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Advances in Mathematical Physics

Nonlinear Fluid Flow and Heat Transfer

Guest Editors: O. D. Makinde, R. J. Moitsheki, R. N. Jana, B. H. Bradshaw-Hajek, and W. A. Khan

Advances in Mathematical Physics


Guest Editors: O. D. Makinde, R. J. Moitsheki, R. N. Jana,B. H. Bradshaw-Hajek, and W. A. Khan

Copyright 2014 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in Advances inMathematical Physics. All articles are open access articles distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the originalwork is properly cited.

Editorial Board

Sergio Albeverio, GermanyStephen C. Anco, CanadaIvan G. Avramidi, USAAngel Ballesteros, SpainViacheslav P. Belavkin, UKLuigi C. Berselli, ItalyCarlo Cattani, ItalyD. chae, Republic of KoreaPierluigi Contucci, ItalyPrabir Daripa, USAManuel De Leon, SpainEmilio Elizalde, SpainJose F. Carinena, SpainShao-Ming Fei, ChinaPartha Guha, IndiaGraham. S. Hall, UK

Nakao Hayashi, JapanM. N. Hounkonnou, BeninXing Biao Hu, ChinaVictor G. Kac, USANiky Kamran, CanadaGiorgio Kaniadakis, ItalyKlaus Kirsten, USAB. G. Konopelchenko, ItalyMaximilian Kreuzer, AustriaPavel Kurasov, SwedenM. Lakshmanan, IndiaMichel Lapidus, USARemi Leandre, FranceDecio Levi, ItalyWen-Xiu Ma, USAJuan C. Marrero, Spain

Willard Miller Jr, USAAndrei D. Mironov, RussiaAndrei Moroianu, FranceHagen Neidhardt, GermanyAnatol Odzijewicz, PolandAlkesh Punjabi, USASoheil Salahshour, IranYulii D. Shikhmurzaev, UKAlexander P. Veselov, UKRicardo Weder, MexicoGongnan Xie, ChinaValentin Zagrebnov, FranceFederico Zertuche, MexicoYao-Zhong Zhang, Australia

Nonlinear Fluid Flow and Heat Transfer, O. D. Makinde, R. J. Moitsheki, R. N. Jana,B. H. Bradshaw-Hajek, and W. A. KhanVolume 2014, Article ID 719102, 2 pages

Spectral Relaxation Method and Spectral Quasilinearization Method for Solving Unsteady BoundaryLayer Flow Problems, S. S. Motsa, P. G. Dlamini, and M. KhumaloVolume 2014, Article ID 341964, 12 pages

A Local Integral Equation Formulation Based on Moving Kriging Interpolation for Solving CoupledNonlinear Reaction-Diffusion Equations, Kanittha Yimnak and Anirut LuadsongVolume 2014, Article ID 196041, 7 pages

A Spectral Relaxation Approach for Unsteady Boundary-Layer Flow and Heat Transfer of a Nanofluidover a Permeable Stretching/Shrinking Sheet, S. S. Motsa, P. Sibanda, J. M. Ngnotchouye, and G. T. MarewoVolume 2014, Article ID 564942, 10 pages

Analysis of Heat Transfer in Berman Flow of Nanofluids with Navier Slip, Viscous Dissipation,and Convective Cooling, O. D. Makinde, S. Khamis, M. S. Tshehla, and O. FranksVolume 2014, Article ID 809367, 13 pages

Dual Approximate Solutions of the Unsteady Viscous Flow over a Shrinking Cylinder with OptimalHomotopy Asymptotic Method, Vasile Marinca and Remus-Daniel EneVolume 2014, Article ID 417643, 11 pages

Simulation of Impinging Cooling Performance with Pin Fins and Mist Cooling Adopted in a SimplifiedGas Turbine Transition Piece, Tao Xu, Hang Xiu, Junlou Li, Haichao Ge, Qing Shao, Guang Yang,and Zhenglei YuVolume 2014, Article ID 327590, 11 pages

Classification of the Group Invariant Solutions for Contaminant Transport in Saturated Soils underRadial UniformWater Flows, M. M. Potsane and R. J. MoitshekiVolume 2014, Article ID 138289, 11 pages

Application of Successive Linearisation Method to Squeezing Flow with Bifurcation, S. S. Motsa,O. D. Makinde, and S. ShateyiVolume 2014, Article ID 410620, 6 pages

Higher Order Compact Finite Difference Schemes for Unsteady Boundary Layer Flow Problems,P. G. Dlamini, S. S. Motsa, and M. KhumaloVolume 2013, Article ID 941096, 10 pages

EditorialNonlinear Fluid Flow and Heat Transfer

O. D. Makinde,1 R. J. Moitsheki,2 R. N. Jana,3 B. H. Bradshaw-Hajek,4 and W. A. Khan5

1 Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa2 School of Computational and Applied Mathematics, University of the Witwatersrand (WITS), Private Bag 3,Johannesburg 2050, South Africa

3 Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India4 School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lake Campus,Mawson Lakes, SA, Australia

5 Department of Engineering Sciences, Pakistan Navy Engineering College, National University of Sciences & Technology,Karachi 75350, Pakistan

Correspondence should be addressed to O. D. Makinde; [email protected]

Received 11 June 2014; Accepted 11 June 2014; Published 24 June 2014

Copyright 2014 O. D. Makinde et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In order to stimulate fluid flow, heat transfer, and otherrelated physical phenomena, it is necessary to describe theassociated physics in mathematical terms. Nearly all thephysical phenomena of interest are obtained by principles ofconservation and are expressed in terms of nonlinear partialor ordinary differential equations expressing these principles.For example, the momentum equations express the conser-vation of linear momentum; the energy equation expressesthe conservation of total energy. These nonlinear differentialequations which model the dynamics of fluid motion andheat transfer do arise in many areas such as optics, plasmaphysics, and even traffic flow. Thus, the underlying mathe-matics has relevance in many branches of science and tech-nology. Generally, the solutions for these complicated non-linear differential equations can be obtained numerically inmost cases; however, analytical solutions for fluid flow andheat transfer problem can still play a very important role inscience and engineering, even in the current age of supercom-puter.This is because analytical solutions have the big advan-tage of revealing directly the parameters which influencethe solution. This special issue is focused on nonlinear anal-ysis and numerical simulation of typical conservation equa-tions modelling physical phenomena with respect to fluidflow and heat transfer.The original papers explored include awide variety of topics such as boundary layer flows, nanoflu-ids dynamics, reactive flows, hydromagnetic flows, phys-iological flows, thermodynamics analysis of fluid flows,

Newtonian and non-Newtonian flows, and nonlinear heattransfer in solids. In Spectral relaxation method and spectralquasilinearizationmethod for solving unsteady boundary layerflow problems, S. S. Motsa et al. numerically solved modelproblems of unsteady boundary layer flow caused by animpulsively stretching sheet and the unsteady MHD flowand mass transfer in a porous space using a spectral relax-ation method (SRM) and spectral quasilinearisation method(SQLM). It was shown that the SRM is significantly morecomputationally efficient than the SQLM which, in turn,is faster than theKeller-boxmethod. In Application of succes-sive linearisation method to squeezing flow with bifurcation,S. S. Motsa et al. employed a computational approachknown as successive linearization method (SLM) to tackle afourth-order nonlinear differential equation modelling thetransient flow of an incompressible viscous fluid between twoparallel plates produced by a simple wall motion. The solu-tion branches as well as a turning point in the flow fieldwere accurately obtained. Moreover, the study revealed thatthe proposed SLM approach converges rapidly to the solutionof the original nonlinear problem and can be used to solvemany other nonlinear equations arising in fluid flow and heattransfer problems. In Higher order compact finite differenceschemes for unsteady boundary layer flow problems, P. G.Dlamini et al. investigated two nonlinear partial differentialequations governing the unsteadyMHD boundary layer flowand heat transfer over an impulsively stretching surface using

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2014, Article ID 719102, 2 pageshttp://dx.doi.org/10.1155/2014/719102

http://dx.doi.org/10.1155/2014/719102

2 Advances in Mathematical Physics

the compact finite difference relaxationmethod (CFDRM). Itwas shown that the CFDRM is computationally faster thanthe Keller-box method and CFDRM gives highly accuratesolutions on coarser grids. Moreover, the results obtained arein good agreement with results obtained using the Keller-box. In Classification of the group invariant solutions forcontaminant transport in saturated soils under radial uniformwater flows, M. M. Potsane and R. J. Moitsheki analysed anonlinear macroscopic deterministic model describing con-taminant transport in saturated soils under uniform radialwater flow backgrounds using classical Lie point symmetries.Anumber of exotic Lie point symmetries are admitted.Groupinvariant solutions are classified according to the elementsof the one-dimensional optimal systems. The symmetrysolutions are obtainable when dispersion coefficient is aconstant or is given by Taylors theory of mixing in soils.In A spectral relaxation approach for unsteady boundary-layer flow and heat transfer of a nanofluid over a permeablestretching/shrinking sheet, S. S. Motsa et al. utilised both thespectral relaxation method and the spectral quasilinearisa-tion method to numerically solve the highly nonlinear equa-tions that describe the unsteady heat transfer in a nanofluidover a permeable stretching or shrinking surface. The resultsrevealed that dual solutions for nanofluid fluid velocityprofiles, temperature profiles, and nanoparticles volumefractions exist for sheet stretching and a critical value ofshrinking velocity exists below which no real solution can befound. In Dual approximate solutions of the unsteady viscousflow over a shrinking cylinder with optimal homotopy asymp-totic method, V. Marinca and R.-D. Ene studied the unsteadyviscous flow over a continuously shrinking surface withmass suction being investigated using the optimal homotopyasymptotic method (OHAM). The nonlinear differentialequation is obtained by means of the similarity transforma-tion. It was revealed that dual solutions exist for a certainrange of mass suction and unsteadiness parameters and theresults obtained using OHAM agreed well with the oneobtained using fourth-order Runge-Kutta iteration schemecoupled with shootingmethod. In Analysis of heat transfer inBerman flow of nanofluids withNavier slip, viscous dissipation,and convective cooling, O. D. Makinde et al. investigatedthe combined effects of viscous dissipation, Navier slip, andconvective cooling on Berman flow and heat transfer of waterbase nanofluids containing Cu and Al

2O3as nanoparticles.

