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Entropy Generation in Viscoelastic Fluid Over a Stretching Surface

Written By

Saouli Salah and Aïboud Soraya

Submitted: April 7th, 2011 Published: October 10th, 2011

DOI: 10.5772/21859

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1. Introduction

Due to the increasing importance in processing industries and elsewhere when materials whose flow behavior cannot be characterized by Newtonian relationships, a new stage in the evolution of fluid dynamics theory is in progress. An intensive effort, both theoretical and experimental, has been devoted to problems of non-Newtonian fluids. The study of MHD flow of viscoelastic fluids over a continuously moving surface has wide range of applications in technological and manufacturing processes in industries. This concerns the production of synthetic sheets, aerodynamic extrusion of plastic sheets, cooling of metallic plates, etc.

(Crane, 1970) considered the laminar boundary layer flow of a Newtonian fluid caused by a flat elastic sheet whose velocity varies linearly with the distance from the fixed point of the sheet. (Chang, 1989; Rajagopal et al., 1984) presented an analysis on flow of viscoelastic fluid over stretching sheet. Heat transfer cases of these studies have been considered by (Dandapat & Gupta, 1989, Vajravelu & Rollins, 1991), while flow of viscoelastic fluid over a stretching surface under the influence of uniform magnetic field has been investigated by (Andersson, 1992).

Thereafter, a series of studies on heat transfer effects on viscoelastic fluid have been made by many authors under different physical situations including (Abel et al., 2002, Bhattacharya et al., 1998, Datti et al., 2004, Idrees & Abel, 1996, Lawrence & Rao, 1992, Prasad et al., 2000, 2002). (Khan & Sanjayanand, 2005) have derived similarity solution of viscoelastic boundary layer flow and heat transfer over an exponential stretching surface.

(Cortell, 2006) have studied flow and heat transfer of a viscoelastic fluid over stretching surface considering both constant sheet temperature and prescribed sheet temperature. (Abel et al., 2007) carried out a study of viscoelastic boundary layer flow and heat transfer over a stretching surface in the presence of non-uniform heat source and viscous dissipation considering prescribed surface temperature and prescribed surface heat flux.

(Khan, 2006) studied the case of the boundary layer problem on heat transfer in a viscoelastic boundary layer fluid flow over a non-isothermal porous sheet, taking into account the effect a continuous suction/blowing of the fluid, through the porous boundary. The effects of a transverse magnetic field and electric field on momentum and heat transfer characteristics in viscoelastic fluid over a stretching sheet taking into account viscous dissipation and ohmic dissipation is presented by (Abel et al., 2008). (Hsiao, 2007) studied the conjugate heat transfer of mixed convection in the presence of radiative and viscous dissipation in viscoelastic fluid past a stretching sheet. The case of unsteady magnetohydrodynamic was carried out by (Abbas et al., 2008). Using Kummer’s funcions, (Singh, 2008) carried out the study of heat source and radiation effects on magnetohydrodynamics flow of a viscoelastic fluid past a stretching sheet with prescribed power law surface heat flux. The effects of non-uniform heat source, viscous dissipation and thermal radiation on the flow and heat transfer in a viscoelastic fluid over a stretching surface was considered in (Prasad et al., 2010). The case of the heat transfer in magnetohydrodynamics flow of viscoelastic fluids over stretching sheet in the case of variable thermal conductivity and in the presence of non-uniform heat source and radiation is reported in (Abel & Mahesha, 2008). Using the homotopy analysis, (Hayat et al., 2008) looked at the hydrodynamic of three dimensional flow of viscoelastic fluid over a stretching surface. The investigation of biomagnetic flow of a non-Newtonian viscoelastic fluid over a stretching sheet under the influence of an applied magnetic field is done by (Misra & Shit, 2009). (Subhas et al., 2009) analysed the momentum and heat transfer characteristics in a hydromagnetic flow of viscoelastic liquid over a stretching sheet with non-uniform heat source. (Nandeppanavar et al., 2010) analysed the flow and heat transfer characteristics in a viscoelastic fluid flow in porous medium over a stretching surface with surface prescribed temperature and surface prescribed heat flux and including the effects of viscous dissipation. (Chen, 2010) studied the magneto-hydrodynamic flow and heat transfer characteristics viscoelastic fluid past a stretching surface, taking into account the effects of Joule and viscous dissipation, internal heat generation/absorption, work done due to deformation and thermal radiation. (Nandeppanavar et al., 2011) considered the heat transfer in viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink in the presence of a magnetic field

