# Unsteady Electro-Hydrodynamic Stagnating Point Flow of Hybridized Nanofluid via a Convectively Heated Enlarging (Dwindling) Surface with Velocity Slippage and Heat Generation

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## Abstract

**:**

_{2}O

_{3}-Cu/H

_{2}O) hybridized nanofluid (HYNF) is an unsteady electro-hydrodynamic stagnation point flow. A stretchable (shrinkable) surface that was convectively heated was studied in the past. In addition to the traditional nonslip surface, the heat generating (absorbing) and the velocity slippage constraints are deliberated in this research. An obtained nonlinear scheme is resolved by the homotopy analysis method. Governing parameters are the electric field parameters, that is, the dimensionless parameters including the magnetic parameter, Prandtl quantity, heat generating factor, Eckert quantity, and unsteady factor. We discuss in detail the effects of these variables on the movement of problems and thermal transmission characteristics. Increasing the values of the magneto and electric force parameters increased the temperature. Increasing the Prandtl number lowered the temperature. For the Eckert parameter, an increase in temperature was recognized. The symmetric form of the geometry model displayed improved the fluid flow by the same amount both above and below the stagnation streamline, while it decreased the flow pressure by the same level. The more heat source uses to increase the temperature of the HYNF over the entire area, the more heat is supplied to the plate, but with a heat sink, the opposite effect is observed.

## 1. Introduction

_{2}O

_{3}magnetically effective nanoparticles to revise the flow of HYNFs through elastic surfaces. This problem was then extended by Devi [2] to a three-dimensional flow that follows Newton’s heating conditions. They exposed that the heat transmission factor of HYNFs was greater than that of regular nanofluids in both studies. Hayat and Nadeem [3] considered the issue of (Ag-CuO/water) and HYNF that does not shake concerning rotating currents. The consequence of rapid slippage on heat transmission in an unsteady stagnating point flow of HYNF throughout a convective-heated enlarging (dwindling) plate was examined by Zainal et al. [4]. Daniel et al. [5] investigated unstable combined natural and forced convective electrical magnetohydrodynamic (MHD) flow and thermal transmission across a transparent stretched layer using the Buongiorno model. Xie et al. [6] investigated the frictional force factor and put it on a quantity of hybridized nanomolecules to evaluate their tribological assets. HYNFs have reduced wear and coefficient of friction compared to pure nanofluids, according to Devi et al. [2] examination on three-dimensional flow of (Cu-Al

_{2}O

_{3}/H

_{2}O) HYNF using the RK Fehlberg integration process. Considering the Lorentz force, the movement is caused by the unidirectional linear elongation of a flat surface. Numerical results suggest that the heat exchange ratio of (Cu-Al

_{2}O

_{3}) HYNF is more advanced than that of the mono nanoliquid. The impact of temperature distribution and nanoparticle attention on the rheological interest of a magnetite ferrofluid silver/ethylene glycol (Fe

_{3}O

_{4}-Ag/EG) HYNF was studied by Afrand et al. [7].

_{2}-Cu/H

_{2}O) HYNF with shape factor. Hussian et al. [9] investigated the flow of HYNF covering (Cu-Al

_{2}O

_{3}/H

_{2}O) combination via an open hollow space with an adiabatic square obstruction in the hollow space. They calculated numerical answers to the usage of the finite detail technique and explored the effect of numerous physical parameters on HYNFs.

## 2. Mathematical Formulation

_{2}O

_{3}-Cu/H

_{2}O) HYNF above a convective-heated stretchable (shrinkable) plate influencing the speed of slippage is deliberated.

_{2}O

_{3}-Cu/H

_{2}O) dynamic viscosity, ${\rho}_{hnf}$ the consistency of (Al

_{2}O

_{3}-Cu/H

_{2}O), $T$ is the HYNF temperature, ${\left(\rho {c}_{p}\right)}_{hnf}$ is HYNF heat capacity, and ${k}_{hnf}$ is the thermal conductance. The boundary conditions, composed through the restricted slippage for rapidity, are established as:

_{2}O

_{3}) and H

_{2}O nanomolecules, are provided in Table 1. Table 2 shows the physical–thermal characteristics of HYNF. The nanomolecules’ volumetric fraction is denoted by $\varphi $, ${\rho}_{f}$ specifies H

_{2}O consistency, ${\rho}_{s}$ is the consistency of the nanosolid particles, ${c}_{p}$ is the continuous pressure of heat capacity, ${k}_{f}$ symbolizes the thermal conductivity of H

_{2}O, and ${k}_{s}$ is the nanoparticles’ thermal conductivity.

