# Magneto-Hydrodynamic Flow above Exponentially Stretchable Surface with Chemical Reaction

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

- (1)
- The negligible effect of gravitation is considered.
- (2)
- Thermal conductivity and specific heat are not depending on temperature.
- (3)
- A uniform magnetic field ${B}_{0}$ is applied normally on the stretching sheet.
- (4)
- The flow is laminar and has constant density.
- (5)
- The adjustable chemical reaction is considered, and the sheet is permeable.
- (6)
- The induced electric and magnetic fields are neglected.

## 3. Numerical Scheme

## 4. Results and Discussion

#### 4.1. Velocity Profiles

#### 4.2. Temperature Profile

#### 4.3. Concentration Profile

#### 4.4. Skin Frictions, Nusselt and Sherwood Numbers

## 5. Conclusions

- The velocity profile increases by a rise in mass and temperature convective parameter while it decays with a rise in porosity and magnetic parameter.
- The temperature profile decays by a rise in temperature exponent and Prandtl number while it increases by a rise in porosity, thermophoresis, and magnetic parameter.
- The concentration profile increases by rising in porosity while it decays by rising in concentration exponent, chemical reaction, and Schmidt number.
- Skin frictions, Nusselt number, and Sherwood number decay by an increase in porosity and magnetic parameter.
- Chemical reaction and thermophoresis parameters increase the Sherwood number.
- Nusselt number increases by rising in mass and temperature convective parameter, temperature exponent, thermophoresis parameter, and concentration exponent.
- Low skin friction and Nusselt and Sherwood number are observed for water as compared to ethanol.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${D}_{B},{D}_{T}$ | Brownian and Thermophoresis diffusion |

${U}_{0},{V}_{0},{T}_{0},{C}_{0}$ | Constants |

$M$ | Magnetic parameter |

$T,C$ | Temperature and concentration |

$N{u}_{x},S{h}_{x}$ | Nusselt and Sherwood number |

$\gamma ,K$ | Chemical reaction, porous medium parameter |

$Pr,Sc$ | Prandtl and Schmidt number |

$\beta $ | Stretching ratio parameter |

${q}_{w},{q}_{m}$ | Heat and mass flux |

$\alpha $ | Temperature diffusivity |

MHD | Magneto-hydrodynamic |

${G}_{r},{G}_{c}$ | Temperature and mass Grashof number |

${C}_{{f}_{x}},{C}_{{f}_{y}}$ | Skin friction coefficients |

${k}_{f}$ | Thermal conductivity |

${\tau}_{wx},{\tau}_{wy}$ | Wall shear stress |

${f}^{\prime},{g}^{\prime}$ | Dimensionless velocity |

$\theta ,\varphi $ | Temperature, and concentration |

${\lambda}_{T},{\lambda}_{M}$ | Temperature and mass convective parameter |

$A$ | Temperature exponent |

${N}_{b},{N}_{t}$ | Brownian diffusion and Thermophoresis parameter |

$\nu $ | Kinematic viscosity |

$L$ | Reference length |

3D | Three dimensional |

$B$ | Concentration exponent |

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**Figure 2.**(

**a**): The influence of $K$ on the velocity profile ${f}^{\prime}\left(\eta \right)$; (

**b**): The influence of $K$ on the velocity profile ${g}^{\prime}\left(\eta \right);$ (

**c**): The influence of $M$ on velocity profile ${g}^{\prime}\left(\eta \right)$; (

**d**): The influence of $M$ on velocity profile ${g}^{\prime}\left(\eta \right)$.

**Figure 3.**(

**a**) The influence of $M$ on temperature profile $\theta \left(\eta \right)$; (

**b**) The influence of $A$ on temperature profile $\theta \left(\eta \right)$; (

**c**) The influence of $K$ on temperature profile $\theta \left(\eta \right)$; (

**d**) The influence of $Pr$ on temperature profile $\theta \left(\eta \right)$.

**Figure 4.**(

**a**) The influence of $Sc$ on concentration profile $\varphi \left(\eta \right)$; (

**b**) The influence of $\gamma $ on concentration profile $\varphi \left(\eta \right)$; (

**c**) The influence of $B$ on concentration profile $\varphi \left(\eta \right)$.

**Figure 5.**(

**a**) Skin friction versus porosity $K$ and magnetic field $M$; (

**b**) Nusselt number versus porosity $K$ and magnetic field $M$; (

**c**) Sherwood number versus porosity $K$ and magnetic field $M$.

**Table 1.**Comparison of present outcomes for skin friction along x-axis for pure fluid with ${\lambda}_{M}=1,Pr=6.2,\gamma =0.2,A=0.5,B=0.8$ with literature.

