# Hybrid Nanofluid Radiative Mixed Convection Stagnation Point Flow Past a Vertical Flat Plate with Dufour and Soret Effects

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## Abstract

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## 1. Introduction

- The model used by the previous study is modified towards the Tiwari and Das [3] nanofluid model.
- New additional effects such as the stagnation point flow, thermal radiation, and convective heated boundary condition are inserted towards the present model.
- The equations of the flow model are solved via a sophisticated solver known as bvp4c in MATLAB that could provide a better numerical solution.
- Two different alternative solutions are provided in the present study and the stability analysis has also been derived and reported to analyze the stability feature of the generated numerical solutions.
- The preferable value of parameters to control the skin friction, heat transfer, and mass transfer rates as well as the boundary layer separation process for the present model are highlighted and discussed in the findings.

## 2. Mathematical Model

## 3. Stability Analysis

## 4. Results and Discussion

## 5. Conclusions

- Two solutions exist but only the first solution is stable, as evaluated through the stability analysis.
- The boundary layer separation is preventable if 2% of copper is used and lesser Dufour and Soret effects are considered.
- Heat transfer performance can be amplified by reducing the volume fraction of copper and lessening the Dufour effect.
- Mass transfer rate is improvable by raising the volume fraction of copper and reducing the Soret effect.
- The skin friction can be reduced by augmenting the Dufour and Soret effects during the opposing flow of mixed convection.
- The flow moves at a higher velocity when the hybrid nanofluid is concentrated but decelerated when stronger Dufour and Soret effects are inserted.
- The fluid temperature is reduceable by considering a greater copper volume fraction and Soret effect, thus, these two effects can be a coolant factor to the fluid.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**${\mathrm{Re}}_{x}^{1/2}{C}_{f}$ for varied ${\varphi}_{2}$ when $N=Rd=Sc=Sr=1$, $Du=0.03$, and $Bi=0.1$.

**Figure 3.**${\mathrm{Re}}_{x}^{-1/2}N{u}_{x}$ for varied ${\varphi}_{2}$ when $N=Rd=Sc=Sr=1$, $Du=0.03$, and $Bi=0.1$.

**Figure 4.**${\mathrm{Re}}_{x}^{-1/2}S{h}_{x}$ for varied ${\varphi}_{2}$ when $N=Rd=Sc=Sr=1$, $Du=0.03$, and $Bi=0.1$.

**Figure 5.**${\mathrm{Re}}_{x}^{1/2}{C}_{f}$ for varied $Du$ when $N=Rd=Sc=Sr=1$, ${\varphi}_{2}=0.01$, and $Bi=0.1$.

**Figure 6.**${\mathrm{Re}}_{x}^{-1/2}N{u}_{x}$ for varied $Du$ when $N=Rd=Sc=Sr=1$, ${\varphi}_{2}=0.01$, and $Bi=0.1$.

**Figure 7.**${\mathrm{Re}}_{x}^{-1/2}S{h}_{x}$ for varied $Du$ when $N=Rd=Sc=Sr=1$, ${\varphi}_{2}=0.01$, and $Bi=0.1$.

**Figure 8.**${\mathrm{Re}}_{x}^{1/2}{C}_{f}$ for varied $Sr$ when $N=Rd=Sc=1$, ${\varphi}_{2}=0.01$, $Du=0.03$, and $Bi=0.1$.

**Figure 9.**${\mathrm{Re}}_{x}^{-1/2}N{u}_{x}$ for varied $Sr$ when $N=Rd=Sc=1$, ${\varphi}_{2}=0.01$, $Du=0.03$, and $Bi=0.1$.

**Figure 10.**${\mathrm{Re}}_{x}^{-1/2}S{h}_{x}$ for varied $Sr$ when $N=Rd=Sc=1$, ${\varphi}_{2}=0.01$, $Du=0.03$, and $Bi=0.1$.

**Figure 11.**Plots of $f{}^{\prime}\left(\eta \right)$ for varied ${\varphi}_{2}$ when $N=Rd=Sc=Sr=1$, $\lambda =-1$, $Du=0.03$, and $Bi=0.1$.

