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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume , Art icle ID , pageshttp://dx.doi.org/ . / /

Research Article A New Numerical Approach of MHD Flow withHeat and Mass Transfer for the UCM Fluid over a Stretching Surface in the Presence of Thermal Radiation

S. Shateyi 1 and G. T. Marewo 2

Department o Mathematics, University o Venda, Private Bag X , Tohoyandou , South A ricaDepartment o Mathematics, University o Swaziland, Private Bag , Kwaluseni, Swaziland

Correspondence should be addressed to S. Shateyi; stan ord.shateyi@univen.ac.za

Received March ; Accepted August

Academic Editor: irivanhu Chinyoka

Copyright S. Shateyi and G. . Marewo. Tis 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 properly cited.

Tis paper numerically investigates the magnetohydrodynamic boundary layer ow with heat and mass trans er o anincompressible upper-convected Maxwell uid over a stretching sheet in the presence o viscous dissipation and thermal radiationas well as chemical reaction. Te governing partial differential equations are trans ormed into a system o ordinary differentialequations by using suitable similarity trans ormations. Te resultant highly nonlinear ordinary differential equations are then

solved usingspectral relaxation method. Te results areobtained orvelocity, temperature, concentration, skin riction, and Nusseltnumber. Te effects o various material parameters on the ow with heat and mass trans er and the dimensionless variables areillustrated graphically and brie y discussed.

1. Introduction

In the past ew decades, the studies o boundary layer owso Newtonian and non-Newtonian uids o stretching sur-

aceshavereceivedgreatattentionbyvirtueo theirnumerousapplications in the eldso metallurgy, chemical engineering,

and biological systems. Tese applications include geother-mal reservoirs, wire and ber coating, ood stuff process-ing, reactor uidization, transpiration cooling, enhanced oilrecovery, packed bed catalytic reactors,andcooling o nuclearreactors. Te prime aim in extrusion is to keep the sur acequality o the extricate. Coating processes demand a smoothglossy sur ace to meet the requirements or the best appear-ance and optimum properties. Sakiadis [ , ] did pioneeringwork on boundary layer ow on a continuously movingsur ace. Afer that many investigators discussed variousaspects o the stretching ow problem (see, e.g., Chiam [ ],Crane [ ], Liao and Pop [ ], Khan and Sanjayanand [ ], Abeland Mahesha [ ], and Fang et al. [ ], among others).

A number o industrial uids such as molten plastics,arti cial bers, blood, polymetric liquids, and ood stuff exhibit non-Newtonian uid behaviour. In many industrialprocesses, cooling continuous strips or laments is done by drawing them through a quiescent uid. Itmust be noted thatduring these processes, the strips are sometimes stretched or

shrunk. Tere ore the properties o the nal product dependto a great extent on the rate o cooling. Te rate o coolingcan be controlled and the desired characteristics o the nalproduct can be obtained by drawing the strips in electrically conducting uids subjected to uni orm magnetic elds.

In recent years, MHD ows o viscoelastic uidsabove stretching sheets have also been studied by variousresearchers (Liu [ ], Cortell [ ], among others). Tis is thesimplest subclass o viscoelastic uid known as the secondgrade uid. However, a non-Newtonian second grade uiddoes not give meaning ul industrial results or highly elastic

uids such as polymer melts, which occur at high Deborahnumber (Hayat et al. [ ]). For theoretical results to become

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o any industrial use, more realistic viscoelastic uid modelssuch as upper-convected Maxwell model should be used inthe analysis.