$\mathit{R}{\mathit{e}}_{\mathit{x}}{}^{1/2}\mathit{C}{\mathit{f}}_{\mathit{x}}$ | ||
---|---|---|

M | Present Results | Nadeem et al. [47] |

$0$ | $1.16353$ | $1.1637$ |

$10$ | $3.37616$ | $3.3772$ |

$100$ | $10.07636$ | $10.07647$ |

**Table 2.**Influence of different study parameters on $C{f}_{x},C{f}_{y},N{u}_{x},$ and $S{h}_{x}$ for Water.

$\mathit{M}$ | $\mathit{K}$ | ${\mathit{\lambda}}_{\mathit{M}}$ | ${\mathit{\lambda}}_{\mathit{T}}$ | $\mathit{A}$ | $\mathit{B}$ | $\mathit{S}\mathit{c}$ | $\mathit{\gamma}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{b}}$ | ${\mathit{N}}_{\mathit{t}}$ | $\mathit{R}{\mathit{e}}_{\mathit{x}}{}^{1/2}\mathit{C}{\mathit{f}}_{\mathit{x}}$ | $\mathit{R}{\mathit{e}}_{\mathit{x}}{}^{1/2}\mathit{C}{\mathit{f}}_{\mathit{y}}$ | $\mathit{R}{\mathit{e}}_{\mathit{x}}{}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$ | $\mathit{R}{\mathit{e}}_{\mathit{x}}{}^{-1/2}\mathit{S}{\mathit{h}}_{\mathit{x}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.3 | 0.2 | 0.6 | 0.3 | 0.3 | 0.4 | 0.5 | 0.2 | 0.6 | 0.5 | 0.6 | −1.55198 | −0.820002 | −0.592402 | 1.50502 |

0.4 | −1.57992 | −0.837306 | −0.609415 | 1.50177 | ||||||||||

0.5 | −1.60741 | −0.854284 | −0.626682 | 1.49859 | ||||||||||

0.6 | −1.66111 | −0.887344 | −0.661989 | 1.49239 | ||||||||||

0.7 | −1.68736 | −0.903459 | −0.680045 | 1.48935 | ||||||||||

0.8 | −1.71324 | −0.919317 | −0.698379 | 1.48636 | ||||||||||

0.7 | −1.55084 | −0.818602 | −0.590103 | 1.50532 | ||||||||||

0.8 | −1.54971 | −0.817207 | −0.587867 | 1.50562 | ||||||||||

0.9 | −1.54858 | −0.815818 | −0.585689 | 1.50591 | ||||||||||

0.4 | −1.48712 | −0.749259 | −0.551301 | 1.51391 | ||||||||||

0.5 | −1.42378 | −0.680671 | −0.515572 | 1.52201 | ||||||||||

0.6 | −1.36203 | −0.613886 | −0.483922 | 1.52952 | ||||||||||

0.4 | −1.55997 | −0.829141 | −0.309853 | 1.51559 | ||||||||||

0.5 | −1.56703 | −0.837203 | −0.047556 | 1.52482 | ||||||||||

0.6 | −1.57333 | −0.844397 | −0.197963 | 1.53293 | ||||||||||

0.5 | −1.55172 | −0.819698 | −0.619867 | 1.52397 | ||||||||||

0.6 | −1.55144 | −0.819417 | −0.646619 | 1.54269 | ||||||||||

0.7 | −1.55119 | −0.819155 | −0.672707 | 1.56119 | ||||||||||

0.6 | −1.54912 | −0.816769 | −0.726352 | 1.55146 | ||||||||||

0.7 | −1.54682 | −0.814234 | −0.844766 | 1.59605 | ||||||||||

0.8 | −1.54501 | −0.812233 | −0.951074 | 1.63912 | ||||||||||

0.3 | −1.55162 | −0.819588 | −0.601766 | 1.50639 | ||||||||||

0.4 | −1.55126 | −0.819171 | −0.611139 | 1.50775 | ||||||||||

0.5 | −1.55092 | −0.818752 | −0.620522 | 1.50911 | ||||||||||

0.7 | −1.60272 | −1.04137 | −0.504336 | 1.52496 | ||||||||||

0.8 | −1.65203 | −1.26967 | −0.420175 | 1.54422 | ||||||||||

0.9 | −1.70002 | −1.50483 | −0.339612 | 1.56228 | ||||||||||

0.6 | −1.55854 | −0.827222 | −0.197169 | 1.35921 | ||||||||||

0.7 | −1.55997 | −0.828592 | −0.004692 | 1.25034 | ||||||||||

0.8 | −1.55932 | −0.827594 | 0.096163 | 1.16727 | ||||||||||

0.3 | −1.59827 | −0.872405 | 1.426472 | 1.07106 | ||||||||||

0.4 | −1.58894 | −0.861872 | 0.948635 | 1.22842 | ||||||||||

0.5 | −1.57538 | −0.846562 | 0.313981 | 1.37673 |

**Table 3.**Influence of different study parameters on $C{f}_{x},C{f}_{y},N{u}_{x},$ and $S{h}_{x}$ for Ethanol.

$\mathit{M}$ | $\mathit{K}$ | ${\mathit{\lambda}}_{\mathit{M}}$ | ${\mathit{\lambda}}_{\mathit{T}}$ | $\mathit{A}$ | $\mathit{B}$ | $\mathit{S}\mathit{c}$ | $\mathit{\gamma}$ | $\mathit{\beta}$ | ${\mathit{N}}_{\mathit{b}}$ | ${\mathit{N}}_{\mathit{t}}$ | $\mathit{R}{\mathit{e}}_{\mathit{x}}{}^{1/2}\mathit{C}{\mathit{f}}_{\mathit{x}}$ | $\mathit{R}{\mathit{e}}_{\mathit{x}}{}^{1/2}\mathit{C}{\mathit{f}}_{\mathit{y}}$ | $\mathit{R}{\mathit{e}}_{\mathit{x}}{}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$ | $\mathit{R}{\mathit{e}}_{\mathit{x}}{}^{-1/2}\mathit{S}{\mathit{h}}_{\mathit{x}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.3 | −1.51799 | −0.785003 | −2.36096 | 1.46091 | ||||||||||

