# Activation Energy and Inclination Magnetic Dipole Influences on Carreau Nanofluid Flowing via Cylindrical Channel with an Infinite Shearing Rate

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## Abstract

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**Background:**The infinite shear viscosity model of Carreau fluid characterizes the attitude of fluid flow at a very high/very low shear rate. This model has the capacity for interpretation of fluid at both extreme levels, and an inclined magnetic dipole in fluid mechanics has its valuable applications such as magnetic drug engineering, cold treatments to destroy tumors, drug targeting, bio preservation, cryosurgery, astrophysics, reaction kinetics, geophysics, machinery efficiency, sensors, material selection and cosmology.

**Novelty:**This study investigates and interprets the infinite shear rate of Carreau nanofluid over the geometry of a cylindrical channel. The velocity is assumed to be investigated through imposing an inclined magnetic field onto cylindrical geometry. Activation energy is utilized because it helps with chemical reactions and mass transport. Furthermore, the effects of thermophoresis, the binary chemical process and the Brownian movement of nanoparticles are included in this attempt.

**Formulation:**The mathematics of the assumed Carreau model is derived from Cauchy stress tensor, and partial differential equations (PDEs) are obtained. Similarity transformation variables converted these PDEs into a system of ordinary differential equations (ODEs). Passing this system under the bvp4c scheme, we reached at numerical results of this research attempt.

**Findings**

**:**Graphical debate and statistical analysis are launched on the basis of the obtained computed numerical results. The infinite shear rate aspect of Carreau nanofluid gives a lower velocity. The inclined magnetic dipole effect shows a lower velocity but high energy. A positive variation in activation energy amplifies the concentration field.

## 1. Introduction

_{3}O

_{4}/Cu/Ag-CH

_{3}OH in MHD stagnation point flow have been utilized over a heated surface and partial slip, activation energy facts which the study of Nandi et al. [11] has also engaged with. Recent studies [12,13] related to the hydromagnetic transport of Casson nanofluid and thermo-solutal convection of a nanofluid have been published with superb results.

## 2. Novelty of Physical Model

## 3. Methodology

## 4. Validity of the Study in Pictorial Form

## 5. Comprehensive Analysis of Outcomes of Attached Parameters

## 6. Conclusions of Debate

- Infinite shear rate aspect of Carreau nanofluid gives lower velocity.
- Inclined magnetic dipole effect shows lower velocity but high energy.
- Unsteadiness parameter is related to time factor due to this magnitude of velocity decreasing.
- Relaxation time of fluid increasing by the virtue of magnification in We, which diminishes the velocity field.
- A positive variation in magnetic parameters diminishes the surface drag phenomenon.
- Amplification in thermal conductivity magnifies Nb and the temperature field.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Ethical Statement

## Nomenclature

${h}_{f}$ | Wall heat transfer coefficient | $Sc=\frac{\nu}{{D}_{B}}$ | Schmidt number |

${k}_{m}$ | Wall mass transfer coefficient | ${\rho}_{f}$ | Density of fluid |

$\tau $ | $\mathrm{Ratio}\mathrm{of}{\left(\rho C\right)}_{p}$$\mathrm{to}{\left(\rho C\right)}_{f}$ | ${C}_{p}$ | Specific heat |

${C}_{\infty}$ | Ambient concentration | ${\left(\rho c\right)}_{p}$ | Operative heat capability |

${T}_{w}$ | Surface temperature | u, v | Velocity components |

$\rho $ | Density of fluid | $\sigma =\frac{{K}_{c}{}^{2}}{\alpha}$ | Reaction rate parameter |

${D}_{T}$ | Thermophoresis diffusion coefficient | ${N}_{t}$ | Thermophoresis parameter |

U | Stretching velocity | Rex | Local Reynolds number |

$\alpha $ | Thermal diffusivity | I | Identity tensor |

$\upsilon $ | Kinematic viscosity | $n$ | Power law index |

$A=\frac{{a}_{0}{}^{2}\beta}{4v}$ | Unsteadiness parameter | A_{1} | First Rivlin–Ericksen tensor |

${D}_{B}$ | Brownian diffusion coefficient | $B$ | Magnetic parameter |

$Sc$ | Schmidt number | $M=\sqrt{\frac{{\sigma}^{*}{B}_{o}{}^{2}}{\rho a}}$ | Magnetic parameter |

