# 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

## References

- Williamson, R.V. The flow of pseudoplastic materials. Ind. Eng. Chem.
**1929**, 21, 1108–1111. [Google Scholar] [CrossRef] - Nadeem, S.; Hussain, S.T.; Lee, C. Flow of a Williamson fluid over a stretching sheet. Braz. J. Chem. Eng.
**2013**, 30, 619–625. [Google Scholar] [CrossRef][Green Version] - Hayat, T.; Shafiq, A.; Alsaedi, A. Hydromagnetic boundary layer flow of Williamson fluid in the presence of thermal radiation and Ohmic dissipation. Alex. Eng. J.
**2016**, 55, 2229–2240. [Google Scholar] [CrossRef][Green Version] - Shawky, H.M.; Eldabe, N.T.; Kamel, K.A.; Abd-Aziz, E.A. MHD flow with heat and mass transfer of Williamson nanofluid over stretching sheet through porous medium. Microsyst. Technol.
**2018**, 25, 1155–1169. [Google Scholar] [CrossRef] - Kho, Y.B.; Hussanan, A.; Mohamed, M.K.A.; Salleh, M.Z. Heat and mass transfer analysis on flow of Williamson nanofluid with thermal and velocity slips: Buongiorno model. Propuls. Power Res.
**2019**, 8, 243–252. [Google Scholar] [CrossRef] - Qureshi, M.A. Numerical simulation of heat transfer flow subject to MHD of Williamson nanofluid with thermal radiation. Symmetry
**2020**, 13, 10. [Google Scholar] [CrossRef] - Khan, M.I.; Alzahrani, F. Cattaneo-Christov Double diffusion (CCDD) and magnetized stagnation point flow of non-Newtonian fluid with internal resistance of particles. Phys. Scr.
**2020**, 95, 125002. [Google Scholar] [CrossRef] - Chandel, S.; Sood, S. Unsteady flow of Williamson fluid under the impact of prescribed surface temperature (PST) and prescribed heat flux (PHF) heating conditions over a stretching surface in a porous enclosure. Z. Angew. Math. Mech.
**2022**, 102, e202100128. [Google Scholar] [CrossRef] - Choi, S.U.S. Enhancing Thermal Conductivity of Fluids with Nanoparticles. In Proceedings of the ASME International Mechanical Engineering Congress and Exposition, San Francisco, CA, USA, 12–17 November 1995; Volume 66, pp. 99–105. [Google Scholar]
- Xiao, B.; Wang, W.; Zhang, X.; Long, G.; Fan, J.; Chen, H.; Deng, L. A novel fractal solution for permeability and Kozeny-Carman constant of fibrous porous media made up of solid particles and porous fibers. Powder Technol.
**2019**, 349, 92–98. [Google Scholar] [CrossRef] - Liang, M.; Fu, C.; Xiao, B.; Luo, L.; Wang, Z. A fractal study for the effective electrolyte diffusion through charged porous media. Int. J. Heat Mass Transf.
**2019**, 137, 365–371. [Google Scholar] [CrossRef] - Ekiciler, R. Effects of novel hybrid nanofluid (TiO
_{2}–Cu/EG) and geometrical parameters of triangular rib mounted in a duct on heat transfer and flow characteristics. J. Therm. Anal. Calorim.**2021**, 143, 1371–1387. [Google Scholar] [CrossRef] - Sundar, L.S.; Ramana, E.V.; Said, Z.; Pereira, A.M.B.; Sousa, A.C.M. Heat transfer of rGO/Co
_{3}O_{4}hybrid nanomaterials-based nanofluids and twisted tape configuration in a tube. J. Therm. Sci. Eng. Appl.**2021**, 13, 031004. [Google Scholar] [CrossRef] - Masuda, H.; Ebata, A.; Teramae, K.; Hishinuma, N. Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Netsu Bussei
**1993**, 7, 227–233. [Google Scholar] [CrossRef] - Choi, S.U.S.; Eastman, J.A. Enhancing Thermal Conductivity of Fluids with Nanoparticles. In ASME International Mechanical Engineering Congress & Exposition, San Francisco, CA, USA, 12–17 November 1995; Argonne National Lab: Argonne, IL, USA, 1995. [Google Scholar]
- Bég, O.A.; Bakier, A.Y.; Prasad, V.R. Numerical modelling of non-similar mixed convection heat and species transfer along an inclined solar energy collector surface with cross diffusion effects. World J. Mech.
**2011**, 4, 185–196. [Google Scholar] [CrossRef][Green Version] - Anbuchezhian, N.; Srinivasan, K.; Chandrasekaran, K.; Kandasamy, R. Thermophoresis and Brownian motion effects on boundary layer flow of nanofluid in presence of thermal stratification due to solar energy. Appl. Math. Mech. English Ed.
**2012**, 33, 765–780. [Google Scholar] [CrossRef] - Nasrin, R.; Alim, M.A. Performance of nanofluids on heat transfer in a wavy solar collector. Int. J. Eng. Sci. Technol.