Although the forgoing research works have covered a wide range of problems involving the flow and heat transfer of viscoelastic fluid over stretching surface they have been restricted, from thermodynamic point of view, to only the first law analysis. The contemporary trend in the field of heat transfer and thermal design is the second law of thermodynamics analysis and its related concept of entropy generation minimization.

Entropy generation is closely associated with thermodynamic irreversibility, which is encountered in all heat transfer processes. Different sources are responsible for generation of entropy such as heat transfer and viscous dissipation (Bejan, 1979, 1982). The analysis of entropy generation rate in a circular duct with imposed heat flux at the wall and its extension to determine the optimum Reynolds number as function of the Prandtl number and the duty parameter were presented by (Bejan, 1979, 1996). (Sahin, 1998) introduced the second law analysis to a viscous fluid in circular duct with isothermal boundary conditions.

In another paper, (Sahin, 1999) presented the effect of variable viscosity on entropy generation rate for heated circular duct. A comparative study of entropy generation rate inside duct of different shapes and the determination of the optimum duct shape subjected to isothermal boundary condition were done by (Sahin, 1998). (Narusawa, 1998) gave an analytical and numerical analysis of the second law for flow and heat transfer inside a rectangular duct. In a more recent paper, (Mahmud & Fraser, 2002a, 2002b, 2003) applied the second law analysis to fundamental convective heat transfer problems and to non-Newtonian fluid flow through channel made of two parallel plates. The study of entropy generation in a falling liquid film along an inclined heated plate was carried out by (Saouli & Aïboud-Saouli, 2004). As far as the effect of a magnetic field on the entropy generation is concerned, (Mahmud et al., 2003) studied the case of mixed convection in a channel. The effects of magnetic field and viscous dissipation on entropy generation in a falling film and channel were studied by (Aïboud-saouli et al., 2006, 2007). The application of the second law analysis of thermodynamics to viscoelastic magnetohydrodynamic flow over a stretching surface was carried out by (Aïboud & Saouli 2010a, 2010b).

The objective of this paper is to study the entropy generation in viscoelastic fluid over a stretching sheet with prescribed surface temperature in the presence of uniform transverse magnetic field.

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2. Formulation of the problem

In two-dimensional Cartesian coordinate system (x,y)we consider magneto-convection, steady, laminar, electrically conduction, boundary layer flow of a viscoelastic fluid caused by a stretching surface in the presence of a uniform transverse magnetic field and a heat source. The x-axis is taken in the direction of the main flow along the plate and the y-axis is normal to the plate with velocity components u,v in these directions.

Under the usual boundary layer approximations, the flow is governed by the following equations:

ux+vy=0E1
uux+vuy=ν2uy2k0(u3uxy2+v3uy3uy2uxy+uy2uy2)σB02ρuE2

The constant k0=α1ρ is the viscoelastic parameter.

The boundary conditions are given by

y=0,u=uP=λx,v=0E3
y=,u=0,uy=0E4

The heat transfer governing boundary layer equation with temperature-dependent heat generation (absorption) is

ρCP(uTx+vTy)=k2Ty2+Q(TT)E5

The relevant boundary conditions are

y=0,T=TP=A(xl)2+TE6
y=,T=TE7
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3. Analytical solution

The equation of continuity is satisfied if we choose a dimensionless stream function Ψ(x,y) such that

u=Ψy,v=ΨxE8

Introducing the similarity transformations

η=yλν,Ψ(x,y)=xνλf(η)E9

Momentum equation (2) becomes

f2(η)f(η)f(η)=f(η)k0λν[2f(η)f(η)f(η)fIV(η)f2(η)]Mnf(η)E10

where

Mn=σB02ρλE11

Now let us seek a solution of Eq. (8) in the form

f(η)=eαη,α0E12

which is satisfied by the following boundary conditions:

