# Rotating Hybrid Nanofluid Flow with Chemical Reaction and Thermal Radiation between Parallel Plates

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*Nanomaterials*

**2022**,

*12*(23), 4177; https://doi.org/10.3390/nano12234177 (registering DOI)

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Continuity Equation:**

**Momentum Equations:**

**Energy Equation:**

**Concentration Equation:**

**Similarity Transformation:**

**Transformed Governing Equations:**

**Quantities of Physical Interest:**

## 3. Solution Methodology

- Obtain the highly non-linear system of PDEs using the boundary layer approximation (BLA) and stress tensor.
- Converting the achieved PDEs into ODEs with the help of suitable similarity transforms.
- Transforming a set of ODEs and associated boundaries into first-order ordinary differential equations so that we can easily call @ex8ode and @ex8bc to compute the problem in MATLAB.
- Achieving the dimensionless form of shear skin relation, Nusselt and Sherwood relations and using them to obtain numeric results.
- Finally, coding the whole problem in MATLAB and obtaining graphical and numeric outcomes and providing analysis of results.

## 4. Results and Discussion

#### 4.1. Velocity Profiles

#### 4.2. Temperature Profile

#### 4.3. Concentration Profile

#### 4.4. Skin Frictions, Nusselt and Sherwood Numbers

## 5. Conclusions

^{−6}. Graphs and tables are used to present the obtained outcomes. The major outcomes of this comparative research are listed below for velocity, concentration, temperature, Nusselt number and Sherwood number.

- The increase in the magnetic field parameter increases the resistance to flow so the velocity profile decays with an increase in the magnetic parameter.
- The porosity and rotation parameter increases the velocity profile while the Reynolds number and mixed convection parameter decay.
- Thermophoresis parameters have a direct relation with the temperature profile, whereas heat source/sink, Prandtl number and Reynolds number have an inverse relation with the temperature profile.
- Heat source/sink does not have a prominent effect on the concentration profile.
- Schmidt number and chemical reaction parameter decay the concentration while the heat source/sink parameter increases.
- Skin friction along the x-axis decreases and the z-axis increases by increasing the Reynolds number for both hybrid nanofluids.
- Reduced skin friction and high Nusselt number are being observed for oxide nanoparticles hybrid nanofluid.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Outcomes and Future Implications

## Nomenclature

${p}^{*}({\mathrm{Kgm}}^{-1}{\mathrm{s}}^{-2})$ | Modified fluid pressure |

$k({\mathrm{Wm}}^{-1}{\mathrm{K}}^{-1})$ | Thermal conductivity |

${B}_{0}\left(\mathrm{A}/\mathrm{m}\right)$ | Constant magnetic field |

$S{h}_{x}$ | Sherwood number |

Kc$\left(\mathrm{mol}/\mathrm{s}\right)$ | Chemical reaction parameter |

$\gamma $ | Mixed convection parameter |

$H,{H}_{1},{H}_{2},{H}_{3},{H}_{4},{H}_{5},{H}_{6}$ | Constants |

$u,v$,w (m${\mathrm{s}}^{-1}$) | Velocity components in $x,y,z$ direction |

Z | Porosity parameter |

$f,g\left({\mathrm{ms}}^{-1}\right)$ | Dimensionless velocity, |

$h$(m) | Distance between plates |

${N}_{b},{N}_{t}$ | Brownian diffusion and Thermophoresis parameter |

${v}_{f}\left({\mathrm{m}}^{2}{\mathrm{s}}^{-1}\right)$ | Kinematic viscosity |

${\mu}_{f}\left({\mathrm{Nsm}}^{-2}\right)$ | Dynamic viscosity |

3D | Three dimensional |

${K}_{r}$ | Rotation parameter |

${C}_{p}\left({\mathrm{Jkg}}^{-1}{\mathrm{K}}^{-1}\right)$ | Specific heat at constant pressure |

$M$ | Magnetic parameter |

${D}_{B},{D}_{T}$ | Brownian and Thermophoresis diffusion |

Q$\left(\mathrm{J}\right)$ | Heat source/sink parameter |

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

$\eta $ | Similarity variable |

$T\left(\mathrm{K}\right),$ $C\left(\mathrm{mol}/{\mathrm{m}}^{3}\right)$ | Temperature and concentration |

$\tilde{Cf},Cf$ | Skin friction coefficients |

$a$(m) | Stretching rate |

$Pr,Ec$ | Prandtl and Eckert number |

$Rd$ | Thermal radiation parameter |

${q}_{w}\left({\mathrm{Wm}}^{-2}\right),$ | Heat flux |

${q}_{m}\left({\mathrm{kgm}}^{-2}{\mathrm{s}}^{-1}\right)$ | Mass flux |

Greek symbols | |

${\tau}_{w}\left({\mathrm{Nm}}^{-2}\right)$ | Shear stress |

$\sigma \left({\mathsf{\Omega}\mathrm{m}}^{-1}\right)$ | Electrical conductivity |

