# Numerical Study of Entropy Generation in Mixed MHD Convection in a Square Lid-Driven Cavity Filled with Darcy–Brinkman–Forchheimer Porous Medium

^{1}

^{2}

^{*}

## Abstract

**:**

^{−3}≤ Da ≤ 1) and for a range of Hartmann number (0 ≤ Ha ≤ 10

^{2}). It was found that entropy generation is affected by the variations of the considered dimensionless physical parameters. Moreover, the form drag related to the Forchheimer effect remains significant until a critical Hartmann number value.

## 1. Introduction

_{2}O

_{3}-water and TiO

_{2}-water nanofluid. They showed that the increase in the solid volume fraction leads to a decrease in the activity of the fluid motion and the fluid temperature. Hydrodynamic mixed convection in a lid-driven cavity heated from the top with a wavy bottom surface was numerically studied by Saha et al. [11], using the Galerkin finite element method. They observed that the variation in the Reynolds number affects the flow and thermal current activities. Mamourian et al. [12] presented the optimum conditions of the mixed convection heat transfer in a wavy surface square cavity filled with Cu-water nanofluid by utilizing the Taguchi method. They observed that the entropy generation and the mean Nusselt number decrease with the increase of the wavelength of the wavy surface for a given Richardson number. Akbar et al. [13] studied the effect of the entropy and the magnetic field in an endoscope filled with Cu-water. They found that the increase of the Brinkman and the Hartmann numbers tends to increase the temperature. Also, the increase of the Brinkman number induced the increase of the entropy generation. Ellahi et al. [14] investigated the shape effect of the nanoparticles suspended in HFE-7100 over a wedge in the mixed convection. They showed that the entropy generation approaches zero near to the free stream region. Akbar et al. [15] presented the effect of the entropy and the magnetic field on the peristaltic flow of the copper water fluid in an asymmetric horizontal channel. They showed that the entropy generation number increases with the Brinkman number. In the middle of the channel, the flow velocity decreases when the magnetic Reynolds number increases.

## 2. Mathematical Formulation

_{0}and is maintained at a constant temperature ${\mathsf{\theta}}_{\mathrm{c}}$, whereas the bottom wall is maintained at a constant temperature ${\mathsf{\theta}}_{\mathrm{h}}$ (${\mathsf{\theta}}_{\mathrm{c}}$ < ${\mathsf{\theta}}_{\mathrm{h}}$). All other remaining walls are adiabatic and insulated. A uniform magnetic field of strength (B

_{0}) is applied in the direction normal to the cavity cross section. The physical properties of the fluid are considered to be constant, except the density variation which is applied in the Boussinesq approximation for the buoyancy term $(\mathsf{\rho}={\mathsf{\rho}}_{0}[1-{\mathsf{\beta}}_{\mathsf{\theta}}(\mathsf{\theta}-{\mathsf{\theta}}_{0})])$. ${\mathsf{\rho}}_{0}$ is the fluid density at temperature ${\mathsf{\theta}}_{0}$ and ${\mathsf{\beta}}_{\mathsf{\theta}}$ is the thermal expansion coefficient.

_{0}, the temperature difference $\left(\mathsf{\theta}-{\mathsf{\theta}}_{0}\right)$ between the two horizontal walls of the cavity, ${\mathsf{\rho}}_{0}{\mathrm{U}}_{0}^{2}$ and $\mathrm{H}/{\mathrm{U}}_{0}$, respectively. Thus $\mathrm{Ra}={\mathrm{g}\mathsf{\beta}}_{\mathsf{\theta}}\left({\mathsf{\theta}}_{\mathrm{h}}-{\mathsf{\theta}}_{\mathrm{c}}\right){\mathrm{L}}^{3}/\left(\mathsf{\nu}\mathsf{\alpha}\right)$ is the Rayleigh number, $\mathrm{Re}={\mathrm{U}}_{0}\mathrm{H}/\mathsf{\nu}$ is the Reynolds number, $\mathrm{Da}=\mathrm{K}/{\mathrm{H}}^{2}$ is the Darcy number, $\mathrm{Ha}={\mathrm{B}}_{0}\mathrm{H}\sqrt{{\mathsf{\sigma}}_{0}/\mathsf{\mu}}$ is the Hartman number, $\mathrm{Ri}=\mathrm{Ra}/\left(\mathrm{PeRe}\right)$ is the Richardson number, $\mathrm{Pe}=\mathrm{RePr}$ is the Peclet number and $\mathrm{Pr}=\mathsf{\nu}/\mathsf{\alpha}$ is the Prandtl number. g is the gravity constant.

## 3. Second Law Formulation

_{l}) can be calculated by following the equation of (Woods [16]):

## 4. Numerical Method

^{3}, 10

^{4}and 10

^{5}. As can be seen in Table 1 and Table 2, a good agreement was obtained.

## 5. Results and Discussion

^{−3}to 1, from 0 to 100, from 10 to 50, from 10

^{4}to 10

^{5}and from 0.25 to 0.87 respectively. The porosity (ɛ) of the media is fixed at 0.5. The modified Brinkman number is fixed at 5 × 10

^{−7}.

^{4}and 10

^{5}. As shown in Figure 7, at a fixed Hartmann number, the entropy generation generally increases as the Rayleigh number increases. This is due to the improvement of the convection in the porous cavity which will, in turn, increase the thermal and the velocity gradients and consequently the entropy generation.