**Figure 12.**Plots of $\theta \left(\eta \right)$ for varied ${\varphi}_{2}$ when $N=Rd=Sc=Sr=1$, $\lambda =-1$, $Du=0.03$, and $Bi=0.1$.

**Figure 13.**Plots of $h\left(\eta \right)$ for varied ${\varphi}_{2}$ when $N=Rd=Sc=Sr=1$, $\lambda =-1$, $Du=0.03$, and $Bi=0.1$.

**Figure 14.**Plots of $f{}^{\prime}\left(\eta \right)$ for varied $Du$ when $N=Rd=Sc=Sr=1$, $\lambda =-1$, ${\varphi}_{2}=0.01$, and $Bi=0.1$.

**Figure 15.**Plots of $\theta \left(\eta \right)$ for varied $Du$ when $N=Rd=Sc=Sr=1$, $\lambda =-1$, ${\varphi}_{2}=0.01$, and $Bi=0.1$.

**Figure 16.**Plots of $h\left(\eta \right)$ for varied $Du$ when $N=Rd=Sc=Sr=1$, $\lambda =-1$, ${\varphi}_{2}=0.01$, and $Bi=0.1$.

**Figure 17.**Plots of $f{}^{\prime}\left(\eta \right)$ for varied $Sr$ when $N=Rd=Sc=1$, $Du=0.03$, $\lambda =-1$, ${\varphi}_{2}=0.01$, and $Bi=0.1$.

**Figure 18.**Plots of $\theta \left(\eta \right)$ for varied $Sr$ when $N=Rd=Sc=1$, $Du=0.03$, $\lambda =-1$, ${\varphi}_{2}=0.01$, and $Bi=0.1$.

**Figure 19.**Plots of $h\left(\eta \right)$ for varied $Sr$ when $N=Rd=Sc=1$, $Du=0.03$, $\lambda =-1$, $Bi=0.1$, and ${\varphi}_{2}=0.01$.

**Figure 20.**Plots of ${\gamma}_{1}$ for selected ${\lambda}_{c}$ when $N=Rd=Sc=Sr=1$, ${\varphi}_{2}=0.01$, $Du=0.03$, and $Bi=0.1$.

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

Density | ${\rho}_{hnf}={\varphi}_{A{l}_{2}{O}_{3}}{\rho}_{A{l}_{2}{O}_{3}}+{\varphi}_{Cu}{\rho}_{Cu}+(1-{\varphi}_{hnf}){\rho}_{f}$ $\mathrm{where}\text{}{\varphi}_{hnf}={\varphi}_{A{l}_{2}{O}_{3}}+{\varphi}_{Cu}$ |

Heat capacity | ${\left(\rho {C}_{p}\right)}_{hnf}={\varphi}_{A{l}_{2}{O}_{3}}{(\rho {C}_{p})}_{A{l}_{2}{O}_{3}}+{\varphi}_{Cu}{(\rho {C}_{p})}_{Cu}+\left(1-{\varphi}_{hnf}\right){(\rho {C}_{p})}_{f}$ |

Dynamic viscosity | ${\mu}_{hnf}={\mu}_{f}{\left(1-{\varphi}_{hnf}\right)}^{-2.5}$ |

Thermal conductivity | $\frac{{k}_{hnf}}{{k}_{f}}=\left[\frac{\left(\frac{{\varphi}_{A{l}_{2}{O}_{3}}{k}_{A{l}_{2}{O}_{3}}+{\varphi}_{Cu}{k}_{Cu}}{{\varphi}_{hnf}}\right)+2{k}_{f}+2\left({\varphi}_{A{l}_{2}{O}_{3}}{k}_{A{l}_{2}{O}_{3}}+{\varphi}_{Cu}{k}_{Cu}\right)-2{\varphi}_{hnf}{k}_{f}}{\left(\frac{{\varphi}_{A{l}_{2}{O}_{3}}{k}_{A{l}_{2}{O}_{3}}+{\varphi}_{Cu}{k}_{Cu}}{{\varphi}_{hnf}}\right)+2{k}_{f}-\left({\varphi}_{A{l}_{2}{O}_{3}}{k}_{A{l}_{2}{O}_{3}}+{\varphi}_{Cu}{k}_{Cu}\right)+{\varphi}_{hnf}{k}_{f}}\right]$ |

