#
Magnetized Flow of Cu + Al_{2}O_{3} + H_{2}O Hybrid Nanofluid in Porous Medium: Analysis of Duality and Stability

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## Abstract

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## 1. Introduction

_{2}in the examination by considering the various sizes and phases of the nanoparticles. Sahoo and Kumar [7] investigated the various mixtures of nanoparticles in order to find the maximum heat transfer rate. In this regard, three nanoparticles Al

_{2}O

_{3}, CuO, and TiO

_{2}were considered, and also Al

_{2}O

_{3}–CuO–TiO

_{2}ternary hybrid nanofluid were examined. Shafiq et al. [8] numerically examined the single-and multi-wall carbon nanotubes. Further, Gireesha et al. [9] used a two-phase nanofluid model to investigate Jeffrey nanofluid in a three-dimensional framework. Lund et al. [10] used the same model of Gireesha et al. [9] for the examination of the micropolar nanofluid. It is worth mentioning here that various mixes of nanoparticles in the different base fluids have been investigated. Nevertheless, nobody concluded which nanoparticle mixture and base fluid may offer superior heat transfer rate improvement (refer to the work of [11,12,13,14,15,16,17,18]).

_{2}–Cu/H

_{2}O, Khan et al. [28] revealed that the lower Nusselt is the Cu-Water nanofluid. Olatundun and Makinde [29] modified the model of Blasius for the hybrid nanofluid in which convective condition had also been considered. Chamkha et al. [30] examined the hybrid nanofluid in the rotating system where they found that “Nusselt number acts as an ascending function of injection and radiation parameters, as well as volume fraction of nanofluid”. Maskeen et al. [31] investigated the hydromagnetic alumina–copper/water hybrid nanofluid. An interesting development of the hybrid nanofluid can be seen in these papers [32,33,34,35,36,37,38].

_{2}O

_{3}–Cu/water hybrid on the exponentially shrinking sheet. Moreover, the effects of the porous medium and viscous dissipation have been taken into account. By applying exponential similarity transformation variables, momentum, and energy conservations are converted to the system of ODEs. The numerical solutions of the resulting equations have been determined by employing the shooting technique. Further, analysis of the stability of solutions has also been performed to specify a stable solution with a bvp4c solver. The effects of the different application parameters are shown graphically on the heat transfer rate and skin friction coefficient. Lastly, this work can be extended in the following directions: (i) considering the vertical exponential surface with thermal radiation effect; (ii) considering the first and second-order slip conditions, and (iii) considering the entire model for the three-dimensional flow.

## 2. Mathematical Modeling

_{0}is constant magnetic strength, $K=2{K}_{0}{e}^{\raisebox{1ex}{$-x$}\!\left/ \!\raisebox{-1ex}{$l$}\right.}$ is considered as the permeability of porous medium where ${K}_{0}$ is the reference permeability, ${\left(\rho {c}_{p}\right)}_{hnf},{\rho}_{hnf},\text{}{\sigma}_{hnf},{k}_{hnf},$ and ${\mu}_{hnf},$ are corresponding effective heat capacity, density, electrical conductivity, thermal conductivity, and viscosity of hybrid nanofluid. Further, ${u}_{w}=-{U}_{w}{e}^{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$l$}\right.}$ is the surface velocity, and ${v}_{w}=\sqrt{\frac{{\vartheta}_{f}{U}_{w}}{2l}}{e}^{\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$2l$}\right.}S$ where $S$ is blowing/suction parameter.

## 3. Stability Analysis

## 4. Results and Discussion

_{1}against suction S is depicted in Figure 12. According to Hamid et al. [60], positive (negative) values of $\gamma $ show initial growth of decay (disturbance), and the solution of flow can be stable (unstable). From Figure 12, the first solution is clearly stable and the second one is unstable. Moreover, the graph of the smallest eigenvalue shows symmetrical behavior.