_{c}= 25, for which the form drag linked to the non-linear velocity term is important. The second is similar to the Darcy regime for (Ha) greater than 50. In this case, the fluid undergoes a deceleration due to the dominant effect of the Lorentz force and consequently the form drag becomes insignificant. The third regime, for (Ha) between 25 and 50, is transitional from the first where the form drag is dominant to the second where the Lorentz force is dominant.

## 6. Conclusions

- (1)
- The entropy generation rate decreases with the decrease of Darcy number and the increase of Hartmann number.
- (2)
- The flow structure strongly depends on the Hartmann number.
- (3)
- At a fixed Hartmann number, the entropy generation increases with the increase of the Reynolds number.
- (4)
- At a fixed and relatively high Reynolds number, the entropy generation remains constant for a Hartmann number smaller than 50.
- (5)
- The Forchheimer effect is significant for a Hartmann number between 0 and 25 and negligible for a Hartmann number greater than 50.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

B_{0} | uniform magnetic field |

Br | Brinkman number $\mathrm{Br}={\mathsf{\mu}\mathrm{U}}_{0}^{2}/\left({\mathsf{\lambda}}_{\mathrm{e}}\mathsf{\Delta}\mathrm{T}\right)$ |

Br* | modified Brinkman number ${\mathrm{Br}}^{*}=\mathrm{Br}/\Omega $ |

C_{p} | specific heat (J∙Kg∙K^{−1}) |

Da | Darcy numberk/H^{2} |

g | gravitational acceleration (m·s^{−2}) |

J_{k} | diffusion flux (k = U, V, T, C) |

K | permeability of the medium (m^{2}) |

Nu | Nusselt number |

P | dimensionless pressure |

Pr | Prandtl number |

Ra | Rayleigh number |

Re | Reynolds number |

Ri | Richardson number |

U | dimensionless velocity vector |

U, V | dimensionless velocity components |

T | Dimensionless temperature $T=\left(\mathsf{\theta}-{\mathsf{\theta}}_{\mathrm{c}}\right)/\left({\mathsf{\theta}}_{\mathrm{h}}-{\mathsf{\theta}}_{\mathrm{c}}\right)$ |

X, Y | dimensionless Cartesian coordinates |

Greek letters | |

τ | dimensionless time |

α | thermal diffusivity ${\mathsf{\lambda}}_{\mathrm{e}}/{\left({\mathsf{\rho}\mathrm{C}}_{\mathrm{p}}\right)}_{\mathrm{f}}$ |

β_{ɵ} | thermal expansion coefficient |

ρ | mass density |

Ɵ | temperature |

μ | dynamic viscosity |

ν | kinematic viscosity (m^{2}·s^{−1}) |

ε | porosity of the media |

$\nabla $ | Nabla vector |

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**Figure 2.**(

**a**) Streamlines; (

**b**) Isotherms; (

**c**) Isentropic lines (Re = 10, Ra = 10

^{5}, Da = 10

^{−2}).

**Figure 3.**Average Nusselt number versus Hartmann number for different Darcy numbers (Ra = 10

^{5}, Re = 10).

**Figure 4.**Entropy generation rate versus Hartmann number for different Darcy numbers (Ra = 10

^{5}, Re = 10).

**Figure 5.**Average Nusselt number versus Hartmann number for different Reynolds numbers (Ra = 10

^{5}, Da = 10

^{−2}).

**Figure 6.**Entropy generation rate versus Hartmann number for different Reynolds numbers (Ra = 10

^{5}, Da = 10

^{−2}).

**Figure 7.**Entropy generation rate versus Hartmann number for Ra = 10

^{4}and Ra = 10

^{5}(Re = 10, Da = 10

^{−2}).

**Figure 8.**Entropy generation rate versus Hartmann number for three Forchheimer parameter values of 0.25, 0.4 and 0.87 (Ra = 10

^{5}, Da = 10

^{−2}).

Ra | 10^{3} | 10^{4} | 10^{5} |
---|---|---|---|

Present study | 1.099 | 2.295 | 4.664 |

Davis (1983) | 1.118 | 2.243 | 4.519 |

Nithyadevi et al. (2009) | 1.123 | 2.304 | 4.899 |

Ra | ε | Present Results | Muthtamilselvan et al. (2009) | June (2001) |
---|---|---|---|---|

10^{5} | 0.4 | 2.9482 | 2.9004 | 2.9435 |

0.6 | 3.1310 | 3.0893 | 3.0877 |

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**MDPI and ACS Style**

Bouabda, R.; Bouabid, M.; Ben Brahim, A.; Magherbi, M.
Numerical Study of Entropy Generation in Mixed MHD Convection in a Square Lid-Driven Cavity Filled with Darcy–Brinkman–Forchheimer Porous Medium. *Entropy* **2016**, *18*, 436.
https://doi.org/10.3390/e18120436

**AMA Style**

Bouabda R, Bouabid M, Ben Brahim A, Magherbi M.
Numerical Study of Entropy Generation in Mixed MHD Convection in a Square Lid-Driven Cavity Filled with Darcy–Brinkman–Forchheimer Porous Medium. *Entropy*. 2016; 18(12):436.
https://doi.org/10.3390/e18120436

**Chicago/Turabian Style**

Bouabda, Rahma, Mounir Bouabid, Ammar Ben Brahim, and Mourad Magherbi.
2016. "Numerical Study of Entropy Generation in Mixed MHD Convection in a Square Lid-Driven Cavity Filled with Darcy–Brinkman–Forchheimer Porous Medium" *Entropy* 18, no. 12: 436.
https://doi.org/10.3390/e18120436