# Numerical Simulation of Heat Transfer Flow Subject to MHD of Williamson Nanofluid with Thermal Radiation

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

#### Model Assumptions and Conditions

- Two-Dimensional laminar steady flow;
- Boundary layer approximation;
- Tiwari and Das model;
- Non-Newtonian Williamson nanofluids;
- Magnetohydrodynamics (MHD);
- variable Thermal conductivity;
- Thermal radiation;
- Nanoparticles shape factor;
- Porous stretching sheet;
- Convective and slip boundary conditions.

## 3. Solution of the Problem

## 4. Verification of Numerical Results

## 5. Numerical Results and Discussion

#### 5.1. Effect of Williamson Parameter $\lambda $

#### 5.2. Effect of Magnetic Parameter M

#### 5.3. Effect of Nanoparticle Concentration $\varphi $

#### 5.4. Effect of Velocity Slip Parameter $\Lambda $

#### 5.5. Effect of Thermal Radiation Parameter $Nr$

#### 5.6. Effect of Biot Number $Bi$

#### 5.7. Effect of the Nanoparticles Shape Factor m

#### 5.8. Effect of the Ehanced Frictional Forces via Reynolds Number $Re$ and Brinkman Number $Br$ on the Entropy

#### 5.9. Effect of Magnetic M and Radiation $Nr$ Parameters on the Physical Quantaties

#### 5.10. Effect of Flow Parameters on Skin Friction and Nusselt Number

## 6. Conclusions

- A decrease in the velocity profile is observed for an increment in the Williamson and volume fraction parameters.
- Nanoparticles are mainly used in fluids to boost up thermal behavior of fluids. Therefore, an increase in nanoparticle concentration enhances the temperature of the nanofluid and hence the thermal boundary layer thickness.
- The $Cu$-water based nanofluid is detected as a superior thermal conductor instead of $Ti{O}_{2}$-water-based nanofluid.
- A rise in magnetic parameter decreases the thickness of the momentum boundary layer whereas it increases the temperature and entropy profile.
- Although slipperiness retards the fluid flow and causes to reduce the velocity and temperature profile, it is significant to observe that the entropy of a system decreases by increasing the slip parameter.

