ARTICLE

MHD mixed convection flow of a viscoelastic fluid over an inclinedsurface with a nonuniform heat source/sinkG.K. Ramesh, Ali J. Chamkha, and B.J. Gireesha

Abstract: The steady mixed convection boundary layer flow over an inclined stretching surface immersed in an incompressibleviscoelastic fluid is considered in this paper. Employing suitable similarity transformations, the governing partial differentialequations are transformed into ordinary differential equations, and the transformed equations are solved numerically usingRunge–Kutta–Fehlberg method. Herein, two different types of heating processes are considered, namely, (i) prescribed surfacetemperature and (ii) prescribed wall heat flux. The effects of the governing parameters on the flow field and heat transfercharacteristics are obtained and discussed. It is found that velocity decreases and temperature increases with an increase in thevalue of angle of inclination.

PACS Nos.: 44.25.+f, 44.27.+g, 47.50.−d.

Résumé : Nous étudions l’écoulement d’une couche limite en convexion mixte sur une surface élastique inclinée plongée dansun fluide viscoélastique incompressible. Utilisant des transformations de similarité appropriées, les équations directrices sonttransformées en équations différentielles ordinaires que nous solutionnons a l’aide de la méthode de Runge–Kutta–Fehlberg.Nous considérons deux types de chauffage, (i) en imposant une température de surface, (ii) en imposant un flux de chaleur dumur. Nous obtenons et analysons les effets des paramètres directeurs sur le champ d’écoulement et le transfert de chaleur. Noustrouvons que l’augmentation de l’angle d’inclinaison cause une diminution de la vitesse et une augmentation de la température.[Traduit par la Rédaction]

1. IntroductionInvestigations on magnetohydrodynamic (MHD) flow and heat

transfer of a non-Newtonian fluid over a stretching sheet haveimportant applications in engineering and industry. For example,in the extrusion of a polymer sheet from a die, the sheet is some-times stretched. The properties of the end product depend consid-erably on the rate of cooling. By drawing such a sheet in aviscoelastic electrically conducting fluid subjected to the action ofa magnetic field, the rate of cooling can be controlled and a finalproduct can be obtained with the desired characteristics. On thebasis of the preceding applications, Sarpakaya [1] was the first tostudy the effect of a magnetic field on flows of non-Newtonianfluids. Rajagopal et al. [2] studied the flow of a viscoelastic fluidover a stretching sheet. Siddapa and Abel [3] considered similarflow analysis without heat transfer in the flow of the non-Newtonian fluid Walter’s liquid. Vajravelu and Rollins [4] ob-tained the analytical solutions for heat transfer in a viscoelasticfluid over a stretching sheet, and showed that there is no bound-ary layer type of solution for small Prandtl number. Andersson [5]investigated the influence of uniform magnetic field on the mo-tion of an electrically conducting fluid past a flat and imperme-able elastic sheet and obtained closed-form solutions of themomentum boundary layer equation; this work does not take intoaccount heat transfer phenomenon. Sam and Nageswara [6] pre-sented a work to analyze momentum and heat transfer phenom-ena in viscoelastic fluid over a stretching surface. Maneschy andMassoudi [7] obtain the numerical solution for flow and heattransfer of a non-Newtonian fluid past a stretching sheet using thequasilinearization method. Vajravelu and Roper [8] analyzed theheat transfer characteristics in a second grade fluid over a stretch-ing sheet with prescribed surface temperature and observed that

the flow problem has a unique solution. Pillai et al. [9] investigatedheat transfer in a viscoelastic boundary layer flowing through aporous medium. Sanjayanand and Khan [10] presented a similaranalysis to investigate the viscoelastic boundary layer flow over anexponentially stretching sheet.

