# Hybrid Nanofluid Flow over a Permeable Shrinking Sheet Embedded in a Porous Medium with Radiation and Slip Impacts

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

_{2}O

_{3}–Cu/water works as the two distinct fluids. The governing ordinary differential equations (ODEs) obtained in this study are converted from a series of partial differential equations (PDEs) by the appropriate use of similarity transformation. Two methods of shooting and bvp4c function are applied to solve the involving physical parameters over the hybrid nanofluid flow. From this study, we conclude that the non-uniqueness of solutions exists through a range of the shrinking parameter, which produces the problem of finding a bigger solution than any other between the upper and lower branches. From the analysis, one can observe the increment of heat transfer rate in hybrid nanofluid versus the traditional nanofluid. The results obtained by the stability of solutions prove that the upper solution (first branch) is stable and the lower solution (second branch) is not stable.

## 1. Introduction

_{2}O

_{3}and Cu as two distinct nanoparticles, and the results reveal that Al

_{2}O

_{3}shows a higher velocity and thermal boundary layer than Cu. Sheikholeslami [5] studied the nanofluid flow and heat transfer over a cylinder with a uniform suction and described the increasing function of Nusselt number alongside nanoparticle volume fraction. Later, a study of magnetohydrodynamics (MHD) flow over a permeable stretching/shrinking sheet with nanofluid and suction/injection was published by Naramgari and Sulochana [6]. Based on their study, they indicated that the magnetic field parameter reduces the boundary layer flow, friction factor, and heat transfer rate on stretching surface. Other works that can be considered are found in [7,8,9,10,11,12,13,14,15].

_{2}O

_{3}/water in a non-Darcy porous medium with thermal dispersion. They found that Cu–Al

_{2}O

_{3}/water has greater heat transfer rate than nanofluid and regular fluid for some of the investigated parameters. In addition, a considerable amount of previous works on hybrid nanofluid over a porous medium have been successfully reported (e.g., [31,32,33,34,35]).

_{2}O

_{3}–Cu/water hybrid nanofluid along a permeable Darcy porous medium is conducted in this present work as the authors are inspired by the above-mentioned literature. We consider shrinking surface, slip factor, and radiation effect in this model. The main objective of this paper is to find the solutions to the current problem, which may benefit other researchers or academicians from the final outcomes.

## 2. Problem Formulation

_{2}O

_{3}) as nano-sized particles and water as a base fluid. Table 1 lists the nanofluids and hybrid nanofluids thermophysical properties. We consider Cu and Al

_{2}O

_{3}in this study as we follow the model introduced by [36], since these two nanoparticles are the most commonly used by many researchers in their experiment works and theoretical studies. It is noted that the basic thermophysical properties of nanofluid are extracted from the standard literature, and their properties of suspended nanoparticles versus fluid at 25° are listed in Table 2. We apply the Darcy equation in this model as it describes the fluid flow over a porous media, as suggested by Rajagopal [37]. Under the above assumptions, the continuity, momentum, and energy of nanoparticles equations based on Darcy flow model (see [38]) are as follows.

## 3. Numerical Soluion

^{−1}-continuous solution that is fourth-order accurate in the specific interval. Hence, the variable ${\eta}_{max}$ is acquired by applying the boundary conditions of the field at the finite value for the similarity variable $\eta $. Thus, we set ${\eta}_{max}=9$ in our analysis to fulfill the far field boundary conditions as in Equation (10) asymptotically.

_{2}O

_{3}against velocity profiles ${f}^{\prime}\left(\eta \right)$ and temperature profiles $\theta \left(\eta \right)$. In these figures, we depict that the upper solution in ${f}^{\prime}\left(\eta \right)$ decreases, while the rest shows promising positive pattern along the flow. These behaviors of increase and decrease can be explained by a contribution of the flow and the conditions of thermal and dispersive elements properties that maximize the heat transfer.

_{2}O

_{3}/water and ${\varphi}_{2}$ for Cu/water, we note that the strength of flow increases, as can be seen from the pattern of the streamlines by alerting that the increase of Cu/water nanoparticle number has a higher heat transfer rate as compared to Al

_{2}O

_{3}/water nanofluid.

