# MHD Boundary Layer Flow of Carreau Fluid over a Convectively Heated Bidirectional Sheet with Non-Fourier Heat Flux and Variable Thermal Conductivity

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Numerical Solutions

## 4. Results and Discussion

## 5. Conclusions

- Strength of homogeneous and heterogeneous reactions show the same decreasing trend on concentration distribution.
- Effects of Prandtl number and Biot number on temperature field are also conflicting.
- The velocity of the fluid is in decline for a stronger magnetic effect.
- Velocity escalates for growing estimates of ratios of stretching rate.
- With an increase in the value of the Schmidt number, the concentration of the fluid is enhanced.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$a,b$ | concentrations of chemical species |

$A,B$ | chemical species |

${a}_{0}$ | positive dimensional constants |

${B}_{0}$ | Magnetic field strength [kg s^{−2} A^{−1}] |

${C}_{p}$ | Specific heat [J/kg K] |

$c,d$ | stretching constants |

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

${D}_{A}$ | diffusion coefficient of species A |

${D}_{B}$ | diffusion coefficient of species B |

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

${h}_{f}$ | Heat transfer coefficient |

h | dimensionless concentration due to heterogeneous reaction |

${K}_{1}$ | thermal relaxation time |

${k}_{\infty}$ | ambient thermal conductivity |

${k}_{c}$ | rate constant of chemical species A |

${k}_{s}$ | rate constant of chemical species B |

${k}_{w}$ | thermal conductivity at wall |

M | Magnetic parameter |

n | power law index |

Pr | Prandtl number |

q | heat flux |

$Sc$ | Schmidth number |

t | time |

${T}_{\infty}$ | Ambient temperature [K] |

T | Temperature of fluid [K] |

${T}_{w}$ | Wall temperature [K] |

${u}_{w}$ | sheet velocity along $x$-axis [m/s] |

${v}_{w}$ | sheet velocity along y-axis [m/s] |

V | Velocity vector |

$(u,v,w)$ | Velocity components [m/s] |

${u}_{w}\left(x\right)$ | Stretching velocity along x-axis [m/s] |

$(x,y,z)$ | Rectangular coordinate axis [m] |

$W{e}_{1}$ | Weissenberg number |

$W{e}_{2}$ | Weissenberg number |

$\alpha $ | variable thermal diffusivity |

${\beta}^{*}$ | ratio of viscosities |

${\gamma}_{1}$ | Thermal Biot number |

${\gamma}_{2}$ | Concentration Biot number |

$\lambda $ | ratio of stretching rates |

${\lambda}_{1}$ | thermal relaxation time coefficient |

$\nu $ | Kinematic viscosity [m^{2}/s] |

$\theta $ | Dimensionless temperature |

$\sigma $ | Electrical conductivity [m^{−3} kg^{−1} s^{3} A^{2}] |

$\mu $ | Dynamic viscosity [kg/m/s] |

$\eta $ | Similarity variable |

$\rho $ | Density of fluid [kg/m^{3}] |

$\delta $ | Deborah number |

$\varphi $ | dimensionless concentration |

$\xi $ | ratio of diffusion coefficients |

▽ | nibla operator |

$\mathrm{\Gamma}$ | material parameter |

$\u03f5$ | variable thermal conductivity |

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**Table 1.**Comparison of $\phantom{\rule{4pt}{0ex}}-{f}^{\u2033}\left(0\right)$ varied estimates of $\lambda $ when $n=3,W{e}_{1}=W{e}_{2}=0$.

$\mathit{\lambda}$ | Khan et al. [11] | Present (bvp4c) |
---|---|---|

0.1 | 1.020264 | 1.020264 |

0.2 | 1.039497 | 1.039497 |

0.3 | 1.057956 | 1.057956 |

0.4 | 1.075788 | 1.075788 |

0.5 | 1.093095 | 1.093095 |

0.6 | 1.109946 | 1.109946 |

0.7 | 1.126397 | 1.126397 |

0.8 | 1.142488 | 1.142488 |

0.9 | 1.158253 | 1.158253 |

1.0 | 1.173720 | 1.173720 |

**Table 2.**Comparison of $-{f}^{\u2033}\left(0\right)$ and $-{g}^{\u2033}\left(0\right)$ for various values of M and $\lambda $.

M | $\mathit{\lambda}=0$ | $\mathit{\lambda}=0.5$ | $\mathit{\lambda}=0.5$ | $\mathit{\lambda}=1.0$ | ||||
---|---|---|---|---|---|---|---|---|

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

[48] | Present | [48] | Present | [48] | Present | [48] | Present | |

0.0 | 1.0042 | 1.0045 | 1.0932 | 1.0930 | 0.4653 | 0.4652 | 1.1748 | 1.1742 |

10 | 3.3165 | 3.3149 | 3.3420 | 3.3137 | 1.6459 | 1.6440 | 3.3667 | 3.3654 |

100 | 10.0498 | 10.0427 | 10.0582 | 10.0531 | 5.0208 | 5.0201 | 10.0663 | 10.0654 |

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## Share and Cite

**MDPI and ACS Style**

Lu, D.; Mohammad, M.; Ramzan, M.; Bilal, M.; Howari, F.; Suleman, M.
MHD Boundary Layer Flow of Carreau Fluid over a Convectively Heated Bidirectional Sheet with Non-Fourier Heat Flux and Variable Thermal Conductivity. *Symmetry* **2019**, *11*, 618.
https://doi.org/10.3390/sym11050618

**AMA Style**

Lu D, Mohammad M, Ramzan M, Bilal M, Howari F, Suleman M.
MHD Boundary Layer Flow of Carreau Fluid over a Convectively Heated Bidirectional Sheet with Non-Fourier Heat Flux and Variable Thermal Conductivity. *Symmetry*. 2019; 11(5):618.
https://doi.org/10.3390/sym11050618

**Chicago/Turabian Style**

Lu, Dianchen, Mutaz Mohammad, Muhammad Ramzan, Muhammad Bilal, Fares Howari, and Muhammad Suleman.
2019. "MHD Boundary Layer Flow of Carreau Fluid over a Convectively Heated Bidirectional Sheet with Non-Fourier Heat Flux and Variable Thermal Conductivity" *Symmetry* 11, no. 5: 618.
https://doi.org/10.3390/sym11050618