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Choi, 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; Argonne National Lab.: DuPage County, IL, USA, 1995; Volume 66, pp. 99–105. [Google Scholar]
- Eastman, J.A.; Choi, S.U.S.; Li, S.; Thompson, L.J.; Lee, S. Enhanced thermal conductivity through the development of nanofluids. In Proceedings of the Fall Meeting of the Materials Research Society, MRS, Boston, MA, USA, 2–6 December 1996; Argonne National Lab.: DuPage County, IL, USA, 1996. [Google Scholar]
- Eastman, J.A.; Choi, S.; Li, S.; Yu, W.; Thompson, L.J. Anomalously inceases effective thermal conductvities of ethylene glycol-bases nanofluids containing copper nanoparticles. Appl. Phys. Lett.
**2001**, 78, 718–720. [Google Scholar] [CrossRef] - Lomascolo, M.; Colangelo, G.; Milanese, M.; Risi, A. Review of heat transfer in nanofluids: Conductive, convective and radiative experimental results. Renew. Sustain. Energy Rev.
**2015**, 43, 1182–1198. [Google Scholar] [CrossRef] - Sakiadis, B.C. Boundary Layer Equations for Two Dimensional and Axisymmetric Flow, Boundary Layer Behavior on Continuous Solid Surfaces; American Institute of Chemical Engineers: New York, NY, USA, 1961; pp. 26–28. [Google Scholar]
- Hakeem, A.k.A.; Ganesh, N.V.; Ganga, B. Magnetic field effect on second order slip flow of nanofluid over a stretching/shrinking sheet with thermal radiation effect. J. Magn. Magn. Mater.
**2015**, 381, 243–257. [Google Scholar] [CrossRef] - Rehman, K.U.; Malik, M.Y.; Bilal, S.; Zehra, I.; Gaffar, S. On Both Magnetohydrodynamics Thermal Stratified and Dual Convection Flow Field Features: A Computational Study. J. Nanofluids
**2019**, 8, 460–465. [Google Scholar] [CrossRef] - Ali, U.; Malik, M.Y.; Alderremy, A.A.; Aly, S.; Rehman, K.U. A generalized findings on thermal radiation and heat generation/absorption in nanofluid flow regime. Physica A
**2020**, 553, 124026. [Google Scholar] [CrossRef] - Hsiao, K.L. Micropolar nanofluid flow with MHD and viscous dissipation effects towards a stretching sheet with multimedia feature. Int. J. Heat Mass Transf.
**2017**, 112, 983–990. [Google Scholar] [CrossRef] - Mukhopadhyay, S. Heat transfer analysis of the unsteady flow of a Maxwell fluid over a stretching surface in the presence of a Heat Source/Sink. Chin. Phys. Soc.
**2012**, 29, 054703. [Google Scholar] [CrossRef] - Crane, L.J. Flow past a stretching plate. J. Appl. Math. Phys.
**1970**, 21, 645–647. [Google Scholar] [CrossRef] - Ishak, A.; Jafarand, K.; Nazar, R.; Pop, I. MHD stagnation point flow towards a stretching sheet, Statistical Mechanics and its Applications. Stat. Mech. Appl.
**2009**, 388, 3377–3383. [Google Scholar] [CrossRef] - Dorrepaal, J.M. Slip flow in converging and diverging channels. J. Eng. Math.
**1993**, 27, 343–356. [Google Scholar] [CrossRef] - Noghrehabadi, A.; Pourrajab, R.; Ghalambaz, M. Effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature. Int. J. Therm. Sci.
**2012**, 54, 253–261. [Google Scholar] [CrossRef] - Ahmed, F.; Iqbal, M.; Ioan, P. Numerical simulation of forced convective power law nanofluid through circular annulus sector. J. Therm. Anal. Calorim.
**2019**, 135, 861–871. [Google Scholar] [CrossRef] - Ahmed, F.; Iqbal, M. Heat transfer Analysis of MHD Power Law Nano fluid flow through Annular Sector Duct. J. Therm. Sci.
**2020**, 29, 169–181. [Google Scholar] [CrossRef] - Sharma, R.; Ishak, A.; Pop, I. Partial slip flow and heat transfer over a stretching sheet in a nanofluid. Math. Probl. Eng.
**2013**, 2013, 724547. [Google Scholar] [CrossRef] - Darjani, S.; Koplik, J. Extracting the equation of state of lattice gases from random sequential adsorption simulations by means of the Gibbs adsorption isotherm. Phys. Rev. E
**2017**, 96, 052803. [Google Scholar] [CrossRef] [Green Version] - Rahmati, A.R.; Akbari, O.A.; Marzban, A.; Karimi, R.; Pourfattah, F. Simultaneous investigations the effects of non-Newtonian nanofluid flow in different volume fractions of solid nanoparticles with slip and no-slip boundary conditions. Therm. Sci. Eng. Prog.
**2018**, 5, 263–277. [Google Scholar] [CrossRef] - Arabpour, A.; Toghraie, D.; Akbari, O.A. Investigation into the effects of slip boundary condition on nanofluid flow in a double-layer microchannel. J. Therm. Anal. Calorim.
