# Chemical MHD Hiemenz Flow over a Nonlinear Stretching Sheet and Brownian Motion Effects of Nanoparticles through a Porous Medium with Radiation Effect

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

- $x$ and $y$ axes are taken as the way of sheet motion and normal to the motion.
- The nonlinear stretching velocity of the flat plate is assumed as ${\overline{u}}_{w}\left(\overline{x}\right)={u}_{0}{\left(\frac{\overline{x}}{l}\right)}^{m}$, where ${u}_{0}$ is a constant indicating the direction of the plate along the positive or negative side of the $x$ axis, depending on whether ${u}_{0}>0$ or ${u}_{0}<0$, and a stationary plate when ${u}_{0}=0$, $m$ is the power-law velocity exponent, and $l$ is the characteristic length.
- The ambient fluid’s moving velocity has the form ${\overline{u}}_{e}\left(\overline{x}\right)={u}_{\infty}{\left(\frac{\overline{x}}{l}\right)}^{m}$, where ${u}_{\infty}$ is a constant.
- A variable magnetic field $B\left(\overline{x}\right)={B}_{0}{\left(\frac{\overline{x}}{l}\right)}^{\frac{m-1}{2}}$ where ${B}_{0}$ is a constant is assumed along the plate.

- Stagnation point flow.
- Micropolar liquid model.
- Joule heat, radiation, source/sink, porous medium and chemical reaction effects are deemed.
- Thermophoresis and Brownian motion effects are taken into account.