$\theta ,\varphi $ | Dimensionless temperature and concentration |

$\Omega \left({\mathrm{ms}}^{-1}\right)$ | Rotational velocity |

$\rho \left({\mathrm{kgm}}^{-3}\right)$ | Density |

$\alpha \left({\mathrm{m}}^{-2}{\mathrm{s}}^{-1}\right)$ | Temperature diffusivity |

## References

- Crane, L.J. Flow past a stretching plate. Z. Angew. Math. Phys. ZAMP
**1970**, 21, 645–647. [Google Scholar] [CrossRef] - Dutta, B.K.; Roy, P.; Gupta, A.S. Temperature field in flow over a stretching sheet with uniform heat flux. Int. Commun. Heat Mass Transf.
**1985**, 12, 89–94. [Google Scholar] [CrossRef] - Hassan, A.; Hussain, A.; Arshad, M.; Karamti, H.; Awrejcewicz, J.; Alharbi, F.M.; Elfasakhany, A.; Galal, A.M. Computational investigation of magneto-hydrodynamic flow of newtonian fluid behavior over obstacles placed in rectangular cavity. Alex. Eng. J. 2022; in press. [Google Scholar] [CrossRef]
- Nadeem, S.; Lee, C. Boundary layer flow of nanofluid over an exponentially stretching surface. Nanoscale Res. Lett.
**2012**, 7, 94. [Google Scholar] [CrossRef] - Sarada, K.; Gowda, R.J.P.; Sarris, I.E.; Kumar, R.N.; Prasannakumara, B.C. Effect of magnetohydrodynamics on heat transfer behaviour of a non-Newtonian fluid flow over a stretching sheet under local thermal non-equilibrium condition. Fluids
**2021**, 6, 264. [Google Scholar] [CrossRef] - Punith Gowda, R.J.; Sarris, I.E.; Naveen Kumar, R.; Kumar, R.; Prasannakumara, B.C. A Three-Dimensional Non-Newtonian Magnetic Fluid Flow Induced Due to Stretching of the Flat Surface with Chemical Reaction. J. Heat Transf.
**2022**, 144, 113602. [Google Scholar] [CrossRef] - Wong, K.V.; De Leon, O. Applications of nanofluids: Current and future. Adv. Mech. Eng.
**2010**, 2, 519659. [Google Scholar] [CrossRef] - Punith Gowda, R.J.; Naveen Kumar, R.; Jyothi, A.M.; Prasannakumara, B.C.; Sarris, I.E. Impact of binary chemical reaction and activation energy on heat and mass transfer of marangoni driven boundary layer flow of a non-Newtonian nanofluid. Processes
**2021**, 9, 702. [Google Scholar] [CrossRef] - Nadeem, S.; Haq, R.U.; Khan, Z.H. Numerical solution of non-Newtonian nanofluid flow over a stretching sheet. Appl. Nanosci.
**2014**, 4, 625–631. [Google Scholar] [CrossRef] - Ghasemi, S.E.; Mohsenian, S.; Gouran, S.; Zolfagharian, A. A novel spectral relaxation approach for nanofluid flow past a stretching surface in presence of magnetic field and nonlinear radiation. Results Phys.
**2022**, 32, 105141. [Google Scholar] [CrossRef] - Arshad, M.; Hussain, A.; Hassan, A.; Shah SA, G.A.; Elkotb, M.A.; Gouadria, S.; Alsehli, M.; Galal, A.M. Heat and mass transfer analysis above an unsteady infinite porous surface with chemical reaction. Case Stud. Therm. Eng.
**2022**, 36, 102140. [Google Scholar] [CrossRef] - Rout, B.C.; Mishra, S.R. Thermal energy transport on MHD nanofluid flow over a stretching surface: A comparative study. Eng. Sci. Technol. Int. J.
**2018**, 21, 60–69. [Google Scholar] [CrossRef] - Reddy, J.R.; Sugunamma, V.; Sandeep, N. Thermophoresis and Brownian motion effects on unsteady MHD nanofluid flow over a slendering stretching surface with slip effects. Alex. Eng. J.
**2018**, 57, 2465–2473. [Google Scholar] [CrossRef] - Raju, C.S.K.; Sandeep, N.; Babu, M.J.; Sugunamma, V. Dual solutions for three-dimensional MHD flow of a nanofluid over a nonlinearly permeable stretching sheet. Alex. Eng. J.
**2016**, 55, 151–162. [Google Scholar] [CrossRef] - Umavathi, J.C.; Prakasha, D.G.; Alanazi, Y.M.; Lashin, M.M.; Al-Mubaddel, F.S.; Kumar, R.; Punith Gowda, R.J. Magnetohydrodynamic squeezing Casson nanofluid flow between parallel convectively heated disks. Int. J. Mod. Phys. B
**2022**, 2350031. [Google Scholar] [CrossRef] - Hussain, A.; Hassan, A.; Al Mdallal, Q.; Ahmad, H.; Rehman, A.; Altanji, M.; Arshad, M. Heat transportation enrichment and elliptic cylindrical solution of time-dependent flow. Case Stud. Therm. Eng.
