# Radiative MHD Nanofluid Flow Due to a Linearly Stretching Sheet with Convective Heating and Viscous Dissipation

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Physical and Graphical Interpretation of Results

## 4. Conclusions

- The increased Maxwell parameter, slip velocity parameter, viscosity parameter, magnetic number and suction parameter diminishes the nanofluid velocity.
- Eckert number and surface-convection parameter values that are larger result in magnifying values for the temperature field.
- The suction parameter, thermal conductivity parameter and magnetic parameter all raise the skin-friction coefficient.
- The results showed that the existence of thermophoresis and Brownian motion makes the heat transmission phenomena more effective.
- Higher radiation and suction parameter values result in a larger Sherwood number, while Maxwell and slip velocity parameter values result in a smaller Sherwood number.
- A larger magnetic number, Brownian motion parameter, viscosity parameter and Maxwell parameter will result in a temperature rise whereas a higher suction parameter and slip velocity parameter will reduce the temperature.
- The concentration of the nanofluid is severely degraded as the viscosity, magnetic number, Maxwell and slip velocity parameters drop.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Greek Symbols

$\rho $ | nanofluid density (kg m${}^{-3}$) |

${\rho}_{\infty}$ | the ambient nanofluid density (kg m${}^{-3}$) |

$\beta $ | the dimensionless Maxwell parameter |

${\beta}_{1}$ | the Maxwell coefficient (S) |

$\mu $ | coefficient of viscosity (kg m${}^{-1}{s}^{-1}$) |

${\mu}_{\infty}$ | the ambient nanofluid viscosity (kg m${}^{-1}{s}^{-1}$) |

$\nu $ | kinematic viscosity (m${}^{2}$ s${}^{-1}$) |

${\nu}_{\infty}$ | the ambient kinematic viscosity (m${}^{2}$ s${}^{-1}$) |

$\theta $ | dimensionless temperature |

$\varphi $ | dimensionless concentration |

${\lambda}_{1}$ | slip velocity factor (m) |

$\lambda $ | slip velocity parameter |

$\sigma $ | electrical conductivity (S m${}^{-1}$) |

${\sigma}^{*}$ | Stefan–Boltzmann constant (W m${}^{-2}$ K${}^{-4}$) |

$\delta $ | the surface convection parameter |

$\eta $ | similarity variable |

$\kappa $ | thermal conductivity (W m${}^{-1}$ K${}^{-1}$) |

${\kappa}_{\infty}$ | the ambient nanofluid thermal conductivity (W m${}^{-1}$ K${}^{-1}$) |

