# Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Solutions to the Problem

#### 3.1. Non-Fractional Casson Fluid Solution

#### 3.2. Skin Friction, Nusselt Number, and Sherwood Number

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AB | Atangana-Baleanu |

CF | Caputo-Fabrizio |

LES | Large Eddy Simulation |

MHD | Magnetohydrodynamic |

Greek Symbols | |

$\alpha $ | Fractional derivative parameter |

$\beta $ | Casson parameter |

${\beta}_{0}$ | Dimensionless Casson parameter |

${\beta}_{C}$ | Volumetric expansion (mol${}^{-1}$· m${}^{3}$) |

${\beta}_{T}$ | Thermal expansion (K${}^{-1}$) |

$\mu $ | Fluid dynamic viscosity (kg· m${}^{-1}$ · s${}^{-1}$) |

$\nu $ | Fluid kinematic viscosity (m${}^{2}$· s${}^{-1}$) |

$\rho $ | Fluid density (kg· m${}^{-3}$) |

$\sigma $ | Electrical conductivity (S· m${}^{-1}$) |

$\sigma $ | Skin friction |

${\sigma}^{*}$ | Stefan-Boltzman constant (W· m${}^{-2}$· K${}^{-4}$) |

Symbols | |

A | Plate acceleration (m$\xb7{\mathrm{s}}^{-2}$) |

${a}_{0\cdots 14}$ | Dimensionless parametric constant |

${B}_{0}$ | Magnetic field (kg$\xb7{\mathrm{s}}^{-2}\xb7{\mathrm{A}}^{-1}$) |

${b}_{1\cdots 4}$ | Dimensionless parametric constant |

C | Fluid concentration (mol$\xb7{\mathrm{m}}^{-3}$) |

${C}_{\infty}$ | Ambient concentration (mol$\xb7{\mathrm{m}}^{-3}$) |

${C}_{P}$ | Specific heat capacity (J$\xb7\mathrm{k}{\mathrm{g}}^{-1}\xb7{\mathrm{K}}^{-1}$) |

${C}_{W}$ | Wall concentration (mol$\xb7{\mathrm{m}}^{-3}$) |

D | Mass diffusion coefficient (m${}^{-3}$) |

${D}_{t}^{\alpha}f$ | Caputo-Fabrizio fractional derivative |

g | Gravitational force (m$\xb7{\mathrm{s}}^{-2}$) |

$Gm$ | Mass Grashof number |

$G{m}_{0}$ | Dimensionless parametric constant |

$Gr$ | Thermal Grashof number |

$G{r}_{0}$ | Dimensionless parametric constant |

${k}^{\prime}$ | Dimensionless porosity parameter |

k | Porosity (m${}^{3}$) |

${k}^{*}$ | Mean absorption coefficient (m${}^{-1}$) |

${k}_{0}^{\prime}$ | Dimensionless parametric constant |

${k}_{1}$ | Thermal conductivity (W$\xb7{\mathrm{m}}^{-1}\xb7{\mathrm{K}}^{-1}$) |

${k}_{2}$ | Chemical reaction (mol$\xb7{\mathrm{m}}^{-3}\xb7{\mathrm{s}}^{-1}$) |

M | Dimensionless magnetic parameter |

${M}_{0}$ | Dimensionless parametric constant |

N | Dimensionless thermal radiation parameter |

$Nu$ | Nusselt number |

$Pr$ | Prandtl number |

$P{r}_{0}$ | Dimensionless parametric constant |

q | Frequency (s${}^{-1}$) |

${q}_{r}$ | Thermal radiation (W· m${}^{-2}$) |

R | Dimensionless chemical reaction parameter |

${R}_{0}$ | Dimensionless parametric constant |

$Sc$ | Schmidt number |

$Sh$ | Sherwood number |

T | Fluid temperature (K) |

t | Time (s) |

${T}_{\infty}$ | Ambient temperature (K) |

${T}_{W}$ | Wall temperature (K) |

U | Fluid velocity (m$\xb7{\mathrm{s}}^{-1}$) |

x | x-coordinate |

y | y-coordinate |

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**Figure 2.**Velocity profile for analytical solution presented in Equation (20) validated with numerical solution using Zakian’s method.

**Figure 3.**Temperature profile for analytical solution presented in Equation (21) validated with numerical solution using Zakian’s method.

**Figure 4.**Concentration profile for analytical solution presented in Equation (22) validated with numerical solution using Zakian’s method.