η=0,f(0)=0,f(0)=1E13
η=,f()=0,f()=0E14

On substituting (9) into (8) and using boundary conditions (12) and (13) the velocity components take the form

u=λxf(η)E15
v=νλf(η)E16

Where k1=k0λν is the viscoelastic parameter, and

α=1+Mn1k1E17

Defining the dimensionless temperature

Θ(η)=TTTPTE18

and using (11), (14), (15), Eq. (17) and the boundary conditions (6) and (7) can be written as

Θ(η)+Prα(1eαη)Θ(η)(2Preαηβ)Θ(η)=0E19
η=0,Θ(0)=1E20
η=,Θ()=0E21

Where and β=Qνλk are respectively the Prandtl number and the heat/sink parameter.

Introducing the variable

ξ=Prα2eαηE22

And inserting (21) in (18) we obtain

ξΘ(ξ)+Prα(1Prα2+ξ)Θ(ξ)(2βα2ξ)Θ(ξ)=0E23

And (19) and (20) transform to

ξ=Prα2,Θ(Prα2)=1E24
ξ=0,Θ(0)=0E25

The solution of Eq. (22) satisfying (23) and (24) is given by

Θ(ξ)=(α2Prξ)a+bM(a+b2,2b+1,ξ)M(a+b2,2b+1,Prα2)E26

The solution of (25) in terms of η is written as

Θ(ξ)=eα(a+b)ηM(a+b2,2b+1,Prα2eαη)M(a+b2,2b+1,Prα2)E27

wherea=Pr2α2, b=Pr24α2β2α2and M(a+b2,2b+1,Prα2eαη) is the Kummer’s function.

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4. Second law analysis

According to (Woods, 1975), the local volumetric rate of entropy generation in the presence of a magnetic field is given by

SG=kT2[(Tx)2+(Ty)2]+μT(uy)2+σB02Tu2E28

Eq. (27) clearly shows contributions of three sources of entropy generation. The first term on the right-hand side of Eq. (27) is the entropy generation due to heat transfer across a finite temperature difference; the second term is the local entropy generation due to viscous dissipation, whereas the third term is the local entropy generation due to the effect of the magnetic field. It is appropriate to define dimensionless number for entropy generation rateNS. This number is defined by dividing the local volumetric entropy generation rate SGto a characteristic entropy generation rateSG0. For prescribed boundary condition, the characteristic entropy generation rate is

SG0=k(ΔT)2l2T2E29

therefore, the entropy generation number is

NS=SGSG0E30

using Eq. (11), (26) and (27), the entropy generation number is given by

NS=4X2Θ2(η)+RelΘ2(η)+RelBrΩf2(η)+BrHa2Ωf2(η)E31

where Rel and Br are respectively the Reynolds number and the Brinkman number. ΩandHa, are respectively the dimensionless temperature difference and the Hartman number. These number are given by the following relationships

Rel=ullν,Br=μuP2kΔT,Ω=ΔTT,Ha=B0lσμE32
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5. Results and discussion

The flow and heat transfer in a viscoelastic fluid under the influence of a transverse uniform magnetic field has been solved analytically using Kummer’s functions and analytic expressions of the velocity and temperature have been used to compute the entropy generation. Figs. 1 and 2 show the variations of the longitudinal velocity f(η)and the transverse velocity f(η) as function of ηfor several values of magnetic parameterMn. It can be observed that f(η) decreases with ηand f(η) increases with ηasymptotically for Mnkeeping constant. For a fixed positionη, both f(η)and f(η)decreases withMn, thus the presence of the magnetic field decreases the momentum boundary layer thickness and increase the power needed to stretch the sheet.

