# MHD Stagnation Point on Nanofluid Flow and Heat Transfer of Carbon Nanotube over a Shrinking Surface with Heat Sink Effect

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## Abstract

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## 1. Introduction

## 2. Methodology

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

## Nomenclature

Symbols | Greek Symbols | ||

c, a | Positive constant | $\epsilon $ | Heat sink/source parameter |

${B}_{0}$ | Magnetic field (induced) | $\sigma $ | Electrical conductivity |

${C}_{f}$ | The coefficient of skin friction | ${\mu}_{nf}$ | Nanofluid viscosity |

M | Magnetic | ${\left(\rho {C}_{p}\right)}_{nf}$ | Nanofluid heat capacity |

$N{u}_{x}$ | Nusselt number; | $\rho $ | Density |

Pr | Prandtl number | $\lambda $ | Stretching/shrinking parameter |

Rd | Thermal radiation | k | Thermal conductivity |

$R{e}_{x}$ | Reynold number | $\phi $ | Volume fraction of CNT |

T | Fluid temperature | Subscript | |

u,v | Velocity component (x- and y-axes) | nf | Nanofluid |

${U}_{w}$ | Velocity (stretching/shrinking sheet) | f | Base fluid |

s | Solid | ||

Superscript | c | Critical | |

(′) | Prime denotes differentiation with respect to η | $\infty $ | Far field condition/ambient |

w | Surface condition |

## References

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**Figure 2.**Variation on $\lambda $ with various values of $\phi $ in SWCNT-water when M = 0.1 and $\epsilon =-1$: (

**a**) Variation on f″ (0); (

**b**) Variation on $-\theta \prime $ (0).

**Figure 3.**Variation on $\lambda $ with various values of M in SWCNT-water when $\phi $ = 0.1 and $\epsilon =-1$: (

**a**) Variation on f″ (0); (

**b**) Variation on $-\theta \prime $ (0).

**Figure 4.**Variation of $-\theta \prime $(0) versus $\lambda $ with various values of $\epsilon $ in SWCNT-water when $\phi $ = 0.1 and M = 0.1.

**Figure 5.**Variation on $\lambda $ with various values of nanomaterial when $\phi $ = 0.1, M = 0.1, and $\epsilon =-1$: (

**a**) Variation on f″ (0); (

**b**) Variation on $-\theta \prime $ (0).

**Figure 6.**Variation on $\lambda $ with various values of $\phi $ in SWCNT-water when $M=$ 0.1: (

**a**) Variation on f″ (0); (

**b**) Variation on $-\theta \prime $ (0).

**Figure 7.**Various CNT and based fluid when $M=$ 0.1, $\lambda $ = $-$ 1.2, $\phi $ = 0.1, and $\epsilon =-1$: (

**a**) Velocity profile; (

**b**) Temperature profile.

Thermophysical Properties | Nanofluid CNT-Water (s = CNT: n = 3) |
---|---|

Density ($kg/{m}^{3}$) | ${\rho}_{nf}=\left(1-\phi \right){\rho}_{f}+\phi {\rho}_{s}$ |

Heat capacity ($J/K$) | ${\left(\rho {C}_{\rho}\right)}_{nf}=\left(1-\phi \right){\left(\rho {C}_{\rho}\right)}_{f}+\phi {\left(\rho {C}_{\rho}\right)}_{s}$ |

Viscosity ($Ns/{m}^{-2}$) | ${\mu}_{nf}=\frac{{\mu}_{f}}{{\left(1-\phi \right)}^{2.5}}$ |

Thermal conductivity ($W/Km$) | $\frac{{k}_{nf}}{{k}_{f}}=\frac{{k}_{s}+\left(n-1\right){k}_{f}-\left(n-1\right)\phi \left({k}_{f}-{k}_{s}\right)}{{k}_{s}+\left(n-1\right){k}_{f}+\phi \left({k}_{f}-{k}_{s}\right)}$ |

Thermophysical Properties | Base Fluid | Nanoparticle | ||
---|---|---|---|---|

Water Pr = 6.2 | Kerosine Pr = 21 | SWCNT | MWCNT | |

$\rho $ ($kg/{m}^{3}$) | 997.1 | 783 | 2600 | 1600 |

${C}_{p}\left(J/kgK\right)$ | 4179 | 2090 | 425 | 796 |

$k\left(W/mK\right)$ | 0.613 | 0.145 | 6600 | 3000 |

**Table 3.**Comparison several of numerical result $f\u2033\left(0\right)$ when stretching/shrinking case and M $=\epsilon =0$ with the change of $\phi $ and $\lambda $ in nanofluid.

$\mathit{\phi}$ | $\mathit{\lambda}$ | ${\mathit{f}}^{\u2033}\left(0\right)$ | |||||
---|---|---|---|---|---|---|---|

Present Result | Bachok et al. [28] | Wang [29] | |||||

CNT-Water | Cu-Water | Water | |||||

First Solution | Second Solution | First Solution | Second Solution | First Solution | Second Solution | ||

0 | 2 | −1.887306668 | −1.887307 | −1.88731 | |||

1 | 0 | 0 | 0 | ||||

0.5 | 0.71329495 | 0.713295 | 0.7133 | ||||

0 | 1.232587647 | 1.232588 | 1.232588 | ||||

−0.5 | 1.495669739 | 1.49567 | 1.49567 | ||||

−1 | 1.328816861 | 0 | 1.328817 | 0 | 1.32882 | 0 | |

−1.2 | 0.932473188 | 0.233649469 | 0.932473 | 0.23365 | |||

0.1 | 2 | −1.78244491 | −2.217106 | ||||

1 | 0 | 0 | |||||

0.5 | 0.673663145 | 0.83794 | |||||

0 | 1.164103115 | 1.447977 | |||||

−0.5 | 1.412567954 | 1.757032 | |||||

−1 | 1.254985673 | 0 | 1.561022 | 0 | |||

−1.2 | 0.880663529 | 0.220667553 | 1.095419 | 0.274479 | |||

0.2 | 2 | −1.641498824 | −2.298822 | ||||

1 | 0 | 0 | |||||

0.5 | 0.620393516 | 0.868824 | |||||

0 | 1.072052147 | 1.501346 | |||||

−0.5 | 1.300869722 | 1.821791 | |||||

−1 | 1.155748188 | 0 | 1.618557 | 0 | |||

−1.2 | 0.811025466 | 0.20321847 | 1.135794 | 0.284596 |

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**MDPI and ACS Style**

Othman, M.N.; Jedi, A.; Bakar, N.A.A.
MHD Stagnation Point on Nanofluid Flow and Heat Transfer of Carbon Nanotube over a Shrinking Surface with Heat Sink Effect. *Molecules* **2021**, *26*, 7441.
https://doi.org/10.3390/molecules26247441

**AMA Style**

Othman MN, Jedi A, Bakar NAA.
MHD Stagnation Point on Nanofluid Flow and Heat Transfer of Carbon Nanotube over a Shrinking Surface with Heat Sink Effect. *Molecules*. 2021; 26(24):7441.
https://doi.org/10.3390/molecules26247441

**Chicago/Turabian Style**

Othman, Mohamad Nizam, Alias Jedi, and Nor Ashikin Abu Bakar.
2021. "MHD Stagnation Point on Nanofluid Flow and Heat Transfer of Carbon Nanotube over a Shrinking Surface with Heat Sink Effect" *Molecules* 26, no. 24: 7441.
https://doi.org/10.3390/molecules26247441