## 5. Conclusions

- The present results show good agreements with the previously published results.
- Dual solutions exist when ${S}_{c}\le S\text{}\mathrm{and}{M}_{c}\le M$, while no solution exists ${S}_{c}>S\text{}\mathrm{and}{M}_{c}M$.
- Shear stress rises in the first solution then declines in the second solution for the rising values of ${\varphi}_{\mathrm{Cu}}$, $M$, $S$, and $\gamma $.
- For the first solution, the heat transfer rate rises as S and M parameters are enhanced, while this is lower when ${\varphi}_{\mathrm{Cu}}$ is up.
- Enhancement in the volume fraction of the nanoparticles pushes forward the boundary layer separation. Therefore, ranges of solutions increase.
- Compared with nanofluid and viscous fluid, hybrid nanofluid seems to be more efficient in cooling processes.
- The first is stable, and the second is unstable.
- The Eckert number and temperature profiles are directly proportional.
- The highest value of Eckert number does not affect the boundary layer separation against suction.
- This model does not function outside the critical points, so there is no solution.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Variation of $f\u2033\left(0\right)$ with ${\varphi}_{\mathrm{Cu}}$ for various values of $M$.

**Figure 5.**Variation of $-\theta \prime \left(0\right)$ with ${\varphi}_{\mathrm{Cu}}$ for various values of $M$.

**Figure 6.**Variation of $f\u2033\left(0\right)$ with ${\varphi}_{\mathrm{Cu}}$ for various values of $S$.

**Figure 7.**Variation of $-\theta \prime \left(0\right)$ with ${\varphi}_{\mathrm{Cu}}$ for various values of $S$.

Names | Properties |
---|---|

Viscosity of Dynamic | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left(1-{\varphi}_{\mathrm{Cu}}\right)}^{2.5}{\left(1-{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right)}^{2.5}}$ |

Density | ${\rho}_{hnf}=\left(1-{\varphi}_{\mathrm{Cu}}\right)\left[\left(1-{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right){\rho}_{f}+{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{\rho}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right]+{\varphi}_{\mathrm{Cu}}{\rho}_{\mathrm{Cu}}$ |

Thermal conductivity | ${k}_{hnf}=\frac{{k}_{\mathrm{Cu}}+2{k}_{nf}-2{\varphi}_{\mathrm{Cu}}\left({k}_{nf}-{k}_{\mathrm{Cu}}\right)}{{k}_{\mathrm{Cu}}+2{k}_{nf}+{\varphi}_{\mathrm{Cu}}\left({k}_{nf}-{k}_{\mathrm{Cu}}\right)}\times \left({k}_{nf}\right)$ where ${k}_{nf}=\frac{{k}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+2{k}_{f}-2{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\left({k}_{f}-{k}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right)}{{k}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+2{k}_{f}+{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\left({k}_{f}-{k}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right)}\times \left({k}_{f}\right)$ |

Heat capacity | ${\left(\rho {c}_{p}\right)}_{hnf}=\left(1-{\varphi}_{\mathrm{Cu}}\right)\left[\left(1-{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right){\left(\rho {c}_{p}\right)}_{f}+{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}{\left(\rho {c}_{p}\right)}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right]+{\varphi}_{\mathrm{Cu}}{\left(\rho {c}_{p}\right)}_{\mathrm{Cu}}$ |

Electrical conductivity | ${\sigma}_{hnf}=\frac{{\sigma}_{\mathrm{Cu}}+2{\sigma}_{nf}-2{\varphi}_{\mathrm{Cu}}\left({\sigma}_{nf}-{\sigma}_{\mathrm{Cu}}\right)}{{\sigma}_{\mathrm{Cu}}+2{\sigma}_{nf}+{\varphi}_{\mathrm{Cu}}\left({\sigma}_{nf}-{\sigma}_{\mathrm{Cu}}\right)}\times \left({\sigma}_{nf}\right)$ where ${\sigma}_{nf}=\frac{{\sigma}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+2{\sigma}_{f}-2{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\left({\sigma}_{f}-{\sigma}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right)}{{\sigma}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}+2{\sigma}_{f}+{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\left({\sigma}_{f}-{\sigma}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}\right)}\times \left({\sigma}_{f}\right)$ |