The effects of the viscoelastic parameter k1on the longitudinal velocity f(η) and the transverse velocity f(η)are illustrated on figs. 3 and 4. As it can be seen, for a fixed value ofη, both f(η)and f(η) decrease as viscoelastic parameter rises. This can be explained by the fact that, as the viscoelastic parameter increases, the hydrodynamic boundary layer adheres strongly to the surface, which in turn retards the flow in the longitudinal and the transverse directions.

Figure 1.

Effect of the magnetic parameter on the longitudinal velocity.

Figure 2.

Effect of the magnetic parameter on the transverse velocity.

Figure 3.

Effect of the viscoelastic parameter on the longitudinal velocity.

Figure 4.

Effect of the viscoelastic parameter on the transverse velocity.

Fig. 5 depicts the temperature profiles Θ(η) as function of ηfor different values of the Prandtl numberPr. As it can be noticed, Θ(η)decreases with ηwhatever is the value of the Prandtl number, for a fixed value ofη, the temperature Θ(η) decreases with an increase in Prandtl number which means that the thermal boundary layer is thinner for large Prandtl number.

Figure 5.

Effect of the Prandtl number on the temperature.

The temperature profiles Θ(η) as function of ηfor different values of the magnetic Mnare plotted in fig. 6. An increase in the magnetic parameter Mnresults in an increase of the temperature; this is due to the fact that the thermal boundary layer increases with the magnetic parameter. Fig. 7 represents graphs of temperature profiles Θ(η) as function of ηfor various values of the heat source/sink parameterβ. For fixed value ofη, the temperature Θ(η) augments with the heat source/ sink parameterβ. This is due to the fact that the increase of the heat source/sink parameter means an increase of the heat generated inside the boundary layer leading to higher temperature profile.

The influence of the magnetic parameter Mnon the entropy generation number NSis shown on fig. 8. The entropy generation number NSdecreases with η for Mnkeeping constant. For

Figure 6.

Effect of the magnetic parameter on the temperature.

Figure 7.

Effect of the heat source/sink parameter on the temperature.

Figure 8.

Effect of the magnetic parameter on the entropy generation number.

fixed value ofη, the entropy generation number increases with the magnetic parameter, because the presence of the magnetic field creates more entropy in the fluid. Moreover, the stretching surface acts as a strong source of irreversibility.

Figure 9.

Effect of the Prandtl number on the entropy generation number.

Fig. 9 illustrates the effect of the Prandtl number Pron the entropy generation numberNS. The entropy generation number is higher for higher Prandtl number near the surface, but as ηincreases, the entropy generation number shows different variation. This is due to the fact that according to fig. 6, the temperature profiles decrease sharply with the increase of the Prandlt number.

Figure 10.

Effect of the Reynolds number on the entropy generation number.

The influence of the Reynolds number Rel on the entropy generation number is plotted on fig. 10. For a given value ofη, the entropy generation number increases as the Reynolds number increases. The augmentation of the Reynolds number increases the contribution of the entropy generation number due to fluid friction and heat transfer in the boundary layer.

Figure 11.

Effect of the dimensionless group on the entropy generation number.

The effect of the dimensionless group parameter BrΩ1on the entropy generation number NSis depicted in fig. 11. The dimensionless group determines the relative importance of viscous effect. For a givenη, the entropy generation number is higher for higher dimensionless group. This is due to the fact that for higher dimensionless group, the entropy generation numbers due to the fluid friction increase.

Figure 12.

Effect of the Hartman number on the entropy generation number.

The effect of the Hartman number Ha on the entropy generation number NSis plotted in fig. 12. For a givenη, as the Hartman number increases, the entropy generation number increases. The entropy generation number is proportional to the square of Hartman number which proportional to the magnetic field. The presence of the magnetic field creates additional entropy.

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6. Conclusion

The velocity and temperature profiles are obtained analytically and used to compute the entropy generation number in viscoelastic magnetohydrodynamic flow over a stretching surface

The effects of the magnetic parameter and the viscoelastic parameter on the longitudinal and transverse velocities are discussed. The influences of the Prandtl number, the magnetic parameter and the heat source/sink parameter on the temperature profiles are presented. As far as the entropy generation number is concerned, its dependence on the magnetic parameter, the Prandlt, the Reynolds, the Hartmann numbers and the dimensionless group are illustrated and analyzed.