## 3. HAM Solution

## 4. Results and Discussion

## 5. Conclusions

- Speed and temperature rise with a rise in an electrical constraint.
- Magnetic constraint has an inverted influence on rapidity and energy parameters.
- Heat generation increases the temperature, while the converse happens with a heat sink.
- Higher values of the unsteadiness constraint decrease the velocity and temperature.
- An augmentation in temperature is observed for Eckert amount, while it decreases for the Prandtl number.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$u$ and $v$ | quickness elements $\left(m/s\right)$ |

${u}_{w}$ | stretching/shrinking rapidity $\left(m/s\right)$ |

${T}_{1}$ | ambient temperature $\left(K\right)$ |

${h}_{f}$ | heat transmission factor |

${k}_{hnf}$ | thermal conductance |

${\mu}_{hnf}$ | viscidness $\left(kg{m}^{-1}{s}^{-1}\right)$ |

$R{e}_{x}$ | Reynolds quantity |

${H}_{1}$ | quickness slippage factor |

${\rho}_{s}$ | nanoparticle density $\left(kg{m}^{-3}\right)$ |

${k}_{s}$ | solid thermal conductance $\left(W{m}^{-1}{K}^{-1}\right)$ |

$\psi $ | stream function |

${M}^{*}$ | magnetic parameter |

$Ec$ | Eckert Number |

$Bi$ | Biot amount |

$\lambda $ | rapidity and heat ratio |

$N{u}_{x}$ | local Nusselt number |

$f\prime $ | non-dimensional rapidity |

$\upsilon $ | kinematic viscidness $\left({m}^{2}{s}^{-1}\right)$ |

$\varphi $ | nanoparticle solid volume fraction |

$x,y$ | plane coordinate axis |

${u}_{e}$ | strength of stagnation flow |

${T}_{0}$ | reference temperature $\left(K\right)$ |

${\rho}_{hnf}$ | density $\left(kg{m}^{-3}\right)$ |

${\left(\rho {c}_{p}\right)}_{hnf}$ | volume heat capacitance $\left({m}^{2}{s}^{-2}{K}^{-1}\right)$ |

$\eta $ | similarity parameter |

$H$ | primary speed slippage |

${c}_{p}$ | constant pressure of heat capacity |

${\rho}_{f}$ | Base fluid density $\left(kg{m}^{-3}\right)$ |

${k}_{f}$ | fluid thermal conductance $\left(W{m}^{-1}{K}^{-1}\right)$ |

$\epsilon $ | unsteady factor |

$E$ | electrical force factor |

$Pr$ | Prandtl number |

$Q$ | heat generating (absorbing) |

${c}_{f}$ | skin friction factor |

$T$ | temperature of fluid $\left(K\right)$ |

${\mu}_{f}$ | dynamical viscidness $\left(kg{m}^{-1}{s}^{-1}\right)$ |

${\tau}_{w}$ | wall shear stress |

${q}_{w}$ | transportation of heat |

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Properties | Cu | Al_{2}O_{3} | H_{2}O |
---|---|---|---|

$\mathit{k}\left(\mathit{W}\mathbf{/}\mathit{m}\mathit{K}\right)$ | 400 | 40 | 0.613 |

$\mathit{\rho}\left(\mathit{k}\mathit{g}\mathbf{/}{\mathit{m}}^{\mathbf{3}}\right)$ | 8933 | 3970 | 9971 |

${\mathit{c}}_{\mathit{p}}\left(\mathit{J}\mathbf{/}\mathit{k}\mathit{g}\mathit{K}\right)$ | 385 | 765 | 4179 |

$\mathit{\beta}\mathbf{\times}{\mathbf{10}}^{\mathbf{5}}\left(\frac{\mathbf{1}}{\mathit{K}}\right)$ | 1.67 | 0.85 | 21 |

