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.
2. Formulation of the problem
In two-dimensional Cartesian coordinate system
Under the usual boundary layer approximations, the flow is governed by the following equations:
The constant
The boundary conditions are given by
The heat transfer governing boundary layer equation with temperature-dependent heat generation (absorption) is
The relevant boundary conditions are
3. Analytical solution
The equation of continuity is satisfied if we choose a dimensionless stream function
Introducing the similarity transformations
Momentum equation (2) becomes
where
Now let us seek a solution of Eq. (8) in the form
which is satisfied by the following boundary conditions:
On substituting (9) into (8) and using boundary conditions (12) and (13) the velocity components take the form
Where
Defining the dimensionless temperature
and using (11), (14), (15), Eq. (17) and the boundary conditions (6) and (7) can be written as
Where
Introducing the variable
And inserting (21) in (18) we obtain
And (19) and (20) transform to
The solution of Eq. (22) satisfying (23) and (24) is given by
The solution of (25) in terms of
where
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
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 rate
therefore, the entropy generation number is
using Eq. (11), (26) and (27), the entropy generation number is given by
where
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
The effects of the viscoelastic parameter
Fig. 5 depicts the temperature profiles
The temperature profiles
The influence of the magnetic parameter
fixed value of
Fig. 9 illustrates the effect of the Prandtl number
The influence of the Reynolds number
The effect of the dimensionless group parameter
The effect of the Hartman number
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:
The velocities depend strongly on the magnetic and the viscoelastic parameters.
The temperature varies significantly with the Prandlt number, the magnetic parameter and the heat source/sink parameter.
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.
The surface acts as a strong source of irreversibility.
7. Nomenclature
References
- 1.
Abbas Z. Wang Y. Hayat T. Oberlack M. 2008 Hydromagnetic flow of a viscoelastic fluid due to the oscillatory stretching surface, , 43,783 793 ,0020-7462 - 2.
Abel M. S. Khan S. K. Prasad K. V. 2002 Study of viscoelastic fluid flow and heat transfer over stretching sheet with variable viscosity, . 37,81 88 ,0020-7462 - 3.
Abel M. S. Siddheshwar P. G. Nandeppanavar M. M. 2007 Heat transfer in a viscoelastic boundary layer low over a stretching sheet with viscous dissipation and non-uniform heat source, 50,960 966 ,0017-9310 - 4.
Abel M. S. Mahesha N. 2008 2008). Heat transfer in MHD viscoelastic fluid over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation, , 32,1965 1983 ,0030-7904 X. - 5.
Abel M. S. Sanjayanand E. Nandeppanavar M. M. 2008 Viscoelastic MHD flow and heat heat transfer over a stretching sheet with viscous and ohmic dissipation, , 13,1808 1821 ,1007-5704 - 6.
Abel M. S. Nandeppanavar M. M. 2009 Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink, , 14,2120 2131 ,1007-5704 - 7.
Aïboud-Saouli S. Saouli S. Settou N. Meza N. 2006 Thermodynamic analysis of gravity-driven liquid film along an inclined heated plate with hydromagnetic and viscous dissipation effects, , 8,188 199 ,1099-4300 - 8.
Aïboud-Saouli S. Settou N. Saouli S. Meza N. 2007 Second-law analysis of laminar fluid flow in a heated channel with hydromagnetic and viscous dissipation effects, , 84,279 289 ,0306-2619 - 9.
Aïboud S. Saouli S. 2010 Second law analysis of viscoelastic fluid over a stretching sheet subject to a transverse magnetic field with heat and mass transfer, Entopy, 12,1867 1884 ,1099-4300 - 10.
Aïboud S. Saouli S. 2010 Entropy analysis for viscoelastic magnetohydrodynamics flow over a stretching surface, . 45,482 489 ,0020-7462 - 11.
Andersson H. D. 1992 MHD flows of a viscoelastic fluid past a stretching surface, ., 95,227 230 ,0001-5970 - 12.
Bejan A. 1982 Second-law analysis in heat transfer and thermal design, Adv. Heat Transfer, 15,1 58 ,10-0-12-020021 - 13.
Bejan A. 1996 Entropy generation minimization. ,9780849396519 New York, USA. - 14.
Bejan A. 1979 A study of entropy generation in fundamental convective heat transfer, J. Heat Transfer, 101,718 725 ,0022-1481 - 15.
Bhattacharya S. Pal A. Gupta A. S. 1998 Heat transfer in the flow of a viscoelastic fluid over a stretching surface, , 34,41 45 ,0947-7411 - 16.
Chang W. D. 1989 The non-uniqueness of the flow of viscoelastic fluid over a stretching sheet, , 47,365 366 ,0003-3569 X. - 17.
Chen C. 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, , 53,4264 4273 ,0017-9310 - 18.
Cortell R. 2006 A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet, 41,78 85 ,0020-7462 - 19.
Crane L. J. 1970 Flow past a stretching sheet, 21,645 647 ,0044-2275 - 20.