The model nonlinear differential equations are tackled bothanalytically using perturbation series method and numeri-cally using Runge-Kutta-Fehlberg integration technique cou-pled with shooting scheme. Pertinent results with respect toeffects of variation in governing parameters on the dimen-sionless velocity, temperature, skin friction, pressure drop,andNusselt numbers are presented graphically and discussedquantitatively. In Simulation of impinging cooling perfor-mancewith pin fins andmist cooling adopted in a simplified gasturbine transition piece, T. Xu et al. employed a commercialcomputational fluid dynamics (CFD) program FLUENT toanalyse the heat transfer and pressure characteristics of theimpinging cooling in the coolant chamber. The simulationresults reveal that the factors of detached space and pin-findiameter ratio have a significant effect on the convective

heat transfer. In A local integral equation formulation basedon moving Kriging interpolation for solving coupled nonlinearreaction-diffusion equations, K. Yimnak and A. Luadsongpresented a meshless local Petrov-Galerkin method (MLPG)with the test function (in view of the Heaviside step function)to solve a system of coupled nonlinear reaction-diffusionequations in two-dimensional spaces subjected to Dirichletand Neumann boundary conditions on a square domain.Two-field velocities are approximated by moving Kriging(MK) interpolation method for constructing nodal shapefunction which holds the Kronecker delta property, therebyenhancing the arrangement nodal shape construction accu-racy, while the Crank-Nicolson method is chosen for tempo-ral discretization. The nonlinear terms are treated iterativelywithin each time step. The numerical experiments revealedthat the solutions are stable and more precise.

Finally, the papers in this special issue represent a broadspectrum of nonlinear fluid flow and heat transfer problemstogether with their analytical and numerical solutions. Theydemonstrate a wide array of new developments with respectto applications. Moreover, articles published in this specialissue will contribute immensely to advancement of knowl-edge in the field of mathematical physics and will providescientists, engineers, industries, research scholars, and prac-titioners latest theoretical and technological achievements influid mechanics and their heat transfer applications.

Acknowledgments

Wewould like to thank the authors for their contribution andthe reviewers for their collaboration.

O. D. MakindeR. J. Moitsheki

R. N. JanaB. H. Bradshaw-Hajek

W. A. Khan

Research ArticleSpectral Relaxation Method and Spectral QuasilinearizationMethod for Solving Unsteady Boundary Layer Flow Problems

S. S. Motsa,1 P. G. Dlamini,2 and M. Khumalo2

1 School of Mathematical Sciences, University of Kwazulu-Natal, Private Bag X01, Pietermaritzburg, Scottsville 3209, South Africa2Department of Mathematics, University of Johannesburg, P.O. Box 17011, Doornfontein 2028, South Africa

Correspondence should be addressed to P. G. Dlamini; [email protected]

Received 20 March 2014; Accepted 26 May 2014; Published 18 June 2014

Academic Editor: Raseelo Joel Moitsheki

Copyright 2014 S. S. Motsa et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Nonlinear partial differential equations (PDEs) modelling unsteady boundary-layer flows are solved by the spectral relaxationmethod (SRM) and the spectral quasilinearization method (SQLM). The SRM and SQLM are Chebyshev pseudospectral basedmethods that have been successfully used to solve nonlinear boundary layer flow problems described by systems of ordinarydifferential equations. In this paper application of these methods is extended, for the first time, to systems of nonlinear PDEs thatmodel unsteady boundary layer flow. The new extension is tested on two problems: boundary layer flow caused by an impulsivelystretching plate and a coupled four-equation system that models the problem of unsteady MHD flow and mass transfer in a porousspace. Numerous simulation experiments are conducted to determine the accuracy and compare the computational performanceof the proposed methods against the popular Keller-box finite difference scheme which is widely accepted as being one of the idealtools for solving nonlinear PDEs that model boundary layer flow problems.The results indicate that the methods are more efficientin terms of computational accuracy and speed compared with the Keller-box.

1. Introduction

Partial differential equations (PDEs) arise in a number ofphysical problems, such as fluid flow, heat transfer, andbiological processes. Finding solutions of the PDEs playsa crucial role in understanding the behaviour of theseproblems. Mostly, the PDEs modelling real-life problems arenonlinear and complex to solve exactly and hence variousanalytical and numerical methods have been employed toapproximate the solutions of these problems. In recent times,many researchers in fluid mechanics have focused theirattention on problems involving boundary layer flows of anincompressible fluid over a stretching surface because of theirsubstantial applications in engineering. A large and growingbody of literature has investigated problems involving steadyflows.However, in some cases the flowfield could be unsteadydue to a sudden stretching of the flat sheet. Unsteadyflows are mostly defined by systems of nonlinear PDEs andare considerably more difficult to solve than steady flowsproblems which are often simplified into system nonlinearODEs using the so-called similarity transformations.

The problem of unsteady boundary layer flow due to animpulsively stretching surface in a viscous fluid has beenconsidered by a number of researchers.These studies includethe work of Seshadri et al. [1] who used the Keller-boxmethod of Cebeci and Bradshaw [2] and a perturbation seriesapproach for the solution of unsteady mixed convection flowalong a heated vertical plate.The Keller-box method was alsoused by Ali et al. [3] to solve a related problem of unsteadyboundary layer flow due to an impulsively stretching surface.Nazar et al. [4, 5] solved the unsteady boundary-layer flowproblem due to an impulsively stretching surface in a rotatingfluid by means of the Keller-box numerical method, and theyobtained a first order perturbation approximation of the solu-tion. Liao [6] noted that a limiting factor of the perturbationapproach is that it gives solutions that are only valid for smalltime. As an alternative approach, Liao [6] suggested the useof the homotopy analysis method (HAM) that was meant toaddress some of the limitations of the perturbation methodsby offering solutions that are uniformly valid for all time. Inrecent years, there has been an increasing amount of literaturethat has adopted Liaos analytic approach in solving unsteady


http://dx.doi.org/10.1155/2014/341964


boundary layer flows. However, there are limits to how faranalytic approaches can be utilised in nonlinear systems ofPDEs involvingmany equations.Nonlinear systems involvingmany coupled equations are very difficult to solve analytically.

In this work, we apply, for the first time, the spectralrelaxation method (SRM) and the spectral quasilineariza-tion method (SQLM) to solve nonlinear PDEs describingunsteady boundary layer flow due to an impulsively stretch-ing surface. The SRM was introduced in [7] for the solutionof the nonlinear ODE system model of von Karman flowof a Reiner-Rivlin fluid. The method has also been used inthe solution of chaotic and hyperchaotic systems [8, 9]. TheSRM is based on simple decoupling and rearrangement ofthe governing nonlinear equations in a Gauss-Seidel manner.The resulting sequence of equations is integrated using theChebyshev spectral collocation method. On the other hand,in the SQLM, the governing nonlinear equations are lin-earised using the Newton-Raphson based quasilinearizationmethod (QLM), developed by Bellman and Kalaba [10], andare then integrated using Chebyshev spectral collocationmethod. A sizeable body of literature now exists on the useof various finite difference based QLM schemes in boundarylayer flows described by both nonlinear ODE and PDE-basedsystems [1115]. Spectral method based quasilinearisationschemes have also been successfully applied to a range offluid mechanics based ODE model problems (see, e.g., [1618]). For problems with smooth solutions, spectral methodsare well known [1921] to be considerably more accuratethan other traditional numerical methods such as finitedifference and finite elements. In this investigation we revisitthe one-dimensional unsteady boundary layer flow due to animpulsively stretching surface that was previously discussedby [6] using the homotopy analysis method and recently in[3] using the Keller-box method. The problem of unsteadythree-dimensional MHD flow and mass transfer in a porousspace [22] is also investigated.Themain purpose of the studyis to investigate the applicability and effectiveness of the newSRM approach to systems of nonlinear PDE-based unsteadyboundary layer flows of varying levels of complexity. Numer-ical simulations are conducted on the sample problems usingthe SRM, SQLM, and Keller-box method.The three methodsare compared in terms of accuracy, computational speed, andeasy implementation.

The rest of the paper is organized as follows. In Section 2,we discuss the development of the SRM and SQLM for thesolution of an unsteady boundary-layer flow caused by animpulsively stretching plate. Section 3 presents the SRM andSQLM implementation of an unsteady three-dimensionalMHD flow and mass transfer in a porous space. Section 4contains the results and discussion, and the conclusions aregiven in Section 5.

2. Unsteady Boundary-Layer Flows Caused byan Impulsively Stretching Plate

The governing partial differential equations can be obtainedby using the standard stream function formulation in con-junction with the transformations suggested by Williams

and Rhyne [23]. The dimensionless governing equation isobtained (see [1, 4, 6] for details) as

3

3+1

2(1 )

2

2+ [

2

2 (

)

2

]

= (1 )2

,

(1)

subject to the boundary conditions

(0, ) = 0,

=0

= 1,

+

= 0, (2)

where the primes denote differentiation with respect to thesimilarity variable . is a nondimensional function thatgives the velocity and [0, 1] is the dimensionless time-scale defined as

= 1 , = , (3)

where is a positive constant and is the time variable. In theanalysis of boundary layer flow problems, a quantity that isof physical interest in the skin friction in this model is given[1, 4, 6], in dimensionless form, as

Re1/2

= 1/2

(, 0) , (4)

where Reis the local Reynolds number.

The initial unsteady solution at = 0 ( = 0) forthe governing equation (1) is obtained as a solution of theequation

+1

2= 0, (5)

(0, 0) = 0, (0, 0) = 1,

(, 0) = 0, (6)

where the primes denote differentiation with respect to .Solving (5) gives

(, 0) = erfc(

2) +

2

[1 exp(

2

4)] , (7)

where erfc() is the standard complementary error functiondefined by

erfc () = 2

exp (2) . (8)

The steady state solution when = 1, corresponding to +, is obtained from

+ ()2

= 0,

(0, 1) = 0, (0, 1) = 1,

(, 1) = 0.