## 3. Methodology (SLM)

## 4. Results and Discussion

## 5. Conclusions

- The temperature profile is significantly driven by the heat source parameter.
- Thermal radiation and thermophoresis parameters lead enhanced temperature.
- The concentration profile lowers as both the Lewis number and the chemical reaction parameters expand.
- The rate of heat transfer elevates with $R$ and $\mathrm{Pr}$.
- The rate of mass transfer elevates with $Le$ and $\gamma $.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Abu-Nada, E.; Oztop, H.F. Effects of inclination angle on natural convection in enclosures filled with Cu-water nanofluid. Int. J. Heat Fluid Flow
**2009**, 30, 669–678. [Google Scholar] [CrossRef] - Zargartalebi, H.; Ghalambaz, M.; Noghrehabadi, A.; Chamkha, A.J. Stagnation point heat transfer of nanofluids toward stretching sheets with variable thermo-physical properties. Adv. Power Technol.
**2015**, 26, 819–829. [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. Therm. Sci.
**2011**, 50, 1326–1332. [Google Scholar] [CrossRef] - Mohamad, A.Q.; Khan, I.; Shafie, S. Unsteady free convection flow of rotating MHD second grade fluid in a porous medium over an oscillating plate. AIP Conf. Proceeding
**2016**, 1750, 030013. [Google Scholar] - Mbeledogu, I.U.; Ogulu, A. Heat and mass transfer of an unsteady MHD natural convection flow of a rotating fluid past a vertical flat plate in the presence of radiative heat transfer. Int. J. Heat Mass Transf.
**2007**, 50, 1902–1908. [Google Scholar] [CrossRef] - Salah, F.; Aziz, Z.A.; Ching, D.L.C. New exact solutions for MHD transient rotating flow of a second-grade fluid in a porous medium. J. Appl. Math.
**2011**, 823034, 2011. [Google Scholar] [CrossRef] - Sivanandam, S.; Marimuthu, B.; Alzahrani, A.K. Numerical study on influence of magnetic field and discrete heating on free convection in a porous container. Sci. Iran.
**2022**, 3063–3071. [Google Scholar] - Sivasankaran, S.; Narrein, K. Influence of geometry and magnetic field on convective flow of nanofluids in trapezoidal microchannel heat sink. Iran. J. Sci. Technol. Trans. Mech. Eng.
**2020**, 44, 373–382. [Google Scholar] [CrossRef] - Sivasankaran, S.; Malleswaran, A.; Bhuvaneswari, M.; Balan, P. Hydro-magnetic mixed convection in a lid-driven cavity with partially thermally active walls. Sci. Iran.
**2017**, 24, 153–163. [Google Scholar] [CrossRef] - Sivasankaran, S.; Ananthan, S.S.; Abdul Hakeem, A.K. Mixed convection in a lid-driven cavity with sinusoidal boundary temperature at the bottom wall in the presence of magnetic field. Sci. Iran.
**2016**, 23, 1027–1036. [Google Scholar] [CrossRef] - Sivasankaran, S.; Chandrapushpam, T.; Bhuvaneswari, M.; Karthikeyan, S.; Alzahrani, A.K. Effect of chemical reaction on double diffusive MHD squeezing copper water nanofluid flow between parallel plates. J. Mol. Liq.
**2022**, 368, 120768. [Google Scholar] [CrossRef] - Devi, L.; Niranjan, H.; Sivasankaran, S. Effects of chemical reactions, radiation, and activation energy on MHD buoyancy induced nano fluidflow past a vertical surface. Sci. Iran.
**2022**, 29, 90–100. [Google Scholar] - Yesodha, P.; Bhuvaneswari, M.; Sivasankaran, S.; Saravanan, K. Convective heat and mass transfer of chemically reacting fluids with activation energy along with Soret and Dufour effects. Mater. Today Proc.
**2021**, 42, 600–606. [Google Scholar] [CrossRef] - Sivanandam, S.; Marimuthu, B.; Arumugam, M.; Bhose, G. Stratification, slip and cross diffusion impacts on time depending convective stream with chemical reaction. Math. Model. Eng. Probl.
**2019**, 6, 581–588. [Google Scholar] [CrossRef] - Ariel, P.D. Hiemenz flow in hydromagnetics. Acta Mech.