$\epsilon $ | thermal conductivity parameter |

## Superscripts

′ | differentiation with respect to $\eta $ |

∞ | free stream condition |

w | wall condition |

## Nomenclature

a | velocity coefficient (s${}^{-1}$) |

A | is a constant (K m${}^{-2}$) |

${B}_{0}$ | strength of a uniform magnetic field (T) |

c | is a constant (mol L${}^{-1}$ m${}^{-2}$) |

${c}_{p}$ | specific heat at constant pressure (J kg${}^{-1}$ K${}^{-1}$) |

C | nanoparticles concentration (mol L${}^{-1}$) |

$C{f}_{x}$ | skin friction coefficient |

${C}_{w}$ | surface nanoparticles concentration (mol L${}^{-1}$) |

${C}_{\infty}$ | ambient nanoparticles concentration (mol L${}^{-1}$) |

${D}_{B}$ | Brownian diffusion coefficient (m${}^{2}$ s${}^{-1}$) |

${D}_{T}$ | thermophoresis diffusion coefficient (m${}^{2}$ s${}^{-1}$) |

$Ec$ | Eckret number |

f | dimensionless stream function |

${f}_{w}$ | suction parameter |

${h}_{f}$ | the heat transfer coefficient (W m${}^{-2}$ K${}^{-1}$) |

${k}^{*}$ | mean absorption coefficient (m${}^{-1}$) |

$Le$ | Lewis parameter |

M | magnetic parameter |

$Nb$ | Brownian motion parameter |

$Nt$ | thermophoresis parameter |

$N{u}_{x}$ | local Nusselt number |

$Pr$ | Prandtl number |

R | radiation parameter |

$R{e}_{x}$ | local Reynolds number |

$S{h}_{x}$ | local Sherwood number |

T | nanofluid temperature (K) |

${T}_{f}$ | convection temperature (K) |

${T}_{\infty}$ | ambient temperature (K) |

u | velocity component in the x-direction (m s${}^{-1}$) |

v | velocity component in the y-direction (m s${}^{-1}$) |

${v}_{w}$ | suction velocity (m s${}^{-1}$) |

$x,y$ | Cartesian coordinates (m) |

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**Figure 2.**(

**a**) The ${f}^{\prime}\left(\eta \right)$ for chosen $\beta $, (

**b**) $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ for chosen $\beta $.

**Figure 3.**(

**a**) The ${f}^{\prime}\left(\eta \right)$ for chosen ${f}_{w}$, (

**b**) $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ for chosen ${f}_{w}$.

**Figure 4.**(

**a**) The ${f}^{\prime}\left(\eta \right)$ for chosen M, (

**b**) $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ for chosen M.

**Figure 5.**(

**a**) The ${f}^{\prime}\left(\eta \right)$ for chosen $\alpha $, (

**b**) $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ for chosen $\alpha $.

**Figure 6.**(

**a**) The ${f}^{\prime}\left(\eta \right)$ for chosen $\lambda $, (

**b**) $\theta \left(\eta \right)$ and $\varphi \left(\eta \right)$ for chosen $\lambda $.

**Figure 7.**(

**a**) $\theta \left(\eta \right)$ for chosen $Ec$ (

**b**) $\theta \left(\eta \right)$ for chosen R.

**Figure 8.**(

**a**) The $\theta \left(\eta \right)$ for chosen $\delta $, (

**b**) $\theta \left(\eta \right)$ for chosen $\epsilon $.

**Figure 9.**(

**a**) The $\theta \left(\eta \right)$ for chosen $Nb$, (

**b**) $\theta \left(\eta \right)$ for chosen $Nt$.

**Table 1.**Comparison of Nusselt number $-{\theta}^{\prime}\left(0\right)$ for different values of ${f}_{w}$ and Pr with the results of Ishak et al. [23] when $\alpha =\beta =M=Ec=\lambda =R=\epsilon =Nt=Nb=0$.

Pr | ${\mathit{f}}_{\mathit{w}}$, | Ishak et al. [23] | Present Work |
---|---|---|---|

0.72 | −1.5 | 0.4570 | 0.457001520 |

1.0 | −1.5 | 0.5000 | 0.500000000 |

10 | −1.5 | 0.6542 | 0.654211910 |

0.72 | 0.0 | 0.8086 | 0.808589088 |

1.0 | 0.0 | 1.0000 | 1.000000000 |

3.0 | 0.0 | 1.9237 | 1.923689985 |

10.0 | 0.0 | 3.7207 | 3.720699510 |

0.72 | 1.5 | 1.4944 | 1.494389791 |

1.0 | 1.5 | 2.0000 | 2.000002010 |

10 | 1.5 | 16.0842 | 16.08419892 |

**Table 2.**Values of $\frac{C{f}_{x}}{2}R{e}_{x}^{\frac{1}{2}}$, $\frac{N{u}_{x}}{\sqrt{R{e}_{x}}}$ and $\frac{S{h}_{x}}{\sqrt{R{e}_{x}}}$ for various values of $\beta ,{f}_{w},M,\alpha ,\lambda ,Ec,R,\delta $ and $\epsilon $ with $Nb=0.8,Le=1.0,Pr=1.0$ and $Nt=0.1$.