The study of heat source–sink effects on heat transfer is veryimportant because its effects are crucial in controlling the heattransfer. Postelnicu et al. [11] examined the effect of variable vis-cosity on forced convection flow past a horizontal flat plate in aporous medium with internal heat generation, but in the heatgeneration part they considered only a space-dependent heatsource. Eldahab and Aziz [12] analyzed the effect of a nonuniformheat source with suction–blowing, but were confined to the caseof viscous fluids only. Bataller [13] examined the effects of heatsource–sink, radiation, and work done by deformation on flowand heat transfer of a viscoelastic fluid over a stretching sheet.Abel et al. [14–16] investigated the effects of viscous dissipationand nonuniform heat source in a viscoelastic boundary layer flowover a stretching sheet. Further they extended the work and stud-ied the MHD viscoelastic fluid flow over a stretching sheet withvariable thermal conductivity, nonuniform heat source, and radi-ation. Later they analyzed the non-Newtonian viscoelastic bound-ary layer flow of Walter’s liquid B past a stretching sheet, takingaccount of nonuniform heat source. In all these papers, the twotypes of general heating processes were studied, namely, pre-scribed surface temperature (PST) and prescribed wall heat flux(PHF) cases, and both analytical and numerical solutions wereobtained. Tsai et al. [17] studied the unsteady stretching surfacewith nonuniform heat source. Hsiao [18] obtained the numericalsolutions for the flow and heat transfer of a viscoelastic fluid overa stretching sheet with electromagnetic effects and nonuniform

Received 10 April 2013. Accepted 12 July 2013.

G.K. Ramesh and B.J. Gireesha. Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta-577451, Shimoga, Karnataka, India.A.J. Chamkha. Manufacturing Engineering Department, Public Authority for Applied Education and Training, Shuweikh, 70654 Kuwait.

Corresponding author: Ali Chamkha (e-mail: achamkha@yahoo.com).

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Can. J. Phys. 91: 1074–1080 (2013) dx.doi.org/10.1139/cjp-2013-0173 Published at www.nrcresearchpress.com/cjp on 22 July 2013.

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heat source–sink using the combination of finite differencemethod, Newton’s method, and the Gauss elimination method. Inall of these papers, it is shown that for effective cooling of astretching sheet, a nonuniform heat source–sink should be used.

This investigation deals with the flow and heat transfer inducedby a horizontal stretching sheet, but there arise some situationswhere the stretching sheet moves vertically in the cooling liquid.Under such circumstances the fluid flow and the heat transfercharacteristics are determined by two mechanisms, namely, themotion of the stretching sheet and the buoyant force. Effects ofthermal buoyancy on the flow and heat transfer under variousphysical situations have been reported by many investigators:Abel et al. [19], Aziz [20], Hayat et al. [21], and Hsiao [22]. Recently,Ramesh et al. [23] obtained the numerical solution on heat trans-fer in MHD dusty boundary layer flow over an inclined stretchingsheet with nonuniform heat source–sink using the Runge–Kutta–Fehlberg (RKF45) method with the help of MAPLE. Qasim [24]studied the radiation effect on the mixed convection flow of aviscoelastic fluid along an inclined stretching sheet.

In this paper, we present an analysis that may be regarded as anextension of the work of Qasim [24], by considering the effects ofMHD and nonuniform heat source–sink. Here are we consider thetwo types of heating process, namely, (i) PST and (ii) PHF. Thegoverning partial differential equations are first transformed intoa system of ordinary differential equations before being solvednumerically. The results are then compared with those obtainedby Vajravelu and Roper [8], Tsai et al. [17], and Hsiao [22] for someparticular cases of the present study, to support their validity. Thisproblem is relevant to several practical applications in the fieldsof metallurgy, chemical engineering, etc.

2. Mathematical analysisA steady two-dimensional flow of an incompressible, electri-

cally conducting viscoelastic fluid of over an impermeable verticalstretching sheet is considered. The sheet is inclined with an acuteangle � and situated in the fluid of ambient temperature T∞, asshown in Fig. 1. The x-axis moves along the stretching surface inthe direction of motion with the slot as the origin, and the y-axisis measured normally from the sheet to the fluid. The flow isgenerated by the action of two equal and opposite forces along thex-axis and the sheet is stretched in such a way that the velocity atany instant is proportional to the distance from the origin. Amagnetic field of strength B0 is applied in the positive y-direction.The magnetic Reynolds number is assumed to be small so that theinduced magnetic field is neglected in comparison to the appliedmagnetic field. The simplified two-dimensional equations govern-ing the flow of a steady, laminar, incompressible, non-Newtonianfluid are