**Table 2.**Utilized relations for physical–thermal characteristics of HYNF [4].

Characteristic | HYNF |
---|---|

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

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

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

$\mathit{k}$ | $\frac{{k}_{hnf}}{{k}_{f}}=\left[\frac{\left(\frac{{\varphi}_{1}{k}_{s1}+{\varphi}_{2}{k}_{s2}}{{\varphi}_{hnf}}\right)+2{k}_{f}+2\left({\varphi}_{1}{k}_{s1}+{\varphi}_{2}{k}_{s2}\right)-2{\varphi}_{hnf}{k}_{f}}{\left(\frac{{\varphi}_{1}{k}_{s1}+{\varphi}_{2}{k}_{s2}}{{\varphi}_{hnf}}\right)+2{k}_{f}-2\left({\varphi}_{1}{k}_{s1}+{\varphi}_{2}{k}_{s2}\right)+{\varphi}_{hnf}{k}_{f}}\right]$ |

**Table 3.**Effect of various physical parameters on skin friction $R{e}_{x}^{1/2}{C}_{f}=\frac{{\mu}_{hnf}}{{\mu}_{f}}{f}^{\u2033}\left(0\right)$.

ε | E | M | $\frac{{\mathit{\mu}}_{\mathit{h}\mathit{n}\mathit{f}}}{{\mathit{\mu}}_{\mathit{f}}}{\mathit{f}}^{\mathbf{\u2033}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$ |
---|---|---|---|

0.3 | 0.1 | 0.4 | 0.72059328 |

0.5 | 0.83542092 | ||

0.7 | 1.03614135 | ||

0.1 | 1.86313569 | ||

0.2 | 1.64385204 | ||

0.3 | 1.76103193 | ||

0.4 | 1.03873708 | ||

0.8 | 1.30863981 | ||

1.0 | 1.58376213 |

**Table 4.**Consequence of various physical parameters on Nusselt number $\left[-\frac{{k}_{hnf}}{{k}_{f}}{\theta}^{\prime}\left(0\right)\right]$.

Ec | Q | Pr | M | E | $\mathbf{-}\frac{{\mathit{k}}_{\mathit{h}\mathit{n}\mathit{f}}}{{\mathit{k}}_{\mathit{f}}}{\mathit{\theta}}^{\mathbf{\prime}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$ |
---|---|---|---|---|---|

0.3 | 0.5 | 4.5 | 0.4 | 0.1 | 1.07386504 |

0.5 | 1.17290347 | ||||

0.7 | 1.23893104 | ||||

0.5 | 2.30319769 | ||||

1.0 | 2.15912307 | ||||

1.5 | 2.02463073 | ||||

4.5 | 0.54354079 | ||||

5.5 | 0.73865302 | ||||

6.5 | 0.93865321 | ||||

0.4 | 1.13159603 | ||||

0.8 | 1.09764384 | ||||

1.0 | 1.05346068 | ||||

0.1 | 1.12183304 | ||||

0.2 | 1.23583931 | ||||

0.3 | 1.30346893 |

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## Share and Cite

**MDPI and ACS Style**

Khan, A.; Jamshed, W.; Eid, M.R.; Pasha, A.A.; Tag El Din, E.S.M.; Khalifa, H.A.E.-W.; Alharbi, S.K.
Unsteady Electro-Hydrodynamic Stagnating Point Flow of Hybridized Nanofluid via a Convectively Heated Enlarging (Dwindling) Surface with Velocity Slippage and Heat Generation. *Symmetry* **2022**, *14*, 2136.
https://doi.org/10.3390/sym14102136

**AMA Style**

Khan A, Jamshed W, Eid MR, Pasha AA, Tag El Din ESM, Khalifa HAE-W, Alharbi SK.
Unsteady Electro-Hydrodynamic Stagnating Point Flow of Hybridized Nanofluid via a Convectively Heated Enlarging (Dwindling) Surface with Velocity Slippage and Heat Generation. *Symmetry*. 2022; 14(10):2136.
https://doi.org/10.3390/sym14102136

**Chicago/Turabian Style**

Khan, Abbas, Wasim Jamshed, Mohamed R. Eid, Amjad Ali Pasha, El Sayed M. Tag El Din, Hamiden Abd El-Wahed Khalifa, and Samaher Khalaf Alharbi.
2022. "Unsteady Electro-Hydrodynamic Stagnating Point Flow of Hybridized Nanofluid via a Convectively Heated Enlarging (Dwindling) Surface with Velocity Slippage and Heat Generation" *Symmetry* 14, no. 10: 2136.
https://doi.org/10.3390/sym14102136