Alizadeh-Pahlavan et al. [ ] investigated using a two-auxiliary-parameter homotopyanalysis method or the prob-lem o laminar MHD ow o an upper-convected Maxwell

uid above a porous isothermal stretching sheet. Aliakbaret al. [ ] analyzed the in uence o Maxwell uids abovestretching sheets. Abel et al. [ ] per ormed an analysis toinvestigate the in uence o MHD and thermal radiationon the two-dimensional steady ow o an incompressible,upper-convected Maxwell uid. Motsa et al. [ ] investigatedthe MHD boundary layer ow o an incompressible upper-convected Maxwell uid over a porous stretching sur ace.Hayat et al. [ ] investigated MHD ow o a more realistic viscoelastic uid model above a porous stretching sheet.Sadeghy et al. [ ] also studied MHD ows o upper-convected Maxwell model uids above porous stretchingsheet using homotopy analysis method. Rafari and Yildrim[ ] approximated an analytical solution o the magneto-hydrodynamic boundary layer ow o an upper-convectedMaxwell uid over a porous stretching sheet. More recently,Alinejad and Samarbakhsh [ ] numerically investigated the

ow and heat trans er characteristics o the incompressible viscous ow over a nonlinear stretching sheet with viscousdissipation.

Hayat et al. [ ] proposed the modi ed decompositionmethod and the Pade approximants or the MHD ow overa nonlinear sheet. In recent years, scientists and engineershave endeavored to develop more accurate and ast con- verging numerical and/or analytical techniques. Methodssuch as homotopy and their hybrid techniques have beenextensively applied to solve nonlinear equations (Liao [ ],He [ ], Domairry and Nadim [ ], Shateyi and Motsa [ ],Sheikholeslami et al. [ ], and Ganji [ ], among others).Recently, successive linearization method has been reportedand success ully utilized in solving boundary value problems(see, e.g., Motsa and Shateyi [ ], Motsa et al. [ ], andShateyi and Motsa [ ]).

Teaim o thisstudyis to investigate the effectso thermalradiation and viscous dissipation on steady MHD ow withheat and mass trans er o an upper-convected Maxwell uidpast a stretching sheet in the presence o a chemical reaction.Secondly, we aimto usea recentlydeveloped iterative methodknown as spectral relaxation method (SRM), and details o this method can be ound in [ ].

2. Mathematical Formulation

We consider the steady and incompressible MHD boundary layer ow with heat and mass trans er o an electrically conducting uid obeyingUCM model over a stretching sheetin the presence o thermal radiation. Te ow is generatingby the stretching o the sheet by applying two equal andopposite orces along the -axis, keeping the origin xed andconsidering the ow to be con ned to the region > 0.We assume that the continuous stretching sheet has a linear velocity, = , with as the stretching rate and being thedistance rom the slit. We impose a uni orm magnetic eld o

strength 0 along the -axis, and the induced magnetic eldis negligible. Tis assumption is valid on a laboratory scaleunder the assumption o small magnetic Reynolds number,and the external electric eld is zero. We also assume that theboundary layer approximations are applicable to all momen-tum, energy, and mass equations. Although this theory is

incomplete or viscoelastic uids, but it is more plausibleor Maxwell uids as compared to other viscoelastic uidmodels (Renardy [ ]). Following Sadeghy et al. [ ] amongothers, in a two-dimensional ow, the equation o continuity,the equation o motion, and the diffusion equations can bewritten as

+ V = 0, ( ) + V + 1 2 2 2 + V 2 2 2 + 2V 2

= ]

2

2 20

,( )

+ V = 2 2 + 2 1 , ( ) + V = 2 2 , ( )

where and V are velocity components in the - and-directions, respectively, is the uid density, is theelectrical conductivity, 0 is the uni orm magnetic eld, is the temperature, is the speci c heat at constantpressure, is the thermal conductivity, is the concentrationo the species diffusion, is the diffusion coefficient o the diffusion species, ] is the kinematic viscosity, is therelaxation time, and denotes the reaction rate constanto the th-order homogeneous and irreversible reaction. Teappropriate boundary conditions are

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

( ,0) = + 2,

( ,0) = + 2

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

( )

By using the Rosseland diffusion approximation (Hossain etal. , among other researchers), the radiative heat ux, ,is given by

= 4

3

4 , ( )

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Mathematical Problems in Engineering

where and are the Ste an-Boltzmann constant andthe Rosseland mean absorption coefficient, respectively. Weassume that the temperature differences within the ow aresufficiently small such that