�u

�x�

�v

�y� 0 (1)

u�u

�x� v

�u

�y� �

�2u

�y2� k0� �

�x�u�2u

�y2� ��u

�y

�2v

�y2� v

�3u

�y3� ��B0

2

u

� g(T � T∞)cos � (2)

where u and v are the velocity components along the x and y axes,respectively. Further, �, the kinematic viscosity; , fluid density;k0, co-efficient of viscoelasticity; g, gravity; , volumetric coeffi-cient of thermal expansion; T, fluid temperature; and T∞ ambientfluid temperature.

We shall assume that the boundary conditions of (1) and (2) are

u � Uw(x) v � 0 at y � 0 u ¡ 0 as y ¡ ∞ (3)

where Uw(x) = cx is the stretching sheet velocity, and c > 0 is knownas stretching rate.

Continuity equation (1) is satisfied by introducing a stream func-tion, �, such that

u ���

�yand v � �

��

�x(4)

The momentum equation can be transformed into correspondingordinary differential equations by the following transformation:

� � �Uw

�x�1/2

y f(�) ��

(x�Uw)1/2(5)

The transformed ordinary differential equations are

f ′2 � ff ′′ � f ′′′ � k1(2f ′f ′′′ � f ′′2 � ff ′′′′) � Mf ′ � Gr cos � (6)

Subject to boundary conditions (3), which become

f � 0 f ′ � 1 at � � 0

f ′¡ 0 f ′′

¡ 0 as � ¡ ∞(7)

where primes denote differentiation with respect to �; k1 = k0c/� isthe viscoelastic parameter; M � �B0

2/c is the magnetic parameter;and Gr = g(Tw − T∞)/c2x is the local Grashof number [25].

The local skin-friction coefficient or the frictional drag isgiven by

Cf ��w

Uw2 /2

� �2Rex�1/2f ′′(0)

where Rex = Uwx/� is the local Reynolds number.

3. Heat transfer analysisThe governing heat transport equations for a viscoelastic fluid

in the presence of nonuniform internal heat source–sink and

Fig. 1. Schematic diagram of the flow.

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neglecting the viscous dissipation and energy due to elastic defor-mation for two-dimensional flow is given by [14]

u�T

�x� v

�T

�y�

k

cp

�2T

�y2�

q′′′

cp

(8)

where k is the thermal conductivity, cp is the specific heat, and q�is the space- and temperature-dependent internal heat source–sink (generation–absorption), which can be expressed as

q ′′′ � �kUw(x)

x���A∗(Tw � T∞)f ′ � B∗(T � T∞)� (9)

where A* and B* are the parameters of the space- and temperature-dependent internal heat source–sink. It is to be noted that A* andB* are positive to internal heat source and negative to internalheat sink.

The solution of (8) is obtained using two different types of heat-ing processes.

3.1. Case 1: PST caseFor this heating process, the boundary conditions in case of

prescribed power law surface temperature are of the form

T � Tw � T∞ � A�x

l�2

at y � 0 T ¡ T∞ as y ¡ ∞ (10)

where Tw and T∞ denote the temperature at the wall and at largedistance from the wall, respectively; A is a positive constant; and

l � ��/c is the characteristic length. Defining the nondimensionaltemperature, (�), as

(�) �T � T∞

Tw � T∞

(11)

where T − T∞ = A(x/l)2 (�).Using (10) and (11) in (9), one can get

′′ � Pr(f ′ � 2f ′ ) � A∗f ′ � B∗ � 0 (12)

Table 1. Comparison results for the wall temperature gradient − =(0) in the case of k1 = A* = M + Gr = 0.

B* PrVajravelu and Roper [8](numerical method)

Tsai et al. [17](Chebyshev finitedifference method)

Hsiao [22] (finitedifference method)

Present result(RKF45) Errors

−1 1 1.7109 1.7109 1.7109 1.7105 0.0004−2 −2 2.4860 2.4859 2.4859 2.4856 0.0003−3 −3 3.0821 3.0821 3.0821 3.0818 0.0003−4 −4 3.5851 3.5851 3.5851 3.5848 0.0003−5 −5 4.0285 4.0285 4.0285 4.0282 0.0003

Fig. 2. Velocity profile for different values of magnetic parameter. Fig. 3. Velocity profile for different values of local Grashof number.