**1994**, 103, 31–43. [Google Scholar] [CrossRef] - Motsa, S.S.; Khan, Y.; Shateyi, S. A new numerical solution of Maxwell fluid over a shrinking sheet in the region of a stagnation point. Math. Probl. Eng.
**2012**, 290615, 2012. [Google Scholar] [CrossRef] - Parand, K.; Lotfi, Y.; Rad, J.A. An efficient analytic approach for solving Hiemenz flow through a porous medium of a non-Newtonian Rivlin-Ericksen fluid with heat transfer. Nonlinear Eng.
**2018**, 7, 287–301. [Google Scholar] [CrossRef] - Waqas, H.; Imran, M.; Muhammad, T.; Sait, S.M.; Ellahi, R. On bio-convection thermal radiation in Darcy–Forchheimer flow of nanofluid with gyrotactic motile microorganism under Wu’s slip over stretching cylinder/plate. Int. J. Numer. Methods Heat Fluid Flow
**2020**, 31, 1520–1546. [Google Scholar] [CrossRef] - Farooq, U.; Waqas, H.; Imran, M.; Alghamdi, M.; Muhammad, T. On melting heat transport and nanofluid in a nozzle of liquid rocket engine with entropy generation. J. Mater. Res. Technol.
**2021**, 14, 3059–3069. [Google Scholar] [CrossRef] - Shahzad, A.; Imran, M.; Tahir, M.; Khan, S.A.; Akgül, A.; Abdullaev, S.; Yahia, I.S. Brownian motion and thermophoretic diffusion impact on Darcy-Forchheimer flow of bioconvective micropolar nanofluid between double disks with Cattaneo–Christov heat flux. Alex. Eng. J.
**2023**, 62, 1–15. [Google Scholar] [CrossRef] - Hamid, A.; Khan, M. Unsteady mixed convective flow of Williamson nanofluid with heat transfer in the presence of variable thermal conductivity and magnetic field. J. Mol. Liq.
**2018**, 260, 436–446. [Google Scholar] - Zhang, C.; Zheng, L.; Zhang, X.; Chen, G. MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction. Appl. Math. Model.
**2015**, 39, 165–181. [Google Scholar] [CrossRef] - Pandey, A.K.; Kumar, M. Natural convection and thermal radiation influence on nanofluid flow over a stretching cylinder in a porous medium with viscous dissipation. Alex. Eng. J.
**2017**, 56, 55–62. [Google Scholar] [CrossRef] - Bhuvaneswari, M.; Sivasankaran, S.; Malarselvi, A.; Ganga, B. Radiation and cross diffusion on unsteady chemically reactive convective flow through an extended surface in heat generating porous medium. Int. J. Energy Technol. Policy
**2021**, 17, 494–509. [Google Scholar] [CrossRef] - Reddy, V.S.; Kandasamy, J.; Sivanandam, S. Impacts of Casson Model on Hybrid Nanofluid Flow over a Moving Thin Needle with Dufour and Soret and Thermal Radiation Effects. Math. Comput. Appl.
**2022**, 28, 2. [Google Scholar] [CrossRef] - Narayanaswamy, M.K.; Kandasamy, J.; Sivanandam, S. Go-MoS2/Water Flow over a Shrinking Cylinder with Stefan Blowing, Joule Heating, and Thermal Radiation. Math. Comput. Appl.
**2022**, 27, 110. [Google Scholar] [CrossRef] - Jagan, K.; Sivasankaran, S. Three-Dimensional Non-Linearly Thermally Radiated Flow of Jeffrey Nanoliquid towards a Stretchy Surface with Convective Boundary and Cattaneo–Christov Flux. Math. Comput. Appl.
**2022**, 27, 98. [Google Scholar] [CrossRef] - Uddin, M.J.; Khan, W.A.; Ismail, A.I. MHD forced convective laminar boundary layer flow of convectively heated moving vertical plate with radiation and transpiration effect. Plos ONE
**2013**, 8, 62664. [Google Scholar] [CrossRef] - Ahmed, M.A.M.; Mohammed, M.E.; Khidir, A.A. On linearization method to MHD boundary layer convective heat transfer with low pressure gradient. Propuls. Power Res.
**2015**, 4, 105–113. [Google Scholar] [CrossRef] - Salah, F.; Alzahrani, A.K.; Sidahmed, A.O.; Viswanathan, K.K. A note on thin-film flow of Eyring-Powell fluid on the vertically moving belt using successive linearization method. Int. J. Adv. Appl. Sci.
**2019**, 6, 17–22. [Google Scholar] - Yih, K.A. MHD forced convection flow adjacent to a non-isothermal wedge. Int. Comm. Heat Mass Transf.
**1999**, 26, 819–827. [Google Scholar] [CrossRef]