From the results the following conclusions could be drawn:

  1. The velocities depend strongly on the magnetic and the viscoelastic parameters.

  2. The temperature varies significantly with the Prandlt number, the magnetic parameter and the heat source/sink parameter.

  3. The entropy generation increases with the increase of the Prandlt, the Reynolds, the Hartmann numbers and also with the magnetic parameter and the dimensionless group.

  4. The surface acts as a strong source of irreversibility.

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7. Nomenclature

Aconstant, KB0uniform magnetic field strength, Wb.m-2BrBrinkman number,Br=μu02kΔT CPspecific heat of the fluid, J.kg-1.K-1fdimensionless functionHaHartman numberHa=B0lσμ kthermal conductivity of the fluid, W.m-1.K-1k1viscoelastic parameter, k1=k0λνk0viscoelastic parameter, m2lcharacteristic length, mMKummer’s functionMnmagnetic parameter, Mn=σB02ρλNSentropy generation number, NS=SGSG0PrPrandlt number, Pr=μCPkQrate of internal heat generation or absorption, W.m-3.K-1 RelReynolds number based on the characteristic length,Rel=ullν SGlocal volumetric rate of entropy generation, W.m-3.K-1 SG0characteristic volumetric rate of entropy generation, W.m-3.K-1Ttemperature, Kuaxial velocity, m.s-1ulplate velocity based on the characteristic length, m.s-1uPplate velocity, m.s-1vtransverse velocity, m.s-1xaxial distance, mXdimensionless axial distance,X=xl ytransverse distance, mαpositive constantβheat source/sink parameter,β=Qνλk λproportional constant, s-1ηdimensionless variable, η=yλνξdimensionless variable, ξ==Prα2eαημdynamic viscosity of the fluid, kg.m-1.s-1νkinematic viscosity of the fluid, m2.s-1ΔTtemperature difference, ΔT=TpTΩdimensionless temperature difference,Ω=ΔTT Θdimensionless temperature, Θ=TTTpTρdensity of the fluid, kg.m-3σelectric conductivity, Ω-1.m-1subscriptsP platefar from the sheet