$\mathit{\beta}$ | ${\mathit{f}}_{\mathit{w}}$ | M | $\mathit{\alpha}$ | $\mathit{\lambda}$ | $\mathit{Ec}$ | R | $\mathit{\delta}$ | $\mathit{\epsilon}$ | $\frac{{\mathit{Cf}}_{\mathit{x}}}{2}{\mathit{Re}}_{\mathit{x}}^{\frac{1}{2}}$ | $\frac{{\mathit{Nu}}_{\mathit{x}}}{\sqrt{{\mathit{Re}}_{\mathit{x}}}}$ | $\frac{{\mathit{Sh}}_{\mathit{x}}}{\sqrt{{\mathit{Re}}_{\mathit{x}}}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

0.0 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 1.010101 | 0.106808 | 1.404140 |

0.2 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.980241 | 0.105987 | 1.361981 |

0.5 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.937128 | 0.104581 | 1.291542 |

0.1 | 0.0 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.845444 | 0.105775 | 1.168950 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.995036 | 0.106410 | 1.383612 |

0.1 | 0.8 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 1.095980 | 0.106979 | 1.536251 |

0.1 | 0.5 | 0.0 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.937012 | 0.107398 | 1.414660 |

0.1 | 0.5 | 0.5 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 1.071621 | 0.105056 | 1.342411 |

0.1 | 0.5 | 1.0 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 1.179310 | 0.103068 | 1.284550 |

0.1 | 0.5 | 0.2 | 0.0 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 1.003950 | 0.106514 | 1.388790 |

0.1 | 0.5 | 0.2 | 1.0 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.958801 | 0.105961 | 1.360981 |

0.1 | 0.5 | 0.2 | 2.5 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.888192 | 0.104945 | 1.308652 |

0.1 | 0.5 | 0.2 | 0.2 | 0.0 | 0.2 | 0.5 | 0.2 | 0.2 | 1.329681 | 0.104874 | 1.515341 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.995036 | 0.106410 | 1.383612 |

0.1 | 0.5 | 0.2 | 0.2 | 0.5 | 0.2 | 0.5 | 0.2 | 0.2 | 0.734743 | 0.106767 | 1.258561 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.0 | 0.5 | 0.2 | 0.2 | 0.997361 | 0.112547 | 1.383421 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.995036 | 0.106410 | 1.383612 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.5 | 0.5 | 0.2 | 0.2 | 0.991547 | 0.097316 | 1.383845 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.0 | 0.2 | 0.2 | 0.994306 | 0.152219 | 1.379111 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.995036 | 0.106410 | 1.383612 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 1.0 | 0.2 | 0.2 | 0.995541 | 0.081922 | 1.386090 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.0 | 0.2 | 0.997938 | 0.057677 | 1.389350 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.995036 | 0.106410 | 1.383612 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.5 | 0.2 | 0.988322 | 0.215834 | 1.370541 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 0.0 | 0.995008 | 0.108859 | 1.383361 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 1.5 | 0.995196 | 0.093315 | 1.384910 |

0.1 | 0.5 | 0.2 | 0.2 | 0.2 | 0.2 | 0.5 | 0.2 | 3.5 | 0.995396 | 0.079346 | 1.386282 |

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## Share and Cite

**MDPI and ACS Style**

Alrihieli, H.; Alrehili, M.; Megahed, A.M.
Radiative MHD Nanofluid Flow Due to a Linearly Stretching Sheet with Convective Heating and Viscous Dissipation. *Mathematics* **2022**, *10*, 4743.
https://doi.org/10.3390/math10244743

**AMA Style**

Alrihieli H, Alrehili M, Megahed AM.
Radiative MHD Nanofluid Flow Due to a Linearly Stretching Sheet with Convective Heating and Viscous Dissipation. *Mathematics*. 2022; 10(24):4743.
https://doi.org/10.3390/math10244743

**Chicago/Turabian Style**

Alrihieli, Haifaa, Mohammed Alrehili, and Ahmed M. Megahed.
2022. "Radiative MHD Nanofluid Flow Due to a Linearly Stretching Sheet with Convective Heating and Viscous Dissipation" *Mathematics* 10, no. 24: 4743.
https://doi.org/10.3390/math10244743