**2021**, 27, 101248. [Google Scholar] [CrossRef] - Naveen Kumar, R.; Suresha, S.; Gowda, R.J.; Megalamani, S.B.; Prasannakumara, B.C. Exploring the impact of magnetic dipole on the radiative nanofluid flow over a stretching sheet by means of KKL model. Pramana
**2021**, 95, 180. [Google Scholar] [CrossRef] - Ziaei-Rad, M.; Saeedan, M.; Afshari, E. Simulation and prediction of MHD dissipative nanofluid flow on a permeable stretching surface using artificial neural network. Appl. Therm. Eng.
**2016**, 99, 373–382. [Google Scholar] [CrossRef] - Zeeshan, A.; Majeed, A.; Ellahi, R. Effect of magnetic dipole on viscous ferro-fluid past a stretching surface with thermal radiation. J. Mol. Liq.
**2016**, 215, 549–554. [Google Scholar] [CrossRef] - Muhammad, S.; Ali, G.; Shah, Z.; Islam, S.; Hussain, S.A. The rotating flow of magneto hydrodynamic carbon nanotubes over a stretching sheet with the impact of non-linear thermal radiation and heat generation/absorption. Appl. Sci.
**2018**, 8, 482. [Google Scholar] [CrossRef] - Jamshed, W.; Nisar, K.S.; Gowda, R.P.; Kumar, R.N.; Prasannakumara, B.C. Radiative heat transfer of second grade nanofluid flow past a porous flat surface: A single-phase mathematical model. Phys. Scr.
**2021**, 96, 064006. [Google Scholar] [CrossRef] - Prasannakumara, B.C.; Gowda, R.P. Heat and mass transfer analysis of radiative fluid flow under the influence of uniform horizontal magnetic field and thermophoretic particle deposition. Waves Random Complex Media
**2022**, 1–12. [Google Scholar] [CrossRef] - Soumya, D.O.; Gireesha, B.J.; Venkatesh, P.; Alsulami, M.D. Effect of NP shapes on Fe
_{3}O_{4}–Ag/kerosene and Fe_{3}O_{4}–Ag/water hybrid nanofluid flow in suction/injection process with nonlinear-thermal-radiation and slip condition; Hamilton and Crosser’s model. Waves Random Complex Media**2022**, 1–22. [Google Scholar] [CrossRef] - Jayaprakash, M.C.; Alsulami, M.D.; Shanker, B.; Varun Kumar, R.S. Investigation of Arrhenius activation energy and convective heat transfer efficiency in radiative hybrid nanofluid flow. Waves Random Complex Media
**2022**, 1–13. [Google Scholar] [CrossRef] - Hussain, A.; Hassan, A.; Al Mdallal, Q.; Ahmad, H.; Rehman, A.; Altanji, M.; Arshad, M. Heat transport investigation of magneto-hydrodynamics (SWCNT-MWCNT) hybrid nanofluid under the thermal radiation regime. Case Stud. Therm. Eng.
**2021**, 27, 101244. [Google Scholar] [CrossRef] - Ramesh, G.K.; Gireesha, B.J. Flow over a stretching sheet in a dusty fluid with radiation effect. J. Heat Transf.
**2013**, 135, 102702. [Google Scholar] [CrossRef] - Hussain, A.; Elkotb, M.A.; Arshad, M.; Rehman, A.; Sooppy Nisar, K.; Hassan, A.; Saleel, C.A. Computational investigation of the combined impact of nonlinear radiation and magnetic field on three-dimensional rotational nanofluid flow across a stretchy surface. Processes
**2021**, 9, 1453. [Google Scholar] [CrossRef] - Tsai, C.J.; Lin, J.S.; Aggarwal, S.G.; Chen, D.R. Thermophoretic deposition of particles in laminar and turbulent tube flows. Aerosol Sci. Technol.
**2004**, 38, 131–139. [Google Scholar] [CrossRef] - Arshad, M.; Hussain, A.; Hassan, A.; Haider, Q.; Ibrahim, A.H.; Alqurashi, M.S.; Almaliki, A.H.; Abdussattar, A. Thermophoresis and brownian effect for chemically reacting magneto-hydrodynamic nanofluid flow across an exponentially stretching sheet. Energies
**2021**, 15, 143. [Google Scholar] [CrossRef] - Hassan, A.; Hussain, A.; Arshad, M.; Alanazi, M.M.; Zahran, H.Y. Numerical and Thermal Investigation of Magneto-Hydrodynamic Hybrid Nanoparticles (SWCNT-Ag) under Rosseland Radiation: A Prescribed Wall Temperature Case. Nanomaterials
**2022**, 12, 891. [Google Scholar] [CrossRef] - Qin, L.; Ahmad, S.; Khan, M.N.; Ahammad, N.A.; Gamaoun, F.; Galal, A.M. Thermal and solutal transport analysis of Blasius–Rayleigh–Stokes flow of hybrid nanofluid with convective boundary conditions. Waves Random