (119910) minus

119894minus1

sum

119899=0

12057910158401015840

119899 (14)


119892 (119910) = minus120582119898(119889119882

119889119910)

2

[1 + 2120574(119889119882

119889119910)

2

] (15)




12057910158401015840

119894+ 119886119894minus1120579119894= 119903119894minus1 (16)


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)



120579 (119910) asymp

119872

sum

119899=0

120579119899(119910) (19)


119894minus1and the



119910119895= cos

120587119895

119873 119895 = 0 1 119873 (20)


1198892119882

1198891199102=

119873

sum

119896=0

D2119896119895119882(119910119896)

1198892120579119894

1198891199102=

119873

sum

119896=0

D2119896119895120579119894(119910119896)

119895 = 0 1 119873

(21)


D2W = F 119882 (1199100) = 119882(119910

119873) = 0 (22)

AT119894= R119894minus1 120579

119894(1199100) = 120579119894(119910119873) = 0 (23)


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)




Velo

city

120574 = 01

120574 = 1

120574 = 5

05

045

04

035

03

025

02

015

01

005


119910

(a)

0

005

01

015

02

025

Tem

pera

ture

minus1 minus05 0 05 1119910

120574 = 01 1 5

(b)


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)



119894(119910) are

obtained as

W = (D2)minus1F

T119894= Aminus1S

119894minus1

(29)

4 Results





Tem

pera

ture


119898 = 15

119898 = 10

119898 = 5

119898 = 0

14

12

1

08

06

04

02


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)




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






119888there are



5 Conclusion



0

005

01

015

02

025

03

035

04

Tem

pera

ture

Lower branch


120576 = 0 1 5 20

119910


0 02 04 06 08 10

1

2

3

4

5

6

7

120582119888 = 0936

120582

minus120579998400

(1)







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Velo

city

120574 = 01

120574 = 1

120574 = 5

05

045

04

035

03

025

02

015

01

005


119910

(a)

0

005

01

015

02

025

Tem

pera

ture

minus1 minus05 0 05 1119910

120574 = 01 1 5

(b)


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)



119894(119910) are

obtained as

W = (D2)minus1F

T119894= Aminus1S

119894minus1

(29)

4 Results





Tem

pera

ture


119898 = 15

119898 = 10

119898 = 5

119898 = 0

14

12

1

08

06

04

02


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)




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






119888there are



5 Conclusion



0

005

01

015

02

025

03

035

04

Tem

pera

ture

Lower branch


120576 = 0 1 5 20

119910


0 02 04 06 08 10

1

2

3

4

5

6

7

120582119888 = 0936

120582

minus120579998400

(1)







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Tem

pera

ture


119898 = 15

119898 = 10

119898 = 5

119898 = 0

14

12

1

08

06

04

02


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)




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






119888there are



5 Conclusion



0

005

01

015

02

025

03

035

04

Tem

pera

ture

Lower branch


120576 = 0 1 5 20

119910


0 02 04 06 08 10

1

2

3

4

5

6

7

120582119888 = 0936

120582

minus120579998400

(1)







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

005

01

015

02

025

03

035

04

Tem

pera

ture

Lower branch


120576 = 0 1 5 20

119910


0 02 04 06 08 10

1

2

3

4

5

6

7

120582119888 = 0936

120582

minus120579998400

(1)







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article on the successive linearisation approach...

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