(9)

The solution to the above equation is

(, 1) = 1 exp () . (10)

Advances in Mathematical Physics 3

2.1. Spectral Relaxation Method (SRM). In this section wediscuss the development of the spectral relaxation methodand its application to solve the partial differential equation(1). It is convenient to reduce the order of (1) from three totwo. To this end, we set = , so that (1) becomes

+1

2 (1 )

+ [

2] = (1 )

,

= .

(11)

The spectral relaxation method [7] algorithm uses theidea of the Gauss-Seidel method to decouple the governingsystems of (11). From the decoupled equations an iterationscheme is developed by evaluating linear terms in the currentiteration level (denoted by + 1) and nonlinear terms in theprevious iteration level (denoted by ). Applying the SRM on(11) gives the following linear partial differential equations:

+1+ 1,

+1+ 2,

= (1 )+1

, (12)

+1

(0, ) = 1, +1

(, ) = 0, (13)

+1= +1

, +1

(0, ) = 0, (14)

where

1,

=1

2 (1 ) +

, 2,

= 2

. (15)

The initial approximation for solving (12)(14) is obtainedas the solutions at = 0. Thus

0(, ) and

0(, ) are given

by

0(, ) = erfc(

2) +

2

[1 exp(

2

4)] ,

0(, ) = erfc(

2) .

(16)

Starting from given initial approximations (16), the iterationschemes (12) can be solved iteratively for

+1(, ) when =

0, 1, 2, . . .. The solution for +1

is used in (14) which is, inturn, solved for

+1. To solve (12) we discretize the equation

using the Chebyshev spectral method in the -direction anduse an implicit finite differencemethod in the -direction. Fordetails of the spectral method, we refer interested readers to[19, 21]. Before applying the spectral method, it is convenientto transform the domain on which the governing equation isdefined to the interval [1, 1] where the spectral method canbe implemented. For convenience, the semi-infinite domainin the space direction is approximated by the truncateddomain [0,

], where

is a finite number selected to be

large enough to represent the behaviour of the flowpropertieswhen is very large. We use the transformation =

( +

1)/2 to map the interval [0, ] to [1, 1]. The basic idea

behind the spectral collocationmethod is the introduction ofa differentiation matrix which is used to approximate the

derivatives of the unknown variables () at the collocationpoints (grid points) as the matrix vector product

=

=0

D () = D, = 0, 1, . . . ,

, (17)

where+1 is the number of collocation points,D = 2/

,

and

= [ (0) , (

1) , . . . , (

)] (18)

is the vector function at the collocation points. Higher orderderivatives are obtained as powers ofD; that is,

()

= D, (19)

where is the order of the derivative. We choose the Gauss-Lobatto collocation points to define the nodes in [1, 1] as

= cos(

) , = 0, 1, . . . , . (20)

The matrix is of size (+ 1) (

+ 1). The grid points

on (, ) are defined as

= cos

, = , = 0, 1, . . . ,

,

= 0, 1, . . . , ,

(21)

where + 1,

+ 1 are the total number of grid points in

the and -directions, respectively, and is the spacing inthe -direction. The finite difference scheme is applied withcentering about amidpoint halfway between +1 and .Thismidpoint is defined as +(1/2) = (+1+)/2.The derivativeswith respect to are defined in terms of the Chebyshevdifferentiationmatrices. Applying the centering about +(1/2)to any function, say (, ) and its associated derivative, weobtain

(, +(1/2)

) = +(1/2)

=+1

+

2,

(

)

+(1/2)

=+1

.

(22)


Before applying the finite differences, we apply the spec-tral method on (12) and (14) to obtain

[D2 + a1,D]+1

+ a2,

= (1 )+1

, (23)

+1

(0, ) = 0,

+1(

, ) = 1, (24)

+1

(, 0) = erfc(

2) , = 0, 1, 2, . . . ,

, (25)

D+1

= +1

, +1

(

, ) = 0, (26)

+1

=

[[[[[[[

[

+1

(0, )

+1

(1, )

...+1

(1, )

+1

(

, )

]]]]]]]

]

, +1

=

[[[[[[[

[

+1

(0, )

+1

(1, )

...+1

(1, )

+1

(

, )

]]]]]]]

]

,

(27)

a2,

=

[[[[[[[

[

2,(0, )

2,(1, )

...2,(1, )

2,(

, )

]]]]]]]

]

, (28)

a1,

=

[[[[[

[

1,(0, )

1,(2, )

dd

1,(

, )

]]]]]

]

. (29)

Next, we apply the finite difference scheme on (23) inthe -direction with centering about the midpoint +(1/2) toobtain

A+1+1

= B+1

+ K, (30)

subject to the following boundary and initial conditions:

+1

(0, ) = 0,

+1(

, ) = 1,

= 0, 1, 2, . . . , ,

+1

(, 0) = erfc(

2) , = 0, 1, 2, . . . ,

,

(31)

where

A = 12(D2 + a+(1/2)

1,D) +

+(1/2)

(1 +(1/2)

)

I,

B = 12(D2 + a+(1/2)

1,D) +

+(1/2)

(1 +(1/2)

)

I,

K = a+(1/2)2,

,

(32)

where I is an (+ 1) (

+ 1).

Starting from the initial condition 0+1

, given by (16),(30) can be solved iteratively to give approximate solutionsfor +1

(, ), = 0, 1, 2, . . ., until a solution that converges towithin a given accuracy level is obtained. The solution

+1is

used in (26) which is, in turn, solved for +1

.

2.2. Spectral Quasilinearization Method (SQLM). In thissection we present the spectral quasilinearization method(SQLM) for solving the partial differential equation (1).The quasilinearization technique is essentially a generalizedNewton-Raphson Method that was originally used by Bell-man andKalaba [10] for solving functional equations.Wefirstset = , so that (1) becomes

+1

2 (1 )

+ [

2] = (1 )

. (33)

Applying the QLM on (33) the nonlinear partial differ-ential equation reduces to the following iterative sequence oflinear partial differential equations:

+1+ 1,

+1+ 2,+1

+ 3,+1

+ 4,

= (1 )+1

,

(34)

+1

(0, ) = 1, +1

(, ) = 0, (35)

+1= +1

, +1

(0, ) = 0, (36)

where

1,

=1

2 (1 ) +

,

2,= 2

,

3,

=

4,

=

+ 2

.

(37)

The indices and + 1 denote the previous and currentiteration levels, respectively.

Starting from given initial approximations, denoted by0(, ) and

0(, ), (34)(36) can be solved iteratively for

+1

(, ) and +1

(, ) ( = 0, 1, 2, . . .). We discretize (34)and (36) using the Chebyshev spectral method in the -direction and we use the implicit finite difference method inthe -direction to discretize (34) as described in the previoussection. Applying the spectral method and finite differenceson (34) and (36) as described previously, we obtain

[A1,1

A1,2

A2,1

A2,2

][

[

+1

+1

+1

+1

]

]

= [B1,1

B1,2

O O ][

[

+1

+1

]

]

+ [K1

K2

] (38)


subject to the boundary and initial conditions (31), where

A1,1

=1

2(D2 + a+(1/2)

1,D + a+(1/2)

2,)

+(1/2)

(1 +(1/2)

)

I,

A1,2

=1

2a+(1/2)3,

,

A2,1

= I, A2,2

= D,

B1,1

= 1

2(D2 + a+(1/2)

1,D + a+(1/2)

2,)

+(1/2)

(1 +(1/2)

)

I,

B1,2

= 1

2a+(1/2)3,

,

K1= a+(1/2)4,

, K2= ,

(39)

where I is an (+ 1) (

+ 1) identity matrix, O is an

(+ 1) (

+ 1) zero matrix, and is an (

+ 1) 1 zero

vector. Starting from the initial condition 0+1

, (38) can besolved iteratively to give approximate solutions for

+1(, )

and +1

(, ), = 0, 1, 2, . . ., until a solution that convergesto within a given accuracy level is obtained.

3. Unsteady Three-Dimensional MHD Flowand Mass Transfer in a Porous Space

We consider the unsteady and three-dimensional flow of aviscous fluid over a stretching surface investigated by Hayatet al. [22]. The fluid is electrically conducting in the presenceof a constant appliedmagnetic field

0.The inducedmagnetic

field is neglected under the assumption of a small magneticReynolds number.The flow is governed by the following fourdimensionless partial differential equations:

+ (1 ) (

2

)

+ [( + ) ()2

2 ] = 0,

(40)

+ (1 ) (

2

)

+ [( + ) ()2

2 ] = 0,

(41)

+ Pr (1 ) (

2

)

+ Pr ( + ) = 0,(42)

+ Sc (1 ) (

2

)

+ Sc ( + ) Sc = 0(43)

with the following boundary conditions:

(, 0) = (, 0) = 0,

(, 0) = (, 0) = (, 0) = 1,

(,) =

(,) = (,) = (,) = 0,

(, 0) = .

(44)

In the above equations prime denotes the derivative withrespect to and the stretching parameter is a positiveconstant. is the local Hartman number, the local porosityparameter, Sc the Schmidt number, Pr the Prandtl number,and the chemical reaction parameter. The initial unsteadysolution can be found exactly by setting = 0 in the aboveequations and solving the resulting equations. The closedform analytical solutions are given by

(0, ) = erfc(

2) +

2

[1 exp(

2

4)] ,

(0, ) = ( erfc(

2) +

2

[1 exp(

2

4)]) ,

(0, ) = erfc(

2) ,

(0, ) = erfc(

2) .

(45)

Quantities of physical interest in this problems are theskin friction coefficients

and

in - and -directions,

local Nusselt number Nu, and local Sherwood number Shwhich are given in [22] in dimensionless form as

Re1/21/2

= (0, ) ,Re1/2

1/2

= (0, ) ,NuRe1/2

1/2

= (0, ) , ShRe1/2

1/2

= (0, ) ,

(46)

where Reand Re

are the local Reynolds numbers, (0, )

and (0, ) are the surface shear stresses in - and -directions, (0, ) is the surface heat transfer parameter, and(0, ) is the surface mass transfer parameter.