Complex Media
**2022**, 1–19. [Google Scholar] [CrossRef] - Madhukesh, J.K.; Ramesh, G.K.; Alsulami, M.D.; Prasannakumara, B.C. Characteristic of thermophoretic effect and convective thermal conditions on flow of hybrid nanofluid over a moving thin needle. Waves Random Complex Media
**2021**, 1–23. [Google Scholar] [CrossRef] - Ullah, A.; Alzahrani, E.O.; Shah, Z.; Ayaz, M.; Islam, S. Nanofluids thin film flow of Reiner-Philippoff fluid over an unstable stretching surface with Brownian motion and thermophoresis effects. Coatings
**2018**, 9, 21. [Google Scholar] [CrossRef] - Shafiq, A.; Sindhu, T.N.; Khalique, C.M. Numerical investigation and sensitivity analysis on bioconvective tangent hyperbolic nanofluid flow towards stretching surface by response surface methodology. Alex. Eng. J.
**2020**, 59, 4533–4548. [Google Scholar] [CrossRef] - Khan, A.S.; Nie, Y.; Shah, Z.; Dawar, A.; Khan, W.; Islam, S. Three-dimensional nanofluid flow with heat and mass transfer analysis over a linear stretching surface with convective boundary conditions. Appl. Sci.
**2018**, 8, 2244. [Google Scholar] [CrossRef] - Arshad, M.; Hussain, A.; Elfasakhany, A.; Gouadria, S.; Awrejcewicz, J.; Pawłowski, W.; Elkotb, M.A.; MAlharbi, F. Magneto-hydrodynamic flow above exponentially stretchable surface with chemical reaction. Symmetry
**2022**, 14, 1688. [Google Scholar] [CrossRef] - Pal, D.; Roy, N.; Vajravelu, K. Thermophoresis and Brownian motion effects on magneto-convective heat transfer of viscoelastic nanofluid over a stretching sheet with nonlinear thermal radiation. Int. J. Ambient Energy
**2022**, 43, 413–424. [Google Scholar] [CrossRef] - Makinde, O.D.; Mabood, F.; Ibrahim, M.S. Chemically reacting on MHD boundary-layer flow of nanofluids over a non-linear stretching sheet with heat source/sink and thermal radiation. Therm. Sci.
**2018**, 22, 495–506. [Google Scholar] [CrossRef] - Hamid, A.; Khan, M.I.; Kumar, R.N.; Gowda, R.P.; Prasannakumara, B.C. Numerical study of bio- convection flow of magneto-cross nanofluid containing gyrotactic microorganisms with effective prandtl number approach. Sci. Rep.
**2021**, 11, 16030. [Google Scholar] [CrossRef] - Varun Kumar, R.S.; Alhadhrami, A.; Punith Gowda, R.J.; Naveen Kumar, R.; Prasannakumara, B.C. Exploration of Arrhenius activation energy on hybrid nanofluid flow over a curved stretchable surface. ZAMM—J. Appl. Math. Mech./Z. Angew. Math. Mech.
**2021**, 101, e202100035. [Google Scholar] [CrossRef] - Shah, S.A.A.; Ahammad, N.A.; Din, E.M.T.E.; Gamaoun, F.; Awan, A.U.; Ali, B. Bio-convection effects on prandtl hybrid nanofluid flow with chemical reaction and motile microorganism over a stretching sheet. Nanomaterials
**2022**, 12, 2174. [Google Scholar] [CrossRef] [PubMed] - Faraz, F.; Imran, S.M.; Ali, B.; Haider, S. Thermo-diffusion and multi-slip effect on an axisymmetric Casson flow over a unsteady radially stretching sheet in the presence of chemical reaction. Processes
**2019**, 7, 851. [Google Scholar] [CrossRef] - Arshad, M.; Hassan, A. A numerical study on the hybrid nanofluid flow between a permeable rotating system. Eur. Phys. J. Plus
**2022**, 137, 1126. [Google Scholar] [CrossRef] - Hassan, A.; Hussain, A.; Arshad, M.; Gouadria, S.; Awrejcewicz, J.; Galal, A.M.; Alharbi, F.M.; Eswaramoorthi, S. Insight into the significance of viscous dissipation and heat generation/absorption in magneto-hydrodynamic radiative casson fluid flow with first-order chemical reaction. Front. Phys.
**2022**, 605. [Google Scholar] [CrossRef] - Krishnamurthy, M.R.; Gireesha, B.J.; Prasannakumara, B.C.; Gorla, R.S.R. Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet. Nonlinear Eng.
**2016**, 5, 147–159. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Ganji, D.D. Numerical investigation for two phase modeling of nanofluid in a rotating system with permeable sheet. J. Mol. Liq.
**2014**, 194, 13–19. [Google Scholar] [CrossRef]