4

43

34

. ( )

Using ( ) and ( ) in the last term o ( ) we obtain

= 16 332

2 . ( ). . Similarity rans ormation. o make the problem amena-

ble we introduce the ollowing nondimensional quantities:

= ] , = ] , = , = ,

( )

where ( ) is the dimensionless stream unction and isthe similarity variable, is the dimensionless temperature,and is the mass concentration. Te continuity equationis automatically satis ed through the variables. Ten intro-ducing the relations ( ) into ( )-( ), we obtain the ollowingnonlinear system o ordinary differential equations:

+ 2 2 + 2 2 = 0, ( )1 + 43 + Pr 2 + PrEc

2 = 0, ( )+ Sc 2 = 0. ( )

Here 2 = 20/ and = are magnetic and elasticparameters, respectively, Pr = / is the Prandtl number,Ec = 2/ is the Eckert number, = 4 3 / isthe thermal radiation parameter,

= 1

0

2

/2 is the

chemical reaction parameter, and Sc = /] is the Schmidt

number. Te boundary conditions are

(0) = 0, (0) = 1, (0) = 1, (0) = 1, 0, 0, 0, as . ( )

3. Method of Solution

Te Successive Relaxation Method (SRM) begins by letting

= ( )

so that = and = . Consequently, ( ) through( ) become+ 2 2 + 2 2 = 0, ( )

1 + 43 +

Pr

2 + PrEc 2

= 0, ( )

+ Sc 2 = 0. ( )Proceeding in a manner similar to the Gauss-Seidel method,( ) and ( ) through ( ) are replaced by the ollowingrecursive ormulae:

+1 = , +1(0) = 0, ( )+1 + +1 +1 2 2 +1

+ 2+1 2+1 = 0,+1

(0) = 1, +1

() = 0,

( )

1Pr 1 + 43 +1 + +1 +1 2+1 +1 + Ec 2+1 = 0,

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

+ Sc +1 2+1 +1 = 0,+1(0) = 1, +1() = 0. ( )

Te Chebyshev spectral collocation method will be used tosolve ( ) through ( ). First, we replace the semi-in niteinterval [0,) with the closed interval [0, ], where issufficiently large. It is convenient to use thechange o variable

() = 1 + 2 ( )to map the interval [0, ] on the -axis onto the interval[1,1]on the -axis.On[1,1]we orm a computational grid1 = < 1 < < 0 = 1, where

= cos , = 0,1,... , ( )are the Chebyshev collocation points. Te derivative

()at

each collocation point is evaluated using ormula

= =0 h = h,

( )

where is the Chebyshev differentiation matrix. Successiveapplication o Chebyshev differentiation reveals the moregeneral ormula

h = h. ( )

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Mathematical Problems in Engineering

Chebyshev differentiation trans orms ( ) through ( ) todiscrete orm:

1f +1 = 1, +1 = 0,2p +1 = 2, +1 = 1, +1 0 = 0,3 +1 = 3, +1 = 1, +1 0 = 0,4 +1 = 4, +1 = 1, +1 0 = 0,

( )

where

1 = , 1 = p ,2 = 2 + diag f +1 D 2 ,

2 = p2 2f +1p p f 2+1p ,3 = 1 + 4

3

2 + Pr diag f +1 2diag p +1 ,3 = PrEcp 2+1 ,

4 = 2 + Sc diag f +1 2diag p +1 ,4 = Sc .