Fig. 4. Velocity profile for different values of viscoelastic parameter.

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where Pr = �cp/k is the Prandtl number. The corresponding bound-ary conditions for (�) as

� 1 at � � 0 ¡ 0 as � ¡ ∞ (13)

The local heat flux can be expressed as

qw � �k��T

�y�

y�0� �k c

�(Tw � T∞) ′(0)

3.2. Case 2: PHF caseThe power law heat flux on the wall surface is considered to be

a quadratic power of x in the form

�k∗�T

�y� qw � D�x

l�2

at y � 0 T ¡ T∞ as y ¡ ∞ (14)

where D is a constant. On the other hand we define a nondimen-sional temperature, g(�), as

g(�) �T � T∞

Tw � T∞

(15)

where

Tw � T∞ �D

k�x

l�2

�

c

Equation (9), on using (14) and (15), can be transformed in terms ofg(�) as

g ′′ � Pr(fg ′ � 2f ′g) � A∗f ′ � B∗g � 0 (16)

Boundary condition (14) becomes

g ′(�) � �1 at � � 0 g(�) ¡ 0 as � ¡ ∞ (17)

Fig. 5. Temperature profile for different values of viscoelastic parameter.

Fig. 6. Temperature profile for different values of Prandtl number.

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The expression for wall temperature in dimensional form is

Tw � T∞ �qw

k �

cg(0)

4. Solutions for some special casesIn the limiting case of cos(�) (i.e., when � = 90°, horizontal)

system of equations (6) and (12) reduces to those of Maneschy andMassoudi [7] in the absence of magnetic field and nonuniformsource–sink, and with those of Bataller [13], when magnetic field,viscous dissipation, and radiation are neglected. In the presenceof magnetic field and nonuniform source–sink, when there is novariable thermal conductivity, viscous dissipation, and radiation,system of equations (6) and (12) reduces to those of Abel andMahesha [15]. Further, when � = 0° (vertical) and the mass transferand the porous medium are absent, the equations are similar tothe ones studied by Hayat et al. [21] and with those of Hsiao [22] inthe absence of unsteadiness.

In the absence of angle of inclination, the MHD boundary layerflow and heat transfer problem degenerates. In this case, the

Fig. 7. Temperature profile for different values of space-dependent heat source–sink.

Fig. 8. Temperature profile for different values of temperature-dependent heat source–sink.

Fig. 9. Variation of f==(0) with k1 for different values of �.

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approximate numerical solutions for the velocity field and tem-perature fields are obtained.

5. Result and discussionThe system of equations (6), (12), or (16) subject to boundary

conditions (7), (13), or (17) has been solved numerically with thehelp of symbolic algebra software Maple, using a procedure usedby Aziz [20]. The numerical approach is based on the RKF45method to solve the coupled nonlinear ordinary differential equa-tions. The RKF45 algorithm in Maple has been well tested for itsaccuracy and robustness, as shown by Aziz [20]. Table 1 shows thecomparison of – =(0) with those of the previous studies [8, 17, 22]for various values of B* and Prandtl number, which shows a fa-vourable agreement thus giving confidence that the numericalresults obtained are accurate.

The variation of velocity profiles for various values of M is de-picted in Fig. 2. This figure shows that the velocity decreases withthe increase of M. This is due to the fact that application of amagnetic field to an electrically conducting fluid produces a drag-like force called Lorentz force. This force causes reduction in thefluid velocity. It is also observed that the velocity increase as theangle of inclination � increases.

Figure 3 depicts the variation in the velocity profiles for differ-ent values of Gr. From this plot it is observed that the effect ofincreasing values of Gr is to increase the velocity distribution.Physically, Gr > 0 indicates heating of the fluid or cooling of theboundary surface, Gr < 0 indicates cooling of the fluid or heatingof the boundary surface, and Gr = 0 corresponds to the absence ofconvection current.