3.1. Spectral Relaxation Method Solution. In this section wediscuss the development of the spectral relaxation method tosolve the system of partial differential equations (40)(43).First, we set = and = V, so that (40) and (41) become

+ (1 ) (

2

)

+ [( + ) 22 ] = 0,

V + (1 ) (

2V

V)

+ [( + ) V V2 2V V] = 0.

(47)


Applying the SRM on the resulting system of nonlinearpartial differential equations gives the following linear partialdifferential equations:

+1+ 1,

+1+ 2,+1

+ 3,

= (1 )+1

, (48)

+1= +1

, (49)

V+1

+ 1,V+1

+ 2,V+1

+ 3,

= (1 )V+1

, (50)

+1= V+1

, (51)

+1+ 1,

+1= (1 )

+1

, (52)

+1+ 1,

+1+ 2,+1

= (1 )+1

, (53)

+1

(0, ) = +1

(0, ) = +1

(0, ) = 1,

+1

(0, ) = +1

(0, ) = 0, V+1

(0, ) = ,

+1

(, ) = V+1

(, ) = +1

(, ) = 0,

+1

(, ) = 0,

(54)

where

1,

=1

2 (1 ) + (

+ ) ,

2,

= (2+ ) ,

3,

= 2

,

1,= 1,,

2,

= 2,,

3,= V2

,

1,

= Pr(12 (1 ) + (

+ )) ,

1,

= Sc(12 (1 ) + (

+ )) ,

2,

= Sc.

(55)

Starting from given initial approximations, denoted by0(, ),

0(, ), V

0(, ),

0(, ),

0(, ), and

0(, ), (48)

(53) can be solved iteratively for the unknown functions. Tosolve the above decoupled system of differential equationswe apply Chebyshev spectral collocation method on thespace variable and finite differences in the time variableas described previously and obtain the following system ofdecoupled equations:

A1+1

+1= B1

+1+ 1,

D+1+1

= +1

+1,

A2+1

+1= B2

+1+ 2,

D+1+1

= +1

+1,

A3+1

+1= B3

+1+ 2,

A4+1

+1= B4

+1+ 4,

+1

(0, ) = +1

(0, ) = +1

(0, ) = 1,

V+1

(0, ) = ,

+1

(0, ) = +1

(0, ) = 0,

+1

(, ) = V+1

(, ) = +1

(, ) = 0,

+1

(, ) = 0,

+1

(, 0) = erfc(

2) +

2

[1 exp(

2

4)] ,

+1

(, 0) = ( erfc(

2) +

2

[1 exp(

2

4)]) ,

+1

(, 0) = erfc(

Pr

2) ,

+1

(, 0) = erfc(

Sc

2) , = 0, 1, 2, . . . , ,

(56)

where the matrices above are defined as

A1=1

2(D2 + a+(1/2)

1,D + a

2,)

+(1/2)

(1 +(1/2)

)

I,

A2=1

2(D2 + b+(1/2)

1,D + b

2,)

+(1/2)

(1 +(1/2)

)

I,

A3=1

2(D2 + c+(1/2)

1,D)

Pr+(1/2) (1 +(1/2))

I,

A4=1

2(D2 + d+(1/2)

1,D + d

2,)

Sc+(1/2) (1 +(1/2))

I,

B1=

1

2(D2 + a+(1/2)

1,D + a

2,)

+(1/2)

(1 +(1/2)

)

I,

B2=

1

2(D2 + b+(1/2)

1,D + b

2,)

+(1/2)

(1 +(1/2)

)

I,

B3=

1

2(D2 + c+(1/2)

1,D)

Pr+(1/2) (1 +(1/2))

I,

B4=

1

2(D2 + d+(1/2)

1,D + d

2,)

Sc+(1/2) (1 +(1/2))

I,

K1= a+(1/2)3,

, K2= b+(1/2)3,

,

K3= , K

4= .

(57)


Above,, , , ,, and are the vectors of the functions, , V, , , and , respectively, when evaluated at the gridpoints

( = 0, 1, . . . ,

).

3.2. Spectral Quasilinearization Method. In this section wediscuss the development of the spectral quasilinearizationmethod to solve the system of partial differential equations(40)(43). First, we set = and = V, so that equations(40) and (41) become

+ (1 ) (

2

)

+ [( + ) 22 ] = 0,

V + (1 ) (

2V

V)

+ [( + ) V V2 2V V] = 0.

(58)

Applying the SQLM on the resulting system of nonlinearpartial differential equations gives the following linear partialdifferential equations:

+1+ 1,

+1+ 2,+1

+ 3,+1

+ 4,+1

+ 5,

= (1 )+1

,

(59)

+1= +1

, (60)

V+1

+ 1,V+1

+ 2,V+1

+ 3,+1

+ 4,+1

+ 5,

= (1 )V+1

,

(61)

+1= V+1

, (62)

+1+ 1,

+1+ 2,+1

+ 3,+1

+ 4,

= (1 )+1

,

(63)

+1+ 1,

+1+ 2,+1

+ 3,+1

+ 4,+1

+ 5,

= (1 )+1

,

(64)

+1

(0, ) = +1

(0, ) = +1

(0, ) = 1,

+1

(0, ) = +1

(0, ) = 0, V+1

(0, ) = ,

+1

(, ) = V+1

(, ) = +1

(, ) = 0,

+1

(, ) = 0,

(65)

where

1,

=1

2 (1 ) + (

+ ) ,

2,

= (2+2+ ) ,

3,

= 4,

=

,

5,

= (2

(+ )

) ,

1,= 1,,

2,

= (2V+2+ ) ,

3,= 4,

= V,

5,

= (V2 (+ ) V) ,

1,

= Pr(12 (1 ) + (

+ )) ,

2,

= 3,

= Pr,

4,

= Pr (+ )

,

1,

= Sc(12 (1 ) + (

+ )) ,

2,

= Sc, 3,

= 4,

= Sc,

5,

= Sc (+ )

.

(66)

Starting from given initial approximations, denoted by0(, ),

0(, ), V

0(, ),

0(, ),

0(, ), and

0(, ), equa-

tions (59)(64) can be solved iteratively for the unknownfunctions. To solve the above decoupled system of differentialequations we apply Chebyshev spectral collocation methodon the space variable andfinite differences in the time variableas described previously.

4. Results and Discussion

In this section we present the SRM and SQLM results forthe two examples described above. Numerical simulationswere carried out to obtain approximate numerical valuesof the quantities of physical interest, namely, the surfaceshear stresses, surface heat transfer, and the surface masstransfer parameter. In all the spectral method based numer-ical simulations a finite computational domain of extent

= 20 was chosen in the -direction. Through numericalexperimentation, this value was found to give accurate resultsfor all the selected governing physical parameters used inthe generation of results. Increasing the value of did notchange the results to a significant extent. The number ofcollocation points used in the spectral method discretizationwas

= 60 in all cases. We remark that both the SRM

and SQLM algorithms are based on the computation ofthe value of some quantity, say +1

+1, at each time step.

This is achieved by iterating using the relaxation method orthe quasilinearization method using a known value at theprevious time step as initial approximation. The iterationcalculations are carried until some desired tolerance level, ,is attained. In this study, the tolerance level was set to be =108. The tolerance level is defined as the maximum values of


the infinity norm of the difference between the values of thecalculated quantities and its first two derivatives at successiveiterations. For example, in calculating+1

+1, the tolerance level

and convergence criteria are defined as

max {+1

+1 +1

,+1

+1 +1

,

+1

+1+1

} < ,

(67)

where = and = . The accuracy of the computedSRM and SQLM approximate results was confirmed againstnumerical results obtained by using the popular Keller-boximplicit finite difference method as described by [2]. TheKeller-box method has been reported in literature to beaccurate, fast, and easier to program for boundary layerflow problems. The algorithm begins with the reduction ofthe governing nonlinear PDEs into a system of first orderequations that are discretized using central differences. Thenonlinear algebraic difference equations are linearised usingNewtons method and written in matrix-vector form. Thelinear matrix systems are solved in an efficient manner usingblock-tridiagonal-elimination technique. The grid spacingin both the -direction and -direction is carefully selectedto ensure that the Keller-box computations yield consistentresults for the governing velocity and temperature distribu-tions to a convergence level of at least 108.

Tables 1 and 2 give the approximate numerical values ofthe skin friction (0, ) for various step sizes , computedusing the SRM and SQLM, respectively, for Example 1. Thetables also give the total computational time for the integra-tion in the whole time domain to be completed. We remarkthat the results were computed using the same number ofcollocation points

and the same

. Reducing the step size

improves the accuracy of the results until the results areconsistent towithin eight decimal digits.The results displayedin the tables are quite revealing in several ways. First, it isclear from the comparison of the computational run timesthat the SRM takes less computer time than the SQLM. Theresults also indicate that the SRM converges more rapidlythan the SQLM results when the step size is reduced. Fullconvergence to within eight decimal digits is reached when is at least 0.0005 in the SRM compared to = 0.0001 inthe case of the SQLM. This means that the SRM convergesfaster than the SQLM with a decrease in the step size .Furthermore, there is good agreement between the two sets ofresults when is very small for all values of . The apparentsuperiority of the SRM in terms of computational efficiencyand accuracy when compared to the SQLMmay be explainedby the fact that the SRM algorithm reduced a coupled systemof equations into smaller sequences of decoupled equationswhich are solved one after the other. Smaller sized matricesare less susceptible to round-off errors and ill-conditioningand take less computer time to invert.

Table 3 gives a comparison of the amount of time it takesfor each method to generate numerical solutions that haveconverged to within 108 at selected time levels. As can beseen from Table 3, the SRM is much faster compared to theSQLM in computing the numerical solutions. For very smalltime steps the SRM appears to be at least twice as fast as the

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Iteratio

ns

100

1000

2000

5000

Nt

Figure 1: Variation of the SRM iterations with time in Example 1.