( )

Initial approximations needed to drive this iterative schememust satis y boundary conditions ( ). Suitable choices are

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

4. Results and Discussion

In this section we give the SRM results or the main parame-ters that have signi cant effects on the uid ow velocity andtemperature. We remark that all the SRM results presented inthis work were obtained using = 50collocation points, andalso convergence was achieved afer as ew as ve iterations.Also the in nity value ( ) was taken to be . It is alsoimportant to note that themagnetic eld is taken quite strongby assigning large values o to ensure the occurrence o steady ownear thesheet. Unless otherwise stated, thede ault values or the parameters are taken as

= 1,

= 0.1,

= 1,

Pr = 0.71, Ec = 0.1, Sc = 0.2, = 0.2, and = 2. Inorderto validate thenumericalmethod, it wascompared withthe MA LAB routine V 4 which is an adaptive Lobattoquadrature iterative scheme. able presents a comparisonbetween SRM approximate results and the V 4 results orselected de ault values o the magnetic parameter . It canbe seen rom this table that there is an excellent agreementbetween the results rom the two methods. Analyzing able shows that an increase in the magnetic eld strength leadsto an increase in the skin- riction but a decrease to theNusselt number. Tis is physically expected as application o a transverse magnetic eld produces a drag orce which thenreduces the ow velocity but generates heat within the uid.

: Comparison o the SRM results o (0), (0) withthose obtained by V 4 or different values o the magneticparameter.(0) (0) V 4 SRM V 4 SRM

. . . . .. . . . .. . . . .

: Comparison o the SRM results o (0), (0) withthose obtained by V 4or different valueso theelastic parameter.(0) (0) V 4 SRM V 4 SRM

. . . . .. . . . .. . . . .

able gives a comparison o the SRM results to thoseobtained by the V 4 or different values o the elasticity parameter . We again observe that the results rom the twomethods agree very well giving con dence to the currentproposed method. Increasing the uid elasticity parameterleads to the increase in the skin riction coefficient but adecrease in the heat trans er coefficient. Figures and show the in uence o the elasticity parameter on the -velocity and V -velocity pro les, respectively. From both these gures,we observe that an increase in the elasticity parameter resultsin the decrease in both velocity components and a decreasein the thickness o the momentum boundary layer.

Figures and depict the effects o the magnetic eldparameter onthe -velocityand V -velocitypro les, respec-tively. From these gures we clearly observe that increas-ing magnetism signi cantly reduces the thickness o theboundary layer, thereby reducing the velocity components.Physically, the application o the transverse magnetic eldpresents a damping effect on the ow velocity by producing adrag orce that opposes the uid motion.

Figure depicts the effect o increasing the elasticity parameter on the temperature distribution. A decrease inthe streamwisevelocity component, , can resultin a decreasein the amount o heat trans erred on the sur ace sheet.Similarly, a decrease in the transverse velocity component, V ,means that the amount o resh uid which is extended romthe lower-temperature region outside the boundary layerand directed towards the sheet is reduced, thereby reducingthe rate o heat trans er. Tese two effects on the velocity components in the same direction rein orce each other. Tus,an increase in the elastic number increases the temperaturedistribution in the uid as depicted in Figure .

Figure represents the dimensionless temperature ordifferentvalues o the magnetic eld parameter . From this

gure we clearly see that the temperature pro les increasewith the increase o the magnetic eld parameter. Tus theapplied magnetic eld tends to heat the uid andthus reduces

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Mathematical Problems in Engineering

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f ( )

= 0, 0.5, 1, 2

F : Graph o the SRM solutions o the -velocity or different values o .

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

f ( )

= 0, 0.5, 1, 2

F : Effect o the on the V -velocity pro le.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f ( )

M = 0 , 0.5, 1

F : In uence o the magnetic parameter on the -velocity pro le.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

f ( )

M = 1

M = 0.5

M = 0

F : Te variation o the V -velocity component or different values o .

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

( )

= 0, 0.5, 1, 2

F : Graph o the SRM solutions o the temperature distribu-tion or different values o .

the heat trans er rom the wall which in turn enlarges thethermal boundary layer thickness.

In Figure we display the effect o the Eckert numberon the temperature pro les. An increase in the values o theEckertnumber is seen to increase the temperature o the uidat any point above the sheet. Increasing the Eckert numberallows energy tobe stored in the uid regionasa consequenceo dissipation due to viscosity and elastic de ormation.

