# Entropy Generation in Magnetohydrodynamic Mixed Convection Flow over an Inclined Stretching Sheet

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation of the Problem

_{w}(x) = cx. A magnetic field B

_{o}is applied normally to the stretching sheet. Figure 1 shows the physical model and coordinates system. For boundary layer flow the continuity, momentum and energy equations take the following forms:

_{∞}shows the free stream temperature, g denotes gravitational acceleration, α* is the thermal diffusivity, ρ is the density of the fluid, v is the kinematic viscosity of the fluid, B

_{o}depicts the imposed magnetic field, C

_{p}represents the specific heat of the fluid at constant pressure, α is the inclination of the stretching sheet and β represents the thermal coefficient. Here we consider the thermal expansion coefficient β = mx

^{–1}[31,32] and the surface temperature of the stretching sheet of the form T

_{w}(x) = T

_{∞}+ ax

^{2}[33,34,35] in order to have a self-similarity equation.

_{w}and u

_{w}respectively represent the temperature and velocity of the stretching boundary.

_{f}and the Nusselt number Nu

_{x}which are defined as:

_{w}is the shear stress and q

_{w}is the heat flux and k is the thermal conductivity. Using variables (6), we obtain:

## 3. Irreversibility Analysis

## 4. Results and Discussion

#### 4.1. Effects of the Magnetic Field Parameter

#### 4.2. Effects of the Prandtl Number

#### 4.3. Effects of the Mixed Convective Parameter

#### 4.4. Effects of Eckert Number

#### 4.5. Effects of the Dimensionless Temperature Parameter

## 5. Conclusions

- (1)
- It is observed that the velocity magnetic parameter M has a decreasing effect on the velocity while the thermal convective parameter λ has an increasing effect on it.
- (2)
- The thickness of the thermal boundary layer increases as the values of magnetic parameter and Eckert number increase, while a decreasing effect has been observed with the variation of thermal convective parameter and Prandtl number.
- (3)
- Increasing in magnetic parameter and Eckert number enhance the entropy generation while entropy generation decreases with the increasing values of the thermal convective parameter and dimensionless temperature function.
- (4)
- The Bejan number decreases by increasing the magnetic parameter, thermal convective parameter and it is decreased by increasing the Prandtl number and Eckert number.
- (5)
- The values of skin friction coefficient and local Nusselt number increases with the increase of Pr. There is a decrease in local Nusselt number for large values of M and Ec (see Table 1).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

a | positive constant |

Bo | applied constant magnetic field |

Be | Bejan number |

c | stretching velocity rate |

Cp | specific heat at constant pressure |

Ec | Eckert number |

g | acceleration due to gravity |

Gr_{x} | local Grashof number |

J | electric current density |

k | thermal conductivity of the fluid |

M | magnetic field parameter |

Ns | entropy generation number |

Pr | Prandtl number |

Re_{x} | local Reynolds number |

Ṡ_{gen}''' | volumetric entropy generation rate |

Ṡ_{o}''' | characteristic volumetric entropy generation rate |

T | temperature of the fluid |

T_{w}(x) | wall temperature |

T_{∞} | ambient temperature |

u | velocity component in x direction |

u_{w}(x) | velocity of the stretching surface |

v | velocity component in y direction |

x, y | Cartessian coordinates along the surface and normal to it respectively |

## Greeks Symbols

α | inclination of the stretching sheet with y-axis |

α* | thermal diffusitivity |

β | coefficient of thermal expansion |

η | similarity variable |

θ | dimensionless temperature |

μ | dynamic viscosity |

ν | kinematic viscosity |

ρ | density of the fluid |

σ | electric conductivity |

λ | mixed convection parameter/buoyancy parameter |

Ω | dimensionless temperature parameter |

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**Figure 2.**(

**a**) Variation of f’(η) with M; (

**b**) variation of θ(η) with M; (

**c**) variation of Ns with M; (

**d**) variation of Be with M.

**Figure 4.**(

**a**) Variation of f’(η) with λ; (

**b**) Variation of θ(η) with λ; (

**c**) Variation of Ns with λ; (

**d**) Variation of Be with λ.

$-\mathbf{R}{\mathbf{e}}_{\mathit{x}}^{1/2}{\mathit{C}}_{\mathit{f}}$ | $\mathbf{R}{\mathbf{e}}_{\mathit{x}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$ | |
---|---|---|

λ | Pr = 0.7; Ec = 1.0; α = π/4; M = 1 | |

0.0 0.5 1.0 1.5 | 1.4142 1.2427 1.0886 0.9439 | 0.5546 0.6976 0.7931 0.8974 |

M | Pr = 0.7; Ec = 1.0; α = π/4; λ = 0.5 | |

0.0 0.5 1.0 1.5 | 0.8155 0.9359 1.2427 1.6479 | 0.9293 0.8659 0.6976 0.4685 |

Pr | M = 1.0; λ = 0.2; α = π/4; Ec = 1.0 | |

0.3 0.7 1.2 1.5 | 1.3284 1.3424 1.3516 1.3552 | 0.3897 0.6219 0.8106 0.8962 |

Ec | M = 1.0; λ = 0.2; α = π/4; Pr = 1.2 | |

0.0 0.3 0.6 0.9 | 1.3603 1.3576 1.3551 1.3525 | 1.3873 1.2125 1.0393 0.8675 |

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**MDPI and ACS Style**

Afridi, M.I.; Qasim, M.; Khan, I.; Shafie, S.; Alshomrani, A.S.
Entropy Generation in Magnetohydrodynamic Mixed Convection Flow over an Inclined Stretching Sheet. *Entropy* **2017**, *19*, 10.
https://doi.org/10.3390/e19010010

**AMA Style**

Afridi MI, Qasim M, Khan I, Shafie S, Alshomrani AS.
Entropy Generation in Magnetohydrodynamic Mixed Convection Flow over an Inclined Stretching Sheet. *Entropy*. 2017; 19(1):10.
https://doi.org/10.3390/e19010010

**Chicago/Turabian Style**

Afridi, Muhammad Idrees, Muhammad Qasim, Ilyas Khan, Sharidan Shafie, and Ali Saleh Alshomrani.
2017. "Entropy Generation in Magnetohydrodynamic Mixed Convection Flow over an Inclined Stretching Sheet" *Entropy* 19, no. 1: 10.
https://doi.org/10.3390/e19010010