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Iteratio

ns

100

1000

2000

5000

Nt

Figure 2: Variation of the SQLM iterations with time in Example1.

SQLM. The results from Tables 1, 2, and 3 indicate that theSRM is much more computationally efficient and gives moreaccurate results than the SQLM under the same conditions.

Figures 1 and 2 display the number of iterations requiredto yield converged results to within the tolerance level of 108against the time for the SRM and SQLM, respectively. Theresults are given for different values of the number of gridpoints

. It can be seen from Figure 1 that more iterations

are required to give the converged results when the numberof grid points is small. For larger values of

, between four

and six iterations are required in the range 0 0.9. For


Table 1: Skin friction (0, ) for various values of computed using SRM.

0.01 0.001 0.0005 0.0002 0.00010.1 0.61046835 0.61046762 0.61046761 0.61046761 0.610467610.3 0.70126751 0.70126681 0.70126680 0.70126680 0.701266800.5 0.78982903 0.78982837 0.78982837 0.78982837 0.789828370.7 0.87626715 0.87626654 0.87626653 0.87626653 0.876266530.9 0.96053875 0.96053800 0.96053800 0.96053800 0.96053800Time 0.66 3.01 5.68 12.54 23.83

Table 2: Skin friction (0, ) for various values of computed using SQLM.

0.01 0.001 0.0005 0.0002 0.00010.1 0.61045544 0.61046674 0.61046742 0.61046758 0.610467610.3 0.70126943 0.70126664 0.70126676 0.70126679 0.701266800.5 0.78981759 0.78982831 0.78982835 0.78982836 0.789828370.7 0.87625663 0.87626652 0.87626653 0.87626653 0.876266530.9 0.96053069 0.96053800 0.96053800 0.96053800 0.96053800Time 0.60 3.29 6.62 20.37 53.92

Table 3: SRM and SQLM computational times for Example 1.

\ 0.01 0.001 0.0005 0.0002 0.0001Spectral relaxation method

0.1 0.07 0.22 0.43 0.95 1.930.3 0.13 0.65 1.38 3.18 6.010.5 0.20 1.14 2.26 5.48 10.690.7 0.27 1.72 3.14 7.70 15.080.9 0.35 2.30 4.25 10.05 19.57

Spectral qausilinearisation method0.1 0.03 0.41 0.76 2.27 4.720.3 0.11 1.26 2.28 6.78 14.670.5 0.20 2.20 3.92 11.01 24.190.7 0.29 3.14 5.47 15.25 33.290.9 0.39 4.09 7.09 20.11 42.65

near 1, the number of iterations required increases.The trendsin the results for SQLM, as seen in Figure 2, are similar tothose for the SRM. However, in the case of the SQLM, fullconvergence is achieved with only four iterations in a widerrange of and for smaller values of

than the SRM. This

observation indicates that the actual convergence rates (withan increase in iterations) of the SQLM are greater than thoseof the SRM.

Tables 4 and 5 give a comparison of the SRM and SQLMapproximate numerical solutions, respectively, against theKeller-box results for the skin frictions, surface heat transferparameter, and surface mass transfer parameters. We remarkthat the results reported in Tables 4 and 5 were generatedusing a tolerance level of 107 in the SRM, SQLM, and Keller-box implementations. It can be noted from Tables 4 and 5that both the SRM and SQLM converge to the Keller-box

Table 4: Comparison between the SRM and Keller-Box approxi-mate numerical values for (0, ), (0, ), (0, ), and (0, )when = 0.5, = 2, = 0.5, Sc = = 1, and Pr = 1.5 in Example 2.

Spectral relaxation method Keller-box \ 0.01 0.002 0.001 0.0005 0.0005

(0, )

0.1 0.851309 0.851259 0.851258 0.851257 0.8512570.3 1.316738 1.316706 1.316705 1.316705 1.3167050.5 1.685327 1.685307 1.685306 1.685306 1.6853060.7 1.992622 1.992609 1.992608 1.992608 1.9926080.9 2.259344 2.259335 2.259335 2.259335 2.259335

(0, )

0.1 0.417173 0.417151 0.417151 0.417150 0.4171500.3 0.639617 0.639603 0.639602 0.639602 0.6396020.5 0.817659 0.817649 0.817649 0.817649 0.8176490.7 0.966610 0.966604 0.966603 0.966603 0.9666040.9 1.095987 1.095983 1.095983 1.095983 1.095983

(0, )

0.1 0.710885 0.710882 0.710882 0.710882 0.7108820.3 0.742845 0.742842 0.742842 0.742842 0.7428430.5 0.765247 0.765244 0.765244 0.765244 0.7652440.7 0.777274 0.777270 0.777270 0.777270 0.7772700.9 0.770819 0.770807 0.770807 0.770807 0.770807

(0, )

0.1 0.634447 0.634443 0.634443 0.634443 0.6344440.3 0.766870 0.766867 0.766867 0.766867 0.7668670.5 0.891209 0.891207 0.891207 0.891207 0.8912070.7 1.010046 1.010045 1.010045 1.010045 1.0100450.9 1.125550 1.125549 1.125549 1.125549 1.125549

results when is sufficiently small. The convergence to six-decimal-digit accurate results is more or less the same for


Table 5: Comparison between the SRM and Keller-Box approxi-mate numerical values for (0, ), (0, ), (0, ), and (0, )when = 0.5, = 2, = 0.5, Sc = = 1, and Pr = 1.5 in Example2.

Spectral quasilinearisation method Keller-Box \ 0.01 0.002 0.001 0.0005 0.0005

(0, )

0.1 0.851309 0.851259 0.851258 0.851257 0.8512570.3 1.316738 1.316706 1.316705 1.316705 1.3167050.5 1.685327 1.685307 1.685306 1.685306 1.6853060.7 1.992622 1.992609 1.992608 1.992608 1.9926080.9 2.259344 2.259335 2.259335 2.259335 2.259335

(0, )

0.1 0.417173 0.417151 0.417151 0.417150 0.4171500.3 0.639617 0.639603 0.639602 0.639602 0.6396020.5 0.817659 0.817649 0.817649 0.817649 0.8176490.7 0.966610 0.966604 0.966603 0.966603 0.9666040.9 1.095987 1.095983 1.095983 1.095983 1.095983

(0, )

0.1 0.710885 0.710882 0.710882 0.710882 0.7108820.3 0.742845 0.742842 0.742842 0.742842 0.7428430.5 0.765247 0.765244 0.765244 0.765244 0.7652440.7 0.777274 0.777270 0.777270 0.777270 0.7772700.9 0.770819 0.770807 0.770807 0.770807 0.770807

(0, )

0.1 0.634447 0.634443 0.634443 0.634443 0.6344440.3 0.766870 0.766867 0.766867 0.766867 0.7668670.5 0.891209 0.891207 0.891207 0.891207 0.8912070.7 1.010046 1.010045 1.010045 1.010045 1.0100450.9 1.125550 1.125549 1.125549 1.125549 1.125549

both SRM and SQLM schemes. We remark that the Keller-box results given in Tables 4 and 5 were calculated usingnonuniform step size in the -direction and a uniform stepsize = 0.0005 in the -direction. Using a nonuniformgrid size significantly improves the computation time of theKeller-box method.Thus, to speed up the computation timesfor the Keller-box method, computations were carried outwith an initial step size of

0= 0.001. This was gradually

increased by the variable grid parameter (VGP) factor of1.005 between successive grid points in accordance with theformula

= 1

+ VGP 1

for = 1, 2, . . . , (where is the number of grid points in the direction). The value of

was fixed at = 10 for the Keller-box implementation.

A comparison of the computational times between theSRM, SQLM, and Keller-box method is given in Table 6 forthe computation of results that converge towithin six decimaldigits. It can be seen from the table that there is a substantialdifference in the computation times of the threemethodswiththe SRM being at least four times faster than the SQLM andthe Keller-box being the slowest method. The demonstratedspeed of the spectral method based methods is primarily dueto the intrinsic property of the spectral collocation methodto be able to give accurate approximate results using onlya few grid points. Only 60 collocation points were used to

500

100 1000

2000

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Iteratio

ns

Nt

Figure 3: Variation of the SRM iterations with time in Example 2.

generate results that converge to at least six decimal digits ofaccuracy. On the other hand, the Keller-box required a lotmore grid points in the direction to give the same amount ofaccuracy.The apparent computational speed of the SRM overthe SQLM is in accord with the observation made earlier inthe case of the one-equation system.As can be seen in Table 6,the superiority in computational efficiency of the SRM overthe SQLM is much more pronounced in the current examplethat involves a system of four coupled equations. Thus, theSRM is a better alternative method that can be used to obtainnumerical solutions of systems of PDEs arising in boundarylayer flow problems.

Figures 3 and 4 show the variation of the SRM and SQLMiterations, respectively, over time for different values of

.

The indicated number of iterations is the total number ofiterations required to obtain results that are consistent towithin a tolerance level of 106. It can be noted from Figure 3that the total number of iterations required for the SRM isbetween 4 and 9 with the total iterations increasing as tendsto 1. In contrast, the range of the required number of iterationsis 3 to 5 in the case of the SQLM. Thus the convergence rateof the SQLM in terms of iterations is higher than that of theSRM.

5. Conclusion

In this paper, we investigated the application of the spectralrelaxation method (SRM) and spectral quasilinearisationmethod (SQLM) in the solution of unsteady boundary layerflows that are described by systems of coupled nonlinear par-tial differential equations.We considered themodel problemsof unsteady boundary layer flow caused by an impulsivelystretching sheet and the unsteady MHD flow and masstransfer in a porous space. The purpose of this study wasto establish the applicability of the SRM, for the first time,


500

100 1000

2000

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Iteratio

ns

Nt

Figure 4: Variation of the SQLM iterations with time in Example2.

Table 6: SRM and SQLM computational times for Example 2.

