# Second Law Analysis for Variable Viscosity Hydromagnetic Boundary Layer Flow with Thermal Radiation and Newtonian Heating

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

_{∞}in the presence of magnetic field and thermal radiation is considered. It is assumed that the lower surface of the plate is heated by convection from a hot fluid at temperature T

_{f}which provides a heat transfer coefficient h

_{f}. The fluid on the upper side of the plate is then subjected to Newtonian heating and its property variations due to temperature are limited to viscosity. A uniform transverse magnetic field B

_{0}is imposed along the y-axis as shown in Figure 1. Both the induced magnetic field due to the motion of the electrically-conducting fluid and the electric field due to the polarization of charges are assumed to be negligible.

_{∞}is the free stream velocity, c

_{p}is the specific heat at constant pressure, α is the thermal diffusivity, ρ is the fluid density, σ is the fluid electrical conductivity. The fluid dynamical viscosity µ is assumed to be an inverse linear function of temperature, Lai and Kulacki [26], as given by:

^{4}may be expressed as a linear function of temperature T using a truncated Taylor series about the free stream temperature ${T}_{\infty}$ i.e.,

#### 2.1. Numerical Procedure

^{−7}in nearly all cases. From the process of numerical computation, the plate surface temperature, the local skin-friction coefficient and the local Nusselt number which are respectively proportional to θ(0), ${f}^{\u2033}(0)$ and $-{\theta}^{\prime}(0)$, are also worked out and their numerical values are presented in a tabular form.

## 3. Entropy Generation Analysis

_{∞}x/ν is the local Reynolds number. We assigned N

_{1}to the first term in Equation (18) due to heat transfer and the second term due to the combined effects of viscous dissipation and magnetic field as N

_{2}, i.e.,

_{2}/N

_{1}. Therefore, whenever 0 ≤ Ф < 1, the heat transfer irreversibility dominates. The fluid friction together with magnetic field dominates when Ф > 1. The contribution of both heat transfer and fluid friction with magnetic field to entropy generation are equal when Ф = 1. However, in many engineering designs and energy optimisation problems, the contribution of heat transfer entropy (N

_{1}) and fluid friction with magnetic field effect (N

_{2}) to overall local entropy generation rate Ns

_{x}is needed. In order to achieve this, we define the following Bejan numbers (Be) mathematically as:

## 4. Results and Discussion

**Table 1.**Computations showing comparison with Aziz [18] results for Ha

_{x}= Br = a = 0, β = 1, Pr = 0.72.

Bi | $\theta (0)$ Aziz [18] | −${\theta}^{\prime}(0)$ Aziz [18] | $\theta (0)$ Present | −${\theta}^{\prime}(0)$ Present |
---|---|---|---|---|

0.05 | 0.1447 | 0.0428 | 0.14466 | 0.04276 |

0.60 | 0.6699 | 0.1981 | 0.66991 | 0.19805 |

1.00 | 0.7718 | 0.2282 | 0.77182 | 0.22817 |

5.00 | 0.9441 | 0.2791 | 0.94417 | 0.27913 |

20.00 | 0.9854 | 0.2913 | 0.98543 | 0.29132 |

**Table 2.**Computation showing ${f}^{\u2033}(0)$, $\theta (0)$, and ${\theta}^{\prime}(0)$ for various values of key parameters.

Bi | a | Br | Ra | Pr | Ha | ${f}^{\u2033}(0)$ | −${\theta}^{\prime}(0)$ | $\theta (0)$ |
---|---|---|---|---|---|---|---|---|