**2013**, 5, 58–77. [Google Scholar] [CrossRef][Green Version] - Kandasamy, R.; Muhaimin, I.; Rosmila, A.K. The performance evaluation of unsteady MHD non-Darcy nanofluid flow over a porous wedge due to renewable (solar) energy. Renew. Energy
**2014**, 64, 1–9. [Google Scholar] [CrossRef][Green Version] - Khan, J.A.; Mustafa, M.; Hayat, T.; Alsaedi, A. Numerical study of Cattaneo-Christov heat flux model for viscoelastic flow due to an exponentially stretching surface. PLoS ONE
**2015**, 10, e0137363. [Google Scholar] - Shehzad, S.A.; Hayat, T.; Alsaedi, A.; Chen, B. A useful model for solar radiation. Energy Ecol. Environ.
**2016**, 1, 30–38. [Google Scholar] [CrossRef] - Zeeshan, A.; Majeed, A. Non Darcy mixed convection flow of magnetic fluid over a permeable stretching sheet with ohmic dissipation. J. Magn.
**2016**, 21, 153–158. [Google Scholar] [CrossRef][Green Version] - Madhukesh, J.K.; Kumar, R.N.; Punith Gowda, R.J.; Prasannakumara, B.C.; Ramesh, G.K.; Khan, M.I.; Khan, S.U.; Chu, Y.-M. Numerical simulation of AA7072-AA7075/water-based hybrid nanofluid flow over a curved stretching sheet with Newtonian heating: A non-Fourier heat flux model approach. J. Mol. Liq.
**2021**, 335, 116103. [Google Scholar] [CrossRef] - Gowda, R.J.P.; Kumar, R.N.; Aldalbahi, A.; Prasannakumara, B.C.; Rahimi-Gorji, M.; Rahaman, M. Thermophoretic particle deposition in time-dependent flow of hybrid nanofluid over rotating and vertically upward/downward moving disk. J. Mol. Liq.
**2021**, 22, 100864. [Google Scholar] - Hamid, M.; Usman, M.; Haq, R.U.; Tian, Z. A Galerkin approach to analyze MHD flow of nanofluid along converging/diverging channels. Arch. Appl. Mech.
**2021**, 91, 1907–1924. [Google Scholar] [CrossRef] - Waqas, H.; Yasmin, S.; Muhammad, T.; Imran, M. Flow and heat transfer of nanofluid over a permeable cylinder with nonlinear thermal radiation. J. Mater. Res. Technol.
**2021**, 14, 2579–2585. [Google Scholar] [CrossRef] - Khan, U.; Zaib, A.; Ishak, A.; Raizah, Z.; Galal, A.M. Analytical approach for a heat transfer process through nanofluid over an irregular porous radially moving sheet by employing KKL correlation with magnetic and radiation effects: Applications to thermal system. Micromachines
**2022**, 13, 1109. [Google Scholar] [CrossRef] - Animasaun, I.L.; Yook, S.-J.; Muhammad, T.; Mathew, A. Dynamics of ternary-hybrid nanofluid subject to magnetic flux density and heat source or sink on a convectively heated surface. Surf. Interfaces
**2022**, 28, 101654. [Google Scholar] [CrossRef] - Pavlov, K.B. Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a plane surface. Magnetohydrodynamics
**1974**, 10, 507–510. [Google Scholar] - Salem, A.M.; El-Aziz, M.A. MHD-mixed convection and mass transfer from a vertical stretching sheet with diffusion of chemically reactive species and space- or temperature-dependent heat source. Can. J. Phys.
**2007**, 85, 359–373. [Google Scholar] - Akbar, N.S.; Ebai, A.; Khan, Z.H. Numerical analysis of magnetic field effects on Eyring Powell fluid flow towards a stretching sheet. J. Magn. Magn. Mater.
**2015**, 382, 355–358. [Google Scholar] [CrossRef] - Khan, M.; Hussain, M.; Azam, M.; Hashim. Magnetohydrodynamic flow of Carreau fluid over a convectively heated surface in the presence of non-linear radiation. J. Magn. Magn. Mater.
**2016**, 412, 63–68. [Google Scholar] [CrossRef] - Moralesa, L.F.; Dasso, S.; Gomez, D.O.; Mininni, P. Hall effect on magnetic reconnection at the Earth's magnetopause. J. Atmosp. Solar-Terr. Phys.
**2005**, 67, 1821–1826. [Google Scholar] [CrossRef] - Su, X.; Zheng, L. Hall effect on MHD flow and heat transfer of nanofluids over a stretching wedge in the presence of velocity slip and Joule heating. Central Euro. J. Phys.
**2013**, 11, 1694–1703. [Google Scholar] [CrossRef][Green Version] - Zaib, A.; Shafie, S. Thermal diffusion and diffusion thermo effects on unsteady MHD free convection flow over a stretching surface considering Joule heating and viscous dissipation with thermal stratification, chemical reaction and hall current. J. Frank. Inst.
**2014**, 351, 1268–1287. [Google Scholar] [CrossRef] - Sreedevi, G.; Rao, R.R.; Prasada Rao, D.R.V.; Chamkha, A.J. Combined influence of radiation absorption and Hall current effects on MHD double-diffusive free convective flow past a stretching sheet. Ain Shams Eng. J.