Fluids | $\mathit{\rho}$ (kg/m^{3})
| ${\mathit{c}}_{\mathit{p}}$ (J/kg K) | k (W/m K) |
---|---|---|---|

Alumina (Al_{2}O_{3}) | 3970 | 765 | 40 |

Copper (Cu) | 8933 | 385 | 400 |

Water (H_{2}O) | 997.1 | 4179 | 0.613 |

**Table 3.**Outcomes of $f\u2033\left(0\right)$ surface where $Pr=6.2,\text{}{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}=0.1$, and $Ec=0.3$.

${\mathit{\varphi}}_{\mathbf{C}\mathbf{u}}$ | M | $\mathit{\gamma}$ | $\mathit{S}$ | ${\mathit{f}}^{\mathbf{\u2033}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$ | |
---|---|---|---|---|---|

First Solution | Second Solution | ||||

0.01 | 0 | 0 | 3 | 2.4863 | −1.1077 |

0.05 | 2.8189 | −1.6261 | |||

0.1 | 3.0749 | −2.0807 | |||

0.1 | 3.1146 | −2.2302 | |||

0.3 | 3.1908 | −2.5230 | |||

0.5 | 3.2633 | −2.8077 | |||

0.1 | 3.2967 | −2.9403 | |||

0.3 | 3.3614 | −3.1999 | |||

0.5 | 3.4236 | −3.4519 | |||

2.75 | 3.0944 | −2.4453 | |||

2.5 | 2.7590 | −1.6382 | |||

2.25 | 2.4142 | −1.0062 |

**Table 4.**The results of $-\theta \prime \left(0\right)$ where $\gamma =0,\text{}{\varphi}_{{\mathrm{Al}}_{2}{\mathrm{O}}_{3}}=0.1$, and $S=3$.

${\mathit{\varphi}}_{\mathbf{C}\mathbf{u}}$ | Pr | M | $\mathit{E}\mathit{c}$ | $\mathbf{-}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$ | |
---|---|---|---|---|---|

First Solution | Second Solution | ||||

0.01 | 6.2 | 0 | 0 | 12.7302 | 12.5387 |

0.05 | 11.2238 | 10.9591 | |||

0.1 | 9.6302 | 9.2758 | |||

5 | 7.6893 | 7.2426 | |||

3 | 4.4876 | 3.7171 | |||

2 | 2.9193 | 1.8444 | |||

6.2 | 0.1 | 9.6319 | 9.2613 | ||

0.3 | 9.6354 | 9.2315 | |||

0.5 | 9.6386 | 9.2000 | |||

0.1 | 8.5867 | 5.9024 | |||

0.3 | 6.4827 | 0.6929 | |||

0.5 | 4.3787 | 7.2883 |

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**MDPI and ACS Style**

Lund, L.A.; Omar, Z.; Dero, S.; Khan, I.; Baleanu, D.; Nisar, K.S.
Magnetized Flow of Cu + Al_{2}O_{3} + H_{2}O Hybrid Nanofluid in Porous Medium: Analysis of Duality and Stability. *Symmetry* **2020**, *12*, 1513.
https://doi.org/10.3390/sym12091513

**AMA Style**

Lund LA, Omar Z, Dero S, Khan I, Baleanu D, Nisar KS.
Magnetized Flow of Cu + Al_{2}O_{3} + H_{2}O Hybrid Nanofluid in Porous Medium: Analysis of Duality and Stability. *Symmetry*. 2020; 12(9):1513.
https://doi.org/10.3390/sym12091513

**Chicago/Turabian Style**

Lund, Liaquat Ali, Zurni Omar, Sumera Dero, Ilyas Khan, Dumitru Baleanu, and Kottakkaran Sooppy Nisar.
2020. "Magnetized Flow of Cu + Al_{2}O_{3} + H_{2}O Hybrid Nanofluid in Porous Medium: Analysis of Duality and Stability" *Symmetry* 12, no. 9: 1513.
https://doi.org/10.3390/sym12091513