# Ramification of Hall and Mixed Convective Radiative Flow towards a Stagnation Point into the Motion of Water Conveying Alumina Nanoparticles Past a Flat Vertical Plate with a Convective Boundary Condition: The Case of Non-Newtonian Williamson Fluid

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## Abstract

**:**

_{2}O

_{3}) nanoparticles, as appropriate in engineering or industry, is investigated. Using pertinent similarity variables, the dominating equations are non-dimensionalized, and after that, via the bvp4c solver, they are numerically solved. We extensively explore the effects of many relevant parameters on axial velocity, transverse velocity, temperature profile, heat transfer, and drag force. In the opposing flow, there are two solutions seen; in the aiding flow, just one solution is found. In addition, the results designate that, due to nanofluid, the thickness of the velocity boundary layer decreases, and the thermal boundary layer width upsurges. The gradients for the branch of stable outcome escalate due to a higher Weissenberg parameter, while they decline for the branch of lower outcomes. Moreover, a magnetic field can be used to influence the flow and the properties of heat transfer.

## 1. Introduction

## 2. Problem Formulation

_{2}O

_{3}/water) nanofluid are written in Table 1.

## 3. Results and Discussion

_{2}O

_{3}/water) nanofluid flow problem. According to Oztop and Abu-Nada [53], the value of the Prandtl number Pr = 6.2 is taken into account throughout the investigation and the volume fraction of nanoparticles ranges from 0 to 0.2, which $\phi =0$ correlates to the base fluid.

_{2}O

_{3}/water) nanoparticles for the unstable and stable branches of outcome are graphically presented in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14, and also in Table 4, Table 5 and Table 6, respectively. The given problem simulations were performed by varying one parameter at a time and fixing the rest of the comprised parameters. The default values of the fixed embraced parameters are the following: $\phi =0.025$, $W{e}_{a}=0.05$, ${\gamma}_{a}=-3.5$, ${\Sigma}_{a}=0.15$, $m=0.5$, ${N}_{r}=2.0$ and $B{i}_{N}=1.3$. In the whole study, the stable and unstable solutions are signified by the solid blue and dashed lines, respectively. Meanwhile, the small, solid black, red, pink, and blue balls symbolize the critical or bifurcation point, where the two solutions converge to one location.

_{2}O

_{3}/water) nanoparticles, respectively. Figure 2 and Figure 3 display that the axial and transverse shear stresses upsurge with increasing ${\Sigma}_{a}$ for the branch of stable solutions. For the branch of unstable solutions, however, the behavior of the results is the same in the directions of transverse shear stress and inverted in the axial shear stress. Physically, there is a drag-type force known as a Lorentz force that tends to impede the velocity of the flow, and, consequently, the thicknesses of the velocity boundary layer become depressed. Consequently, the velocity profiles behave inversely to the coefficient of shear stresses. Hence, the shear stress escalates due to the larger consequences of ${\Sigma}_{a}$. As illustrated in Figure 4, the heat transmission decreases with ${\Sigma}_{a}$ for the branch of stable solutions, but it increases with ${\Sigma}_{a}$ for the branch of unstable solutions. Since there is a large resistance to the motion of fluid due to the Lorentz force, which in turn produces the heat, as an artefact, the thickness or width of the thermal boundary layer, as well as the Nusselt number profiles, increases. Further, the temperature of fluid disappears from the flat plate at some large distance. This is not surprising, since there is a magnetic field that exerts retarding force on the free convection flow. Several physical properties of such fluids can be adjusted by varying the magnetic field. The obtained solutions undoubtedly demonstrate that the flow and the characteristics of heat transfer can be controlled using the magnetic field. Moreover, the changing critical values are obtained for each selected default values of ${\Sigma}_{a}$ (see Figure 2, Figure 3 and Figure 4). With growing values of ${\Sigma}_{a}$, the critical or bifurcation value elevates magnitude-wise. As a result, this pattern suggests that the flow separation decreases as the magnetic parameter’s effects increase. Moreover, for negative values of ${\gamma}_{a}$, there exists a critical value ${\gamma}_{a}C1$, with dual solutions for ${\gamma}_{a}>{\gamma}_{a}C1$: a saddle node bifurcation at ${\gamma}_{a}={\gamma}_{a}C1$, and no outcome for ${\gamma}_{a}<{\gamma}_{a}C1$. The separation of the boundary layer occurs at ${\gamma}_{a}={\gamma}_{a}C1$; therefore, it is not possible to obtain the solution beyond this amount.