\ 0.01 0.001 0.0005 0.0002 0.0001Spectral relaxation method

0.1 0.13 0.91 1.52 3.92 8.520.3 0.37 3.02 5.68 12.07 26.130.5 0.62 5.24 10.09 21.32 43.580.7 0.91 7.40 14.54 32.06 62.840.9 1.20 9.97 18.90 42.86 87.48

Spectral qausilinearisation method0.1 0.57 4.53 8.76 21.19 43.050.3 1.50 14.18 27.28 69.15 140.010.5 2.43 23.33 45.72 119.98 235.450.7 3.39 32.57 64.30 168.09 326.960.9 4.66 41.87 83.24 216.00 418.40

Keller-box method0.1 5.30 47.67 100.590.3 17.62 143.90 294.220.5 30.27 241.03 486.320.7 42.94 337.97 677.870.9 55.68 464.56 900.30

to systems of PDEs that model unsteady boundary layerflows.The investigation also sought to assess the accuracy andefficiency of the SRM compared to the SQLM and Keller-boxmethod.

The most obvious finding to emerge from this study isthat the SRM is significantly more computationally efficientthan the SQLM which, in turn, is faster than the Keller-box method. For sufficiently small step-sizes, all the threemethods yield results that are consistent to within a giventolerance level. The SRM was observed to convergence fasterthan the SQLM with a reduction of the step size. It is

this feature that makes the SRM computationally efficientas accurate results are obtained using fewer grid points inthe time direction. In addition, the SRM algorithm involvesthe solution of a sequence of smaller sized matrix equationscompared to the SQLM. The numerical results presentedin this study clearly demonstrate the potential of the SRMscheme for the simulation of numerical solutions of the classof unsteady boundary layer flows equations related to themodel equations discussed in this study. The evidence of theaccuracy and efficiency of the SRM from this study suggeststhat the method can be used as a more practical tool forsolving unsteady boundary layer flows and for validatingthe results generated using other numerical methods in thesolution of similar boundary layer flow equations. The pre-sented SRM approach adds to a growing body of literature onpractical numerical methods for solving complex nonlinearPDEs in some fluid mechanics applications.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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[6] S. Liao, An analytic solution of unsteady boundary-layer flowscaused by an impulsively stretching plate, Communications inNonlinear Science and Numerical Simulation, vol. 11, no. 3, pp.326339, 2006.

[7] S. S.Motsa andZ.G.Makukula, On spectral relaxationmethodapproach for steady von Karman flow of a Reiner-Rivlin fluidwith Joule heating, viscous dissipation and suction/injection,Central European Journal of Physics, vol. 11, no. 3, pp. 363374,2013.

[8] S. S. Motsa, P. Dlamini, and M. Khumalo, A new multistagespectral relaxation method for solving chaotic initial valuesystems, Nonlinear Dynamics, vol. 72, pp. 265283, 2013.

[9] S. S. Motsa, P. G. Dlamini, and M. Khumalo, Solving hyper-chaotic systems using the spectral relaxation method, AbstractandAppliedAnalysis, vol. 2012, Article ID 203461, 18 pages, 2012.

[10] R. E. Bellman and R. E. Kalaba, Quasilinearization and Non-linear Boundary-Value Problems, Elsevier, New York, NY, USA,1965.


[11] K. Bhattacharyya, Effects of heat source/sink on mhd flowand heat transfer over a shrinking sheet with mass suction,Chemical Engineering Research Bulletin, vol. 15, no. 1, pp. 1217,2011.

[12] P. M. Patil, S. Roy, and A. J. Chamkha, Double diffusive mixedconvection flow over a moving vertical plate in the presenceof internal heat generation and a chemical reaction, TurkishJournal of Engineering and Environmental Sciences, vol. 33, no.3, pp. 193205, 2009.

[13] S. Roy, Non-uniform mass transfer or wall enthalpy into acompressible flow over yawed cylinder, International Journal ofHeat and Mass Transfer, vol. 44, no. 16, pp. 30173024, 2001.

[14] P. Saikrishnan and S. Roy, Non-uniform slot injection (suction)into water boundary layers over (i) a cylinder and (ii) a sphere,International Journal of Engineering Science, vol. 41, no. 12, pp.13511365, 2003.

[15] S. Ponnaiah, Boundary layer flow over a yawed cylinder withvariable viscosity role of non-uniform double slot suction(injection), International Journal of Numerical Methods forHeat and Fluid Flow, vol. 22, no. 3, pp. 342356, 2012.

[16] S. S. Motsa, T. Hayat, and O. M. Aldossary, MHD flow ofupper-convected Maxwell fluid over porous stretching sheetusing successive Taylor series linearization method, AppliedMathematics and Mechanics, vol. 33, no. 8, pp. 975990, 2012.

[17] S. S. Motsa and S. Shateyi, Successive linearization analysisof the effects of partial slip, thermal diffusion, and diffusion-thermo on steady MHD convective flow due to a rotating disk,Mathematical Problems in Engineering, vol. 2012, Article ID397637, 15 pages, 2012.

[18] F. G. Awad, P. Sibanda, S. S. Motsa, and O. D. Makinde,Convection from an inverted cone in a porous mediumwith cross-diffusion effects, Computers & Mathematics withApplications, vol. 61, no. 5, pp. 14311441, 2011.

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[23] J. C.Williams andT. B. Rhyne, Boundary layer development ona wedge impulsively set into motion, SIAM Journal on AppliedMathematics, vol. 38, no. 2, pp. 215224, 1980.

Research ArticleA Local Integral Equation Formulation Based onMoving Kriging Interpolation for Solving CoupledNonlinear Reaction-Diffusion Equations

Kanittha Yimnak and Anirut Luadsong

Department of Mathematics, Faculty of Science, King Mongkuts University of TechnologyThonburi (KMUTT), 126 Pracha-utid Road,Bangmod, Toongkru, Bangkok 10140, Thailand

Correspondence should be addressed to Anirut Luadsong; [email protected]

Received 5 April 2014; Revised 21 May 2014; Accepted 21 May 2014; Published 4 June 2014

Academic Editor: Oluwole Daniel Makinde

Copyright 2014 K. Yimnak and A. Luadsong.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Themeshless local Pretrov-Galerkin method (MLPG) with the test function in view of the Heaviside step function is introduced tosolve the system of coupled nonlinear reaction-diffusion equations in two-dimensional spaces subjected to Dirichlet and Neumannboundary conditions on a square domain. Two-field velocities are approximated by moving Kriging (MK) interpolation methodfor constructing nodal shape function which holds the Kronecker delta property, thereby enhancing the arrangement nodalshape construction accuracy, while the Crank-Nicolson method is chosen for temporal discretization. The nonlinear terms aretreated iteratively within each time step. The developed formulation is verified in two numerical examples with investigating theconvergence and the accuracy of numerical results.Thenumerical experiments revealing the solutions by the developed formulationare stable and more precise.

1. Introduction

Reaction-diffusion systems, which are proposed by AlanTuring [1], have an important application in mathematics,physics, chemistry, and biology. Turing or diffusion-driveninstability is initiated by arbitrary random deviations of thestationary state and results in stationary spatially periodicvariations in the chemical concentration, that is, chemicalpatterns. Intuitively turning instability can be understood byconsidering the long-range effects of chemicals, which are notequal due to the difference in the pace of diffusion and thusthe instability arises [2].The Prigogine principle of minimumentropy production from [3] is not in general a necessarycondition for the steady state and the most favorable state ofthe system cannot be determined based on the behavior in thevicinity of the steady state, but one must consider the globalnonequilibrium dynamics. The reaction-diffusion equationsare often solved by numerical methods and usually diffusionis thought to be stabilizing.The idea that diffusion couldmakea stable and uniform chemical state unstable was innovative.

Shirzadi et al. [4] developed meshless local Petrov-Galerkin (MLPG) formulation for numerical solution of thenonlinear reaction-diffusion equations.The spatial variationsare approximated by moving least squares and the nonlinearterms are treated iteratively within each time step. Construct-ing shape functions is one of the most important issues intheMLPGmethod.There aremanymethods for constructingshape functions such as the moving least squares (MLS) andthe weighted least squares (WLS) method. The most popularmethod isMLS.Although theMLPGmethod andmany othermeshless methods have been gradually applied to differentfields, there exists inconvenience because of the difficultyin implementing some essential boundary conditions; theshape function constructed by MLS approximation does notsatisfy theKronecker delta function property. In this research,the MLPG method based on moving Kriging approximationis developed to solve the system of two nonlinear partialdifferential equations of the parabolic type. The movingKriging, which was proposed by Gu [5] for constructingshape function, has the Kronecker delta property that is a


http://dx.doi.org/10.1155/2014/196041


good property for constructing the shape function. Thesesystems are solved by local integral equation formulation andone-step time discretizationmethod.The nonlinear terms aretreated iteratively within each time step. The boundary anddomain integrals are calculated using the Simpson and theGauss-Legendre quadrature rules. Two numerical examplesare considered in order to verify the proposed method withtesting its accuracy and convergence.

2. Governing Equation

The numerical simulations of the coupled pair of nonlinearpartial differential equations are as follows [4]:

= 12 + 1 + (, V) +

1(x, ) ,

V= 22V +

2V + + (, V) +

2(x, )

(1)

given initial and Dirichlet and/or Neumann boundary con-ditions in the two-dimensional region , where

1, 2, 1,

2, and are given constants, and are functions of the

field variables and V, and 1and

2are assumed to be

prescribed sources. In the case of two-component reactionsystem, (x, ) and V(x, ) stand for concentrations and

1,

2stand for the diffusion coefficients of chemical species.

3. Moving Kriging Interpolation Method

The Kriging interpolation is a well-known geostatic tech-nique for spatial interpolation in geology andmining [5].Theformulation of the construction of meshless shape functionby moving Kriging (MK) interpolation is introduced brieflyin the following.

Similar to the MLS approximation, consider the function(x) defined in the domain discretized by a set of properlyscattered nodes x

, ( = 1, 2, . . . , ), where is the total

number of nodes in the whole domain. It is assumed that only nodes surrounding point x have the effect on (x).