The graph of velocity profile f=(�) verses � for different values ofk1 is plotted in Fig. 4. It can be seen that the velocity increases withincrease of k1. Further, it is observed that the temperature profile, (�) for PST and g(�) for PHF, decreases with increasing k1. We inferfrom these figures that for higher value of k1, the viscoelastic forcecan remove the heat form the fluid, therefore temperature de-creases. Also, temperature increases with the angle of inclination,�, which is clearly seen from Fig. 5.

Figure 6, depicts the effect of Pr on temperature distributionsfor both PST and PHF cases. This figure illustrates that the thermalboundary layer thickness increases with decreasing Pr. Physically,for higher Prandtl number, the fluid has a relatively low thermal

conductivity, which reduces conduction. Here also one can ob-serve that the temperature profile increases with �.

Figure 7, illustrates the temperature profiles (�) for PST andg(�) for PHF versus � for different values of A*. When A* is positive,it can be observed that the thermal boundary layer generates theenergy, and this causes the temperature in the thermal boundarylayer to increase with increasing A*. Whereas, when A* is negativeit leads to a decrease in the thermal boundary layer. Similar pre-dictions are valid for B* also as is shown in Fig. 8.

Figure 9 shows that the variation of skin-friction coefficientf==(0) with the changes of viscoelastic parameter k1, for differentvalues of �. From this graph we observe that the skin-frictioncoefficient f==(0) decreases with increasing �. The skin friction co-efficient f==(0) is negative for all values of k1. Physically, negativevalue of f==(0) means the surface exerts a drag force on the fluid,and positive value means the opposite. This is not surprising be-cause in the present problem, we consider the case of a stretchingsheet, which induces the flow. Variation of local Nusselt number =(0) (PST) and wall temperature gradient g(0) (PHF) with thechanges of viscoelastic parameter k1, different values of � areshown in Fig. 10. It is observed form these figures that, =(0) andg(0) increase with �. Negative value of =(0) means the heat flowsfrom the fluid to the solid surface. This is not surprising becausethe fluid is hotter than the solid surface. Positive values of g(0)mean the opposite.

6. ConclusionWe have numerically studied the similarity solutions for the

steady viscoelastic fluid over an inclined surface with the effect ofnonuniform heat source–sink. The governing partial differentialequations are converted into ordinary differential equations bysimilarity transformation, before being solved numerically usingthe RKF45 method. Results for the skin friction coefficient, localNusselt number, wall temperature gradient, velocity profiles, aswell as temperature profiles are presented for different values ofthe governing parameters. Effects of the magnetic parameter, an-gle of inclination, nonuniform heat source–sink parameter, andPrandtl number on the flow and heat transfer characteristics arethoroughly examined. It is found that the effects of � on the skinfriction coefficient, local Nusselt number, and wall temperature

Fig. 10. Variation of =(0) and g(0) with k1 for different values of �.

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function reveal that f==(0) decreases and =(0) and g(0) increase as �increases.

References1. T. Sarpakaya. AIChE J. 7, 324 (1961). doi:10.1002/aic.690070231.2. K.R. Rajagopal, T.Y. Na, and A.S. Gupta. Rheol. Acta, 23, 213 (1984). doi:10.

1007/BF01332078.3. B. Siddappa and A. Subhas. ZAMP, 36, 890 (1985). doi:10.1007/BF00944900.4. K. Vajravelu and D. Rollins. J. Math. Anal. Appl. 158(1), 241 (1991). doi:10.1006/

jmaa.1998.5940.5. H.I. Andersson. Acta Mech. 95, 227 (1992). doi:10.1007/BF01170814.6. P. Sam Lawrence and B. Nageswara Rao. Acta Mech. 93, 53 (1992). doi:10.1007/

BF01182572.7. C.E. Maneschy and M. Massoudi. Int. J. Math. Math. Sci. 18(4), 765 (1995).

doi:10.1155/S0161171295000986.8. K. Vajravelu and T. Roper. Int. J. Non-Linear Mech. 34(6), 1031 (1999). doi:10.

1016/S0020-7462(98)00073-0.9. K.M.C. Pillai, K.S. Sai, N.S. Swamy, H.R. Nataraja, S.B. Tiwari, and B.N. Rao.

