# Second Law Analysis of Viscoelastic Fluid over a Stretching Sheet Subject to a Transverse Magnetic Field with Heat and Mass Transfer

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation and Solution

## 3. Second Law Analysis

## 4. Results and Discussion

_{s}is shown on Figure 11. The entropy generation number N

_{s}(Eqation 31) decreases with η for Mn keeping constant. For fixed value of η, the entropy generation number increases with the magnetic parameter, because the presence of the magnetic field creates more entropy in the fluid. Moreover, the entropy generation number is higher near the surface where both temperature and velocity are at their maximum values. This means that the surface acts as a strong source of irreversibility.

_{s}(Equation 31). The entropy generation number is higher for higher Prandtl number near the surface, and then, the situation is inverted as η increases.

_{s}(Equation 31) is illustrated in Figure 13. The entropy generation number is lower for higher Schmidt number near the surface, and then, the situation is inverted as η increases.

_{L}on the entropy generation number N

_{s}(second, third and sixth term of Equation 31) is plotted on Figure 14. For a given value of η, 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, heat and mass transfer in the boundary layer.

^{-1}on the entropy generation number N

_{s}(third and fourth term of Equation 31) is depicted in Figure 15. The dimensionless group determines the relative importance of viscous effect. For a given η, 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 and to the magnetic field increase.

_{s}(fourth term of Equation 31) is plotted in Figure 16. For a given η, as the Hartmann number increases, the entropy generation number increases. The entropy generation number is proportional to the Hartmann number which proportional to the magnetic field. The presence of the magnetic field creates additional entropy.

^{-1}which is the ratio of the dimensionless concentration difference to the dimensionless temperature difference, on the entropy generation number N

_{s}(fifth and sixth term of Equation 31) is plotted in Figure 17. For a given η, as this parameter increases, the entropy generation number increases. This augmentation is due to the contribution of the mass transfer to the entropy generation number.

_{s}(fifth and sixth term of Equation 31) is plotted in Figure 18. For a given η, as the constant parameter increases, the entropy generation number increases. This increase is the contribution of the mass transfer to the entropy generation number.

**Figure 17.**Effect of the ratio of the dimensionless concentration difference to the dimensionless temperature difference on the entropy generation number.

## 5. Conclusions

- (a)
- The longitudinal and the transverse velocities decrease as the magnetic parameter and the viscoelastic paramaeter increase.
- (b)
- The temperature increases as the magnetic parameter and the heat source sink parameter increases, but it decreases as the Prandtl number increases.
- (c)
- The concentration augmentes as the magnetic parameter increases, however it dimishes as the Schmidt number increases.
- (d)
- The entropy generation increases with the increase of the magnetic parameter, the Prandlt number, The Schmidt number, the Reynolds number, the dimensionless group, the Hartmann number and also with ratio of the dimensionless concentration difference to the dimensionless temperature difference and the constant parameter.
- (e)
- The surface acts as a strong source of irreversibility.