$\mathit{\alpha}$ | Pr | N | Nu | Result |
---|---|---|---|---|

0.2 | 5 | 0.5 | 1.532 | - |

0.8 | 5 | 0.5 | 0.802 | ↓ |

1.0 | 5 | 0.5 | −0.515 | ↓ |

0.2 | 15 | 0.5 | 2.653 | ↑ |

0.2 | 5 | 2.5 | 0.950 | ↓ |

$\mathit{\alpha}$ | Sc | R | Sh | Result |
---|---|---|---|---|

0.2 | 0.2 | 0.5 | 0.505 | - |

0.8 | 0.2 | 0.5 | 0.349 | ↓ |

1.0 | 0.2 | 0.5 | −0.037 | ↓ |

0.2 | 0.8 | 0.5 | 1.010 | ↑ |

0.2 | 0.2 | 1.5 | 0.674 | ↑ |

$\mathit{\alpha}$ | t | $\mathit{\beta}$ | Pr | Gr | Gm | Sc | N | R | k’ | M | $\mathit{\gamma}$ | Result |
---|---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 2 | 0.1 | 5.0 | 5.0 | 5.0 | 0.2 | 1.5 | 0.5 | 0.5 | 0.5 | 0.376 | - |

0.8 | 2 | 0.1 | 5.0 | 5.0 | 5.0 | 0.2 | 1.5 | 0.5 | 0.5 | 0.5 | 0.149 | ↓ |

1.0 | 2 | 0.1 | 5.0 | 5.0 | 5.0 | 0.2 | 1.5 | 0.5 | 0.5 | 0.5 | −0.847 | ↓ |

0.2 | 3 | 0.1 | 5.0 | 5.0 | 5.0 | 0.2 | 1.5 | 0.5 | 0.5 | 0.5 | 0.855 | ↑ |

0.2 | 2 | 0.5 | 5.0 | 5.0 | 5.0 | 0.2 | 1.5 | 0.5 | 0.5 | 0.5 | 0.208 | ↓ |

0.2 | 2 | 0.1 | 15.0 | 5.0 | 5.0 | 0.2 | 1.5 | 0.5 | 0.5 | 0.5 | 0.467 | ↑ |

0.2 | 2 | 0.1 | 5.0 | 10.0 | 5.0 | 0.2 | 1.5 | 0.5 | 0.5 | 0.5 | 0.107 | ↓ |

0.2 | 2 | 0.1 | 5.0 | 5.0 | 10.0 | 0.2 | 1.5 | 0.5 | 0.5 | 0.5 | −0.063 | ↓ |

0.2 | 2 | 0.1 | 5.0 | 5.0 | 5.0 | 0.8 | 1.5 | 0.5 | 0.5 | 0.5 | 0.521 | ↑ |

0.2 | 2 | 0.1 | 5.0 | 5.0 | 5.0 | 0.2 | 3.0 | 0.5 | 0.5 | 0.5 | 0.327 | ↓ |

0.2 | 2 | 0.1 | 5.0 | 5.0 | 5.0 | 0.2 | 1.5 | 1.5 | 0.5 | 0.5 | 0.437 | ↑ |

0.2 | 2 | 0.1 | 5.0 | 5.0 | 5.0 | 0.2 | 1.5 | 0.5 | 1.5 | 0.5 | 0.038 | ↓ |

0.2 | 2 | 0.1 | 5.0 | 5.0 | 5.0 | 0.2 | 1.5 | 0.5 | 0.5 | 1.5 | 0.749 | ↑ |

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**MDPI and ACS Style**

Reyaz, R.; Mohamad, A.Q.; Lim, Y.J.; Saqib, M.; Shafie, S.
Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects. *Fractal Fract.* **2022**, *6*, 38.
https://doi.org/10.3390/fractalfract6010038

**AMA Style**

Reyaz R, Mohamad AQ, Lim YJ, Saqib M, Shafie S.
Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects. *Fractal and Fractional*. 2022; 6(1):38.
https://doi.org/10.3390/fractalfract6010038

**Chicago/Turabian Style**

Reyaz, Ridhwan, Ahmad Qushairi Mohamad, Yeou Jiann Lim, Muhammad Saqib, and Sharidan Shafie.
2022. "Analytical Solution for Impact of Caputo-Fabrizio Fractional Derivative on MHD Casson Fluid with Thermal Radiation and Chemical Reaction Effects" *Fractal and Fractional* 6, no. 1: 38.
https://doi.org/10.3390/fractalfract6010038