# Heat and Mass Transfer Impact on Differential Type Nanofluid with Carbon Nanotubes: A Study of Fractional Order System

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

#### Preliminaries

## 3. Solutions via Caputo

#### 3.1. Temperature Profile

#### 3.2. Concentration Profile

#### 3.3. Velocity Profile

## 4. Solutions via Caputo–Fabrizio

#### 4.1. Temperature Profile

#### 4.2. Concentration Profile

#### 4.3. Velocity Profile

## 5. Solutions via Atangana Baleanu

#### 5.1. Temperature Profile

#### 5.2. Concentration Profile

#### 5.3. Velocity Profile

## 6. Results and Discussion

#### 6.1. Effect of $\chi $

#### 6.2. Effect of $Gm$

#### 6.3. Effect of $Gr$

#### 6.4. Effect of P

#### 6.5. Effect of ${\hslash}_{2}$

#### 6.6. Effect of $\kappa $

#### 6.7. Effect of ${M}_{o}$

#### 6.8. Effect of $Pr$

#### 6.9. Effect of ${A}_{o}$

#### 6.10. Effect of ℵ

#### 6.11. Effect of $Sc$

## 7. Conclusions

- Fluid flow descends for $\chi $, P, ${\hslash}_{2}$, $Pr$, ℵ and $Sc$.
- Presence of resistive forces due to ${M}_{o}$, fluid flow decelerates.
- Increment in flow field has been generated when $Gr$, $Gm$, $\kappa $ and ${A}_{o}$.
- For $\aleph =0$, velocity of nano-fluid escalates. Whereas for $\aleph >0$, flow’s velocity de-escalates.
- Velocity of flow increases with increment in volumetric fraction for Caputo, Caputo–Fabrizio and Atangana–Baleanu models.
- Velocity profile is maximum when $d\left(\tau \right)={e}^{\tau}$ and minimum when $d\left(\tau \right)=1$.
- Among C, CF and ABC, flow profile is maximum for ABC.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Quantity |

${\kappa}_{r}$ | Chemical reaction parameter |

h | Coefficient of heat transfer (Wm${}^{-2}$K${}^{-1}$) |

${\tilde{\mathrm{\Lambda}}}_{w}$ | Concentration level on the plate (kgm${}^{-3}$) |

${\tilde{\mathrm{\Lambda}}}_{\infty}$ | Concentration of the fluid far away from the plate (kgm${}^{-3}$) |

${\nu}_{o}$ | Constant suction velocity (s) |

${\rho}_{nf}$ | Density of nano fluid (kgm${}^{-3}$) |

${\mu}_{nf}$ | Dynamic viscosity of nano fluid (kgm${}^{-1}$s${}^{-1}$) |

$\tilde{\mathrm{\Lambda}}$ | Fluid concentration (kgm${}^{-3}$) |

$\tilde{\mathrm{Y}}$ | Fluid temperature (K) |

$\tilde{W}$ | Fluid velocity (ms${}^{-1}$) |

$\chi $ | Fractional parameter |

g | Gravitational acceleration (ms${}^{-2}$) |

${A}_{o}$ | Heat generation |

${\nu}_{nf}$ | Kinematic viscosity of nano fluid (m${}^{2}$s${}^{-1}$) |

u | Laplace transforms parameter |

${M}_{o}$ | Magnetic field |

$Gm$ | Mass Grashof number |

${\hslash}_{1}$ | Material constant or 2nd grade parameter |

${P}_{1}$ | Permitivity of medium |

℘ | Porosity |

$Pr$ | Prandtl number |

$Sc$ | Schmidt number |

$Sh$ | Sherwood number |

${C}_{p}$ | Specific heat at constant temperature (jkg${}^{-1}$K${}^{-1}$) |

ℵ | Suction |

${\tilde{\mathrm{Y}}}_{w}$ | Temperature of fluid at the plate (K) |

${\tilde{\mathrm{Y}}}_{\infty}$ | Temperature of fluid far away from the plate (K) |

k | Thermal conductivity of the fluid (Wm${}^{-1}$K${}^{-1}$) |

$Gr$ | Thermal Grashof number |

${\beta}_{{\displaystyle \tilde{\bigwedge}}}$ | Volumetric coefficient of expansion for mass concentration (m${}^{3}$kg${}^{-1}$) |

${\beta}_{\tilde{\mathrm{Y}}}$ | Volumetric coefficient of thermal expansion (K${}^{-1}$) |

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**Figure 2.**Graphs for two different cases of velocity with variable fractional parameter and ${M}_{o}=2$, $Sc=1.2$, $Pr=1$, $Gm=5$, $Gr=5$, ${A}_{o}=0.5$, ${\hslash}_{2}=0.7$, $\aleph =0.2$, $P=0.3$ and $\kappa =0.9$.