## Nomenclature

$a$ | constant |

$A$ | constant |

$b$ | constant |

${\overrightarrow{B}}_{0}$ | uniform magnetic field strength |

$Br$ | Brinkman number |

${C}_{P}$ | specific heat of the fluid |

$C$ | concentration |

$D$ | diffusion coefficient |

$f$ | dimensionless function |

$Ha$ | Hartmann number |

$k$ | thermal conductivity of the fluid |

${k}_{1}$ | viscoelastic parameter |

${k}_{0}$ | viscoelastic parameter |

$l$ | characteristic length |

$M$ | Kummer’s function |

$Mn$ | magnetic parameter |

${N}_{S}$ | entropy generation number |

$\mathit{Pr}$ | Prandlt number |

${\left(q\right)}_{n}$ | Pochhammer’s symbol |

$Q$ | rate of internal heat generation or absorption |

${\left(r\right)}_{n}$ | Pochhammer’s symbol |

$R$ | ideal gas constant |

${\mathit{Re}}_{l}$ | Reynolds number based on the characteristic length |

$Sc$ | Schmidt number |

${S}_{G}$ | local volumetric rate of entropy generation |

${S}_{G0}$ | characteristic volumetric rate of entropy generation |

$T$ | temperature |

$u$ | axial velocity |

${u}_{l}$ | plate velocity based on the characteristic length |

${u}_{0}$ | plate velocity |

$v$ | transverse velocity |

$x$ | axial distance |

$X$ | dimensionless axial distance |

$y$ | transverse distance |

$\alpha $ | positive constant |

$Br{\Omega}^{-1}$ | dimensionless group |

$\beta $ | heat source/sink parameter |

$\lambda $ | proportional constant |

$\eta $ | dimensionless variable |

$\epsilon $ | constant parameter |

$\xi $ | dimensionless variable |

$\mu $ | dynamic viscosity of the fluid |

$\nu $ | kinematic viscosity of the fluid |

$\Delta C$ | concentration difference |

$\Delta T$ | temperature difference |

$\Omega $ | dimensionless temperature difference |

$\Theta $ | dimensionless temperature |

$\Phi $ | dimensionless concentration |

$\Sigma $ | dimensionless concentration difference |

$\rho $ | density of the fluid |

$\sigma $ | electric conductivity |

## subscripts

$0$ | plate |

$\infty $ | far from the sheet |

## References and Notes

- Crane, L.J. Flow past a stretching sheet. Z. Angew. Math. Phys.
**1970**, 21, 645–647. [Google Scholar] [CrossRef] - Rajagopal, K.R.; Na, T.Y.; Gupta, A.S. Flow of a viscoelastic fluid over a stretching sheet. Rheo. Acta
**1984**, 23, 213–215. [Google Scholar] [CrossRef] - Chang, W.D. The non-uniqueness of the flow of viscoelastic fluid over a stretching sheet. Q. Appl. Math.
**1989**, 47, 365–366. [Google Scholar] - Dandapat, B.S.; Gupta, A.S. Flow and heat transfer in a viscoelastic fluid over a stretching sheet. Int. J. Non-Linear Mech.
**1989**, 24, 215–219. [Google Scholar] [CrossRef] - Vajravelu, K.; Rollins, D. Heat transfer in a viscoelastic fluid over a stretching sheet. J. Math. Anal. Appl.
**1991**, 158, 241–255. [Google Scholar] [CrossRef] - Andersson, H.I. MHD flows of a viscoelastic fluid past a stretching surface. Acta Mech.
**1992**, 95, 227–230. [Google Scholar] [CrossRef] - Lawrence, P.S.; Rao, B.N. Heat transfer in the flow of viscoelastic fluid over stretching sheet. Acta Mech.
**1992**, 93, 53–61. [Google Scholar] [CrossRef] - Idrees, M.K.; Abel, M.S. Viscoelastic flow past a stretching sheet in porous meadia and heat transfer with internal heat source. Indian J. Theor. Phys.
**1996**, 44, 233–244. [Google Scholar] - Bhattacharya, B.; Pal, A.; Gupta, A.S. Heat transfer in the flow of a viscoelastic fluid over a stretching surface. Heat Mass Transfer
**1998**, 34, 41–45. [Google Scholar] [CrossRef] - Prasad, K.V.; Abel, M.S.; Khan, S.K. Momentum and heat transfer in viscoelastic fluid flow in a porous medium over a non-isothermal stretching sheet. Int. J. Numer. Method Heat flow
**2000**, 10, 786–801. [Google Scholar] [CrossRef] - Abel, M.S.; Khan, S.K.; Prasad, K.V. Study of viscoelastic fluid flow and heat transfer over stretching sheet with variable viscosity. Int. J. Non-Linear Mech.
**2002**, 37, 81–88. [Google Scholar] [CrossRef] - Prasad, K.V.; Abel, M.S.; Khan, S.K.; Datti, P.S. Non-Darcy forced convective heat transfer in a viscoelastic fluid flow over a non-Isothermal stretching sheet. J. Porous Media
**2002**, 5, 41–47. [Google Scholar] [CrossRef] - Datti, P.S.; Prasad, K.V.; Abel, M.S.; Joshi, A. MHD viscoelastic fluid flow over a non-isothermal stretching sheet. Int. J. Eng. Sci.
