# MHD Flow of a Hybrid Nano-Fluid in a Triangular Enclosure with Zigzags and an Elliptic Obstacle

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}/EG hybrid nano-fluid inside a porous annulus between a zigzagged triangle and different cylinders and under the influence of an inclined magnetic field. The effect of numerous factors is detailed, including Rayleigh number (103 ≤ Ra ≤ 106), Hartmann number (0 ≤ Ha ≤ 100), volume percent of the nano-fluid (0.02 ≤ ϕ ≤ 0.08), and the rotating speed of the cylinder (−4000 ≤ w ≤ 4000). Except for the Hartmann number, which decelerates the flow rate, each of these parameters has a positive impact on the thermal transmission rate.

## 1. Introduction

_{2}/EG hybrid nano-fluid and a spinning cylinder. The findings of this research will aid in achieving a better understanding of the effect of different cylinders inside the cavity, as well as expanding the contribution of sophisticated triangular geometries, which already have wide spread applications in industry [54,55,56] and are unquestionably adaptable in real-world applications, such as cooling electronic devices, solar heating systems, heat exchangers, solar collectors, etc. Aiming for high heat transmission efficiency under unusual settings, this work provides a significant contribution to future applications.

## 2. Physical Model

_{0}, detailed in Table 1, and filled with the Cu-TiO

_{2}/EG hybrid nano-fluid featured in Table 2. The right-angled wall is cold with a fixed temperature, T

_{c}, while the inclined wall is subject to a heating source and set as T

_{h}. The base of the triangle and the cylinder are both adiabatic.

_{c};

_{h};

## 3. Grid Test

^{5}and ϕ = 0.04 to calculate the Nusselt number. Table 3 shows the obtained results, which reveal that the deviations in Nusselt number decreased as the mesh quality increased, and we can conclude that the mesh with the highest quality, “extra fine” ensured accurate outcomes. Therefore, the extra-fine mesh presented in Figure 2 was selected for our study.

## 4. Formulation of the Problem

#### 4.1. Equations

_{avg}is the average temperature:

- Dimensionless numbers

- Dimensionless variables

#### 4.2. Validation

#### 4.3. Properties of the Hybrid Nano-Fluid

_{2}nanoparticles, respectively, were obtained from [63,64] and can be calculated as follows:

## 5. Results and Discussion

^{3}≤ Ra ≤ 10

^{6}), to study the convective heat transfer in the laminar regime and explore its features near the transition mode; Hartmann number (0 ≤ Ha ≤ 100), in order to investigate the relation between magnetic-field strength heat-transfer efficiency; and the volume fraction of the hybrid nano-fluid (0.02 ≤ ϕ ≤ 0.08), to evaluate the presence of nanoparticle in a porous medium with constant properties: Darcy number, Da = 0.1; porosity, ε=0.4. Additionally the following geometrical factors are discussed: the impact of the rotation of the internal cylinder with a speed, w, of (−4000 ≤ w ≤ 4000), as well as the placement of the cylinder and several shaped obstacles (square, circle, elliptic, and triangle).

#### 5.1. Impact of the Nano-Fluid Volume Fraction

_{2}, that present enhanced thermo-physical characteristics compared to classical fluids, as presented in Table 4, particularly their thermal conductivity. It worth mentioning that these properties improve the thermal conductivity of the hybrid nano-fluid and also increment the surface area of the nanoparticles [58]. Therefore, Nu

_{avg}is proportional to the presence and the volume fraction of the hybrid nano-fluid, and such correlation contributes to convective transfer.

#### 5.2. Impact of Rayleigh Number

_{max}at Ra = ${10}^{4}$ and ten times greater than Ψ

_{max}at Ra = ${10}^{5}.$ These results show that approaching the critical Rayleigh value (almost ${10}^{9}$) engenders significant values of heat transfer.

#### 5.3. Impact of Hartmann Number

#### 5.4. Impact of the Geometrical Features

#### 5.4.1. Effect of Cylinder Placement

#### 5.4.2. Effect of the Rotation of the Cylinder

#### 5.4.3. Effect of the Different Obstacles

_{avg}values when compared to the other cylinders. The geometrical features of the triangle and the uniform space provided around it make it easier for the hybrid nano-fluid to disperse, which can help amplify and alter the average Nu number and therefore convective transfer [68].

## 6. Conclusions

_{2}/ EG hybrid nano-fluid under magnetic-field influence reveal that incrementing the concentration of the working fluid from 0.02 to 0.08 improves the Nusselt number by 19%. Enhancing the Rayleigh number also accelerates the flow and strengthens the velocity field.