**Table 1.**Convergence of SLM solution when $M=m=0.5,V=2.0,Nb=0.3,Nt=0.2,R=Q=0.1,\mathrm{Pr}=Le=1.0,\gamma =0.2,\Omega =0.1$.

Order | $-{\mathit{f}}^{\u2033}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{\varphi}}^{\prime}(0)$ |
---|---|---|---|

1 | 1.615243564 | 0.591309575 | 0.807258968 |

2 | 1.615264958 | 0.588594177 | 0.808486096 |

5 | 1.615264959 | 0.588587567 | 0.808488879 |

10 | 1.615264959 | 0.588587567 | 0.808488879 |

20 | 1.615264958 | 0.588587567 | 0.808488879 |

30 | 1.615264958 | 0.588587567 | 0.808488879 |

35 | 1.615264958 | 0.588587567 | 0.808488879 |

40 | 1.615264958 | 0.588587567 | 0.808488879 |

45 | 1.615264958 | 0.588587567 | 0.808488879 |

m | M | V | Uddin et al. [28] | SLM (Present) |
---|---|---|---|---|

−0.6 | 1 | −1.2465 | 1.831134 | 1.829965702 |

−0.6 | 1 | −0.3 | 0.754875 | 0.754875083 |

−0.6 | 1 | 0.5 | 0.147122 | 0.147122313 |

−0.6 | 1 | 1.1 | 0.008662 | 0.008662160 |

−0.7 | 2 | −1.2465 | 2.8552230 | 2.855222907 |

−0.7 | 2 | −0.3 | 1.4340566 | 1.434056645 |

−0.7 | 2 | 0.5 | 0.4700876 | 0.470087557 |

−0.7 | 2 | 1.1 | −0.0799084 | −0.079908545 |

−0.6 | 2 | −1.2465 | 2.8886778 | 2.888677771 |

−0.6 | 2 | −0.3 | 1.5013537 | 1.501353678 |

−0.6 | 2 | 0.5 | 0.5167744 | 0.516774431 |

−0.6 | 2 | 1.1 | −0.093447 | −0.093446596 |

m | Uddin et al. [28] | Yih [31] (Finite Difference) | SLM (Present) |
---|---|---|---|

−0.05 | 0.213483 | 0.213484 | 0.213483741 |

0.0 | 0.33206 | 0.332057 | 0.332057336 |

1/3 | 0.75745 | 0.757448 | 0.757447581 |

1.0 | 1.23259 | 1.232588 | 1.232587657 |

**Table 4.**Skin friction coefficients, local Nusselt numbers and local Sherwood numbers for various values of parameters involved using SLM.

m | M | V | Nb | Nt | R | Q | Pr | Le | $\mathit{\gamma}$ | $\mathbf{\Omega}$ | $\left|{\mathit{f}}^{\u2033}\left(0\right)\right|$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\mathit{\varphi}}^{\prime}\left(0\right)$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 0.5 | 2 | 0.3 | 0.2 | 0.1 | 0.1 | 1 | 1 | 0.2 | 0.1 | 1.036818487 | 0.466696358 | 0.727197663 |

0.5 | 1.615264958 | 0.588587567 | 0.808488879 | ||||||||||

1 | 2.037891791 | 0.691219392 | 0.884779820 | ||||||||||

2 | 2.691867471 | 0.861726041 | 1.022496729 | ||||||||||

1 | 2.782432421 | 0.858913310 | 1.020208245 | ||||||||||

2 | 2.955494943 | 0.853763691 | 1.016051769 | ||||||||||

3 | 3.119202610 | 0.849151404 | 1.012365107 | ||||||||||

4 | 3.274896834 | 0.844984821 | 1.009063514 | ||||||||||

−0.6 | 3 | 3.065259197 | 0.239179732 | 0.626117442 | |||||||||

4 | 4.274052296 | 0.291595855 | 0.647177685 | ||||||||||

5 | 5.241191366 | 0.344251921 | 0.669459499 | ||||||||||

0.2 | 5.388545930 | 0.341608410 | 0.668515318 | ||||||||||

0.3 | 5.532218947 | 0.339072211 | 0.667615489 | ||||||||||

0.7 | 6.074769939 | 0.329858959 | 0.664392561 | ||||||||||

1.0 | 6.453114860 | 0.662297603 | 0.662297603 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Salah, F.; Sidahmed, A.O.M.; Viswanathan, K.K.
Chemical MHD Hiemenz Flow over a Nonlinear Stretching Sheet and Brownian Motion Effects of Nanoparticles through a Porous Medium with Radiation Effect. *Math. Comput. Appl.* **2023**, *28*, 21.
https://doi.org/10.3390/mca28010021

**AMA Style**

Salah F, Sidahmed AOM, Viswanathan KK.
Chemical MHD Hiemenz Flow over a Nonlinear Stretching Sheet and Brownian Motion Effects of Nanoparticles through a Porous Medium with Radiation Effect. *Mathematical and Computational Applications*. 2023; 28(1):21.
https://doi.org/10.3390/mca28010021

**Chicago/Turabian Style**

Salah, Faisal, Abdelmgid O. M. Sidahmed, and K. K. Viswanathan.
2023. "Chemical MHD Hiemenz Flow over a Nonlinear Stretching Sheet and Brownian Motion Effects of Nanoparticles through a Porous Medium with Radiation Effect" *Mathematical and Computational Applications* 28, no. 1: 21.
https://doi.org/10.3390/mca28010021