References

  1. 1. AbbasZ.WangY.HayatT.OberlackM. 2008 Hydromagnetic flow of a viscoelastic fluid due to the oscillatory stretching surface, Int. J. Non-Linear Mech., 43, 783793 , 0020-7462
  2. 2. AbelM. S.KhanS. K.PrasadK. V. 2002 Study of viscoelastic fluid flow and heat transfer over stretching sheet with variable viscosity, Int. J. Non-Linear Mech. 37, 8188 , 0020-7462
  3. 3. AbelM. S.SiddheshwarP. G.NandeppanavarM. M. 2007 Heat transfer in a viscoelastic boundary layer low over a stretching sheet with viscous dissipation and non-uniform heat source, Int. J. Heat Mass Transfer, 50, 960966 , 0017-9310
  4. 4. AbelM. S.MaheshaN. 2008 2008). Heat transfer in MHD viscoelastic fluid over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, Appl. Math. Modelling, 32, 19651983 , 0030-7904X.
  5. 5. AbelM. S.SanjayanandE.NandeppanavarM. M. 2008 Viscoelastic MHD flow and heat heat transfer over a stretching sheet with viscous and ohmic dissipation, Comm. Nonlinear. Sci. and Num. Simulation, 13, 18081821 , 1007-5704
  6. 6. AbelM. S.NandeppanavarM. M. 2009 Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink, Comm. NonLinear Sci. and Num. Simu., 14, 21202131 , 1007-5704
  7. 7. Aïboud-SaouliS.SaouliS.SettouN.MezaN. 2006 Thermodynamic analysis of gravity-driven liquid film along an inclined heated plate with hydromagnetic and viscous dissipation effects, Entropy, 8, 188199 , 1099-4300
  8. 8. Aïboud-SaouliS.SettouN.SaouliS.MezaN. 2007 Second-law analysis of laminar fluid flow in a heated channel with hydromagnetic and viscous dissipation effects, Applied Energy, 84, 279289 , 0306-2619
  9. 9. AïboudS.SaouliS. 2010 Second law analysis of viscoelastic fluid over a stretching sheet subject to a transverse magnetic field with heat and mass transfer, Entopy, 12, 18671884 , 1099-4300
  10. 10. AïboudS.SaouliS. 2010 Entropy analysis for viscoelastic magnetohydrodynamics flow over a stretching surface, Int. J. Non-Linear Mech. 45, 482489 , 0020-7462
  11. 11. AnderssonH. D. 1992 MHD flows of a viscoelastic fluid past a stretching surface, Acta Mech., 95, 227230 , 0001-5970
  12. 12. BejanA. 1982 Second-law analysis in heat transfer and thermal design, Adv. Heat Transfer, 15, 158 , 10-0-12-020021
  13. 13. BejanA. 1996 Entropy generation minimization. CRC Press, Boca Raton, 9780849396519New York, USA.
  14. 14. BejanA. 1979 A study of entropy generation in fundamental convective heat transfer, J. Heat Transfer, 101, 718725 , 0022-1481
  15. 15. BhattacharyaS.PalA.GuptaA. S. 1998 Heat transfer in the flow of a viscoelastic fluid over a stretching surface, Heat mass Transfer, 34, 4145 , 0947-7411
  16. 16. ChangW. D. 1989 The non-uniqueness of the flow of viscoelastic fluid over a stretching sheet, Q. Appl. Math., 47, 365366 , 0003-3569X.
  17. 17. ChenC. H. 2010 On the analytic solution of MHD flow and heat transfer for two types of viscoelastic fluidc over a stretching sheet with energy dissipation internal heat source and thermal radiation, Int. J. Heat mass Transfer, 53, 42644273 , 0017-9310
  18. 18. CortellR. 2006 A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet, Int. J. Non-Linear Mech., 41, 7885 , 0020-7462
  19. 19. CraneL. J. 1970 Flow past a stretching sheet, Z. Angew. Math. Phys., 21, 645647 , 0044-2275
  20. 20. DandapatB. S.GuptaA. S. 1998 Flow and heat transfer in a viscoelastic fluid over a stretching sheet, Int. J. Non-Linear Mech., 24, 215219 , 0020-7462
  21. 21. DattiP. S.PrasadK. V.AbelM. S.JoshiA. 2004 MHD viscoelastic fluid flow over a non-isothermal stretching sheet, Int. J. Eng. Sci., 42, 935946 , 0020-7225
  22. 22. HayatT.SajidM.PopI. 2008 Three-dimensional flow over a stretching sheet in a viscoelastic fluid, Nonlinear. Ana. Real World Appl., 9, 18111822 , 1468-1218
  23. 23. HsiaoK. L. 2007 Conjugate heat transfer of magnetic mixed convection with viscous dissipation effects for second-grade viscoelastic fluid past a stretching sheet, Appl. Therm. Eng., 27, 18951903 , 1359-4311
