# Hybrid Nanofluid Flow Past a Shrinking Cylinder with Prescribed Surface Heat Flux

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Stability Analysis

## 4. Results and Discussion

## 5. Conclusions

- For both shrinking cylinder and flat plate surfaces with the prescribed surface heat flux, the steady flow solutions are obtainable when the suction parameter is $S>2.6$. No second solution was observed when considering the stretching surface.
- The separation of boundary layer can be decelerated by the extension of the critical value when $K=0$. The flat plate surface also contributes to the maximum heat transfer rate.
- Among the three sets of hybrid Al${}_{2}$O${}_{3}$-Cu nanoparticle concentrations such that $({\varphi}_{1}=0.5\%$, ${\varphi}_{2}=1.5\%)$, $({\varphi}_{1}=1\%,{\varphi}_{2}=1\%)$ and $({\varphi}_{1}=1.5\%,{\varphi}_{2}=0.5\%)$, the hybrid nanofluids with concentration $({\varphi}_{1}=0.5\%,{\varphi}_{2}=1.5\%)$ provided the greatest heat transfer rate and skin friction coefficient.
- The stability analysis mathematically supports the reliability of the first solution.
- The hybrid nanofluid flow due to the shrinking surfaces is a reverse (opposite) flow from the stretching surfaces. The velocity profile for the shrinking case $(\lambda <0)$ shows a negative value and contradicts the positive velocity profile for the stretching case $(\lambda >0)$.
- The hybrid nanofluid temperature for the stretching case is lower than the shrinking case.

## 6. Recommendations for Future Work

- Different hybrid nanofluids (other stable combinations based on the experimental work of hybrid nanofluids);
- Stagnation point flow (exclusion of the wall mass suction parameter);
- Other physical parameters, such as magnetic field, thermal radiation, viscous dissipation and Joule heating.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The coordinate system and physical models of (

**a**) the stretching cylinder and (

**b**) the shrinking cylinder.

**Figure 2.**Re${}_{x}^{1/2}$C${}_{f}$ towards S when $\lambda =-1.9$, ${\varphi}_{1}={\varphi}_{2}=0.01$ and disparate values of the curvature parameter K.

**Figure 3.**Re${}_{x}^{-1/2}$Nu${}_{x}$ towards S when $\lambda =-1.9$, ${\varphi}_{1}={\varphi}_{2}=0.01$ and disparate values of the curvature parameter K.

**Figure 4.**Re${}_{x}^{1/2}$C${}_{f}$ towards $\lambda $ when $S=3$, ${\varphi}_{1}={\varphi}_{2}=0.01$ and disparate values of the curvature parameter K.

**Figure 5.**Re${}_{x}^{-1/2}$Nu${}_{x}$ towards $\lambda $ when $S=3$, ${\varphi}_{1}={\varphi}_{2}=0.01$ and disparate values of the curvature parameter K.

**Figure 6.**Re${}_{x}^{1/2}$C${}_{f}$ towards $\lambda $ when $S=3$, $K=0.2$ and different volumetric concentrations of nanoparticles.

**Figure 7.**Re${}_{x}^{-1/2}$Nu${}_{x}$ towards $\lambda $ when $S=3$, $K=0.2$ and different volumetric concentrations of nanoparticles.

Properties | Nanofluids |
---|---|

Density $\left(\rho \right)$ | $\rho}_{nf}=\left(1-\varphi \right){\rho}_{bf}+\varphi {\rho}_{s$ |

$\rho}_{hnf}=(1-{\varphi}_{hnf}){\rho}_{bf}+{\varphi}_{1}{\rho}_{s1}+{\varphi}_{2}{\rho}_{s2$ | |

Heat Capacity $\left(\rho {C}_{p}\right)$ | $\left(\rho {C}_{p}\right)}_{nf}=(1-\varphi ){\left(\rho {C}_{p}\right)}_{bf}+\varphi {\left(\rho {C}_{p}\right)}_{s$ |

$\left(\rho {C}_{p}\right)}_{hnf}=\left(1-{\varphi}_{hnf}\right){\left(\rho {C}_{p}\right)}_{bf}+{\varphi}_{1}{\left(\rho {C}_{p}\right)}_{s1}+{\varphi}_{2}{\left(\rho {C}_{p}\right)}_{s2$ | |

Dynamic Viscosity $\left(\mu \right)$ | $\frac{{\mu}_{nf}}{{\mu}_{bf}}=\frac{1}{{(1-\varphi )}^{2.5}}$ |