## 4. Stability Analysis

## 5. Conclusions

_{2}O

_{3}–Cu/water is employed in this study as a model of hybrid nanofluid. From our observation, we conclude that the skin friction coefficient ${C}_{fx}R{e}_{x}^{1/2}$ is expanding when we increase the number of involving parameters and most of the parameters used in this investigation show an increasing pattern on boundary layer flow, in either upper or lower solution. Moreover, two branches of solutions are found to exist within a range of negative numbers in shrinking parameter $\alpha $. Due to this, the most stable solution between these two is identified via a work of stability analysis. It is then concluded that the first branch (upper solution) is stable and physically realizable, while the second branch (lower solution) is unstable.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Al_{2}O_{3} | Aluminium Oxide |

bvp4c | Boundary value problem – fourth-order method |

Cu | Copper |

CuO | Copper Oxide |

KKL | Koo–Kleinstreuer–Li approach |

MHD | Magnetohydrodynamics |

ODE | Ordinary differential equations |

PDE | Partial differential equations |

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**Figure 2.**${C}_{fx}R{e}_{x}^{1/2}$ and $N{u}_{x}R{e}_{x}^{-1/2}$ for Al

_{2}O

_{3}–Cu/water with S and $\alpha $.

**Figure 3.**${C}_{fx}R{e}_{x}^{1/2}$ and $N{u}_{x}R{e}_{x}^{-1/2}$ for Al

_{2}O

_{3}–Cu/water with $\delta $ and $\alpha $.

**Figure 4.**Local Nusselt number $N{u}_{x}R{e}_{x}^{-1/2}$ for Al

_{2}O

_{3}–Cu/water with R and $\alpha $.

**Figure 5.**The flow of ${f}^{\prime}\left(\eta \right)$ and $\theta \left(\eta \right)$ against ${\varphi}_{1}$.

**Figure 6.**The flow of ${f}^{\prime}\left(\eta \right)$ and $\theta \left(\eta \right)$ against ${m}_{1}$.

Properties | Nanofluid | Hybrid Nanofluid |
---|---|---|

Density | ${\rho}_{nf}=(1-\varphi ){\rho}_{f}+\varphi {\rho}_{s}$ | ${\rho}_{hnf}=(1-{\varphi}_{2})[(1-{\varphi}_{1}){\rho}_{f}+{\varphi}_{1}{\rho}_{s1}]+{\varphi}_{2}{\rho}_{s2}$ |

Heat capacity | ${\left(\rho {C}_{p}\right)}_{nf}=(1-\varphi ){\left(\rho {C}_{p}\right)}_{f}+\varphi {\left(\rho {C}_{p}\right)}_{s}$ | ${\left(\rho {C}_{p}\right)}_{hnf}=(1-{\varphi}_{2})[(1-{\varphi}_{1}){\left(\rho {C}_{p}\right)}_{f}+$ |

${\varphi}_{1}{\left(\rho {C}_{p}\right)}_{s1}]+{\varphi}_{2}{\left(\rho {C}_{p}\right)}_{s2}$ | ||

Dynamic viscosity | ${\nu}_{nf}={\displaystyle \frac{{\nu}_{f}}{{(1-\varphi )}^{2.5}}}$ | ${\nu}_{hnf}={\displaystyle \frac{{\nu}_{f}}{{(1-{\varphi}_{1})}^{2.5}{(1-{\varphi}_{2})}^{2.5}}}$ |

Thermal conductivity | $\frac{{k}_{f}}{{k}_{nf}}}={\displaystyle \frac{{k}_{s}+2{k}_{f}+\varphi ({k}_{f}-{k}_{s})}{{k}_{s}+2{k}_{f}-2\varphi ({k}_{f}-{k}_{s})}$ | $\frac{{k}_{nf}}{{k}_{hnf}}}={\displaystyle \frac{{k}_{s2}+2{k}_{nf}+{\varphi}_{2}({k}_{nf}-{k}_{s2})}{{k}_{s2}+2{k}_{nf}-2{\varphi}_{2}({k}_{nf}-{k}_{s2})}$ |

where | ||

$\frac{{k}_{f}}{{k}_{nf}}}={\displaystyle \frac{{k}_{s1}+2{k}_{f}+{\varphi}_{1}({k}_{f}-{k}_{s1})}{{k}_{s1}+2{k}_{f}-2{\varphi}_{1}({k}_{f}-{k}_{s1})}$ |