**2018**, 131, 2975–2991. [Google Scholar] [CrossRef] - Koriko, O.K.; Animasaun, I.L.; Reddy, M.G.; Sandeep, N. Scrutinization of thermal stratification, nonlinear thermal radiation and quartic autocatalytic chemical reaction effects on the flow of three-dimensional Eyring-Powell Alumina-water nanofluid. Multidiscip. Model. Mater. Struct.
**2018**, 14, 261–283. [Google Scholar] [CrossRef] - Tlili, I.; Khan, W.A.; Khan, I. Multiple slips effects on MHD SA–Al
_{2}O_{3}and SA–Cu non-Newtonian nanofluid flow over a stretching cylinder in porous medium with radiation and chemical reaction. Results Phys.**2018**, 8, 213–222. [Google Scholar] [CrossRef] - Barnoon, P.; Toghraie, D. Numerical investigation of laminar flow and heat transfer of non-Newtonian nanofluid within a porous medium. J. Mol. Liq.
**2018**, 325, 78–91. [Google Scholar] [CrossRef] - Sheikholeslami, M. CuO-water nanofluid flow due to magnetic field inside a porous media considering Brownian motion. J. Mol. Liq.
**2018**, 249, 921–929. [Google Scholar] [CrossRef] - Ghasemi, H.; Darjani, S.; Mazloomi, H.; Mozaffari, S. Preparation of stable multiple emulsions using food-grade emulsifiers: Evaluating the effects of emulsifier concentration, W/O phase ratio, and emulsification process. SN Appl. Sci.
**2020**, 2, 1–9. [Google Scholar] [CrossRef] - Ghasemi, H.; Aghabarari, B.; Alizadeh, M.; Khanlarkhani, A.; Zahra, N.A. High efficiency decolorization of wastewater by Fenton catalyst: Magnetic iron-copper hybrid oxides. J. Water Process Eng.
**2020**, 37, 101540. [Google Scholar] [CrossRef] - Mukhtar, T.; Jamshed, W.; Aziz, A.; Kouz, W.A. Computational investigation of heat transfer in a flow subjected to magnetohydrodynamic of Maxwell nanofluid over a stretched flat sheet with thermal radiation. Numer. Methods Part. Differ. Equ.
**2020**. [Google Scholar] [CrossRef] - Hussain, S.; Oztop, H.F.; Qureshi, M.A.; Hamdeh, N.A. Magnetohydrodynamic flow and heat transfer of ferrofluid in a channel with non symmetric cavities. J. Therm. Anal. Calorim.
**2020**, 140, 811–823. [Google Scholar] [CrossRef] - Ibrahim, W.; Negera, M. MHD slip flow of upper-convected Maxwell nanofluid over a stretching sheet with chemical reaction. J. Egypt. Math. Soc.
**2020**, 28, 7. [Google Scholar] [CrossRef] [Green Version] - Patel, H.R.; Mittal, A.S.; Darji, R.R. MHD flow of micropolar nanofluid over a stretching/shrinking sheet considering radiation. Int. Commun. Heat Mass Transf.
**2019**, 108, 104322. [Google Scholar] [CrossRef] - Ghobadi, A.H.; Hassankolaei, M.G. Numerical treatment of magneto Carreau nanofluid over a stretching sheet considering Joule heating impact and nonlinear thermal ray. Heat Transf. Asian Res.
**2019**, 48, 4133–4151. [Google Scholar] [CrossRef] - Aziz, A.; Jamshed, W.; Aziz, T. Mathematical model for thermal and entropy analysis of thermal solar collectors by using Maxwell nanofluids with slip conditions, thermal radiation and variable thermal conductivity. Open Phys.
**2018**, 16, 123–136. [Google Scholar] [CrossRef] - Khan, M.; Irfan, M.; Khan, W.A.; Alshomrani, A.S. A new modeling for 3D Carreau fluid flow considering nonlinear thermal radiation. Results Phys.
**2017**, 7, 2692–2704. [Google Scholar] [CrossRef] - Naseem, F.; Shafiq, A.; Zhao, L.; Naseem, A. MHD biconvective flow of Powell–Eyring nanofluid over stretched surface. AIP Adv.
**2017**, 7, 065013. [Google Scholar] [CrossRef] [Green Version] - Reddy, C.S.; Kishan, N.; Madhu, M. Finite Element Analysis of Eyring–Powell Nano Fluid Over an Exponential Stretching Sheet. Int. J. Appl. Comput. Math.
**2017**, 4, 8. [Google Scholar] [CrossRef] - Ansari, M.S.; Motsa, S.S.; Trivedi, M. A New Numerical Approach to MHD Maxwellian nanofluid Flow Past an Impulsively Stretching Sheet. J. Nanofluids
**2018**, 7, 449–459. [Google Scholar] [CrossRef] - Mahmood, A.; Jamshed, W.; Aziz, A. Entropy and heat transfer analysis using Cattaneo-Christov heat flux model for a boundary layer flow of Casson nanofluid. Result Phys.
**2018**, 10, 640–649. [Google Scholar] [CrossRef] - Khan, S.A.; Nie, Y.; Ali, B. Multiple Slip Effects on Magnetohydrodynamic Axisymmetric Buoyant nanofluid Flow above a Stretching Sheet with Radiation and Chemical Reaction. Symmetry
**2019**, 11, 1171. [Google Scholar] [CrossRef] [Green Version] - Hussain, S.; Oztop, H.F.; Qureshi, M.A.; Hamdeh, N.A. Double diffusive buoyancy induced convection in stepwise open porous cavities filled nanofluid. Int. Commun. Heat Mass Transf.