## 4. Conclusions

- For the stable branch solutions, the magnetic parameter causes an increase in the axial shear stress and the rate of heat transfer, while it has an entirely different effect on the branch of unstable solutions.
- Due to the higher values of the magnetic parameter, the transverse shear stress increases for upper and lower branches.
- Due to the bigger values of the magnetic parameter, the magnitude of the bifurcation values was increased.
- The radiation parameter is more significantly impacted by the escalating temperature distribution in both solution branches.
- Due to the greater Hall parameter, the axial shear stress falls, while the transverse shear stress increases.
- The heat transfer rate is increased by the nanoparticles for stable branch solutions, while it is decreased for unstable branch solutions.
- The Williamson constraint augments the shear stresses, as well as the heat transfer.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Table 1.**The thermophysical features of the base fluid and nanoparticles [52].

Properties | $\mathsf{\rho}\left(\mathit{k}\mathit{g}/{\mathit{m}}^{3}\right)$ | ${\mathit{c}}_{\mathit{p}}\left(\mathit{J}\mathit{k}\mathit{g}\mathit{K}\right)$ | $\mathit{k}\left(\mathit{W}/\mathit{m}\mathit{k}\right)$ | $\mathsf{\sigma}\left(\mathit{S}/\mathit{m}\right)$ | $\mathsf{\beta}\times {10}^{-5}\left(1/\mathit{K}\right)$ |
---|---|---|---|---|---|

Water | 997.1 | 4179 | 0.613 | 5.5 × 10^{−6} | 21 |

Alumina | 3970 | 765 | 40 | 3.5 × 10^{7} | 0.85 |

**Table 2.**Comparison of ${C}_{fx}{\mathrm{Re}}_{x}^{1/2}$ and $N{u}_{x}{\mathrm{Re}}_{x}^{-1/2}$ for various values of ${\Sigma}_{a}$.

${\mathbf{\Sigma}}_{\mathit{a}}$ | Ishak et al. [54] | Present Results | ||
---|---|---|---|---|

${\mathit{C}}_{\mathit{f}\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{1/2}\text{}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-1/2}$ | ${\mathit{C}}_{\mathit{f}\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{1/2}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-1/2}$ | |

0.00 | −0.5607 | 1.0873 | −0.560723 | 1.087336 |

0.01 | −0.5658 | 1.0863 | −0.565813 | 1.086323 |

0.04 | −0.5810 | 1.0833 | −0.581056 | 1.083329 |

0.25 | −0.6830 | 1.0630 | −0.683035 | 1.063014 |

1.00 | −1.0000 | 1.0000 | −1.000000 | 1.000000 |

4.00 | −1.8968 | 0.8311 | −1.896819 | 0.831129 |

Pr | Grubka and Bobba [55] | Ali [56] | Yih [57] | Present |
---|---|---|---|---|

0.01 | 0.0197 | - | 0.0197 | 0.019743 |

0.72 | 0.8086 | 0.8058 | 0.8086 | 0.808656 |

1.00 | 1.0000 | 0.9961 | 1.0000 | 1.00000 |

3.00 | 1.9237 | 1.9144 | 1.9237 | 1.923734 |

10.0 | 3.7207 | 3.7006 | 3.7207 | 3.720712 |

1000 | 12.2940 | - | 12.2940 | 12.294023 |

**Table 4.**Numerical values of the skin friction coefficient along the $x-$ axis direction for the several values of $\phi $, ${\Sigma}_{a}$, $W{e}_{a}$ and $m$ when ${N}_{r}=2.0$, $B{i}_{N}=1.3$, and ${\gamma}_{a}=-3.5$.

Parameters | ${\mathit{C}}_{\mathit{f}\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{1/2}$ | ||||
---|---|---|---|---|---|

$\mathsf{\phi}$ | ${\mathbf{\Sigma}}_{\mathit{a}}$ | $\mathit{m}$ | $\mathit{W}{\mathit{e}}_{\mathit{a}}$ | Stable Branch Solution | Unstable Branch Solution |

0.025 | 0.015 | 0.5 | 0.05 | 0.219198 | −0.599912 |

0.030 | - | - | - | 0.221735 | −0.606152 |

0.035 | - | - | - | 0.224611 | −0.612562 |

0.025 | 0.05 | 0.5 | 0.05 | 0.122133 | −0.540843 |

- | 0.10 | - | - | 0.173385 | −0.573070 |

- | 0.15 | - | - | 0.219198 | −0.599912 |

0.25 | 0.15 | 0.5 | 0.05 | 0.219198 | −0.599912 |

- | - | 0.7 | - | 0.195805 | −0.586155 |

- | - | 0.9 | - | 0.173778 | −0.572856 |

0.25 | 0.15 | 0.5 | 0.05 | 0.219198 | −0.599912 |

- | - | - | 0.10 | 0.224448 | −0.598987 |

- | - | - | 0.15 | 0.229606 | −0.597892 |

**Table 5.**Numerical values of the skin friction coefficient along the $z-$ axis direction for the several values of $\phi $, ${\Sigma}_{a}$, $W{e}_{a}$ and $m$ when ${N}_{r}=2.0$, $B{i}_{N}=1.3$, and ${\gamma}_{a}=-3.5$.