0.4 | −1.54297 | −0.798889 | −2.38491 | 1.45422 | ||||||||||

0.5 | −1.56693 | −0.811822 | −2.41134 | 1.44686 | ||||||||||

0.6 | −1.61086 | −0.833746 | −2.47464 | 1.42883 | ||||||||||

0.7 | −1.62967 | −0.841405 | −2.51481 | 1.41643 | ||||||||||

0.8 | −1.64362 | −0.843741 | −2.56741 | 1.39779 | ||||||||||

0.7 | −1.51710 | −0.783881 | −2.358312 | 1.46152 | ||||||||||

0.8 | −1.51621 | −0.782751 | −2.35577 | 1.46213 | ||||||||||

0.9 | −1.51531 | −0.781613 | −2.353323 | 1.46272 | ||||||||||

0.4 | −1.45107 | −0.713161 | −2.300743 | 1.47852 | ||||||||||

0.5 | −1.38618 | −0.643651 | −2.254673 | 1.49178 | ||||||||||

0.6 | −1.32305 | −0.576174 | −2.216883 | 1.50272 | ||||||||||

0.4 | −1.54659 | −0.817391 | −1.822983 | 1.49851 | ||||||||||

0.5 | −1.56493 | −0.837983 | −1.352733 | 1.52192 | ||||||||||

0.6 | −1.57848 | −0.853128 | −0.925557 | 1.53891 | ||||||||||

0.5 | −1.51773 | −0.784734 | −2.387893 | 1.47742 | ||||||||||

0.6 | −1.51753 | −0.784527 | −2.414123 | 1.49382 | ||||||||||

0.7 | −1.51738 | −0.784371 | −2.439782 | 1.51012 | ||||||||||

0.6 | −1.51176 | −0.777979 | −2.491912 | 1.49683 | ||||||||||

0.7 | −1.50813 | −0.773919 | −2.597682 | 1.53383 | ||||||||||

0.8 | −1.50597 | −0.771544 | −2.689922 | 1.57121 | ||||||||||

0.3 | −1.51664 | −0.783461 | −2.373752 | 1.46095 | ||||||||||

0.4 | −1.51527 | −0.781891 | −2.386522 | 1.46096 | ||||||||||

0.5 | −1.51387 | −0.780293 | −2.399192 | 1.46093 | ||||||||||

0.7 | −1.58203 | −1.021231 | −2.229782 | 1.49738 | ||||||||||

0.8 | −1.64036 | −1.258831 | −2.106492 | 1.52717 | ||||||||||

0.9 | −1.69486 | −1.500221 | −1.990282 | 1.55305 | ||||||||||

0.6 | −1.55436 | −0.825628 | −1.338712 | 1.35338 | ||||||||||

0.7 | −1.56124 | −0.832921 | −0.866811 | 1.24892 | ||||||||||

0.8 | −1.56217 | −0.833555 | −0.592008 | 1.16614 | ||||||||||

0.7 | −1.62695 | −0.906515 | 1.388572 | 1.08242 | ||||||||||

0.8 | −1.61242 | −0.890505 | 0.465142 | 1.24341 | ||||||||||

0.9 | −1.58869 | −0.864355 | −0.698724 | 1.38837 |

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**MDPI and ACS Style**

Arshad, M.; Hussain, A.; Elfasakhany, A.; Gouadria, S.; Awrejcewicz, J.; Pawłowski, W.; Elkotb, M.A.; M. Alharbi, F.
Magneto-Hydrodynamic Flow above Exponentially Stretchable Surface with Chemical Reaction. *Symmetry* **2022**, *14*, 1688.
https://doi.org/10.3390/sym14081688

**AMA Style**

Arshad M, Hussain A, Elfasakhany A, Gouadria S, Awrejcewicz J, Pawłowski W, Elkotb MA, M. Alharbi F.
Magneto-Hydrodynamic Flow above Exponentially Stretchable Surface with Chemical Reaction. *Symmetry*. 2022; 14(8):1688.
https://doi.org/10.3390/sym14081688

**Chicago/Turabian Style**

Arshad, Mubashar, Azad Hussain, Ashraf Elfasakhany, Soumaya Gouadria, Jan Awrejcewicz, Witold Pawłowski, Mohamed Abdelghany Elkotb, and Fahad M. Alharbi.
2022. "Magneto-Hydrodynamic Flow above Exponentially Stretchable Surface with Chemical Reaction" *Symmetry* 14, no. 8: 1688.
https://doi.org/10.3390/sym14081688