Comput. Mech. 34, 27 (2004). doi:10.1007/s00466-004-0550-8.10. S. Emmanuel and S.K. Khan. Int. J. Therm. Sci. 45, 819 (2006). doi:10.1016/j.

ijthermalsci.2005.11.002.11. A. Postelnicu, T. Grosan, and I. Pop. Mech. Res. Com. 28, 331 (2001). doi:10.

1016/S0093-6413(01)00181-1.12. E.M. Abo-Eladahab and M.A. El Aziz. Int. J. Therm. Sci. 43, 709 (2004). doi:10.

1016/j.ijthermalsci.2004.01.005.13. R.C. Bataller. Comput. Math. Appl. 53, 305 (2007). doi:10.1016/j.camwa.2006.

02.041.14. M.S. Abel, P.G. Siddheshwar, and M.M. Nandeppanavar. Int. J. Heat Mass

Transfer, 50, 960 (2007). doi:10.1016/j.ijheatmasstransfer.2006.08.010.15. M.S. Abel and N. Mahesha. Appl. Math. Model. 32(10), 1965 (2008). doi:10.1016/

j.apm.2007.06.038.16. M.S. Abel and M.M Nandeppanavar. Commun. Nonlinear Sci. Numer. Simu-

lat. 14, 2120 (2009). doi:10.1016/j.cnsns.2008.06.004.17. R. Tsai, K.H. Huang, and J.S Haung. Int. Comm. Heat Mass Trans. 35, 1340

(2008). doi:10.1016/j.icheatmasstransfer.2008.07.001.18. K.-L. Hsiao. Viscoelastic Fluid over a Stretching Sheet with Electromagnetic

Effects and Nonuniform Heat Source/Sink. Mathematical Problems in Engi-neering, Vol. 2010, Article ID 740943, 14 pages. doi: 10.1155/2010/740943.

19. M.S. Abel, S.K. Khan, and K.V. Prasad. Int. J. Numer. Meth. Heat Fluid Flow,11(8), 779 (2001). doi:10.1108/09615530110409420.

20. A. Aziz. Commun Nonlinear Sci. Numer. Simulat. 14, 1064 (2009). doi:10.1016/j.cnsns.2008.05.003.

21. T. Hayat, M. Mustafa, and I. Pop. Commun Nonlinear Sci. Numer. Simulat. 15,1183 (2010). doi:10.1016/j.cnsns.2009.05.062.

22. K.-L. Hsiao. Int. J. Phys. Sci. 6, 5080 (2011). doi:10.5897/IJPS10.610.23. G.K. Ramesh, B.J. Gireesha, and C.S. Bagewadi. Heat Transfer in MHD Dusty

Boundary Layer Flow over an Inclined Stretching Sheet with Non-Uniform

Heat Source/Sink. Advances in Mathematical Physics, Vol. 2012, Article ID657805, 13 pages. doi: 10.1155/2012/657805.

24. M. Qasim, T. Hayat, and S. Obaidat. Z. Naturforsch. 67a, 195 (2012). doi:10.5560/ZNA.2012-0006.

25. W.T. Kierkus. Int. J. Heat Mass Transfer, 11, 241 (1968). doi:10.1016/0017-9310(68)90153-1.

List of symbols

A, D prescribed constantsA* space-dependent internal heat source–sinkB* temperature-dependent internal heat source–sinkB0 magnetic field strength

c stretching rateCf skin frictioncp specific heatg gravity

Gr local Grashof numberk0 elastic parameterk1 viscoelastic parameterk thermal conductivityl characteristic length

M magnetic parameter�Pr Prandtl numberq== rate of internal heat source–sinkqw heat flux

T temperature of the fluidTw temperature at the wallT∞ temperature at large distance from the wall

u, v velocity components along x and y directionsUw stretching sheet velocity

x coordinate along the stretching sheety distance normal to the stretching sheet� angle of inclination volumetric coefficient of thermal expansion� dimensionless space variable dimensionless temperature� kinematic viscosity fluid density� electrical conductivity

�w wall shearing stress

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