Parameters | ${\mathit{C}}_{\mathit{f}\mathit{z}}{\mathbf{Re}}_{\mathit{x}}^{1/2}$ | ||||
---|---|---|---|---|---|

$\mathsf{\phi}$ | ${\mathbf{\Sigma}}_{\mathit{a}}$ | $\mathit{m}$ | $\mathit{W}{\mathit{e}}_{\mathit{a}}$ | Stable Branch Solution | Unstable Branch Solution |

0.025 | 0.015 | 0.5 | 0.05 | 0.030567 | 0.014645 |

0.030 | - | - | - | 0.030987 | 0.014891 |

0.035 | - | - | - | 0.031418 | 0.015136 |

0.025 | 0.05 | 0.5 | 0.05 | 0.010414 | 0.006121 |

- | 0.10 | - | - | 0.020619 | 0.010942 |

- | 0.15 | - | - | 0.030567 | 0.014645 |

0.25 | 0.15 | 0.5 | 0.05 | 0.030567 | 0.014645 |

- | - | 0.7 | - | 0.036079 | 0.018213 |

- | - | 0.9 | - | 0.038359 | 0.020299 |

0.25 | 0.15 | 0.5 | 0.05 | 0.030567 | 0.014645 |

- | - | - | 0.10 | 0.030573 | 0.014599 |

- | - | - | 0.15 | 0.030578 | 0.014556 |

**Table 6.**Numerical values of the heat transfer for the numerous values of $\phi $, ${N}_{r}$ and $B{i}_{N}$ when ${\Sigma}_{a}=0.15$, $W{e}_{a}=0.05$, $m=0.5$, ${\gamma}_{a}=-3.5$.

Parameters | ${\mathit{C}}_{\mathit{f}\mathit{z}}{\mathbf{Re}}_{\mathit{x}}^{1/2}$ | |||
---|---|---|---|---|

$\mathsf{\phi}$ | ${\mathit{N}}_{\mathit{r}}$ | $\mathit{B}{\mathit{i}}_{\mathit{N}}$ | Stable Branch Solution | Unstable Branch Solution |

0.025 | 2.0 | 1.3 | 1.732743 | 1.065175 |

0.030 | - | - | 1.728290 | 1.068399 |

0.035 | - | - | 1.723942 | 1.071205 |

0.025 | 1.0 | 1.3 | 1.284758 | 0.607092 |

- | 1.5 | - | 1.524764 | 0.829616 |

- | 2.0 | - | 1.732743 | 1.065175 |

0.25 | 2.0 | 0.5 | 0.963059 | 0.428501 |

- | - | 0.9 | 1.330421 | 0.651489 |

- | - | 1.3 | 1.732743 | 1.065175 |

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**MDPI and ACS Style**

Khan, U.; Zaib, A.; Ishak, A.; Waini, I.; M. Sherif, E.-S.; Boonsatit, N.; Pop, I.; Jirawattanapanit, A.
Ramification of Hall and Mixed Convective Radiative Flow towards a Stagnation Point into the Motion of Water Conveying Alumina Nanoparticles Past a Flat Vertical Plate with a Convective Boundary Condition: The Case of Non-Newtonian Williamson Fluid. *Lubricants* **2022**, *10*, 192.
https://doi.org/10.3390/lubricants10080192

**AMA Style**

Khan U, Zaib A, Ishak A, Waini I, M. Sherif E-S, Boonsatit N, Pop I, Jirawattanapanit A.
Ramification of Hall and Mixed Convective Radiative Flow towards a Stagnation Point into the Motion of Water Conveying Alumina Nanoparticles Past a Flat Vertical Plate with a Convective Boundary Condition: The Case of Non-Newtonian Williamson Fluid. *Lubricants*. 2022; 10(8):192.
https://doi.org/10.3390/lubricants10080192

**Chicago/Turabian Style**

Khan, Umair, Aurang Zaib, Anuar Ishak, Iskandar Waini, El-Sayed M. Sherif, Nattakan Boonsatit, Ioan Pop, and Anuwat Jirawattanapanit.
2022. "Ramification of Hall and Mixed Convective Radiative Flow towards a Stagnation Point into the Motion of Water Conveying Alumina Nanoparticles Past a Flat Vertical Plate with a Convective Boundary Condition: The Case of Non-Newtonian Williamson Fluid" *Lubricants* 10, no. 8: 192.
https://doi.org/10.3390/lubricants10080192