**Figure 2.**(

**a**) Influence of magnetic field M on velocity constitute $f\left(\eta \right)$; (

**b**) influence of magnetic field M on velocity constitute $g\left(\eta \right)$; (

**c**) influence of mixed convection constraint γ on velocity constitute $f\left(\eta \right)$; (

**d**) influence of rotation constraint $Kr$ on velocity constitute $f\left(\eta \right)$; (

**e**) influence of reynolds number R on velocity constitute $f\left(\eta \right)$; (

**f**) influence of reynolds number R on velocity constitute $g\left(\eta \right)$; and (

**g**) influence of rotation parameter $Kr$ on velocity constitute $g\left(\eta \right)$.

**Figure 3.**(

**a**) Influence of Prandtl number $Pr$ on temperature constitute $\theta \left(\eta \right)$; (

**b**) influence of thermophoresis parameter ${N}_{t}$ on temperature constitute $\theta \left(\eta \right)$; (

**c**) influence of heat source/sink constraint $Q$ on temperature constitute $\theta \left(\eta \right)$; and (

**d**) influence of Reynold number R on temperature constitute $\theta \left(\eta \right)$.

**Figure 4.**(

**a**) Influence of Schmidt number $Sc,$ on concentration constitute $\varphi \left(\eta \right)$; (

**b**) influence of heat source/sink constraint Q on concentration constitute $\varphi \left(\eta \right)$; and (

**c**) influence of chemical reaction $Kc$ on concentration constitute $\varphi \left(\eta \right)$.

$\mathit{\lambda}$ | $\mathbf{Present}\mathbf{Outcomes}\mathbf{for}\mathit{K}\mathit{r}$ | $\mathbf{Sheikholeslami}\mathbf{and}\mathbf{Ganji}\mathit{K}\mathit{r}$ [46] | ||||
---|---|---|---|---|---|---|

0.5 | 2 | 4 | 0.5 | 2 | 4 | |

1 | 2.633501 | 2.633502 | 2.633515 | 2.63350 | 2.63350 | 2.63351 |

2 | 3.271112 | 3.271754 | 3.274182 | 3.27111 | 3.27175 | 3.27418 |

3 | 3.745803 | 3.746081 | 3.747421 | 3.74580 | 3.74607 | 3.74742 |

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

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

$\mathrm{Dynamic}\mathrm{Viscosity}$ | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left[1-\left({\varphi}_{1}+{\varphi}_{2}\right)\right]}^{5/2}}$ |

$\mathrm{Heat}\mathrm{Capacity}$ | ${\left(\rho {C}_{p}\right)}_{hnf}=\left[1-\left({\varphi}_{1}+{\varphi}_{2}\right)\right]{\left(\rho {c}_{p}\right)}_{f}+{\varphi}_{1}{\left(\rho {c}_{p}\right)}_{s1}+{\varphi}_{2}{\left(\rho {c}_{p}\right)}_{s2}$ |

$\mathrm{Thermal}\mathrm{Conductivity}$ | $\frac{{k}_{hnf}}{{k}_{f}}=\frac{\left({k}_{s1}+{k}_{s2}\right)+2{k}_{f}\left(1-\left({\varphi}_{1}+{\varphi}_{2}\right)\right)+2{\varphi}_{1}{k}_{s1}+2{\varphi}_{2}{k}_{s2}}{\left({k}_{s1}+{k}_{s2}\right)+\left(2+\left({\varphi}_{1}+{\varphi}_{2}\right)\right){k}_{f}-\left({\varphi}_{1}{k}_{s1}+{\varphi}_{2}{k}_{s2}\right)}$ |