**2016**, 7, 383–397. [Google Scholar] [CrossRef][Green Version] - Pal, D.; Mandal, G. Influence of Lorentz force and thermal radiation on heat transfer of nanofluids over a stretching sheet with velocity–thermal slip. Int. J. Appl. Comput. Math.
**2017**, 3, 3001–3020. [Google Scholar] [CrossRef] - Khan, M.; Ali, W.; Ahmed, J. A hybrid approach to study the influence of Hall current in radiative nanofluid flow over a rotating disk. Appl. Nanosci.
**2017**, 10, 5167–5177. [Google Scholar] [CrossRef] - Rana, P.; Mahanthesh, B.; Thriveni, K.; Muhammad, T. Significance of aggregation of nanoparticles, activation energy, and Hall current to enhance the heat transfer phenomena in a nanofluid: A sensitivity analysis. Waves Rand. Complex Media
**2022**, 1–23. [Google Scholar] [CrossRef] - Rana, P.; Gupta, G. FEM solution to quadratic convective and radiative flow of Ag-MgO/H
_{2}O hybrid nanofluid over a rotating cone with Hall current: Optimization using response surface methodology. Math. Comp. Simul.**2022**, 201, 121–140. [Google Scholar] [CrossRef] - Aziz, A. A similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. Commun. Nonlinear Sci. Numer. Simul.
**2009**, 14, 1064–1068. [Google Scholar] [CrossRef] - Makinde, O.D.; Aziz, A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int. J. Thermal Sci.
**2010**, 49, 1813–1820. [Google Scholar] [CrossRef] - Ishak, A. Similarity solutions for flow and heat transfer over a permeable surface with a convective boundary condition. Appl. Math. Comp.
**2010**, 217, 837–842. [Google Scholar] [CrossRef] - Yao, S.; Fang, T.; Zhong, Y. Heat transfer of a generalized stretching/shrinking wall problem with convective boundary conditions. Commun. Nonlinear Sci. Numer. Simul.
**2011**, 16, 752–760. [Google Scholar] [CrossRef] - Rahman, M.M.; Merkin, J.H.; Pop, I. Mixed convection boundary-layer flow past a vertical flat plate with a convective boundary condition. Acta Mech.
**2015**, 226, 2441–2460. [Google Scholar] [CrossRef] - Mustafa, M.; Khan, J.A.; Hayat, T.; Alsaedi, A. Simulations for Maxwell fluid flow past a convectively heated exponentially stretching sheet with nanoparticles. AIP Adv.
**2015**, 5, 037133. [Google Scholar] [CrossRef][Green Version] - Ibrahim, W.; Haq, R.U. Magnetohydrodynamic (MHD) stagnation point flow of nanofluid past a stretching sheet with convective boundary condition. J. Braz. Soc. Mech. Sci. Eng.
**2015**, 38, 1155–1164. [Google Scholar] [CrossRef] - Makinde, O.D.; Khan, W.A.; Khan, Z.H. Stagnation point flow of MHD chemically reacting nanofluid over a stretching convective surface with slip and radiative heat. Proc. Inst. Mech E Part E J. Process Mech. Eng.
**2017**, 231, 695–703. [Google Scholar] [CrossRef] - Abo-Eldahab, E.M.; El Aziz, M.A. Hall and Ion-Slip effects on MHD free convective heat generating flow past a semi-infinite vertical flat plate. Phys. Scr. A
**2000**, 61, 344. [Google Scholar] [CrossRef] - Sutton, G.W.; Sherman, A. Engineering Magnetohydrodynamics; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Takabi, B.; Salehi, S. Augmentation of the heat transfer performance of a sinusoidal corrugated enclosure by employing hybrid nanofluid. Adv. Mech. Eng.
**2014**, 2014, 147059. [Google Scholar] [CrossRef] - Arifuzzaman, M.; Uddin, M.J. Convective flow of alumina–water nanofluid in a square vessel in presence of the exothermic chemical reaction and hydromagnetic field. Res. Eng.
**2021**, 10, 100226. [Google Scholar] [CrossRef] - Oztop, H.F.; Abu-Nada, E. Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids. Int. J. Heat Fluid Flow
**2008**, 29, 1326–1336. [Google Scholar] [CrossRef] - Ishak, A.; Nazar, R.; Pop, I. Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. Heat Mass Transf.
**2008**, 44, 921. [Google Scholar] [CrossRef] - Grubka, L.J.; Bobba, K.M. Heat transfer characteristics of a continuous, stretching surface with variable temperature. ASME J. Heat Transf.
**1985**, 107, 248–250. [Google Scholar] [CrossRef] - Ali, M.E. On thermal boundary layer on a power-law stretched surface with suction or injection. Int. J. Heat Fluid Flow
**1995**, 16, 280–290. [Google Scholar] [CrossRef] - Yih, K.A. Free convection effect on MHD coupled heat and mass transfer of a moving permeable vertical surface. Int. Commun. Heat Mass Transf.
**1999**, 26, 95–104. [Google Scholar] [CrossRef]

**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