Dandapat B. S. Gupta A. S. 1998 Flow and heat transfer in a viscoelastic fluid over a stretching sheet, 24,215 219 ,0020-7462 - 21.
Datti P. S. Prasad K. V. Abel M. S. Joshi A. 2004 MHD viscoelastic fluid flow over a non-isothermal stretching sheet, 42,935 946 ,0020-7225 - 22.
Hayat T. Sajid M. Pop I. 2008 Three-dimensional flow over a stretching sheet in a viscoelastic fluid, , 9,1811 1822 ,1468-1218 - 23.
Hsiao K. L. 2007 Conjugate heat transfer of magnetic mixed convection with viscous dissipation effects for second-grade viscoelastic fluid past a stretching sheet, , 27,1895 1903 ,1359-4311 - 24.
Idrees M. K. Abel M. S. 1996 Viscoelastic flow past a stretching sheet in porous meadia and heat transfer with internal heat source, , 44,233 244 ,0019-5693 - 25.
Khan S. K. Sanjayanand E. 2005 2005). Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet, , 48,1534 1542 ,0017-9310 - 26.
Khan S. K. 2006 Heat transfer in a viscoelastic fluid over a stretching surface with source/sink, suction/blowing and radiation, , 49,628 639 ,0017-9310 - 27.
Lawrence P. S. Rao B. N. 1992 Heat transfer in the flow of viscoelastic fluid over stretching sheet, Acta Mech., 93,53 61 ,0001-5970 - 28.
Mahmud S. Fraser R. A. 2003 The second law analysis in fundamental convective heat transfer problems, 42,177 186 ,1290-0729 - 29.
Mahmud S. Fraser R. A. 2002 Thermodynamic analysis of flow and heat transfer inside channel with two parallel plates, 2,140 146 ,1164-0235 - 30.
Mahmud S. Fraser R. A. 2002 Inherent irreversibility of channel and pipe flows for non-Newtonian fluids, 29,577 587 ,0947-7411 - 31.
Mahmud S. Tasnim S. H. Mamun H. A. A. 2003 Thermodynamic analysis of mixed convection in a channel with transverse hydromagnetic effect, 42,731 740 ,1290-0729 - 32.
Misra J. C. Shit G. C. 2009 Biomagnetic viscoelastic fluid flow over a stretching sheet, , 210,350 361 ,0096-3003 - 33.
Nandeppanavar M. M. Abel M. S. Vajravelu K. 2010 Flow and heat transfer characteristics of a viscoelastic fluid in a porous medium over an impermeable stretching sheet with viscous dissipation, , 53,4707 4713 ,0017-9310 - 34.
Nandeppanavar M. M. Vajravelu K. Abel M. S. 2011 Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with thermal radiation and non-uniform heat source/sink, , 16,3578 3590 ,1007-5704 - 35.
Narusawa U. 1998 The second-law analysis of mixed convection in rectangular ducts, 37,197 203 ,0947-7411 - 36.
Prasad K. V. Abel M. S. Khan S. K. Datti P. S. 2002 Non-Darcy forced convective heat transfer in a viscoelastic fluid flow over a non-Isothermal stretching sheet, , 5,41 47 ,0109-1028 X. - 37.
Prasad K. V. Abel M. S. Khan S. K. 2000 Momentum and heat transfer in viscoelastic fluid flow in a porous medium over a non-isothermal stretching sheet, , 10,786 801 ,0961-5539 - 38.
Prasad K. V. Pal D. Umesh V. Prasanna Rao. N. S. 2010 The effect of variable viscosity on MHD viscoelastic fluid flow and heat transfer over a stretching sheet, , 15,331 344 ,1007-5704 - 39.
Rajagopal K. R. . Na T. Y. Gupta A. S. 1984 Flow of a viscoelastic sheet over a stretching sheet, , 23,213 221 ,0035-4511 - 40.
Sahin A. Z. 1998 Second law analysis of laminar viscous flow through a duct subjected to constant wall temperature, 120,76 83 ,0022-1481 - 41.
Sahin A. 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, 35,499 506 ,0947-7411 - 42.
Sahin A. Z. 1998 A second law comparison for optimum shape of duct subjected to constant wall temperature and laminar flow, 33,425 430 ,0947-7411 - 43.
Saouli S. Aïboud-Saouli S. 2004 Second law analysis of laminar falling liquid film along an inclined heated plate, 31,879 886 ,0947-7411 - 44.
Singh A. K. 2008 Heat source and radiation effects on magneto-convection flow of a viscoelastic fluid past a stretching sheet: Analysis with Kummer’s functions, 35,637 642 ,0947-7411 - 45.
Vajravelu K. Rollins D. 1991 Heat transfer in a viscoelastic fluid over a stretching sheet, 158,241 255 ,0002-2247 X. - 46.
Woods L. C. 1975 Thermodynamics of fluid systems,0198561806 Oxford, UK.