The subdomain x that encompasses these surroundingnodes is called the interpolation domain of point x. The MKinterpolation (x) at point x is defined as presented in [6].Therefore, the formulation of the meshless shape functionusing MK interpolation is given by

(x) =

=1

(x) = (x)u, x x, (2)

where u = [(x1) (x

2) (x

)] is a vector value of

the function in the domain.(x) is a 1 vector of shapefunctions, expressed as

(x) = p (x)A + r (x)B, (3)

where matrices A and B are defined as

A = (PR1P)1

PR1,

B = R1 (I PA) ,(4)

in which I is a unit matrix of size and vector p(x) is

p (x) = [1 (x1) (x)] . (5)

In general, a linear basis in two-dimensional space is

p (x) = (1, , ) , = 3, (6)

a quadratic basis is given as

p (x) = (1, , , 2, , 2) , = 6, (7)

and a cubic basis is

p (x) = (1, , , 2, , 2, 3, 2, 2, 3) , = 10.(8)

For matrix P with the size , values of the polynomialbasis functions (5) at the given set of nodes are collected asfollows:

P = [[[

1(x1)

(x1)

......

...1(x)

(x)

]]

]

. (9)

MatricesR and vector r(x) are defined by the following equa-tions:

R = [[[

(x1, x1) (x

1, x)

......

... (x, x1) (x

, x)

]]

]

,

r (x) = [ (x, x1) (x, x

)] ,

(10)

where (x, x) is the correlation function between any pair of

nodes located at xand x, representing the covariance of the

field value (x); that is,

(x, x) = [ (x

) (x

)] . (11)

Similarly, the covariance [(x) (x

)] can be replaced

by (x, x). It can be seen from the foregoing formulations

that the values of matrices R and r play important roles inthe computation. A simple and frequently used correlationfunction is a Gaussian function as

(x, x) = 2

, (12)

where = x x and > 0 are the correlation parameters

used to fit the model and are assumed to be given.The first-order partial derivatives of the shape function(x) against the coordinates x

, = 1, 2, can be easily obtained

from (3) as

,(x) = p

,(x)A + r

,(x)B, (13)

where (),denotes ()/.


4. Local Integral Equations

The local integral formulation of (1) can be written as

= 1

(2) +

1

+

(, V) +

1(x, )

,

V

= 2

(2V) +

2

V

+

+

(, V) +

2(x, )

,

(14)

where is a Heaviside step used as the test function:

(x) = {1, x

0, x .

(15)

and V are trial functions, and instead of the entire domain

we have considered a subdomain

located entirely at

domain , x = (, ) 2. The domain is enclosedby =

, with boundary conditions

= , V = V on , (16)

n = , n V = V on , (17)

where n = (1, 2) is an outward unit normal vector of the

boundary and n /n. Condition (16) is often referredto as the Dirichlet boundary condition and (17) is referred toas the Neumann boundary condition. Let (x, ) and V(x, ),which substitute (x, ) and V(x, ), respectively, be the trialsolutions:

(x, ) =

=1

(x) () , V

(x, ) =

=1

(x) V() ,

= ( (x, ) , V (x, )) , = ( (x, ) , V (x, )) .(18)

For internal nodes, from local integral equation (14), andusing theMK (2), we have the following nonlinear equations:

=1

[

(x) ]

=

=1

[1(

,(x) +

,(x) )

+(1

(x) )]

+1

+

+

1(x, ) ,

=1

[

(x) ]

V

=

=1

[2(

,(x) +

,(x) )

+(2

(x) )] V

+

=1

[

(x) ]

+ 2

V +

+

2(x, ) .

(19)

The following abbreviations have been used for the integralterm:

=

(x) ,

1

= 1(

((x)

n) +

((x)

n))

+ 1

(x) ,

2

= 2(

((x)

n) +

((x)

n))

+ 2

(x) ,

=

(x) ,

1= 1

+

+

1(x, ) ,

2= 2

V +

+

2(x, ) .

(20)


The boundary and domain integrals are calculated using theSimpson and the Gauss-Legendre quadrature rules.

We can rewrite (19) as

=1

=

n

j=11

+ 1,

=1

V

=

n

j=1(2

V+) + 2.

(21)

4.1. Temporal Discretization. Equation (21) can be rewrittenas

KU

= H1U + F1. (22)

Similarly, we have

KV

= H2V + LU + F2, (23)

whereK = [

]

, H1 = [1

]

, H2 = [2

]

,

L = []

, F1 = [1]1, F2 = [2]1,

U = [1 2 ]

, V = [V1 V2 V]

.

(24)

The finite-difference approximation of the time deriva-tives of (22) and (23) in Crank-Nicolson method is given asfollows:(2K H1) U

k+1= (2K + H1) U

k+ (F+1

1+ F1) ,

(2K H2) Vk+1

= (2K + H2) Vk

+ (LU+1 + LU)

+ (F+12

+ F2) .

(25)

Because and are nonlinear functions of and V, we solvethem iteratively in each time step with replacing +1 and+1 by and , respectively, at zeroth iteration. Equation

(25) converted into a set of nonlinear algebraic equations forunknowns U+1 and V+1.

5. Numerical Experiments

The analyzed domain is = [0, 1] [0, 1].The error of andV, which are presented in the numerical results, is representedby maximum relative error (MRE) and root mean square ofrelative error (RMSRE) of and V, respectively, where

MRE= max

, = 1, 2, . . . , ,

RMSRE=

1

=1

(

)

2

, = 1, 2, . . . , .

(26)

and

are the exact and computed values of at point x

,

respectively, and is the number of nodes.

Example 1. The system of nonlinear PDEs in the region =[0, 1] [0, 1] is given as follows:

= (

2

2+2

2) +

2V2 + 1(, , ) ,

V= (

2V2

+2V2

) + V 2 + V + 2(, , ) ,

(27)

where

1(, , ) =

46 52++

,

2(, , ) = 4

++ 4+2+2

23

.

(28)

The initial and Dirichlet boundary conditions are chosen insuch a way that the exact solution is

(, , ) = 2++

, V (, , ) = +. (29)

In this example, the governing equations resemble those of(1) with = 0 and

1= 2= 1. The shown results have

been obtained using = 25, 81, 256, and 441 nodal points,respectively, and = 0.1 at time instant = 1. Figures 1(a)and 1(b) show that theMRE of and V increases as a functionof the number of nodal points using = 3; meanwhile, using = 6 and 10, the MRE increases gradually as a function ofthe number of nodal points. Figures 2(a) and 2(b) show thatthe RMSRE of and V decreases as a function of the numberof nodal points using = 3, 6, and 10. Moreover, the errorsof and V using = 6, 10 are less than the RMSRE of and V when using = 3. The profile of trial solution of and V is similar to the profile of exact solution of and V (seeFigures 3(a), 3(b), 3(c), and 3(d)). Figures 3(e) and 3(f) revealcorresponding error profile of and V. The errors of and Vsatisfy the boundary conditions as well as the Kronecker deltaproperty.

Example 2 (application for Brusselator system). The devel-oped formulation from this research can solve a real worldapplication example. Let us consider the nonlinear reaction-diffusion Brusselator system in the two-dimensional region[7], = [0, 1] [0, 1]. Consider

= (

2

2+2

2) ( + 1) +

2V + ,

V= (

2V2

+2V2

) + 2V,

(30)

with = 0.002, = 1/2, = 1, initial conditions

(, , 0) =1

221

33, V (, , 0) =

1

221

33, (31)


0 50 100 150 200 250 300 350 400 4500

0.005

0.01

0.015

0.02

0.025M

RE o

fu

N

m = 3

m = 6

m = 10

(a)

0 50 100 150 200 250 300 350 400 4500

1

2

3

4

5

6

7

8

MRE

of

N

103

m = 3

m = 6

m = 10

(b)

Figure 1: Both errors of and V against the number of nodal points: = 10, = 1, and = 0.1. (a) MRE of ; (b) MRE of V.

0 50 100 150 200 250 300 350 400 4500

1

2

3

4

5

6

RMSR

E of

u

104

N

m = 3

m = 6

m = 10

(a)

0 50 100 150 200 250 300 350 400 4500

1

2

3

RMSR

E of

104

N

m = 3

m = 6

m = 10

(b)

Figure 2: Both errors of and V against the number of nodal points: = 10, = 1, and = 0.1. (a) RMSRE of ; (b) RMSRE of V.

and Neumann boundary conditions

(0, , )

= (1, , )

= (, 0, )

= (, 1, )

= 0,

V (0, , )

=V (1, , )

=V (, 0, )

=V (, 1, )

= 0,

(32)

for which the exact solution is unknown. For small valuesof the diffusion coefficient , if 1 + 2 > 0 then thenumerical solution of the Brusselator system converges to anequilibriumpoint (, /) (see [7]).The experimental resultsfor maximum and minimum values of the exact solution arepresented in Table 1. According to Table 1, it is found that thevalues of the exact solution tend to the steady state valuesof (, V) = (, /) = (1, 1/2). Figure 4 shows how thesolution changes from initial condition to the steady state as


0 0.20.4 0.6

0.8 1

00.2

0.40.6

0.810

0.5

1

1.5

yx

u(x,y,1)

Exac

t sol

utio

n

(a)

0 0.20.4 0.6

0.8 1

00.2

0.40.6

0.810

0.2

0.4

0.6

0.8

1

y x

Exac

t sol

utio

n(x,y,1)

(b)

0 0.20.4 0.6

0.8 1

00.2

0.40.6

0.810

0.5

1

1.5

y x

u(x,y,1)

Tria

l sol

utio

n

(c)

0 0.20.4 0.6

0.8 1

00.2

0.40.6

0.810

0.2

0.4

0.6

0.8

1

yx

(x,y,1)

Tria

l sol

utio

n

(d)

0 0.20.4 0.6

0.8 1

00.2

0.40.6

0.810

1

2

3

4

RE o

fu

yx

103

(e)

0 0.20.4 0.6

0.8 1

00.2

0.40.6

0.81

0

0.5

1

1.5

2

RE o

f

y x

103

(f)

Figure 3: Exact and trial solutions of and V using = 441, = 10, = 1, and = 0.1. (a) exact solution of ; (b) exact solution of V; (c)trial solution of ; (d) trial solution of V; (e) corresponding error profile of ; (f) corresponding error profile of V.