**2020**, 119, 104949. [Google Scholar] [CrossRef] - Shah, Z.; Alzahrani, E.O.; Dawar, A.; Ullah, A.; Khan, I. Influence of Cattaneo-Christov model on Darcy-Forchheimer flow of Micropolar Ferrofluid over a stretching/shrinking sheet. Int. Commun. Heat Mass Transf.
**2020**, 220, 104385. [Google Scholar] [CrossRef] - Abdal, S.; Ali, B.; Younas, S.; Ali, L.; Mariam, A. Thermo-Diffusion and Multislip Effects on MHD Mixed Convection Unsteady Flow of Micropolar nanofluid over a Shrinking/Stretching Sheet with Radiation in the Presence of Heat Source. Symmetry
**2020**, 12, 49. [Google Scholar] [CrossRef] [Green Version] - Shafiq, A.; Rasool, G.; Khalique, C.M. Significance of Thermal Slip and Convective Boundary Conditions in Three Dimensional Rotating Darcy-Forchheimer nanofluid Flow. Symmetry
**2020**, 12, 741. [Google Scholar] [CrossRef] - Jamshed, W.; Aziz, A. A comparative entropy based analysis of Cu and Fe
_{3}O_{4}/methanol Powell-Eyring nanofluid in solar thermal collectors subjected to thermal radiation, variable thermal conductivity and impact of different nanoparticles shape. Result Phys.**2018**, 9, 195–205. [Google Scholar] [CrossRef] - Tiwari, R.J.; Das, M.K. heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf.
**2007**, 50, 2002–2018. [Google Scholar] [CrossRef] - Dapra, S.G. Perturbation solution for pulsatile flow of a non-Newtonian Williamson fluid in a rock fracture. Int. J. Rock Mech. Min. Sci.
**2007**, 44, 271–278. [Google Scholar] [CrossRef] - Nadeem, S.; Hussain, S.T. Flow and heat transfer analysis of Williamson nanofluid. Appl. Nanosci.
**2013**, 4, 1005–1012. [Google Scholar] [CrossRef] [Green Version] - Brewster, M.Q. Thermal Radiative Transfer and Properties; John Wiley and Sons: Hoboken, NJ, USA, 1992. [Google Scholar]
- Bhaskar, N.; Reddy, T.P.; Sreenivasulu, P. Influence of variable thermal conductivity on MHD boundary layar slip flow of ethylene-glycol based CU nanofluids over a stretching sheet with convective boundary condition. Int. J. Eng. Math.
**2014**, 10, 1–10. [Google Scholar] [CrossRef] [Green Version] - Asif, M.; Aziz, A.; Jamshed, W.; Hussian, S. Mathematical model for thermal solar collectors by using magnetohydrodynamic Maxwell nanofluids with slip conditions, thermal radiation and variable thermal conductivity. Result Phys.
**2017**, 7, 3425–3433. [Google Scholar] - Hamilton, R.L.; Crosser, O.K. Thermal conductivity of heterogeneous two-component systems. Ind. Eng. Chem. Fundam.
**1962**, 13, 27–40. [Google Scholar] [CrossRef] - Xu, X.; Chen, S. Cattaneo Christov Heat Flux Model for Heat Transfer of Marangoni Boundary Layer Flow in a Copper Water Nanofluid; Wiley: Hoboken, NJ, USA, 2017; pp. 1–13. [Google Scholar]
- Mutuku, W.N. Ethylene glycol (EG) based nanofluids as a coolant for automotive radiator. Int. J. Comput. Electr. Eng.
**2016**, 3, 1–15. [Google Scholar] [CrossRef] [Green Version] - Minea, A.A. A review on the Thermophysical properties of water-based nanofluids and their hybrids. Ann. "Dunarea de Jos" Univ. Galati Fascicle IX Metall. Mater. Sci.
**2016**, 39, 35–47. [Google Scholar] - Aziz, A.; Jamshed, W.; Aziz, T.; Bahaidarah, H.M.S.; Rehman, K.U. Entropy analysis of Powell-Eyring hybrid nanofluid including effect of linear thermal radiation and viscous dissipation. J. Therm. Anal. Calorim.
**2020**, 1–13. [Google Scholar] [CrossRef] - Keller, H.B. A New Difference Scheme for Parabolic Problems. In Numerical Solutions of Partial Differential Equations; Hubbard, B., Ed.; Academic Press: New York, NY, USA, 1971; Volume 2, pp. 327–350. [Google Scholar]
- Abolbashari, M.H.; Freidoonimehr, N.; Nazari, F.; Rashidi, M.M. Entropy analysis for an unsteady MHD flow past a stretching permeable surface in nano-fluid. Powder Technol.
**2014**, 267, 256–267. [Google Scholar] [CrossRef] - Das, S.; Chakraborty, S.; Jana, R.N.; Makinde, O.D. Entropy analysis of unsteady magneto-nanofluid flow past accelerating stretching sheet with convective boundary condition. Appl. Math. Mech.
**2015**, 36, 1593–1610. [Google Scholar] [CrossRef] - Jamshed, W.; Aziz, A. Cattaneo-Christov based study of TiO
_{2}–CuO/H_{2}O Casson hybrid nanofluid flow over a stretching surface with entropy generationn. Appl. Nanosci.**2018**, 8, 1–14. [Google Scholar] [CrossRef] - Aziz, A.; Jamshed, W.; Ali, Y.; Shams, M. Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation. Discret. Contin. Dyn. Syst. S
**2019**, 13, 2667. [Google Scholar] [CrossRef] [Green Version]

