# Magnetohydrodynamic Effects on Third-Grade Fluid Flow and Heat Transfer with Darcy–Forchheimer Law over an Inclined Exponentially Stretching Sheet Embedded in a Porous Medium

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

#### Flow Analysis

## 3. Solution Methodology

#### 3.1. Similarity Formulation

#### 3.2. Solution Technique

## 4. Results and Discussion

#### 4.1. Influence of Involved Parameters on Velocity Profile ${f}^{\prime}\left(\eta \right),$ Temperature Profile $\theta \left(\eta \right)$, and Mass Concentration $\varphi \left(\eta \right)$

#### 4.2. Influence of Parameters on Skin Friction Coefficient $R{e}^{1/2}{C}_{f}$, Nusselt Number $R{e}^{-1/2}Nu$, and Sherwood Number $R{e}^{-1/2}Sh$

## 5. Conclusions

- ${f}^{\prime}\left(\eta \right)$ rises as $Ri,K$ and $\beta $ rises, but the reverse scenario is noted against the elevating values of $L,{K}^{*},Fr,$ and $Pr$.
- $\theta \left(\eta \right)$ is elevated as $N$ and $Fr$ are raised, but the reverse scenario is noted owing to increase in $\beta $ and $Pr$.
- $\varphi \left(\eta \right)$ is intensified as ${K}^{*}$ is augmented but falls down as $\beta $ and $K$ are elevated.
- It is concluded that the skin friction coefficient increases with increasing values of ${K}^{*}$ and $Fr$, but the opposite trend is seen for increasing magnitudes of $\beta $.
- The Nusselt number is elevated against elevating values of $\beta $ and the opposite action is viewed for increasing ${K}^{*}$ and $Fr$.
- The Sherwood number fleshes out an increasing attitude for increasing $\beta $ and shows the decreasing trend for augmenting ${K}^{*}$ and $Fr$.
- The present results for such a special case are compared with the previously published results and there is nice agreement between them, which validates the accuracy of the current results.
- In the future, this model can be extended to nanofluid and hybrid nanofluid with different fluid characteristics.
- This can also be extended to analyze the impact of reduced gravity on this model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Nomenclature | $Fr$ Local inertial coefficient |

${T}_{\infty}\left(K\right)$ Ambient temperature | ${D}_{m}\left({m}^{2}{s}^{-1}\right)$ Mass diffusion coefficient |

${C}_{w}$ Wall concentration | $Nu$ Nussle number |

${C}_{\infty}$ Ambient concentration | ${T}_{m}$ Mean temperature of fluid |

${T}_{w}$ Wall temperature | Greeks |

$\alpha $ Angle of inclination | ${\beta}_{C}\left({K}^{-1}\right)$ Volumetric coefficient concentration expansion |

$u,v$ Velocity components in $x$ and $y$ directions | $\theta $ Dimensionless temperature |

$N$ Buoyancy ratio parameter | ${\beta}_{T}\left({K}^{-1}\right)$ Volumetric coefficient thermal expansion |

$K$ Viscoelastic parameter | $\varphi $ Dimensionless mass concentration |

$C\left(kg{m}^{-3}\right)$ Concentration in boundary layer | $\kappa \left(W{m}^{-1}.{K}^{-1}\right)$ Thermal conductivity |

$Sc$ Schmidt number | $\mu \left(Pa.s\right)$ Dynamic viscosity |

$F$ Coefficient of inertia | $\beta $ Third-grade fluid parameter |

${C}_{P}\left(Jk{g}^{-1}{K}^{-1}\right)$ Specific heaconstant pressure | $\sigma $ Electrical conductivity |

$x,y$ Coordinates | ${\alpha}_{m}\left(m{s}^{-1}\right)$Thermal diffusivity |

$Sh$ Sherwood number | $\nu \left({m}^{2}{s}^{-1}\right)$ Kinematic viscosity |

$L$ Cross viscous parameter | ρ$\left(kg{m}^{-3}\right)$ Fluid density |

${C}_{f}$ Skin friction | ${\alpha}_{1},{\alpha}_{2},{\beta}_{3}$ Material moduli |

$x,y$ Coordinates | Subscripts |

$Ri$ Richardson number | $\infty $ Ambient conditions |

${C}_{b}$ Drag Coefficient | $w$ Wall conditions |

$Re$ Reynolds number | |

T$\left(K\right)$ Fluid temperature in boundary layer | |

$Pr$ Prandtl number | |

$g\left(m{s}^{-2}\right)$ Gravitational acceleration | |

${K}^{*}$ Permeability parameter | |

$Gr$ Grashof number | |

${K}_{o}$ Porous medium permeability |

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**Table 1.**Comparison of results for $-{\theta}^{\prime}\left(0\right)$ when $Ri=0,N=0,Sc=0,M=0,\beta =0,L=0,K=0,{K}^{*}=0,Fr=0$.

Pr | Magyari and Keller [31] | Present Study |
---|---|---|

1.0 | 0.9547 | 0.9551 |

3 | 1.8691 | 1.8121 |

5 | 2.5001 | 2.5577 |

10 | 3.6604 | 3.6868 |

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**MDPI and ACS Style**

Abbas, A.; Jeelani, M.B.; Alharthi, N.H.
Magnetohydrodynamic Effects on Third-Grade Fluid Flow and Heat Transfer with Darcy–Forchheimer Law over an Inclined Exponentially Stretching Sheet Embedded in a Porous Medium. *Magnetochemistry* **2022**, *8*, 61.
https://doi.org/10.3390/magnetochemistry8060061

**AMA Style**

Abbas A, Jeelani MB, Alharthi NH.
Magnetohydrodynamic Effects on Third-Grade Fluid Flow and Heat Transfer with Darcy–Forchheimer Law over an Inclined Exponentially Stretching Sheet Embedded in a Porous Medium. *Magnetochemistry*. 2022; 8(6):61.
https://doi.org/10.3390/magnetochemistry8060061

**Chicago/Turabian Style**

Abbas, Amir, Mdi Begum Jeelani, and Nadiyah Hussain Alharthi.
2022. "Magnetohydrodynamic Effects on Third-Grade Fluid Flow and Heat Transfer with Darcy–Forchheimer Law over an Inclined Exponentially Stretching Sheet Embedded in a Porous Medium" *Magnetochemistry* 8, no. 6: 61.
https://doi.org/10.3390/magnetochemistry8060061