  24. 24. IdreesM. K.AbelM. S. 1996 Viscoelastic flow past a stretching sheet in porous meadia and heat transfer with internal heat source, Indian J. Theor. Phys., 44, 233244 , 0019-5693
  25. 25. KhanS. K.SanjayanandE. 2005 2005). Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet, Int. J. Heat Mass Transfer, 48, 15341542 , 0017-9310
  26. 26. KhanS. K. 2006 Heat transfer in a viscoelastic fluid over a stretching surface with source/sink, suction/blowing and radiation, Int. J. Heat Mass Transfer, 49, 628639 , 0017-9310
  27. 27. LawrenceP. S.RaoB. N. 1992 Heat transfer in the flow of viscoelastic fluid over stretching sheet, Acta Mech., 93, 5361 , 0001-5970
  28. 28. MahmudS.FraserR. A. 2003 The second law analysis in fundamental convective heat transfer problems, Int. J. Therm. Sci., 42, 177186 , 1290-0729
  29. 29. MahmudS.FraserR. A. 2002 Thermodynamic analysis of flow and heat transfer inside channel with two parallel plates, Exergy 2, 140146 , 1164-0235
  30. 30. MahmudS.FraserR. A. 2002 Inherent irreversibility of channel and pipe flows for non-Newtonian fluids, Int. Comm. Heat Mass Transfer, 29, 577587 , 0947-7411
  31. 31. MahmudS.TasnimS. H.MamunH. A. A. 2003 Thermodynamic analysis of mixed convection in a channel with transverse hydromagnetic effect, In. J. Therm. Sci., 42, 731740 , 1290-0729
  32. 32. MisraJ. C.ShitG. C. 2009 Biomagnetic viscoelastic fluid flow over a stretching sheet, Appl. Math. And Compu, 210, 350361 , 0096-3003
  33. 33. NandeppanavarM. M.AbelM. S.VajraveluK. 2010 Flow and heat transfer characteristics of a viscoelastic fluid in a porous medium over an impermeable stretching sheet with viscous dissipation, Int. J. Heat Mass Transfer, 53, 47074713 , 0017-9310
  34. 34. NandeppanavarM. M.VajraveluK.AbelM. S. 2011 Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink, Comm. Nonlinear. Sci. and Num. Simulation, 16, 35783590 , 1007-5704
  35. 35. NarusawaU. 1998 The second-law analysis of mixed convection in rectangular ducts, Heat Mass Transfer, 37, 197203 , 0947-7411
  36. 36. PrasadK. V.AbelM. S.KhanS. K.DattiP. S. 2002 Non-Darcy forced convective heat transfer in a viscoelastic fluid flow over a non-Isothermal stretching sheet, J. Porous Media, 5, 4147 , 0109-1028X.
  37. 37. PrasadK. V.AbelM. S.KhanS. K. 2000 Momentum and heat transfer in viscoelastic fluid flow in a porous medium over a non-isothermal stretching sheet, Int. J. Numer. Method Heat flow, 10, 786801 , 0961-5539
  38. 38. PrasadK. V.PalD.UmeshV.PrasannaRao. N. S. 2010 The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet, Comm. Nonlinear Sci. and Num. Simulation, 15, 331344 , 1007-5704
  39. 39. RajagopalK. R. .NaT. Y.GuptaA. S. 1984 Flow of a viscoelastic sheet over a stretching sheet, Rheo. Acta, 23, 213221 , 0035-4511
  40. 40. SahinA. Z. 1998 Second law analysis of laminar viscous flow through a duct subjected to constant wall temperature, J. Heat Transfer, 120, 7683 , 0022-1481
  41. 41. SahinA. Z. 1999 Effect of variable viscosity on the entropy generation and pumping power in a laminar fluid flow through a duct subjected to constant heat flux, Heat Mass Transfer, 35, 499506 , 0947-7411
  42. 42. SahinA. Z. 1998 A second law comparison for optimum shape of duct subjected to constant wall temperature and laminar flow, Heat Mass Transfer, 33, 425430 , 0947-7411
  43. 43. SaouliS.Aïboud-SaouliS. 2004 Second law analysis of laminar falling liquid film along an inclined heated plate, Int. Comm. Heat Mass Transfer, 31, 879886 , 0947-7411
  44. 44. SinghA. K. 2008 Heat source and radiation effects on magneto-convection flow of a viscoelastic fluid past a stretching sheet: Analysis with Kummer’s functions, Int. Comm. Heat Mass Transfer, 35, 637642 , 0947-7411
  45. 45. VajraveluK.RollinsD. 1991 Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl. 158, 241255 , 0002-2247X.
  46. 46. WoodsL. C. 1975 Thermodynamics of fluid systems, Oxford University Press, 0198561806Oxford, UK.

Written By

Saouli Salah and Aïboud Soraya

Submitted: April 7th, 2011 Published: October 10th, 2011