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

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

Density $\left(\rho \right)$ | ${\rho}_{nf}$ = $(1-\varphi ){\rho}_{f}$ + $\varphi {\rho}_{s}$ |

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

Thermal Conductivity $\left(\kappa \right)$ | $\frac{{\kappa}_{nf}}{{\kappa}_{f}}$ = $\left[\frac{({\kappa}_{s}+2{\kappa}_{f})-2\varphi ({\kappa}_{f}-{\kappa}_{s})}{({\kappa}_{s}+2{\kappa}_{f})+\varphi ({\kappa}_{f}-{\kappa}_{s})}\right]$ |

Electrical Conductivity $\left(\sigma \right)$ | $\frac{{\sigma}_{nf}}{{\sigma}_{f}}$ = $\left[1+\frac{3(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1)\varphi}{(\frac{{\sigma}_{s}}{{\sigma}_{f}}+2)-(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1)\varphi}\right]$ |

Thermophysical | $\mathit{\rho}$ (kg/m${}^{3}$) | ${\mathit{c}}_{\mathit{p}}$ (J/kgK) | k (W/mK) | $\mathit{\sigma}$ (S/m) |
---|---|---|---|---|

Ethylene glycol ($EG$) | 1114 | 2415 | 0.252 | $5.5\times {10}^{-6}$ |

Water (${H}_{2}O$) | 997.1 | 4179 | 0.613 | $0.5\times {10}^{-6}$ |

Methanol ($MeOH$) | 792 | 2545 | 0.2035 | $0.5\times {10}^{-6}$ |

Ferro ($F{e}_{3}{O}_{4}$) | 5180 | 670 | 9.7 | $0.74\times {10}^{6}$ |

Copper ($Cu$) | 8933 | 385.0 | 401 | $5.96\times {10}^{7}$ |

Copper oxide ($CuO$) | 6510 | 540 | 18 | $5.96\times {10}^{7}$ |

Alumina ($A{l}_{2}{O}_{3}$) | 3970 | 765.0 | 40 | $3.5\times {10}^{7}$ |

Titanium oxide ($Ti{O}_{2}$) | 4250 | 686.2 | 8.9538 | $2.38\times {10}^{6}$ |

**Table 3.**Numerical results of heat transfer rate (Nusselt Number) for various values of the Prandtl Number.

Pr | Abolbashari | Das | Jamshed | Aziz | Present |
---|---|---|---|---|---|

Results [56] | Results [57] | Results [58] | Results [59] | Results | |

0.72 | 0.80863135 | 0.80876122 | 0.80876181 | 0.80876181 | 0.80876176 |

1.0 | 1.0000000 | 1.0000000 | 1.0000000 | 1.0000000 | 1.0000000 |

3.0 | 1.92368259 | 1.92357431 | 1.92357420 | 1.92357420 | 1.92357403 |

7.0 | 3.07225021 | 3.07314679 | 3.07314651 | 3.07314651 | 3.07314652 |

10 | 3.72067390 | 3.72055436 | 3.72055429 | 3.72055429 | 3.72055417 |

**Table 4.**Values of Skin Friction $={C}_{f}R{e}_{x}^{\frac{1}{2}}$ and Nusselt Number $={N}_{u}R{e}_{x}^{\frac{-1}{2}}$ for $Pr=6.2$, $m=3$.