$\frac{{\mu}_{hnf}}{{\mu}_{bf}}=\frac{1}{{(1-{\varphi}_{hnf})}^{2.5}}$ | |

Thermal Conductivity $\left(k\right)$ | $\frac{{k}_{nf}}{{k}_{bf}}=\left[\frac{{k}_{s}+2{k}_{bf}-2\varphi \left({k}_{bf}-{k}_{s}\right)}{{k}_{s}+2{k}_{bf}+\varphi \left({k}_{bf}-{k}_{s}\right)}\right]$ |

$\frac{{k}_{hnf}}{{k}_{bf}}=\left[\frac{\frac{{\varphi}_{1}{k}_{1}+{\varphi}_{2}{k}_{2}}{{\varphi}_{hnf}}+2{k}_{bf}+2\left({\varphi}_{1}{k}_{1}+{\varphi}_{2}{k}_{2}\right)-2{\varphi}_{hnf}kbf}{\frac{{\varphi}_{1}{k}_{1}+{\varphi}_{2}{k}_{2}}{{\varphi}_{hnf}}+2{k}_{bf}-\left({\varphi}_{1}{k}_{1}+{\varphi}_{2}{k}_{2}\right)+{\varphi}_{hnf}{k}_{bf}}\right]$ |

**Table 2.**Thermo-physical properties of the nanoparticles and water (see Oztop and Abu-Nada [59]).

Thermophysical Properties | Base Fluid | Nanoparticles | |
---|---|---|---|

Pure Water | Alumina | Copper | |

${\mathbf{C}}_{\mathbf{p}}$(J/kgK) | 4179 | 765 | 385 |

$\mathbf{\rho}$ (kg/m${}^{\mathbf{3}}$) | 997.1 | 3970 | 8933 |

$\mathbf{k}$(W/mK) | 0.6130 | 40 | 400 |

**Table 3.**Comparison values of $\theta \left(0\right)$ when $S=0$, $\lambda =1$ and ${\varphi}_{1},{\varphi}_{2}\approx 0$ (viscous fluid).

K | Pr | Present | Qasim et al. [57] | Bachok et al. [66] |
---|---|---|---|---|

0.0 | 0.72 | 1.23666 | 1.23664 | 1.2367 |

1.0 | 1.00000 | 1.00000 | 1.0000 | |

6.7 | 0.33330 | 0.33330 | 0.3333 | |

10 | 0.26877 | 0.26876 | 0.2688 | |

1.0 | 0.72 | 0.87058 | 0.87018 | 0.8701 |

1.0 | 0.74395 | 0.74406 | 0.7439 | |

6.7 | 0.29653 | 0.29661 | 0.2966 | |

10.0 | 0.24412 | 0.24217 | 0.2442 |

**Table 4.**Comparison values of Re${}_{x}^{1/2}$C${}_{fx}$ when $K=0$, $S=2.5$, $\lambda =-1$, Pr$=6.2$, ${\varphi}_{1}=0.1$ and various ${\varphi}_{2}$.

${\mathit{\varphi}}_{2}$ | Present | Khashi’ie et al. [65] | ||
---|---|---|---|---|

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

0 | 2.594177 | 0.645222 | 2.594178 | 0.645222 |

0.01 | 2.781516 | 0.655350 | 2.781516 | 0.655350 |

0.02 | 2.967257 | 0.666893 | 2.967257 | 0.666894 |

0.03 | 3.151544 | 0.679725 | 3.151544 | 0.679725 |

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**MDPI and ACS Style**

Khashi’ie, N.S.; Waini, I.; Zainal, N.A.; Hamzah, K.; Mohd Kasim, A.R.
Hybrid Nanofluid Flow Past a Shrinking Cylinder with Prescribed Surface Heat Flux. *Symmetry* **2020**, *12*, 1493.
https://doi.org/10.3390/sym12091493

**AMA Style**

Khashi’ie NS, Waini I, Zainal NA, Hamzah K, Mohd Kasim AR.
Hybrid Nanofluid Flow Past a Shrinking Cylinder with Prescribed Surface Heat Flux. *Symmetry*. 2020; 12(9):1493.
https://doi.org/10.3390/sym12091493

**Chicago/Turabian Style**

Khashi’ie, Najiyah Safwa, Iskandar Waini, Nurul Amira Zainal, Khairum Hamzah, and Abdul Rahman Mohd Kasim.
2020. "Hybrid Nanofluid Flow Past a Shrinking Cylinder with Prescribed Surface Heat Flux" *Symmetry* 12, no. 9: 1493.
https://doi.org/10.3390/sym12091493