**2004**, 42, 935–946. [Google Scholar] [CrossRef] - Khan, S.K.; Sanjayanand, E. Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet. Int. J. Heat Mass Transfer
**2005**, 48, 1534–1542. [Google Scholar] [CrossRef] - Cortell, R. A note on flow and heat transfer of a viscoelastic fluid over a stretching sheet. Int. J. Non-Linear Mech.
**2006**, 41, 78–85. [Google Scholar] - Abel, M.S.; Siddheshwar, P.G.; Nandeppanavar, M.M. 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
**2007**, 50, 960–966. [Google Scholar] - Bejan, A. Second-law analysis in heat transfer and thermal design. Adv. Heat Transfer
**1982**, 15, 1–58. [Google Scholar] - Bejan, A. Entropy Generation Minimization; CRC Press: Boca Raton, FL, USA, 1996. [Google Scholar]
- Bejan, A. A study of entropy generation in fundamental convective heat transfer. J. Heat Transfer
**1979**, 101, 718–725. [Google Scholar] [CrossRef] - Sahin, A.Z. Second law analysis of laminar viscous flow through a duct subjected to constant wall temperature. J. Heat Transfer
**1998**, 120, 76–83. [Google Scholar] [CrossRef] - Sahin, A.Z. 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
**1999**, 35, 499–506. [Google Scholar] - Sahin, A.Z. A second law comparison for optimum shape of duct subjected to constant wall temperature and laminar flow. Heat Mass Transfer
**1998**, 33, 425–430. [Google Scholar] - Narusawa, U. The second-law analysis of mixed convection in rectangular ducts. Heat Mass Transfer
**1998**, 37, 197–203. [Google Scholar] [CrossRef] - Mahmud, S.; Fraser, R.A. The second law analysis in fundamental convective heat transfer problems. Int. J. Therm. Sci.
**2003**, 42, 177–186. [Google Scholar] [CrossRef] - Mahmud, S.; Fraser, R.A. Thermodynamic analysis of flow and heat transfer inside channel with two parallel plates. Exergy
**2002**, 2, 140–146. [Google Scholar] [CrossRef] - Mahmud, S.; Fraser, R.A. Inherent irreversibility of channel and pipe flows for non-Newtonian fluids. Int. Comm. Heat Mass Transfer
**2002**, 29, 577–587. [Google Scholar] [CrossRef] - Saouli, S.; Aïboud-Saouli, S. Second law analysis of laminar falling liquid film along an inclined heated plate. Int. Comm. Heat Mass Transfer
**2004**, 31, 879–886. [Google Scholar] [CrossRef] - Mahmud, S.; Tasnim, S.H.; Mamun, S.A.A. Thermodynamic analysis of mixed convection in a channel with transverse hydromagnetic effect. In. J. Therm. Sci.
**2003**, 42, 731–740. [Google Scholar] [CrossRef] - Aïboud-Saouli, S.; Saouli, S.; Settou, N.; Meza, N. Thermodynamic analysis of gravity-driven liquid film along an inclined heated plate with hydromagnetic and viscous dissipation effects. Entropy
**2006**, 8, 188–199. [Google Scholar] [CrossRef] - Aïboud-Saouli, S.; Saouli, S.; Settou, N.; Meza, N. Second-law analysis of laminar fluid flow in a heated channel with hydromagnetic and viscous dissipation effects. Appl. Ener.
**2007**, 84, 279–289. [Google Scholar] [CrossRef] - Woods, L.C. Thermodynamics of Fluid Systems; Oxford University Press: Oxford, UK, 1975. [Google Scholar]
- Megherbi, M.; Abassi, H.; Hidouri, N.; Ben Brahim, A. Second law analysis in convective heat and mass transfer. Entropy
**2006**, 8, 1–17. [Google Scholar] [CrossRef]

© 2010 by the authors; licensee MDPI, Basel, Switzerland. This article is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Aïboud, S.; Saouli, S.
Second Law Analysis of Viscoelastic Fluid over a Stretching Sheet Subject to a Transverse Magnetic Field with Heat and Mass Transfer. *Entropy* **2010**, *12*, 1867-1884.
https://doi.org/10.3390/e12081867

**AMA Style**

Aïboud S, Saouli S.
Second Law Analysis of Viscoelastic Fluid over a Stretching Sheet Subject to a Transverse Magnetic Field with Heat and Mass Transfer. *Entropy*. 2010; 12(8):1867-1884.
https://doi.org/10.3390/e12081867

**Chicago/Turabian Style**

Aïboud, Soraya, and Salah Saouli.
2010. "Second Law Analysis of Viscoelastic Fluid over a Stretching Sheet Subject to a Transverse Magnetic Field with Heat and Mass Transfer" *Entropy* 12, no. 8: 1867-1884.
https://doi.org/10.3390/e12081867