**Table 2.**Fluid and nanoparticles thermophysical characteristics (see [36]).

Physical Characteristics | Water (f) | Al_{2}O_{3} (s1) | Cu (s2) |
---|---|---|---|

Density, $\rho $ (kg/m${}^{3}$) | 997.0 | 3970 | 8933 |

Thermal expansion, $\beta $ (K${}^{-1}$) | $21\times {10}^{-5}$ | $0.85\times {10}^{-5}$ | $1.67\times {10}^{-5}$ |

Thermal conductivity, k (W/m K) | 0.6071 | 40 | 400 |

Thermal capacity, ${C}_{p}$ (J/kg K) | 4180 | 765 | 385 |

$\mathit{\alpha}$ | ${\mathit{C}}_{\mathit{f}\mathit{x}}{\mathit{R}\mathit{e}}_{\mathit{x}}^{1/2}$ | ||
---|---|---|---|

Present Study | Bhattacharyya et al. [43] | Wang [42] | |

−0.50 | 1.49566 | 1.49655 | 1.49567 |

−0.625 | 1.52071 | 1.50715 | – |

−0.75 | 1.48929 | 1.48929 | 1.48930 |

−1.00 | 1.32882 (0) | 1.32881 (0) | 1.32282 (0) |

−1.15 | 1.08223 (0.11670) | 1.08223 (0.11670) | 1.08223 (0.11670) |

−1.20 | 0.93247 (0.23364) | 0.93247 (0.23364) | – |

${\mathit{\varphi}}_{1}$ | $\mathit{\alpha}$ | Upper Solution | Lower Solution |
---|---|---|---|

0.2 | −1.6 | 0.87412 | −0.56313 |

−1.7 | 0.90673 | −0.56422 | |

−1.8 | 0.94051 | −0.56540 | |

0.3 | −1.6 | 0.91186 | −0.59207 |

−1.7 | 0.93004 | −0.61358 | |

−1.8 | 0.96728 | −0.63776 |

${\mathit{\varphi}}_{2}$ | $\mathit{\alpha}$ | Upper Solution | Lower Solution |
---|---|---|---|

0.1 | −1.6 | 0.51620 | −0.47993 |

−1.7 | 0.55783 | −0.48015 | |

−1.8 | 0.59022 | −0.48154 | |

0.2 | −1.6 | 0.73326 | −0.52197 |

−1.7 | 0.74748 | −0.52244 | |

−1.8 | 0.81905 | −0.52293 |

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**MDPI and ACS Style**

Abu Bakar, S.; Md Arifin, N.; Khashi’ie, N.S.; Bachok, N.
Hybrid Nanofluid Flow over a Permeable Shrinking Sheet Embedded in a Porous Medium with Radiation and Slip Impacts. *Mathematics* **2021**, *9*, 878.
https://doi.org/10.3390/math9080878

**AMA Style**

Abu Bakar S, Md Arifin N, Khashi’ie NS, Bachok N.
Hybrid Nanofluid Flow over a Permeable Shrinking Sheet Embedded in a Porous Medium with Radiation and Slip Impacts. *Mathematics*. 2021; 9(8):878.
https://doi.org/10.3390/math9080878

**Chicago/Turabian Style**

Abu Bakar, Shahirah, Norihan Md Arifin, Najiyah Safwa Khashi’ie, and Norfifah Bachok.
2021. "Hybrid Nanofluid Flow over a Permeable Shrinking Sheet Embedded in a Porous Medium with Radiation and Slip Impacts" *Mathematics* 9, no. 8: 878.
https://doi.org/10.3390/math9080878