0.1 | 0.1 | 0.1 | 0.7 | 0.72 | 0.1 | 0.46167359 | 0.06417756 | 0.35822438 |

1.0 | 0.1 | 0.1 | 0.7 | 0.72 | 0.1 | 0.47460353 | 0.16490527 | 0.83509472 |

10 | 0.1 | 0.1 | 0.7 | 0.72 | 0.1 | 0.47847240 | 0.19577586 | 0.98042241 |

0.1 | 0.5 | 0.1 | 0.7 | 0.72 | 0.1 | 0.49847466 | 0.06463133 | 0.35368667 |

0.1 | 1.0 | 0.1 | 0.7 | 0.72 | 0.1 | 0.53980796 | 0.06510562 | 0.34894370 |

0.1 | −0.1 | 0.1 | 0.7 | 0.72 | 0.1 | 0.44170340 | 0.06391808 | 0.36081912 |

0.1 | −0.5 | 0.1 | 0.7 | 0.72 | 0.1 | 0.39774382 | 0.06331171 | 0.36688289 |

0.1 | 0.1 | 1.0 | 0.7 | 0.72 | 0.1 | 0.46903194 | 0.03335240 | 0.66647595 |

0.1 | 0.1 | 10 | 0.7 | 0.72 | 0.1 | 0.52982570 | 0.24134507 | 3.41345073 |

0.1 | 0.1 | 0.1 | 5.0 | 0.72 | 0.1 | 0.46104375 | 0.06820145 | 0.31798541 |

0.1 | 0.1 | 0.1 | 10.0 | 0.72 | 0.1 | 0.46100454 | 0.06856032 | 0.31439674 |

0.1 | 0.1 | 0.1 | 0.7 | 3.00 | 0.1 | 0.45924014 | 0.07604897 | 0.23951024 |

0.1 | 0.1 | 0.1 | 0.7 | 7.10 | 0.1 | 0.45801360 | 0.08146073 | 0.18539265 |

0.1 | 0.1 | 0.1 | 0.7 | 0.72 | 0.5 | 0.78699049 | 0.06242200 | 0.37577998 |

0.1 | 0.1 | 0.1 | 0.7 | 0.72 | 1.0 | 1.06625574 | 0.06065042 | 0.39349571 |

0.1 | 0.1 | 0.1 | 0.7 | 0.72 | 2.0 | 1.47684794 | 0.05773749 | 0.42262501 |

#### 4.1. Effect of Parameter Variation on Velocity Profiles

#### 4.2. Effects of Parameter Variation on Temperature Profiles

#### 4.3. Effects of Parameter Variation on Entropy Generation Rate

^{−1}on the entropy generation number is represented in Figure 11 and Figure 12. It seems that these parameters have similar effect on the entropy generation number by creating more entropy in the fluid. However, it is interesting to note that the entropy generated in the fluid attained it maximum value at a distance near the plate surface within the boundary layer region. The influence of the Prandtl number on the entropy generation number is shown in Figure 13. As the Prandtl number increases, the entropy generation number increases gradually from the plate surface, then to its highest value within the boundary layer and decreases to its lowest zero value at the free stream. It is noteworthy that the entropy generation rate peak value for Pr = 0.72 (Air) is higher than that of Pr = 7.1 (Water). Similar trend is observed in Figure 14 and Figure 15 with increasing value of parameters a and Ra. As the fluid viscosity decreases and thermal radiation increases, the entropy generation rate increases from the plate surface, reaching its peak values and then decreases to zero value at free stream. The peak value for Ra = 0.7 is higher to that of Ra = 3.

**Figure 10.**Entropy generation rate for Pr = 0.72, BrΩ

^{−1}= 0.1, Re= 0.1, Bi = 0.1, a = 0.1, Ra = 0.7.

**Figure 11.**Entropy generation rate for Pr = 0.72, BrΩ

^{−1}= 0.1, Re = 0.1, Ha = 0.1, Ra = 0.7, a = 0.1.

**Figure 13.**Entropy generation rate for Bi = 0.1, BrΩ

^{−1}= 0.1, Ra = 0.7, Re = 0.1, Ha = 0.1, a = 0.1.

