# Application of Fractional Derivative Without Singular and Local Kernel to Enhanced Heat Transfer in CNTs Nanofluid Over an Inclined Plate

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## Abstract

**:**

## 1. Introduction

## 2. Description of the Proposed Model

_{b}, and ${p}_{r}$. By virtue of Equation (1) along with the momentum equation, the Maxwell set of equations [37], Darcy’s law [38], Fourier law of heat conduction [39], and Boussinesq approximation [40] the governing equations of the proposed problem are given by [36]

## 3. Methodology

## 4. Solutions of the Problem

#### 4.1. Solution of Energy Equation

#### 4.2. Solutions of Momentum Equation

## 5. Discussion of Results

## 6. Concluding Remarks

- The fractional solutions for temperature and velocity fields are more general, reliable, and flexible, with memory and heredity properties that can be numerically reduced for any values of $0<\alpha \le 1$.
- The temperature profile increase with an increasing volume fraction of CNTs and decreases with increasing fractional parameters (for both cases of CNTs) because of variation in the thermal boundary layer.
- The velocity profile increases with increased permeability of the porous medium and thermal Grashof number, due to the improvement in the velocity boundary layer.
- Nanofluid motion (SWCNTs and MWCTs) retarded with increment in volume concentration of CNTs and magnetic parameters. The normal magnetic field has the strongest resistance to the motion.
- The trends and features of all the physical flow parameters are in excellent agreement with the published work.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Of $\alpha $ on $\theta \left(y,t\right)$ when $t=0.5$, $\mathrm{Pr}=21$ and $\varphi =0.04$.

**Figure 4.**Of $\alpha $ on $u\left(y,t\right)$ when $t=0.5$, $\beta =0.5$, $M=0.5$, $\gamma =\pi /2$, $K=0.5$ $Gr=7.0$, $\mathrm{Pr}=21$ and $\varphi =0.04$.

**Figure 5.**Of $\varphi $ on $\theta \left(y,t\right)$ when $t=0.5$, $\mathrm{Pr}=21$ and $\alpha =0.5$.

**Figure 6.**Of $\varphi $ on $u\left(y,t\right)$ when $t=0.5$, $\beta =0.5$, $M=0.5$, $\gamma =\pi /2$, $K=0.5$ $Gr=7.0$, $\mathrm{Pr}=21$ and $\alpha =0.5$.

**Figure 8.**Of $\beta $ on $u\left(y,t\right)$ when $t=0.5$, $\varphi =0.04$, $M=0.5$, $\gamma =\pi /2$, $K=0.5$ $Gr=7.0$, $\mathrm{Pr}=21$ and $\alpha =0.5$.

**Figure 9.**Of $M$ on $u\left(y,t\right)$ when t = 0.5, $\beta =0.5$, $\varphi =0.04$, $\gamma =\pi /2$, $K=0.5$ $Gr=7.0$, $\mathrm{Pr}=21$ and $\alpha =0.5$.

**Figure 10.**Of $\gamma $ on $u\left(y,t\right)$ when $t=0.5$, $\varphi =0.04$ $\beta =0.5$, $M=0.5$, $K=0.5$ $Gr=7.0$, $\mathrm{Pr}=21$ and $\alpha =0.5$.

**Figure 11.**Of $K$ on $u\left(y,t\right)$ when $t=0.5$, $\varphi =0.04$ $\beta =0.5$, $M=0.5$, $\gamma =\pi /2$, $Gr=7.0$, $\mathrm{Pr}=21$ and $\alpha =0.5$.

**Figure 12.**Of $Gr$ on $u\left(y,t\right)$ when $t=0.5$, $\varphi =0.04$ $\beta =0.5$, $M=0.5$, $\gamma =\pi /2$, $K=0.5$, $\mathrm{Pr}=21$ and $\alpha =0.5$.

**Table 1.**Model for thermophysical properties of nanofluid [41].

Physical Quantity | Mathematical Model |
---|---|

Density | ${\rho}_{nf}=\left(1-\varphi \right){\rho}_{f}+\varphi {\rho}_{s}$ |

Dynamic viscosity | ${\mu}_{nf}=\frac{{\mu}_{f}}{{\left(1-\varphi \right)}^{2.5}}$ |

Electrical conductivity | ${\sigma}_{nf}=\left\{1+\frac{3\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\varphi}{\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}+2\right)-\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\varphi}\right\}{\sigma}_{f}$ |

Thermal expansion | ${\left(\rho {\beta}_{T}\right)}_{nf}=\left(1-\varphi \right){\left(\rho {\beta}_{T}\right)}_{f}+\varphi {\left(\rho {\beta}_{T}\right)}_{s}\text{}$ |

Heat capacitance | ${\left(\rho {C}_{p}\right)}_{nf}=\left(1-\varphi \right){\left(\rho {C}_{p}\right)}_{f}+\varphi {\left(\rho {C}_{p}\right)}_{s}\text{}$ |

Thermal conductivity | ${k}_{nf}=\left\{\frac{\left(1-\varphi \right)+2\varphi \frac{{k}_{s}}{{k}_{s}-{k}_{f}}\mathrm{ln}\frac{{k}_{s}-{k}_{f}}{2{k}_{f}}}{\left(1-\varphi \right)+2\varphi \frac{{k}_{s}}{{k}_{s}-{k}_{f}}\mathrm{ln}\frac{{k}_{s}+{k}_{f}}{2{k}_{f}}}\right\}{k}_{f}$ |

**Table 2.**Values of thermophysical properties of base fluid and nanoparticles [36].

Material | Base Fluid | Nanoparticles | |
---|---|---|---|

Human Blood | SWCNTs | MWCMTs | |

$\rho \left(\mathrm{kg}/{\mathrm{m}}^{3}\right)$ | 1053 | 2600 | 1600 |

${C}_{p}\left(\mathrm{J}/\mathrm{kg}\text{}\mathrm{K}\right)$ | 3594 | 425 | 796 |

$k\left(\mathrm{W}/\mathrm{m}\text{}\mathrm{K}\right)$ | 0.492 | 6600 | 3000 |

${\beta}_{T}\times {10}^{-5}({K}^{-1})$ | 0.8 | 10^{−6}−10^{−7} | 1.9 × 10^{−4} |

$\sigma $ | 0.18 | 21 | 44 |

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**MDPI and ACS Style**

Saqib, M.; Mohd Kasim, A.R.; Mohammad, N.F.; Chuan Ching, D.L.; Shafie, S.
Application of Fractional Derivative Without Singular and Local Kernel to Enhanced Heat Transfer in CNTs Nanofluid Over an Inclined Plate. *Symmetry* **2020**, *12*, 768.
https://doi.org/10.3390/sym12050768

**AMA Style**

Saqib M, Mohd Kasim AR, Mohammad NF, Chuan Ching DL, Shafie S.
Application of Fractional Derivative Without Singular and Local Kernel to Enhanced Heat Transfer in CNTs Nanofluid Over an Inclined Plate. *Symmetry*. 2020; 12(5):768.
https://doi.org/10.3390/sym12050768

**Chicago/Turabian Style**

Saqib, Muhammad, Abdul Rahman Mohd Kasim, Nurul Farahain Mohammad, Dennis Ling Chuan Ching, and Sharidan Shafie.
2020. "Application of Fractional Derivative Without Singular and Local Kernel to Enhanced Heat Transfer in CNTs Nanofluid Over an Inclined Plate" *Symmetry* 12, no. 5: 768.
https://doi.org/10.3390/sym12050768