# Magnetohydrodynamic and Thermal Performance of Electrically Conducting Fluid along the Symmetrical and Vertical Magnetic Plate with Thermal Slip and Velocity Slip Effects

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## Abstract

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## 1. Introduction and Literature Review

## 2. The Flow Geometry and Mathematical Formulation

## 3. Stream Functions and Similarity Variables

## 4. Numerical Technique and Simulation

## 5. Matrix Form of Vector Equations

## 6. The Results and Discussions

## 7. Conclusions

- At each level of Pr, the temperature plot demonstrates significant temperature slip with significant variance. The maximum slip effect is obtained for a small value of Pr. This is because slip effects increase as Pr decreases due to lower viscosity of the fluid in the presence of velocity slip and thermal slip.
- It is concluded that the magnetic profile is improved at small values of $\delta $ = 0.1 and is minimal at large values of $\delta $ = 2.5. The good slip effect in the temperature graph is observed at each value of Pm along the utilized shape with good agreement.
- It is determined that the Prandtl number indicates the connection between thermal and momentum diffusivity. The magnetic effects are significantly detected exactly at the surface because of conducting processes, but they are zero for all values below the surface.
- In the domains of magnetic resonance imaging (MRI) resonance patterns, artificial heart wolves, interior heart cavities, and nanoburning systems, the present thermodynamic and magnetohydrodynamic issues are significant.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**(

**a**–

**c**): Physical sequence of ${f}^{\prime}\left(\eta \right),$ temperature $\theta $, magnetic/field ${g}^{\prime}\left(\eta \right)$ for Pr.

**Figure 3.**(

**a**–

**c**): Physical sequence of ${f}^{\prime}\left(\eta \right),$ temperature $\theta $, magnetic/field ${g}^{\prime}\left(\eta \right)$ for $\delta $.

**Figure 4.**(

**a**–

**c**): Physical sequence of ${f}^{\prime}\left(\eta \right),$ temperature $\theta $, magnetic/field ${g}^{\prime}\left(\eta \right)$ for Pm.

**Figure 5.**(

**a**–

**c**): Physical sequence of ${f}^{\prime}\left(\eta \right),$ temperature $\theta $, magnetic/field ${g}^{\prime}\left(\eta \right)$ for $\beta $.

**Figure 6.**(

**a**–

**c**): Physical sequence of ${f}^{\prime}\left(\eta \right),$ temperature $\theta $, magnetic/field ${g}^{\prime}\left(\eta \right)$ for $\lambda $.

**Table 1.**The numerical outcomes of skin friction ${\mathit{f}}^{\u2033}\left(\mathbf{0}\right),$ magneticflux $-{\mathit{g}}^{\u2033}\left(\mathbf{0}\right)$ and heat transfer $-{\mathit{\theta}}^{\prime}\left(\mathbf{0}\right)$ with some choices of $\mathit{\delta}=\mathbf{0.1},\mathbf{0.7},\mathbf{1.5},\mathbf{2.5}$.

$\mathit{\delta}=$ | ${\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{g}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ |
---|---|---|---|

$0.1$ | $1.989774405441794$ | $1.452028440838210$ | $0.381378974954840$ |

$0.7$ | $1.452028440838210$ | $0.402289137201107$ | $0.426522228527584$ |

$1.5$ | $1.025010274138276$ | $0.390450947704641$ | $0.449802888251293$ |

$2.5$ | $0.740080079359131$ | $0.383745624147633$ | $0.462288381003255$ |

**Table 2.**The numerical outcomes of skin friction ${\mathit{f}}^{\u2033}\left(\mathbf{0}\right),$ magneticflux $-{\mathit{g}}^{\u2033}\left(\mathbf{0}\right)$ and heat transfer $-{\mathit{\theta}}^{\prime}\left(\mathbf{0}\right)$ with some choices of $\mathit{\beta}=\mathbf{0.5},\mathbf{1.5},\mathbf{2.5},\mathbf{3.5}$.

$\mathit{\beta}$ | ${\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{g}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ |
---|---|---|---|

$0.5$ | $0.816233852426368$ | $1.152414669813850$ | $0.213488501588350$ |

$1.5$ | $0.875996451145098\text{}$ | $0.655892431891026$ | $0.216760098451247$ |

$2.5$ | $0.903041872750674$ | $0.502439580088601$ | $0.218263512085719$ |

$3.5$ | $0.919880101214037$ | $0.421244286666454$ | $0.219185569281904$ |

**Table 3.**The numerical outcomes of skin friction ${\mathit{f}}^{\u2033}\left(\mathbf{0}\right)$ various choices of $\mathrm{Pm}=\mathbf{1.0},\mathbf{10.0},\mathbf{100.0}$ for $\mathit{\delta}=\mathbf{7.0},\mathit{\beta}=\mathbf{0.1},\mathit{M}=\mathbf{3.7},Pr=\mathbf{7.0},\mathit{\lambda}=\mathbf{0.1}$ at the leading edge.

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## Share and Cite

**MDPI and ACS Style**

Alharbi, K.A.M.; Ullah, Z.; Jabeen, N.; Ashraf, M.
Magnetohydrodynamic and Thermal Performance of Electrically Conducting Fluid along the Symmetrical and Vertical Magnetic Plate with Thermal Slip and Velocity Slip Effects. *Symmetry* **2023**, *15*, 1148.
https://doi.org/10.3390/sym15061148

**AMA Style**

Alharbi KAM, Ullah Z, Jabeen N, Ashraf M.
Magnetohydrodynamic and Thermal Performance of Electrically Conducting Fluid along the Symmetrical and Vertical Magnetic Plate with Thermal Slip and Velocity Slip Effects. *Symmetry*. 2023; 15(6):1148.
https://doi.org/10.3390/sym15061148

**Chicago/Turabian Style**

Alharbi, Khalid Abdulkhaliq M., Zia Ullah, Nawishta Jabeen, and Muhammad Ashraf.
2023. "Magnetohydrodynamic and Thermal Performance of Electrically Conducting Fluid along the Symmetrical and Vertical Magnetic Plate with Thermal Slip and Velocity Slip Effects" *Symmetry* 15, no. 6: 1148.
https://doi.org/10.3390/sym15061148