$We=\frac{\Gamma 8\nu xr}{{a}_{o}^{4}}$ | Local Weissenberg number | $\delta =\frac{{T}_{s}-{T}_{\infty}}{{T}_{\infty}}$ | Temperature constant |

$\sigma =\frac{{K}_{c}{}^{2}}{\alpha}$ | Reaction rate parameter | Pr =$\frac{\mu {c}_{p}}{\alpha}$ | Prandtl number |

${N}_{b}$ | Brownian motion parameter | $E=\frac{{E}_{a}}{k{T}_{\infty}}$ | Activation energy |

$x$ | Distance along the axial direction | $\beta $ | Expansion or contraction strength |

$r$ | Distance along the radial direction | ${f}^{\prime}$ | Dimensionless velocity |

${\theta}^{\prime}$ | Dimensionless temperature | $a(t)$ | Radius of cylinder |

${\varphi}^{\prime}$ | Dimensionless concentration | ${\gamma}_{1}=\frac{{h}_{f}a(t)}{2{k}_{\infty}}$ | Thermal Biot number |

${\gamma}_{2}=\frac{{k}_{m}a(t)}{2{D}_{m}}$ | Concentration Biot number | $t$ | Time |

We | Local Weissenberg number | V | Velocity field |

$\epsilon $ | Thermal conductivity parameter | C | Concentration field |

$Cf$ | Skin friction coefficient | $\tau $ | Cauchy stress tensor |

$hf$ | Wall heat transfer coefficient | $p$ | Pressure |

${q}_{m}$ | Wall mass flux | $\mu $ | Apparent viscosity |

${q}_{w}$ | Wall heat flux | ${\mu}_{0}$ | Zero shear rate viscosity |

${\tau}_{rx}$ | Wall shear stress | ${\mu}_{\infty}$ | Infinite shear rate viscosity |

${T}_{w}$ | Surface temperature | $\stackrel{\u2022}{r}$ | Shear rate |

$Nb=\frac{\tau {D}_{B}\left({C}_{f}-{C}_{\infty}\right)}{\nu}$ | Brownian motion parameter | $\Gamma $ | Time material constant |

$Nt=\frac{\tau {D}_{T}\left({T}_{f}-{C}_{\infty}\right)}{\nu {T}_{\infty}}$ | Thermophoresis parameter | $\eta $ | Local similarity variable |

$K(T)$ | Variable thermal conductivity |

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**Figure 2.**Comparison of this outcome vs. previous literature [1].

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## Share and Cite

**MDPI and ACS Style**

Ayub, A.; Sajid, T.; Jamshed, W.; Zamora, W.R.M.; More, L.A.V.; Talledo, L.M.G.; Rodríguez Ortega de Peña, N.I.; Hussain, S.M.; Hafeez, M.B.; Krawczuk, M.
Activation Energy and Inclination Magnetic Dipole Influences on Carreau Nanofluid Flowing via Cylindrical Channel with an Infinite Shearing Rate. *Appl. Sci.* **2022**, *12*, 8779.
https://doi.org/10.3390/app12178779

**AMA Style**

Ayub A, Sajid T, Jamshed W, Zamora WRM, More LAV, Talledo LMG, Rodríguez Ortega de Peña NI, Hussain SM, Hafeez MB, Krawczuk M.
Activation Energy and Inclination Magnetic Dipole Influences on Carreau Nanofluid Flowing via Cylindrical Channel with an Infinite Shearing Rate. *Applied Sciences*. 2022; 12(17):8779.
https://doi.org/10.3390/app12178779

**Chicago/Turabian Style**

Ayub, Assad, Tanveer Sajid, Wasim Jamshed, William Rolando Miranda Zamora, Leandro Alonso Vallejos More, Luz Marina Galván Talledo, Nélida Isabel Rodríguez Ortega de Peña, Syed M. Hussain, Muhammad Bilal Hafeez, and Marek Krawczuk.
2022. "Activation Energy and Inclination Magnetic Dipole Influences on Carreau Nanofluid Flowing via Cylindrical Channel with an Infinite Shearing Rate" *Applied Sciences* 12, no. 17: 8779.
https://doi.org/10.3390/app12178779