# Influence of Ramped Wall Temperature and Ramped Wall Velocity on Unsteady Magnetohydrodynamic Convective Maxwell Fluid Flow

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## Abstract

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## 1. Introduction

## 2. Mathematical Modeling

## 3. Analytical Solutions

## 4. Special Cases

#### 4.1. Case 1

#### 4.2. Case 2

## 5. Parametric Study

## 6. Conclusions

- Momentum boundary layer thickness increases with increasing values of the parameters Gr (due to the strong buoyancy force) and K (due to the decrease in viscous force). Meanwhile, it reduces with increasing values of M (due to the strong drag force) and $\lambda $ (due to the increase in resistance).
- An elevation in the relaxation time $\lambda $ leads to a decrease in velocity on the plate (in terms of skin friction).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\rho $ | Fluid density |

$\lambda $ | Maxwell parameter |

$\tau $ | Dimensionless time |

$\mu $ | Dynamic viscosity |

u | Fluid velocity |

$\eta $ | Space variable |

$\sigma $ | Electrical conductivity |

$\beta $ | Coefficient of thermal expansion |

$\varphi $ | Porosity parameter |

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

g | Acceleration due to gravity |

T | Fluid temperature |

${T}_{\infty}$ | Free stream temperature |

S | Shear stress |

k | Thermal conductivity |

${c}_{p}$ | Heat capacitance |

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

Gr | Grashof number |

M | Magnetic parameter |

K | Dimensionless porosity parameter |

Pr | Prandtl number |

s | Laplace parameter |

$\mathcal{L}$ | Laplace transform coefficient |

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**Figure 2.**Effect of different values of M on the momentum profile for Gr $=1$, $\lambda =0.2$ and K $=0.5$.

**Figure 3.**Effect of different values of Gr on momentum profile for M $=2$, $\lambda =0.2$ and K $=0.5$.

**Figure 4.**Effect of different values of K on momentum profile for Gr $=1$, $\lambda =0.2$ and M $=2$.

**Figure 5.**Effect of different values of Pr on momentum profile for Gr $=1$, $\lambda =0.2$, M $=2$, and K $=0.5$.

**Figure 6.**Effect of different values of $\lambda $ on momentum profile for Gr $=1$, M $=2$ and K $=0.5$.

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**MDPI and ACS Style**

Anwar, T.; Kumam, P.; Watthayu, W.; Asifa.
Influence of Ramped Wall Temperature and Ramped Wall Velocity on Unsteady Magnetohydrodynamic Convective Maxwell Fluid Flow. *Symmetry* **2020**, *12*, 392.
https://doi.org/10.3390/sym12030392

**AMA Style**

Anwar T, Kumam P, Watthayu W, Asifa.
Influence of Ramped Wall Temperature and Ramped Wall Velocity on Unsteady Magnetohydrodynamic Convective Maxwell Fluid Flow. *Symmetry*. 2020; 12(3):392.
https://doi.org/10.3390/sym12030392

**Chicago/Turabian Style**

Anwar, Talha, Poom Kumam, Wiboonsak Watthayu, and Asifa.
2020. "Influence of Ramped Wall Temperature and Ramped Wall Velocity on Unsteady Magnetohydrodynamic Convective Maxwell Fluid Flow" *Symmetry* 12, no. 3: 392.
https://doi.org/10.3390/sym12030392