Te in uence o thermal radiation on the temperaturepro les is shown in Figure . It can be seen that the thermalboundary layer thickness increases as increases. Tisinduces the decrease in the absolute value o the temperaturegradient at the sur ace. Tus, the heat trans er rate at thesur ace decreases with increasing , thereby causing thetemperature pro les to increase.

Figure depicts the effect o increasing the Prandtlnumber on the uid temperature distribution. An increase in

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Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

( )

M = 0 , 0.5, 1

F : Variation o the temperature distribution or different values o .

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

( )

Ec = 0, 1, 2

F : Graph o the SRM solutions o the temperaturepro le ordifferent values o Ec.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

( )

R = 0, 0.5, 1

F : In uence o thermal radiation o the uid temperature.

0 2 4 6 8 10 12 14 16 18 20 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

( )

Pr = 0.71, 1, 2

F : Te effect o the Prandtl number on the temperaturedistribution.

0 5 10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 0, 0.5, 1

( )

F : Graph o the SRM solutions o the concentration pro leor different values o .

the Prandtl number as expected is seen to reduce the uidtemperature above the sheet. Tis is because as the Prandtlnumber increases, the thermal boundary layer becomesthinner. Tus the rate o thermal diffusion drops, resultingin the uid temperature dropping as well.

Figure shows how the elastic parameter affects theconcentrationpro les.As the elasticityparameter reduces theow velocity, itmeans thatless uid is taken awayat any given

point resulting in the concentration pro les increasing. Tesame phenomenon happens when the values o the magneticparameter increase as can be clearly seen in Figure . Tereduction o ow velocity as the result o increasing thestrength o the magnetic eld causes the uid concentrationto increase as less uid is taken downstream at any givenpoint.

Figure presents the pro les o the concentration orselected de ault values o the chemical reaction parameter

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Mathematical Problems in Engineering

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

M = 0 , 0.5, 1

( )

F : Graph o how the magnetic parameter affects theconcentration distribution.

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

= 0, 0.1, 0.2 ( )

F : In uence o the chemical reaction on the concentrationdistribution.

0. It is observed rom this gure that an increase in the values o chemical reaction parameter leads to a decrease inthe concentration pro les. Te concentration boundary layerbecomes thin as the reaction parameter increases.

Figure shows graphically the effect o increasing thereaction-order parameter . Te effect o is seen as toincrease the uid concentration.5. Conclusion

Te present work analyzed the MHD ow with heat andmass trans er within a boundary layer o an upper-convectedMaxwell uid above a stretching sheet in the presence o viscous dissipation, thermal radiation, and chemical reaction.Numerical results are presented in tabular/graphical ormto elucidate the details o ow with heat and mass trans ercharacteristics and their dependence on the various physical

0 2 4 6 8 10 12 14 16 18 20 220

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n = 1.2, 2, 3

( )

F : Graph o the SRM solutions o the concentration ordifferent orders o the homogeneous reaction.

parameters. Te accuracy o the SRM is validated against theMA LAB in-built V 4 routine or solving boundary valueproblems.

( ) We observe that the ow velocity is decreased whenthe magnetic parameter increases. Also an increasein the elasticity parameter results in velocity decre-ments. However, both the temperature and concen-tration pro les are enhanced by increasing the valueso the magnetic parameter as well as the elasticity parameter.

( ) Te dimensionless temperature increases withincrease in the thermal radiation but decreases withincreasing Prandtl number.( ) Te effect o the chemical reaction is to decrease the

uid concentration while the concentration pro lesare increased as the order o reaction is increased.

( ) It is shown that the skin- riction increases with anincrease o the magnetic parameter and elasticity parameter, but the Nusselt number and Sherwoodnumber (not shown or brie ) are ound to bedecreased as these parameters are increased.

( ) Finally the Nusselt number increases with increasing

values o the Prandtl number but decreases as thermalradiation increases.

Acknowledgments

Te authors wish to acknowledge nancial support rom theUniversity o Venda and NRF.

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