Electrical Conductivity | $\frac{{\sigma}_{hnf}}{{\sigma}_{f}}=1+\frac{3\left[\frac{\sigma {s}_{1}{\varphi}_{1}+{\sigma}_{s2}{\varphi}_{2}}{{\sigma}_{f}}-\left({\varphi}_{1}+{\varphi}_{2}\right)\right]}{\left(2+\frac{{\sigma}_{s1}+{\sigma}_{s2}}{{\sigma}_{f}}\right)-\left[\frac{{\sigma}_{s1}{\varphi}_{1}+{\sigma}_{s2}{\varphi}_{2}}{{\sigma}_{f}}\right]+\left({\varphi}_{1}+{\varphi}_{2}\right)}$ |

Properties | ρ (kg/m^{3}) | Cp (J/kg K) | K (W/m K) | σ (Ω·m)^{−1} | $\mathit{\beta}{\left(\mathbf{K}\right)}^{-1}$ |
---|---|---|---|---|---|

Water | 997.1 | 4179 | 0.613 | 5 × 10^{−2} | 21 × 10^{−5} |

Cu | 8933 | 385 | 400 | 5.96 × 10^{7} | 1.67 × 10^{−5} |

$A{l}_{2}{O}_{3}$ | 3970 | 765 | 40 | 1 × 10^{−9} | 0.85 × 10^{−5} |

$Ti{O}_{2}$ | 4250 | 686.2 | 8.96 | 6.27 × 10^{−5} | 0.9 × 10^{−5} |

**Table 4.**The reduced skin frictions $\tilde{C{f}_{x}}$ and$\tilde{C{f}_{z}}$ for $A{l}_{2}{O}_{3}/Ti{O}_{2}$-water and $Cu/Ti{O}_{2}$-water hybrid nanofluid when $Pr=6.3,Nt=\mathrm{Q}=Kc=\mathrm{Rd}=Sc=0.5$.

$\mathit{R}$ | $\mathit{K}\mathit{r}$ | $\mathit{M}$ | $\mathit{Z}$ | $\mathit{\gamma}$ | $\tilde{\mathit{C}{\mathit{f}}_{\mathit{x}}}$ $(\mathit{C}\mathit{u}/\mathit{T}\mathit{i}{\mathit{O}}_{2}-\mathit{Water})$ | $\tilde{\mathit{C}{\mathit{f}}_{\mathit{x}}}$ $(\mathit{A}{\mathit{l}}_{2}{\mathit{O}}_{3}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ | $\tilde{\mathit{C}{\mathit{f}}_{\mathit{z}}}$ $(\mathit{C}\mathit{u}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ | $\tilde{\mathit{C}{\mathit{f}}_{\mathit{z}}}$ $(\mathit{A}{\mathit{l}}_{2}{\mathit{O}}_{3}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ |
---|---|---|---|---|---|---|---|---|

$0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.5$ | $-2.11328$ | $-1.93874$ | $0.29138$1 | $0.277788$ |

$0.6$ | $-2.11820$ | $-1.94371$ | $0.290821$ | $0.277224$ | ||||

$0.7$ | $-2.12311$ | $-1.94866$ | $0.290251$ | $0.276662$ | ||||

$0.8$ | $-2.12801$ | $-1.95360$ | $0.289694$ | $0.276103$ | ||||

$0.5$ | $02$ | $0.5$ | $0.5$ | $0.5$ | $-2.08725$ | $-1.91522$ | $1.16450$ | $1.11024$ |

$04$ | $-2.00328$ | $-1.83948$ | $2.32261$ | $2.21487$ | ||||

$06$ | $-1.86147$ | $-1.71183$ | $3.46923$ | $3.30924$ | ||||

$08$ | $-1.66003$ | $-1.53085$ | $4.60146$ | $4.39035$ | ||||

$0.5$ | $0.5$ | $0.2$ | $0.5$ | $0.5$ | $-1.66849$ | $-1.56202$ | $0.365014$ | $0.339169$ |

$0.4$ | $-1.97204$ | $-1.81824$ | $0.311316$ | $0.294825$ | ||||

$0.6$ | $-2.24845$ | $-2.05475$ | $0.274541$ | $0.263144$ | ||||

$0.8$ | $-2.50282$ | $-2.27474$ | $0.24751$ | $0.239182$ | ||||

$0.5$ | $0.5$ | $0.5$ | $05$ | $0.5$ | $-2.30332$ | $-2.13388$ | $0.330034$ | $0.320293$ |