- Rayleigh number and the volume fraction of the nanoparticles can be considered crucial features in modulating convection.
- The existence of a magnetic field, and therefore increasing Hartmann number, restricts heat transfer.
- Thermal transmission can be improved by using triangular obstacles.
- The angular velocity of the cylinder can alter the efficiency of the convective flow.
- The location of the obstacle is a key parameter to adjust the thermal transfer.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

u, v | Velocity components (m·s^{−1}) |

U, V | Dimensionless velocity components |

x, y | Cartesian coordinates (m) |

X, Y | Dimensionless Cartesian coordinates |

p | Pressure (N·m^{−2}) |

P | Dimensionless pressure |

ρ | Density (Kg·m^{−3}) |

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

T | Temperature (K) |

T_{avg} | Average temperature (K) |

θ | Dimensionless temperature |

α | Thermal diffusivity (m²·s^{−1}) |

υ | Kinematic viscosity (m²·s^{−1}) |

K | Permeability (H·m^{−1}) |

ε | Porosity |

σ | Electric conductivity (Ohm m)^{−1} |

B_{0} | Magnetic field density (Tesla) |

k | Thermal conductivity ratio (W K^{−1} m^{−1}) |

Cp | Specific heat (J K^{−1} Kg^{−1}) |

β | Thermal expansion (K^{−1}) |

µ | Dynamic viscosity (Kg·m^{−1}·s^{−1}) |

ϕ | Volume fraction of the nanoparticles |

γ | Inclination angle of the magnetic field |

w | Velocity of rotation(rad/s) |

Ψ | Stream function |

L | Length of the enclosure (m) |

Subscripts | |

h | Hot |

c | Cold |

EG | Ethylene glycol |

Cu | Copper |

TiO_{2} | Titanium dioxide |

MHD | Magneto-hydrodynamic |

Nf | Nano-fluids |

hnf | Hybrid nano-fluid |

Bf | Base fluid |

np | Nanoparticle |

Max | Maximum |

Fc | Forcheimer coefficient |

Ra | Rayleigh |

Nu | Nusselt |

Ha | Hartmann |

Da | Darcy |

Pr | Prandtl |

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**Figure 3.**Isotherm and streamline comparison: previous work (left) and current work (right) for Ha = 0 (

**a**), Ra = 10

^{4}(

**b**), and Ra = 10

^{6}.

Ha | 2550 | 100 | |

B_{0} (Tesla) | 1351 | 2702 | 5404 |

Mesh | Extra Coarse | Coarse | Fine | Extra Fine |
---|---|---|---|---|

Maximum element size (m) | 0.13 | 0.067 | 0.035 | 0.013 |

Minimum element size (m) | 0.005 | 0.003 | 0.001 | 0.00015 |

Curvature factor | 0.8 | 0.4 | 0.3 | 0.25 |

Growth rate | 1.3 | 1.2 | 1.13 | 1.08 |

Number of elements | 840 | 1984 | 3944 | 22184 |

Average quality | 0.7110 | 0.7736 | 0.7803 | 0.8003 |

Mesh Quality | Nu | Nu Deviation % |
---|---|---|

0.7110 | 3.9 | 12.05% |

0.7736 | 4.37 | 7.09% |

0.7803 | 4.68 | 0.64% |

0.8003 | 4.71 | / |

Cu | TiO_{2} | EG | |
---|---|---|---|

C_{P} (J. K^{−1}·Kg^{−1}) | 385 | 686.2 | 2415 |

ρ(Kg·m^{−3}) | 8933 | 4250 | 1114 |

k(W. K^{−1}·m^{−1}) | 401 | 8.95 | 0.252 |

$\beta $ (K^{−1}) | 1.67 × ${10}^{-5}$ | 0.9 × ${10}^{-5}$ | 57 × ${10}^{-5}$ |

$\sigma $(Ohm·m)^{−1} | 5.96 × ${10}^{-7}$ | 2.38 × ${10}^{-6}$ | 5.5 × ${10}^{-6}$ |

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Chabani, I.; Mebarek-Oudina, F.; Ismail, A.A.I.
MHD Flow of a Hybrid Nano-Fluid in a Triangular Enclosure with Zigzags and an Elliptic Obstacle. *Micromachines* **2022**, *13*, 224.
https://doi.org/10.3390/mi13020224

**AMA Style**

Chabani I, Mebarek-Oudina F, Ismail AAI.
MHD Flow of a Hybrid Nano-Fluid in a Triangular Enclosure with Zigzags and an Elliptic Obstacle. *Micromachines*. 2022; 13(2):224.
https://doi.org/10.3390/mi13020224

**Chicago/Turabian Style**

Chabani, Ines, Fateh Mebarek-Oudina, and Abdel Aziz I. Ismail.
2022. "MHD Flow of a Hybrid Nano-Fluid in a Triangular Enclosure with Zigzags and an Elliptic Obstacle" *Micromachines* 13, no. 2: 224.
https://doi.org/10.3390/mi13020224