# Thermodynamic Analysis of Gravity-driven Liquid Film along an Inclined Heated Plate with Hydromagnetic and Viscous Dissipation Effects

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## Abstract

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## Introduction

## Problem formulation and analytical solution

## Results and discussion

**Figure 4.**Comparison between the temperature profiles across the liquid film without the hydomagnetic effect, with hydromagnetic effect and with hydromagnetic and viscous dissipation effects.

**Figure 6.**Temperature profiles as function of the transverse distance at different Brinkman numbers.

**Figure 7.**Entropy generation number as function of thetransverse distance at different Hartman number.

**Figure 8.**Entropy generation number as function of thetransverse distance at different Brinkman numbers.

**Figure 9.**Entropy generation number as function of the transversedistance at different dimensionless group.

## Conclusion

## Nomenclature

$A$ | area, (m ^{2}) |

$B$ | magnetic induction, (Wb.m ^{-2}) |

$Br$ | Brinkman number,
$\mu {u}_{m}^{2}{C}_{P}^{2}/\lambda \Delta T$ |

${C}_{P}$ | specific heat, (J.kg ^{-1}.K^{-1}) |

$Ha$ | Hartman number,
$B\delta \sqrt{\sigma /\mu}$ |

${N}_{B}$ | entropy generation number, magnetic induction |

${N}_{C}$ | entropy generation, axial conduction |

${N}_{F}$ | entropy generation, fluid friction |

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

${N}_{Y}$ | entropy generation number, transverse conduction |

$Pe$ | Peclet number,$\rho {u}_{m}{C}_{P}\delta /\lambda $ |

$q$ | wall heat flux, (W.m ^{-2}) |

$Q$ | liquid mass flow rate, (kg.m ^{-1}.s^{-1}) |

${Q}_{0}$ | liquid mass flow rate in absence of the magnetic field, (kg.m ^{-1}.s^{-1}) |

$Re$ | Reynolds magnetic number,
$\eta \sigma {u}_{m}\delta $ |

${S}_{G}$ | entropy generation rate, (W.m ^{-3}.K^{-1}) |

$T$ | temperature, (K) |

$u$ | axial velocity, (m.s ^{-1}) |

$U$ | dimensionless axial velocity |

$x$ | axial distance, (m) |

$X$ | dimensionless axial distance |

$y$ | transverse distance, (m) |

$Y$ | dimensionless transverse distance |

## Greek symbols

$\alpha $ | scalar constant |

$\delta $ | thickness of the liquid film (m) |

$\Delta T$ | reference temperature difference,
$\Delta T=\frac{q\delta}{\lambda}$ |

$\eta $ | magnetic permeability, (H.m ^{-1}) |

$\mu $ | dynamic viscosity, (kg.m ^{-1}.s^{-1}) |

$\lambda $ | thermal conductivity, (W.m ^{-1}.K^{-1}) |

$\Theta $ | dimensionless temperature,
$\left(T\left(x,y\right)-{T}_{0}\right)/\Delta T$ |

$\Omega $ | dimensionless temperature difference,
$\Delta T/{T}_{0}$ |

$\rho $ | density of the fluid, (kg.m ^{-3}) |

$\sigma $ | electric conductivity, (Ω ^{-1}.m^{-1}) |

## Subscripts

$b$ | bulk value |

$m$ | maximum value |

$0$ | inlet value, reference value |

## References

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/e8040188

**AMA Style**

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(4):188-199.
https://doi.org/10.3390/e8040188

**Chicago/Turabian Style**

Aïboud-Saouli, Soraya, Salah Saouli, Noureddine Settou, and Nouredine Meza.
2006. "Thermodynamic Analysis of Gravity-driven Liquid Film along an Inclined Heated Plate with Hydromagnetic and Viscous Dissipation Effects" *Entropy* 8, no. 4: 188-199.
https://doi.org/10.3390/e8040188