$06$ | $-2.34519$ | $-2.17685$ | $0.34121$ | $0.332983$ | ||||

$07$ | $-2.38690$ | $-2.21966$ | $0.353697$. | $0.347387$ | ||||

$08$ | $-2.42847$ | $-2.26231$ | $0.36775$ | $0.36389$ | ||||

$0.5$ | $0.5$ | $0.5$ | $0.5$ | $0.6$ | $-2.12701$ | $-1.94437$ | $0.291221$ | $0.277724$ |

$0.7$ | $-2.14073$ | $-1.95011$ | $0.291056$ | $0.277661$ | ||||

$0.8$ | $-2.15446$ | $-1.95562$ | $0.290891$ | $0.277598$ | ||||

$0.9$ | $-2.16819$ | $-1.96125$ | $0.290725$ | $0.277535$ |

**Table 5.**Outcomes of Nusselt and Sherwood number $N{u}_{x}$ and$S{h}_{x}$ for ${\mathrm{Al}}_{2}{\mathrm{O}}_{3}/{\mathrm{TiO}}_{2}$-water and $Cu/Ti{O}_{2}$-water hybrid nanofluid when $Pr=6.3,Nt=Q=Rd=Sc=0.5$.

$\mathit{R}$ | $\mathit{K}\mathit{r}$ | $\mathit{M}$ | $\mathit{Z}$ | $\mathit{\gamma}$ | $\mathit{K}\mathit{c}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}$ $(\mathit{C}\mathit{u}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}$ $(\mathit{A}{\mathit{l}}_{2}{\mathit{O}}_{3}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}$ $(\mathit{C}\mathit{u}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}$ $(\mathit{A}{\mathit{l}}_{2}{\mathit{O}}_{3}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ |
---|---|---|---|---|---|---|---|---|---|

0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 1.67292 | 1.68783 | −50.8965 | −45.0512 |

0.6 | 1.71408 | 1.72957 | −50.8515 | −45.0124 | |||||

0.7 | 1.75548 | 1.77157 | −50.8063 | −44.9735 | |||||

0.8 | 1.79711 | 1.81378 | −50.7609 | −44.9344 | |||||

0.5 | 02 | 0.5 | 0.5 | 0.5 | 0.5 | 1.6727 | 1.68763 | −50.8967 | −45.0513 |

04 | 1.67202 | 1.68702 | −50.8972 | −45.0517 | |||||

06 | 1.67098 | 1.68607 | −50.8980 | −45.0523 | |||||

08 | 1.66969 | 1.68488 | −50.8991 | −45.0532 | |||||

0.5 | 0.5 | 0.2 | 0.5 | 0.5 | 0.5 | 1.67782 | 1.69208 | −50.8937 | −45.049 |

0.4 | 1.67444 | 1.68916 | −50.8956 | −45.0505 | |||||

0.6 | 1.67151 | 1.68657 | −50.8974 | −45.0518 | |||||

0.8 | 1.66894 | 1.68426 | −50.8989 | −45.053 | |||||

0.5 | 0.5 | 0.5 | 05 | 0.5 | 05 | 1.671 | 1.68583 | −50.8976 | −45.0522 |

06 | 1.67057 | 1.68539 | −50.8979 | −45.0524 | |||||

07 | 1.67015 | 1.68494 | −50.8981 | −45.0526 | |||||

08 | 1.66972 | 1.6845 | −50.8984 | −45.0528 | |||||

0.5 | 0.5 | 0.5 | 0.5 | 0.6 | 0.5 | 1.67279 | 1.68777 | −50.8966 | −45.0512 |

0.7 | 1.67266 | 1.68772 | −50.8967 | −45.0512 | |||||

0.8 | 1.67253 | 1.68766 | −50.8968 | −45.0512 | |||||

0.9 | 1.67239 | 1.68761 | −50.8968 | −45.0513 | |||||

0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.2 | 1.59652 | 1.62653 | −70.1406 | −60.6065 |

0.3 | 1.67905 | 1.69262 | −53.0924 | −46.205 | |||||

0.4 | 1.69542 | 1.69995 | −26.6378 | −22.5092 | |||||

0.5 | 1.71541 | 1.71557 | −0.885862 | 2.5687 |

**Table 6.**Outcomes of Nusselt and Sherwood number $N{u}_{x}$ and$S{h}_{x}$ for $A{l}_{2}{O}_{3}/Ti{O}_{2}$-water and $Cu/Ti{O}_{2}$-water hybrid nanofluid when $R=Kr=M=Z=0.5$.