Thermal expansion | ${\left(\rho \beta \right)}_{hnf}=\left(1-{\varphi}_{hnf}\right){\left(\rho \beta \right)}_{f}+{\varphi}_{A{l}_{2}{O}_{3}}{\rho}_{A{l}_{2}{O}_{3}}{\beta}_{A{l}_{2}{O}_{3}}+{\varphi}_{Cu}{\rho}_{Cu}{\beta}_{Cu}$ |

Properties | $\mathbf{Water}\text{}({H}_{2}O)$ | $\mathbf{Alumina}\text{}\left(\mathit{A}{\mathit{l}}_{2}{\mathit{O}}_{3}\right)$ | $\mathbf{Copper}\text{}\left(\mathit{C}\mathit{u}\right)$ |
---|---|---|---|

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

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

$k\left(\mathrm{W}/\mathrm{mK}\right)$ | 0.613 | 40 | 400 |

$\beta \left(1/\mathrm{K}\right)$ | 21 × 10^{−5} | 0.85 × 10^{−5} | 1.67 × 10^{−5} |

$\mathrm{Pr}$ | 6.2 | - | - |

**Table 3.**Values of $f{}^{\prime}{}^{\prime}\left(0\right)$ when $N=Rd=Du=Sc=Sr={\varphi}_{1}={\varphi}_{2}=0$ and $Bi\to \infty $.

$\mathbf{Pr}$ | $\mathit{\lambda}=1$ | $\mathit{\lambda}=-1$ | ||
---|---|---|---|---|

Present | Khashi’ie et al. [59]; Ishak et al. [61] | Present | Roşca et al. [62]; Ramachandran et al. [63] | |

0.7 | 1.706322692 (1.238727738) | 1.7063 (1.2387) | 0.691661306 (−0.285049030) | 0.6917 |

6.2 | 1.526774663 (0.613170553) | - | 0.913106146 (−0.371891985) | - |

7 | 1.517912618 (0.582400958) | 1.5179 (0.5824) | 0.923481290 (−0.375336817) | 0.9235 |

20 | 1.448482926 (0.343640272) | 1.4485 (0.3436) | 1.003108154 (−0.400012699) | 1.0031 |

**Table 4.**Values of $-\theta {}^{\prime}\left(0\right)$ when $N=Rd=Du=Sc=Sr={\varphi}_{1}={\varphi}_{2}=0$ and $Bi\to \infty $.

$\mathbf{Pr}$ | $\mathit{\lambda}=1$ | $\mathit{\lambda}=-1$ | ||
---|---|---|---|---|

Present | Khashi’ie et al. [59]; Ishak et al. [61] | Present | Roşca et al. [62]; Ramachandran et al. [63] | |

0.7 | 0.764063389 (1.022631377) | 0.7641 (1.0226) | 0.633247080 (−0.222165242) | 0.6332 |

7 | 1.722381598 (2.219194096) | 1.7224 (2.2192) | 1.546031855 (−1.285559433) | 1.5403 |

20 | 2.457590047 (3.164608405) | 2.4576 (3.1647) | 2.268272410 (−2.573646060) | 2.2683 |

**Table 5.**Values of ${\mathrm{Re}}_{x}^{1/2}{C}_{f}$ when $N=Rd=Du=Sc=Sr=\lambda =0$ and $Bi\to \infty $.

$\mathit{\varphi}$ | Present | Wahid et al. [60]; Khashi’ie et al. [59] | ||
---|---|---|---|---|