**Figure 3.**Graphs for two different cases of velocity with variable mass Grashof number and ${M}_{o}=2$, $Sc=1.2$, $Pr=1$, $\chi =0.1$, $Gr=5$, ${A}_{o}=0.5$, ${\hslash}_{2}=0.7$, $\aleph =0.2$, $P=0.3$ and $\kappa =0.9$.

**Figure 4.**Graphs for two different cases of velocity with variable thermal Grashof number and ${M}_{o}=2$, $Sc=1.2$, $Pr=1$, $\chi =0.1$, $Gm=5$, ${A}_{o}=0.5$, ${\hslash}_{2}=0.7$, $\aleph =0.2$, $P=0.3$ and $\kappa =0.9$.

**Figure 5.**Graphs for two different cases of velocity with variable permittivity of medium and ${M}_{o}=2$, $Sc=1.2$, $Pr=1$, $Gm=5$, $Gr=5$, ${A}_{o}=0.5$, ${\hslash}_{2}=0.7$, $\aleph =0.2$, $\chi =0.1$ and $\kappa =0.9$.

**Figure 6.**Graphs for two different cases of velocity with variable 2nd grade parameter and ${M}_{o}=1$, $Sc=1.2$, $Pr=1$, $Gm=5$, $Gr=5$, ${A}_{o}=0.5$, $P=2$, $\aleph =0.2$, $\chi =0.1$ and $\kappa =0.9$.

**Figure 7.**Graphs for two different cases of velocity with variable chemical reaction parameter and ${M}_{o}=2$, $Sc=1.2$, $Pr=1$, $Gm=5$, $Gr=5$, ${A}_{o}=0.5$, ${\hslash}_{2}=0.7$, $\aleph =0.2$, $\chi =0.1$ and $P=0.3$.

**Figure 8.**Graphs for two different cases of velocity with variable magnetic field and $P=0.3$, $Sc=1.2$, $Pr=1$, $Gm=5$, $Gr=5$, ${A}_{o}=0.5$, ${\hslash}_{2}=0.7$, $\aleph =0.2$, $\chi =0.1$ and $\kappa =0.9$.

**Figure 9.**Graphs for two different cases of velocity with variable Prandtl number and ${M}_{o}=2$, $Sc=1.2$, $P=0.3$, $Gm=5$, $Gr=5$, ${A}_{o}=0.5$, ${\hslash}_{2}=0.7$, $\aleph =0.2$, $\chi =0.1$ and $\kappa =0.9$.

**Figure 10.**Graphs for two different cases of velocity with variable heat generation and ${M}_{o}=2$, $Sc=1.2$, $Pr=1$, $Gm=5$, $Gr=5$, $P=0.3$, ${\hslash}_{2}=0.7$, $\aleph =0.2$, $\chi =0.1$ and $\kappa =0.9$.

**Figure 11.**Graphs for two different cases of velocity with variable Suction and ${M}_{o}=2$, $Sc=1.2$, $Pr=1$, $Gm=5$, $Gr=5$, ${A}_{o}=0.5$, ${\hslash}_{2}=0.7$, $P=0.3$, $\chi =0.1$ and $\kappa =0.9$.

**Figure 12.**Graphs for two different cases of velocity with variable Schmidt number and ${M}_{o}=2$, $P=0.3$, $Pr=1$, $Gm=5$, $Gr=5$, ${A}_{o}=0.5$, ${\hslash}_{2}=0.7$, $\aleph =0.2$, $\chi =0.1$ and $\kappa =0.9$.

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**MDPI and ACS Style**

Javed, F.; Riaz, M.B.; Iftikhar, N.; Awrejcewicz, J.; Akgül, A.
Heat and Mass Transfer Impact on Differential Type Nanofluid with Carbon Nanotubes: A Study of Fractional Order System. *Fractal Fract.* **2021**, *5*, 231.
https://doi.org/10.3390/fractalfract5040231

**AMA Style**

Javed F, Riaz MB, Iftikhar N, Awrejcewicz J, Akgül A.
Heat and Mass Transfer Impact on Differential Type Nanofluid with Carbon Nanotubes: A Study of Fractional Order System. *Fractal and Fractional*. 2021; 5(4):231.
https://doi.org/10.3390/fractalfract5040231

**Chicago/Turabian Style**

Javed, Fatima, Muhammad Bilal Riaz, Nazish Iftikhar, Jan Awrejcewicz, and Ali Akgül.
2021. "Heat and Mass Transfer Impact on Differential Type Nanofluid with Carbon Nanotubes: A Study of Fractional Order System" *Fractal and Fractional* 5, no. 4: 231.
https://doi.org/10.3390/fractalfract5040231