0 0.20.4 0.6

0.8 1

00.2

0.40.60.8

11

1.0002

1.0004

1.0006

1.0008

u(x,y,10)

yx

(a)

0 0.20.4 0.6

0.8 1

00.2

0.40.6

0.81

0.4992

0.4993

0.4994

0.4995

0.4996

(x,y,10)

y x

(b)

Figure 4: The approximate solutions profiles of and V for the Brusselator model at = 10 obtained by using Crank-Nicolson method: = 10, = 25, and = 0.1. (a) at = 10; (b) V at = 10.


Table 1: The maximum and minimum values of the exact solutionby using Crank-Nicolson method: = 10, = 0.1, and = 25.

V

max min Vmax Vmin1 0.5871 0.5258 0.3173 0.13842 0.7555 0.6944 0.4658 0.34583 0.8701 0.8060 0.5328 0.47944 0.9499 0.8985 0.5395 0.53065 0.9918 0.9631 0.5309 0.52126 1.0053 0.9953 0.5163 0.50727 1.0060 1.0048 0.5052 0.50078 1.0046 1.0031 0.5003 0.49909 1.0024 1.0011 0.4991 0.499010 1.0008 1.0001 0.4995 0.4993

1 1 0.5 0.5

tends to the infinity. The experimental results are similar tothose previously reported [7, 8].

6. Conclusions

The developed formulation has been proposed for couplednonlinear reaction-diffusion equation by using MK nodalshape function with temporal discretization by the Crank-Nicolson method. Cubic polynomial basis is the best forconstructing the nodal shape function. The robust methodworks well in the sense of accuracy and satisfies the boundarycondition. Moreover, the solutions by the developed for-mulation with temporal discretization by Crank-Nicolsonwith iterative methods are stable and more precise, so theincreased can be chosen for studying. The convergencetesting is shown in the Brusselator model for which the exactsolution was unknown, showing that the proposed methodis competent at simulating coupled nonlinear reaction-diffusion problems.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This research is partially supported by Dhurakij PunditUniversity, Thailand.

References

[1] A. Turing, The chemical basis ofmorphogenesis,PhilosophicalTransactions of the Royal Society B, vol. 237, pp. 3772, 1952.

[2] T. Leppanen, Computational studies of pattern formation inturing systems [Ph.D. thesis], Helsinki University of Technology,Espoo, Finland, 2004.

[3] I. Prigogine, Etude Thermodynamics des Phenomenes Irre-versibles (Study of the thermodymamics of irreversible phe-nomenon), in Presented to the Science Faculty at the FreeUniversity of Brussels (1945), Dunod, Paris, France, 1947.

[4] A. Shirzadi, V. Sladek, and J. Sladek, A local integral equa-tion formulation to solve coupled nonlinear reaction-diffusionequations by using moving least square approximation, Engi-neering Analysis with Boundary Elements, vol. 37, no. 1, pp. 814,2013.

[5] L. Gu, Moving kriging interpolation and element-freeGalerkinmethod, International Journal for Numerical Methods in Engi-neering, vol. 56, no. 1, pp. 111, 2003.

[6] L. Chen and K. M. Liew, A local Petrov-Galerkin approachwith moving Kriging interpolation for solving transient heatconduction problems, Computational Mechanics, vol. 47, no. 4,pp. 455467, 2011.

[7] E. H. Twizell, A. B. Gumel, and Q. Cao, A second-orderscheme for the Brusselator reaction-diffusion system, Journalof Mathematical Chemistry, vol. 26, no. 4, pp. 297316, 1999.

[8] Siraj-ul-Islam, A. Ali, and S. Haq, A computational modelingof the behavior of the two-dimensional reaction-diffusionBrusselator system, Applied Mathematical Modelling, vol. 34,no. 12, pp. 38963909, 2010.

Research ArticleA Spectral Relaxation Approach for UnsteadyBoundary-Layer Flow and Heat Transfer of a Nanofluidover a Permeable Stretching/Shrinking Sheet

S. S. Motsa,1 P. Sibanda,1 J. M. Ngnotchouye,1 and G. T. Marewo2

1 School of Mathematical Sciences, University of KwaZulu-Natal, Private Bag X01, Scottsville, Pietermaritzburg 3209, South Africa2University of Swaziland, Private Bag 4, Kwaluseni, Swaziland

Correspondence should be addressed to S. S. Motsa; [email protected]

Received 20 March 2014; Accepted 10 April 2014; Published 28 April 2014

Academic Editor: Raseelo Joel Moitsheki

Copyright 2014 S. S. Motsa et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper introduces two novel numerical algorithms for the efficient solution of coupled systems of nonlinear boundary valueproblems. The methods are benchmarked against existing methods by finding dual solutions of the highly nonlinear system ofequations that model the flow of a viscoelastic liquid of Oldroyd-B type in a channel of infinite extent. The methods discussed hereare the spectral relaxation method and spectral quasi-linearisation method. To verify the accuracy and efficiency of the proposedmethods a comparative evaluation of the performance of the methods against established numerical techniques is given.

1. Introduction

Exact solutions to a wide class of problems in engineeringand science are generally available only for a limited range ofproblems. For this reason the quest for new techniques andthe improvement of existing techniques for finding solutionsof nonlinear equations are an ongoing challenge in engi-neering and science. In addition to the classical numericalmethods, such as those based on finite differences, finiteelements, and finite volume techniques, there is currently awide variety of methods for nonlinear equations, such as,among others, linearisation methods [14] and the trans-form methods of Fokas [57]. Advances in decompositionand variational methods in recent years have added to therepertoire of available techniques for finding solutions ofnonlinear BVPs. In recent years these methods have beenfurther augmented by Hes [8, 9] and Liaos [10, 11] homotopybasedmethods and their various variants such as the spectral-homotopy analysis of Motsa et al. [12, 13] and Liaos [14]generalized boundary element method.

This paper introduces two novel techniques based on acombination of linearisation techniques and spectral meth-ods and that allow for simple and straightforward integration

of systems of ordinary differential equations on finite andinfinite domains.We present an overview of these techniquesand provide a comparative evaluation of the two methodsagainst results in the literature.

A recent study by Bachok et al. [15] investigated the two-component convection in a viscoelastic liquid of Oldroyd-Btype occupying a horizontal channel of infinite extent anddepth . The flow is governed by the coupled system ofequations

+ ( 2

) + +

2= 0, (1)

1

Pr+ ( +

2) +Nb +Nt()

2

= 0, (2)

+ Le( +

2) +

NtNb

= 0, (3)

where (), (), and () represent, respectively, the nondi-mensional stream function, temperature, and nanoparticlevolume fraction, Pr is the Prandtl number, Le is the Lewisnumber, Nb is the Brownian motion number, and and Ntare constant dimensionless parameters.


http://dx.doi.org/10.1155/2014/564942


Equations (1)(3) are solved subject to the boundaryconditions

(0) = , (0) = , (0) = 1, (0) = 1,

(4)

() 0, () 0, () 0, as ,

(5)

where is the stretching/shrinking velocity and is a constantmass flux. In Bachok et al. [15] it was shown that dualsolutions of (1)(3) exist for >

, where

(< 0) is some

critical value of . No solutions exist for < .

In the present study we revisit the solution of the systemof nonlinear equations (1)(5) using the spectral relaxationmethod (Motsa [16]) and the spectral quasi-linearizationmethod (SQLM).The objective of this study is to give a com-parative analysis of the performance of the two techniquesin finding solutions of coupled highly nonlinear problemsin fluid mechanics. We determine, inter alia, the accuracyof each method and demonstrate how dual solutions canbe obtained using the SQLM. In addition, we present asystematic approach of obtaining critical parameter valuesand multiple solutions of the governing equations.

2. The Spectral Relaxation Method

In this section we describe the development of the spectralrelaxation method (SRM) for the solution of the nonlinearsystem (1)(3). The system of equations is decoupled andthe resulting subsystems are solved in a sequential manner.The method appears to be particularly effective for nonlinearsystems of differential equations in which some of theunknown functions have exponentially decaying profiles.Thealgorithm for the method when applied to the system (1)(3)may be summarized as follows.

(1) Reduce the order of the momentum equation fromthree to one by using the transformation

() = () . (6)

(2) Assume that is known from some initial guess andarrange the transformed equations in a particularorder, placing the equations with the least unknownsat the top of the equations list. This gives

+ (

2) + +

2= 0, (0) = , () = 0,

(7)

= , (0) = , (8)

with (2) and (3) placed below (8) in their originalform.

(3) Assign the labels , , , and to the order list of (7),(8), (2), and (3), respectively.

(4) In the equation for (1st equation), the iterationscheme is developed by assuming that only linear

terms in are to be evaluated at the current iterationlevel (denoted by + 1) and all other terms (linearand nonlinear) in, , and are assumed to be knownfrom the previous iteration (denoted by ). In additionnonlinear terms in are also evaluated at the previousiteration.

(5) In developing the iteration scheme for the nextequation , only linear terms in are evaluated atthe current iteration level ( + 1) with all other termsevaluated at the previous level, except which is nowknown from the solution of the first equation.

(6) This process is repeated in the equations for usingthe updated solutions for , . The same procedure iseffected on the equation for , now using the updatedsolutions for , , and .

The strategy used for decoupling the system of equationsis analogous to the Gauss-Seidel relaxation method which isnormally used in solving linear algebraic system of equations.In the context of the algorithmdescribed above, we obtain thefollowing iteration scheme:

+1+

+1+ +1

+

2

+1= 2

,

+1

(0) = , +1

() = 0,

+1= +1

, +1

(0) = ,

1

Pr

+1+ (

+1+

2)

+1+Nb

+1

= Nt(

)2

,

+1+ Le(

+1+

2)

+1=

NtNb

+1,

(9)

subject to the boundary conditions

+1

(0) = 1, +1

(0) = 1,

+1

() = 0, +1

() = 0.

(10)

Given a set of initial approxi

nonlinear fluid flow and heat transfer - hindawi...

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