#### 4.4. Effects of Parameter Variation on Bejan Number

^{−1}, variable viscosity parameter (a) and Prandtl number (Pr) as illustrated in Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20. In Figure 21, we notice that the effect of heat transfer irreversibility increases as the local Biot number (Bi) parameter increase due to Newtonian heating of the plate surface. Hence, the plate surface acts as a strong source of heat transfer irreversibility.

## 5. Conclusions

- The skin friction and the Nusselt number increase with increasing values of Bi, Br, a. An increase Ha causes a decrease in the Nusselt number and an increase in the skin friction. As Ra and Pr increase, the skin friction decreases while the Nusselt number increases.
- The velocity boundary layer thickness decreases with Ha, a.
- The thermal boundary layer thickness increases with Ha, Br, Bi and decreases with Ra, Pr, a.
- The plate surface act as a strong source of entropy generation and heat transfer irreversibility. However, the peak of entropy generation rate is attained within the boundary layer region.
- The entropy generation number increases as Ha, Bi and BrΩ
^{−1}increase while it decreases as the Pr, Ra and a increase. - The optimum design and efficient performance of the flow system can be enhanced by the ability to clearly identify the source and location of entropy generation. The present results show that minimum entropy generation in the flow system can be achieved with appropriate choice and combination of the various thermophysical parameters.