$\mathit{\lambda}$ | M | $\mathit{\varphi}$ | $\mathbf{\Lambda}$ | $\mathit{\u03f5}$ | $\mathit{N}\mathit{r}$ | $\mathit{B}\mathit{i}$ | S | ${\mathit{C}}_{\mathit{f}}{\mathit{Re}}_{\mathit{x}}^{\frac{1}{2}}$ | ${\mathit{C}}_{\mathit{f}}{\mathit{Re}}_{\mathit{x}}^{\frac{1}{2}}$ | ${\mathit{N}}_{\mathit{u}}{\mathit{Re}}_{\mathit{x}}^{\frac{-1}{2}}$ | ${\mathit{N}}_{\mathit{u}}{\mathit{Re}}_{\mathit{x}}^{\frac{-1}{2}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{Cu}$-$\mathit{Water}$ | ${\mathit{T}}_{\mathit{i}}{\mathit{O}}_{\mathbf{2}}$-$\mathit{Water}$ | $\mathit{Cu}$-$\mathit{Water}$ | ${\mathit{T}}_{\mathit{i}}{\mathit{O}}_{\mathbf{2}}$-$\mathit{Water}$ | ||||||||

0.1 | 0.6 | 0.2 | 0.3 | 0.2 | 0.3 | 0.2 | 0.1 | 2.2694 | 2.0679 | 0.1426 | 0.1288 |

0.3 | 1.5869 | 1.4599 | 0.1412 | 0.1274 | |||||||

0.5 | 1.4912 | 0.3741 | 0.1409 | 0.1271 | |||||||

0.1 | 1.5313 | 1.3881 | 0.1270 | 0.1074 | |||||||

0.6 | 2.2694 | 2.0679 | 0.1426 | 0.1288 | |||||||

1.6 | 3.6174 | 3.4425 | 0.1632 | 0.1352 | |||||||

0.1 | 1.8477 | 1.7234 | 0.1809 | 0.1717 | |||||||

0.15 | 1.9601 | 1.8335 | 0.1674 | 0.1539 | |||||||

0.2 | 2.2694 | 2.0679 | 0.1426 | 0.1288 | |||||||

0.0 | 3.1094 | 2.7299 | 0.1443 | 0.1306 | |||||||

0.2 | 2.6158 | 2.3491 | 0.1434 | 0.1296 | |||||||

0.3 | 2.2694 | 2.0679 | 0.1426 | 0.1288 | |||||||

0.2 | 2.2694 | 2.0679 | 0.1426 | 0.1288 | |||||||

1.2 | 2.2694 | 2.0679 | 0.1237 | 0.1123 | |||||||

2.2 | 2.2694 | 2.0679 | 0.1153 | 0.1047 | |||||||

0.1 | 2.2694 | 2.0679 | 0.0762 | 0.0691 | |||||||

0.3 | 2.2694 | 2.0679 | 0.1426 | 0.1288 | |||||||

0.5 | 2.2694 | 2.0679 | 0.2009 | 0.1810 | |||||||

0.1 | 2.2694 | 2.0679 | 0.1082 | 0.1056 | |||||||

0.2 | 2.2694 | 2.0679 | 0.1426 | 0.1288 | |||||||

0.3 | 2.2694 | 2.0679 | 0.2384 | 0.1725 | |||||||

0.1 | 2.2694 | 2.0679 | 0.1426 | 0.1288 | |||||||

0.2 | 2.4864 | 2.2181 | 0.1470 | 0.1331 | |||||||

0.5 | 2.7112 | 2.3755 | 0.1503 | 0.1362 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Amer Qureshi, M.
Numerical Simulation of Heat Transfer Flow Subject to MHD of Williamson Nanofluid with Thermal Radiation. *Symmetry* **2021**, *13*, 10.
https://doi.org/10.3390/sym13010010

**AMA Style**

Amer Qureshi M.
Numerical Simulation of Heat Transfer Flow Subject to MHD of Williamson Nanofluid with Thermal Radiation. *Symmetry*. 2021; 13(1):10.
https://doi.org/10.3390/sym13010010

**Chicago/Turabian Style**

Amer Qureshi, Muhammad.
2021. "Numerical Simulation of Heat Transfer Flow Subject to MHD of Williamson Nanofluid with Thermal Radiation" *Symmetry* 13, no. 1: 10.
https://doi.org/10.3390/sym13010010