$\mathit{N}\mathit{t}$ | $\mathit{P}\mathit{r}$ | $\mathit{R}\mathit{d}$ | $\mathit{S}\mathit{c}$ | Q | $\mathit{N}{\mathit{u}}_{\mathit{x}}$ $(\mathit{C}\mathit{u}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}$ $(\mathit{A}{\mathit{l}}_{2}{\mathit{O}}_{3}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}$ $(\mathit{C}\mathit{u}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$$-\mathit{Water})$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}$$(\mathit{A}{\mathit{l}}_{2}{\mathit{O}}_{3}/\mathit{T}\mathit{i}{\mathit{O}}_{2}$ $-\mathit{Water})$ |
---|---|---|---|---|---|---|---|---|

$0.5$ | $6.3$ | $0.5$ | $0.5$ | $0.5$ | $1.67614$ | $1.6911$ | $-50.8951$ | $-45.0499$ |

$0.6$ | $1.6262$ | $1.64022$ | $-50.9590$ | $-45.1011$ | ||||

$0.7$ | $1.57829$ | $1.59145$ | $-51.0421$ | $-45.1688$ | ||||

$0.8$ | $1.53233$ | $1.54471$ | $-51.144$ | $-45.2526$ | ||||

$0.5$ | $10$ | $0.5$ | $0.5$ | $0.5$ | $1.79301$ | $1.80991$ | $-50.8421$ | $-45.0038$ |

$15$ | $1.95798$ | $1.97754$ | $-50.7675$ | $-44.9381$ | ||||

$20$ | $2.12475$ | $2.14689$ | $-50.6937$ | $-44.8726$ | ||||

$25$ | $2.29210$ | $2.3167$ | $-50.6222$ | $-44.8086$ | ||||

$0.5$ | $6.3$ | $0.3$ | $0.5$ | $0.5$ | $1.8433$0 | $1.86321$ | $-50.9273$ | $-45.0761$ |

$0.4$ | $1.89737$ | $1.9159$ | $-50.9104$ | $-45.0623$ | ||||

$0.6$ | $1.9911$ | $2.00716$ | $-50.8851$ | $-45.0421$ | ||||

$0.8$ | $2.06923$ | $2.08319$ | $-50.8679$ | $-45.0286$ | ||||

$0.5$ | $6.3$ | $0.5$ | $05$ | $0.5$ | $1.74568$ | $1.74682$ | $-22.6591$ | $-19.7974$ |

$06$ | $1.69653$ | $1.70747$ | $-49.0909$ | $-43.5017$ | ||||

$07$ | $1.69988$ | $1.71023$ | $-48.694$ | $-43.1596$ | ||||

$08$ | $1.70299$ | $1.71279$ | $-48.2778$ | $-42.8004$ | ||||

$0.5$ | $6.3$ | $0.5$ | $0.5$ | $0.1$ | $1.57487$ | $1.59186$ | $-50.8721$ | $-45.0312$ |

$0.2$ | $1.59961$ | $1.61607$ | $-50.8782$ | $-45.0362$ | ||||

$0.3$ | $1.62419$ | $1.64013$ | $-50.8843$ | $-45.0412$ | ||||

$0.4$ | $1.64863$ | $1.66405$ | $-50.8904$ | $-45.0462$ |

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Arshad, M.; Hassan, A.; Haider, Q.; Alharbi, F.M.; Alsubaie, N.; Alhushaybari, A.; Burduhos-Nergis, D.-P.; Galal, A.M. Rotating Hybrid Nanofluid Flow with Chemical Reaction and Thermal Radiation between Parallel Plates. *Nanomaterials* **2022**, *12*, 4177.
https://doi.org/10.3390/nano12234177

**AMA Style**

Arshad M, Hassan A, Haider Q, Alharbi FM, Alsubaie N, Alhushaybari A, Burduhos-Nergis D-P, Galal AM. Rotating Hybrid Nanofluid Flow with Chemical Reaction and Thermal Radiation between Parallel Plates. *Nanomaterials*. 2022; 12(23):4177.
https://doi.org/10.3390/nano12234177

**Chicago/Turabian Style**

Arshad, Mubashar, Ali Hassan, Qusain Haider, Fahad M. Alharbi, Najah Alsubaie, Abdullah Alhushaybari, Diana-Petronela Burduhos-Nergis, and Ahmed M. Galal. 2022. "Rotating Hybrid Nanofluid Flow with Chemical Reaction and Thermal Radiation between Parallel Plates" *Nanomaterials* 12, no. 23: 4177.
https://doi.org/10.3390/nano12234177