# Numerical Investigation of MWCNT and SWCNT Fluid Flow along with the Activation Energy Effects over Quartic Auto Catalytic Endothermic and Exothermic Chemical Reactions

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Numerical Solution

## 4. Step-by-Step Graphical Detail of the Problem

#### 4.1. Problem Formulation

#### 4.2. Modeling

#### 4.3. Numerical Process

#### 4.4. Numerical Results

#### 4.5. Analysis

## 5. Results and Discussions

## 6. Conclusions

- ▪
- Larger magnetic parameters, slip parameters, and velocity ratio factors all cause fluid flow to speed up, but the solid volume fraction causes it to slow down.
- ▪
- As with the measurements of the heat generation and solid volume ratio, the system is observed to gradually cool down.
- ▪
- When increasing the slip parameters and velocity ratio, fluid tends to flow smoothly, whereas for the solid volume fractions, surface roughness increased.
- ▪
- The concentration profile decreases for the larger values of activation energy and exothermic/endothermic parameters.
- ▪
- The process of heat transmission inside the system was influenced in opposing ways by the velocity ratio parameter as well as the thermal expansion parameter.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | |

$u,v$ | Velocity component along the x and y directions |

${Q}_{0}$ | Volumetric rate of a heat source |

$Pr$ | Prandtl number |

${U}_{\infty}\left(x\right)$ | Free-stream velocity of the fluid |

${S}_{c}$ | Schmidt number |

${C}_{f}$ | Surface drag force |

${N}_{u}$ | Local heat transfer |

${f}^{\prime}$ | Dimensionless stream velocity |

E | Activation energy |

${D}_{c}$ | Dimensionless heat generation parameter |

$m,n$ | Unitless rate constants |

Greek Symbols | |

${\rho}_{nf}$ | Density of nanofluid |

${\rho}_{f}$ | Density of fluid |

${\gamma}^{*}$ | Navier slip length density |

$\u03f5$ | Velocity ratio parameter |

${\mu}_{nf}$ | Dynamic viscosity shear stress |

${\mu}_{f}$ | Dynamic viscosity shear stress |

${\tau}_{w}$ | Dynamic viscosity shear stress |

${\alpha}_{nf}$ | Thermal diffusivity of nanofluid |

$\tau $ | Ratio of specific heats |

${\xi}^{*}$ | Reciprocal of some critical shear rate |

$\xi $ | Critical shear rate |

${\left(\rho {C}_{p}\right)}_{nf}$ | Heat capacity of nanofluid |

${\beta}_{f},{\beta}_{s}$ | Coefficient of thermal expansion |

${\left(\rho {C}_{p}\right)}_{f}$ | Heat capacity of fluid |

${\gamma}_{1}$ | Non-dimensional slip velocity parameter |

${\sigma}_{nf}$ | Electric conductivity of fluid |

${\sigma}_{f}$ | Electric conductivity of fluid |

${\sigma}_{f}$ | Electric conductivity of nanofluid |

${\beta}_{CNT}$ | Coefficient of thermal expansion of carbon nanotubes |

$\varphi $ | Nanofluid volume fraction |

$\gamma $ | Dimensionless thermal relaxation time |

$\beta $ | Exothermic/endothermic parameter |

$\sigma ,{\lambda}_{1}$ | Dimensionless chemical reaction rate |

${k}_{nf}$ | Thermal conductivity of nanofluid |

${k}_{f}$ | Thermal conductivity of fluid |

${\nu}_{nf}$ | Kinematic viscosity of nanofluid |

${k}_{CNT}$ | Thermal conductivity of carbon nanotubes |

${\rho}_{CNT}$ | Density of carbon nanotubes |

${\left(\rho {C}_{p}\right)}_{CNT}$ | Heat capacity of carbon nanotubes |

${B}^{2}\left(x\right)$ | Magnetic field strength |

## References

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**Figure 1.**Geometry of the model [25].

Physical Attributes | Base Fluid $\left({\mathit{H}}_{2}\mathit{O}\right)$ | MWCNT | SWCNT |
---|---|---|---|

${C}_{p}(J/kgK)$ | 4179 | 796 | 425 |

$\rho (kg/{m}^{3})$ | 997 | 1600 | 2600 |

$K(W/mK)$ | 0.613 | 3000 | 6600 |

$\sigma {\left(\mathrm{\Omega}m\right)}^{-1}$ | $5.5\times {10}^{-6}$ | ${10}^{7}$ | ${10}^{6}$ |

**Table 2.**The work of Ramzan et al. [7]’s limited case is compared with statistical data on surface drag force as well as local Nusselt number versus Prandtl number.