## Acknowledgements

## References

- Woods, L.C. Thermodynamics of Fluid Systems; Oxford University Press: Oxford, UK, 1975. [Google Scholar]
- Aıboud, S.; Saouli, S. Entropy analysis for viscoelastic magnetohydrodynamic flow over a stretching surface. Int. J. Non-Linear Mech.
**2010**, 45, 482–489. [Google Scholar] [CrossRef] - Sparrow, E.M.; Cess, R.D. Radiation Heat Transfer; Harpercollins College Div: New York, NY, USA, 1978; pp. 7–10. [Google Scholar]
- Raptis, A.; Perdikis, C.; Takhar, H.S. Effect of thermal radiation on MHD flow. Appl. Math. Comput.
**2004**, 153, 645–649. [Google Scholar] [CrossRef] - Mbeledogu, I.U.; Amakiri, A.R.C.; Ogulu, A. Unsteady MHD free convection flow of a compressible fluid past a moving vertical plate in the presence of radiative heat transfer. Int. J. Heat Mass Transf.
**2007**, 50, 1668–1674. [Google Scholar] [CrossRef] - Bataller, R.C. Radiation effects for the Blasius and Sakiadis flows with a convective surface boundary condition. Appl. Math. Comput.
**2008**, 206, 832–840. [Google Scholar] [CrossRef] - Makinde, O.D.; Aziz, A. MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition. Int. J. Therm. Sci.
**2010**, 49, 1813–1820. [Google Scholar] [CrossRef] - Makinde, O.D. On MHD heat and mass transfer over a moving vertical plate with a convective surface boundary condition. Can. J. Chem. Eng.
**2010**, 88, 983–990. [Google Scholar] [CrossRef] - Narusawa, U. The second-law analysis of mixed convection in rectangular ducts. Heat Mass Transf.
**1998**, 37, 197–203. [Google Scholar] [CrossRef] - Sahin, A.Z. Second law analysis of laminar viscous flow through a duct subjected to constant wall temperature. J. Heat Transf.
**1998**, 120, 76–83. [Google Scholar] [CrossRef] - Bejan, A. Second-law analysis in heat transfer and thermal design. Adv. Heat Transf.
**1982**, 15, 1–58. [Google Scholar] - Bejan, A. Entropy Generation Minimization; CRC: Boca Raton, FL, USA, 1996. [Google Scholar]
- Kobo, N.S.; Makinde, O.D. Second law analysis for a variable viscosity reactive Couette flow under Arrhenius kinetics. Math. Probl. Eng.
**2010**, 2010, 278104. [Google Scholar] [CrossRef] - Jalaal, M.; Ganji, D.D.; Ahmadi, G. Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media. Adv. Powder Technol.
**2010**, 21, 298–304. [Google Scholar] [CrossRef] - Jalaal, M.; Ganji, D.D. An analytical study on motion of a sphere rolling down an inclined plane submerged in a Newtonian fluid. Powder Technol.
**2010**, 198, 82–92. [Google Scholar] [CrossRef] - Makinde, O.D.; Aziz, A. Second law analysis for a variable viscosity plane Poiseuille flow with asymmetric convective cooling. Comput. Math. Appl.
**2010**, 60, 3012–3019. [Google Scholar] [CrossRef] - Makinde, O.D. Thermodynamic second law analysis for a gravity driven variable viscosity liquid film along an inclined heated plate with convective cooling. J. Mech. Sci. Technol.
**2010**, 24, 899–908. [Google Scholar] [CrossRef] - Mahmud, S; Tasnim, S.H.; Mamun, H.A.A. Thermodynamic analysis of mixed convection in a channel with transverse hydromagnetic effect. Int. J. Therm. Sci.
**2003**, 42, 731–740. [Google Scholar] [CrossRef] - Makinde, O.D.; Beg, O.A. On inherent irreversibility in a reactive hydromagnetic channel flow. J. Therm. Sci.
**2010**, 19, 72–79. [Google Scholar] [CrossRef] - Chen, S.; Tolke, J.; Krafczyk, M. Numerical investigation of double-diffusive (natural) convection in vertical annuluses with opposing temperature and concentration gradients. Int. J. Heat Fluid Flow
**2010**, 31, 217–226. [Google Scholar] [CrossRef] - Chen, S.; Zheng, C. Entropy generation in impinging flow confined by planar opposing jets. Int. J. Therm. Sci.
**2010**, 49, 2067–2075. [Google Scholar] [CrossRef] - Chen, S.; Du, R. Entropy generation of turbulent double-diffusive natural convection in a rectangle cavity. Energy
**2011**, 36, 1721–1734. [Google Scholar] [CrossRef] - Chen, S. Analysis of entropy generation in counter-flow premixed hydrogen-air combustion. Int. J. Hydrogen Energy
**2010**, 35, 1401–1411. [Google Scholar] [CrossRef] - Chen, S.; Liu, Z.; Liu, J.; Li, J.; Wang, L.; Zheng, C. Analysis of entropy generation in hydrogen-enriched ultra-lean counter-flow methane–air non-premixed combustion. Int. J. Hydrogen Energy
**2010**, 35, 12491–12501. [Google Scholar] [CrossRef] - Aziz, A. A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simul.
**2009**, 14, 1064–1068. [Google Scholar] [CrossRef] - Lai, F.C.; Kulacki, F.A. The effect of variable viscosity on convective heat and mass transfer in saturated porous media. Int. J. Heat Mass Transf.
**1991**, 33, 1028–1031. [Google Scholar] [CrossRef] - Nachtsheim, P.R.; Swigert, P. Satisfaction of the Asymptotic Boundary Conditions in Numerical Solution of the System of Nonlinear Equations of Boundary Layer Type; NASA TND-3004; NASA: Washington, DC, USA, 1965. [Google Scholar]

© 2011 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**

Makinde, O.D. Second Law Analysis for Variable Viscosity Hydromagnetic Boundary Layer Flow with Thermal Radiation and Newtonian Heating. *Entropy* **2011**, *13*, 1446-1464.
https://doi.org/10.3390/e13081446

**AMA Style**

Makinde OD. Second Law Analysis for Variable Viscosity Hydromagnetic Boundary Layer Flow with Thermal Radiation and Newtonian Heating. *Entropy*. 2011; 13(8):1446-1464.
https://doi.org/10.3390/e13081446

**Chicago/Turabian Style**

Makinde, Oluwole Daniel. 2011. "Second Law Analysis for Variable Viscosity Hydromagnetic Boundary Layer Flow with Thermal Radiation and Newtonian Heating" *Entropy* 13, no. 8: 1446-1464.
https://doi.org/10.3390/e13081446