Alumina–Water | Copper–Water | Alumina–Water | Copper–Water | |

0.05 | 1.408762990 | 1.553849593 | 1.4088 | 1.5538 |

0.10 | 1.602056737 | 1.884323749 | 1.6020 | 1.8843 |

0.15 | 1.816825555 | 2.236903962 | 1.8168 | 2.2369 |

0.20 | 2.058324533 | 2.622743101 | 2.0583 | 2.6227 |

**Table 6.**Values of ${\mathrm{Re}}_{x}^{-1/2}N{u}_{x}$ when $N=Rd=Du=Sc=Sr=\lambda =0$ and $Bi\to \infty $.

$\mathit{\varphi}$ | Present | Wahid et al. [60]; Khashi’ie et al. [59] | ||
---|---|---|---|---|

Alumina–Water | Copper–Water | Alumina–Water | Copper–Water | |

0.05 | 1.716899309 | 1.775765930 | 1.7169 | 1.7758 |

0.10 | 1.860326121 | 1.969206054 | 1.8603 | 1.9692 |

0.15 | 2.004503652 | 2.159313050 | 2.0045 | 2.1593 |

0.20 | 2.150196604 | 2.349362585 | 2.1502 | 2.3494 |

**Table 7.**Values of ${\mathrm{Re}}_{x}^{1/2}{C}_{f}$, ${\mathrm{Re}}_{x}^{-1/2}N{u}_{x}$ and ${\mathrm{Re}}_{x}^{-1/2}S{h}_{x}$ for different $Bi$ and $Sc$ when $N=Rd=Sr=1$, $Du=0.03$, $\lambda =-1$, $\mathrm{Pr}=6.2$ and ${\varphi}_{1}={\varphi}_{2}=0.01$.

$\mathit{B}\mathit{i}$ | $\mathit{S}\mathit{c}$ | ${\mathbf{Re}}_{\mathit{x}}^{1/2}{\mathit{C}}_{\mathit{f}}$ | ${\mathbf{Re}}_{\mathit{x}}^{-1/2}\mathit{N}{\mathit{u}}_{\mathit{x}}$ | ${\mathbf{Re}}_{\mathit{x}}^{-1/2}\mathit{S}{\mathit{h}}_{\mathit{x}}$ |
---|---|---|---|---|

0.1 | 1 | 0.721828291 (−0.337285448) | 0.203868995 (0.220230916) | 0.687806632 (−0.379048912) |

0.5 | 1 | 0.568251544 (−0.236016413) | 0.751675151 (1.321437252) | 0.553722971 (−0.940825682) |

0.7 | 1 | 0.513295773 (−0.101330845) | 0.924275190 (2.265412293) | 0.510747744 (−1.465072866) |

0.1 | 0.5 | 0.642507875 (−0.318062728) | 0.204276266 (0.216204343) | 0.524234821 (−0.221922364) |

0.1 | 0.1 | 0.450361082 (−0.322012008) | 0.203299825 (0.208272113) | 0.267859211 (−0.072123986) |

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**MDPI and ACS Style**

Wahid, N.S.; Arifin, N.M.; Khashi’ie, N.S.; Pop, I.; Bachok, N.; Hafidzuddin, M.E.H.
Hybrid Nanofluid Radiative Mixed Convection Stagnation Point Flow Past a Vertical Flat Plate with Dufour and Soret Effects. *Mathematics* **2022**, *10*, 2966.
https://doi.org/10.3390/math10162966

**AMA Style**

Wahid NS, Arifin NM, Khashi’ie NS, Pop I, Bachok N, Hafidzuddin MEH.
Hybrid Nanofluid Radiative Mixed Convection Stagnation Point Flow Past a Vertical Flat Plate with Dufour and Soret Effects. *Mathematics*. 2022; 10(16):2966.
https://doi.org/10.3390/math10162966

**Chicago/Turabian Style**

Wahid, Nur Syahirah, Norihan Md Arifin, Najiyah Safwa Khashi’ie, Ioan Pop, Norfifah Bachok, and Mohd Ezad Hafidz Hafidzuddin.
2022. "Hybrid Nanofluid Radiative Mixed Convection Stagnation Point Flow Past a Vertical Flat Plate with Dufour and Soret Effects" *Mathematics* 10, no. 16: 2966.
https://doi.org/10.3390/math10162966