Ramzan et al. [7] ${\mathit{f}}^{\u2033}\left(0\right)$ | Ramzan et al. [7] $-{\mathit{\theta}}^{\prime}\left(0\right)$ | Present ${\mathit{f}}^{\u2033}\left(0\right)$ | Results | Present $-{\mathit{\theta}}^{\prime}\left(0\right)$ | Results | |||
---|---|---|---|---|---|---|---|---|

$\varphi $ | SWCNT | MWCNT | SWCNT | MWCNT | SWCNT | MWCNT | SWCNT | MWCNT |

0.01 | 0.338910 | 0.337270 | 1.105710 | 1.079040 | 0.338995 | 0.337276 | 1.105710 | 1.079043 |

0.1 | 0.408120 | 0.390070 | 4.806290 | 4.277160 | 0.408107 | 0.390084 | 4.806290 | 4.277160 |

0.2 | 0.504530 | 0.464660 | 12.30352 | 10.56796 | 0.504522 | 0.464669 | 12.30358 | 10.56796 |

${\mathit{R}}_{\mathit{e}}^{1/2}{\mathit{C}}_{\mathit{f}}$ | ||||
---|---|---|---|---|

Parameters | Comparison Analysis | |||

$\mathbf{\varphi}$ | $\mathbf{\u03f5}$ | Othman et al. [28] | Wang [29] | Current |

0 | 2 | −1.887306668 | −1.88731 | −1.88795 |

0 | 1 | 0 | 0 | 0 |

0 | 0.5 | 0.71329495 | 0.7133 | 0.7136 |

0 | 0 | 1.232587647 | 1.232588 | 1.232600 |

0 | −0.5 | 1.495669739 | 1.49567 | 1.49590 |

0 | −1 | 1.328816861 | 1.32882 | 1.32900 |

**Table 4.**Rheological numerics of $-(1+\beta ){f}^{\u2033}\left(0\right)$ and $-{\theta}^{\prime}\left(0\right)$.

${\mathit{R}}_{\mathit{e}}^{1/2}{\mathit{C}}_{\mathit{f}}$ | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{\u03f5}$ | ${\mathit{\gamma}}_{\mathbf{1}}$ | ${\mathit{\varphi}}_{\mathbf{2}}$ | $\mathit{\lambda}$ | ${\mathit{S}}_{\mathit{c}}$ | $\mathit{\beta}$ | $\mathit{E}$ | $\mathit{m}$ | $\mathit{\sigma}$ | $\mathit{n}$ | SWCNT | MWCNT |

0.1 | 0.1 | 0.01 | 0.1 | 0.5 | 0.5 | 1 | 0.5 | 0.1 | 0.1 | 1.118505 | 1.113907 |

0.3 | 0.909755 | 0.906056 | |||||||||

0.5 | 0.678612 | 0.675895 | |||||||||

0.2 | 0.1 | 1.178579 | 1.175287 | ||||||||

0.2 | 1.037153 | 1.034589 | |||||||||

0.3 | 0.924627 | 0.922584 | |||||||||

0.5 | 0.01 | 0.719589 | 0.718489 | ||||||||

0.03 | 0.753620 | 0.750208 | |||||||||

0.05 | 0.790372 | 0.784373 | |||||||||

0.01 | 0.2 | 0.709207 | 0.707394 | ||||||||

0.3 | 0.721306 | 0.719536 | |||||||||

0.4 | 0.733302 | 0.731576 | |||||||||

0.1 | 0.1 | 0.796844 | 0.793677 | ||||||||

0.3 | 0.908853 | 0.905215 | |||||||||

0.5 | 1.015271 | 1.011183 | |||||||||

0.5 | 0.1 | 1.045206 | 1.044045 | ||||||||

0.5 | 0.933344 | 0.921626 | |||||||||

0.5 | 0.815149 | 0.805263 | |||||||||

0.9 | 0.1 | 0.631555 | 0.618866 | ||||||||

0.5 | 0.755968 | 0.743733 | |||||||||

1 | 0.875441 | 0.863706 | |||||||||

1 | 0.2 | 0.835110 | 0.825900 | ||||||||

0.3 | 0.745517 | 0.737508 | |||||||||

0.4 | 0.655203 | 0.648064 | |||||||||

0.1 | 0.1 | 0.559715 | 0.552366 | ||||||||

0.5 | 0.465787 | 0.459695 | |||||||||

0.9 | 0.364019 | 0.365172 | |||||||||

0.1 | 0.2 | 0.745179 | 0.737164 | ||||||||

0.3 | 0.654127 | 0.647323 | |||||||||

0.4 | 0.555421 | 0.5558216 |

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## Share and Cite

**MDPI and ACS Style**

Mehmood, Y.; Shafqat, R.; Sarris, I.E.; Bilal, M.; Sajid, T.; Akhtar, T.
Numerical Investigation of MWCNT and SWCNT Fluid Flow along with the Activation Energy Effects over Quartic Auto Catalytic Endothermic and Exothermic Chemical Reactions. *Mathematics* **2022**, *10*, 4636.
https://doi.org/10.3390/math10244636

**AMA Style**

Mehmood Y, Shafqat R, Sarris IE, Bilal M, Sajid T, Akhtar T.
Numerical Investigation of MWCNT and SWCNT Fluid Flow along with the Activation Energy Effects over Quartic Auto Catalytic Endothermic and Exothermic Chemical Reactions. *Mathematics*. 2022; 10(24):4636.
https://doi.org/10.3390/math10244636

**Chicago/Turabian Style**

Mehmood, Yasir, Ramsha Shafqat, Ioannis E. Sarris, Muhammad Bilal, Tanveer Sajid, and Tasneem Akhtar.
2022. "Numerical Investigation of MWCNT and SWCNT Fluid Flow along with the Activation Energy Effects over Quartic Auto Catalytic Endothermic and Exothermic Chemical Reactions" *Mathematics* 10, no. 24: 4636.
https://doi.org/10.3390/math10244636