# Applications of Fractional Partial Differential Equations for MHD Casson Fluid Flow with Innovative Ternary Nanoparticles

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Preliminaries of Fractional Calculus

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

## 4. Solution of the Problem Based on Generalized Law with Ternary Nanoparticles

#### 4.1. Computation of Temperature Field

#### 4.2. Computation of Concentration Field

#### 4.3. Computation of Velocity Field

## 5. Graphical Outcomes

## 6. Discussion

## 7. Conclusions

- The model based on ternary is stronger approach than the hybrid and mono nanoparticle.
- Enhancement in temperature and velocity can be achieved for larger values of fractional parameters.
- Model based on generalized laws are reliable way for fractional modeling and can be accurately fitted any experimental data.
- Maximum temperature is achievable for ternary nanoparticles instead of hybrid and mono nanoparticles, respectively.

## 8. Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Explanation | Unit |

$\rho $ | Density | kg/m${}^{3}$ |

$\tau $ | Shear stress | Pascal (Pa) N/s${}^{2}$ |

${C}_{p}$ | Specific heat capacity | JK${}^{-1}$kg${}^{-1}$ |

$\mu $ | Fluid viscosity | kg/ms |

${\beta}_{0}$ | Casson parameter | no unit |

$\mathrm{Gr}$ | Thermal Grashof number | no unit |

Pr | Prandtl number | no unit |

$\mathrm{Gm}$ | Mass Grashof number | no unit |

$\mathrm{Sc}$ | Schimdt number | no unit |

g | Gravitational force | m/s${}^{2}$ |

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

D | mass diffusion coefficient | m${}^{3}$ |

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

$\alpha $ | Fractional derivative parameter | no unit |

$\beta $ | Fractional derivative parameter | no unit |

$\gamma $ | Fractional derivative parameter | no unit |

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

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

${T}_{w}$ | Wall temperature | K |

${C}_{w}$ | Wall concentration | mol·m${}^{-3}$ |

${C}_{\infty}$ | Ambient concentration | mol·m${}^{-3}$ |

q | Frequency | ${\mathrm{s}}^{-1}$ |

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

$M$ | Dimensionless magnetic parameter | no unit |

A | Accelration | m/s${}^{2}$ |

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**Figure 1.**Physical model of the problem [49].

**Figure 2.**Comparison of temperature assessment across y as $\alpha =0.5$, $\beta =0.5$, and $\gamma =0.5$ when t = 0.04, $\mathrm{Sc}=1.5$, and $\mathrm{Pr}=21$.

**Figure 3.**Comparison of temperature assessment across y as volume fraction ${\varphi}_{1}={\varphi}_{2}={\varphi}_{3}=0.01$ when $\alpha =\beta =\gamma =0.5$, t = 0.04, $\mathrm{Sc}=1.5$, and $\mathrm{Pr}=21$.

**Figure 4.**Comparison of temperature assessment with Pr across y as $\alpha =0.5$, $\beta =0.5$, and $\gamma =0.5$ when t = 0.04 and $\mathrm{Sc}=1.5$.

**Figure 5.**Comparison of velocity assessment with $\alpha $$\beta $$\gamma $ across y as $\mathrm{Sc}=9.2$ when t = 1, $\alpha =\beta =\gamma =0.2$, $M=1.5$, and $\mathrm{Pr}=21$.

**Figure 6.**Comparison of velocity assessment with $\mathrm{Gm}$ across y as $\alpha =\beta =\gamma =0.5$ when t = 1, $\mathrm{Sc}=9.2$, ${\varphi}_{1}=0.02$, ${\varphi}_{2}=0.03$, ${\varphi}_{3}=0.04$, $\mathrm{Pr}=21$, and $M=1.5$.

**Figure 7.**Comparison of velocity assessment with $\mathrm{Gr}$ across y as $\alpha =0.5$, $\beta =0.5$, $\gamma =0.5$ when t = 1, $\mathrm{Sc}=9.2$, ${\varphi}_{1}=0.02$, ${\varphi}_{2}=0.03$, ${\varphi}_{3}=0.04$, $\mathrm{Pr}=21$, and $M=1.5$.

**Figure 8.**Comparison of velocity assessment with volume fraction across y as $\alpha =\beta =\gamma =0.5$ when t = 1, $\mathrm{Pr}=21$, and $M=1.5$.

**Figure 9.**Comparison of velocity assessment with magnetic parameter $M$ across y as $\alpha =0.3$, $\beta =0.3$, $\gamma =0.3$ when t = 1, ${\varphi}_{1}=0.03$, ${\varphi}_{2}=0.03$, ${\varphi}_{3}=0.03$, $\mathrm{Pr}=21$, $\mathrm{Gm}=1.4$, and $\mathrm{Gr}=1.5$.

**Table 1.**Thermophysical possessions of nanoparticles and base fluid [48].

Material | Base Fluid Kerosene Oil | Silver (Ag) | Copper (Cu) | Titanium Dioxide (TiO${}_{2}$) |
---|---|---|---|---|

$\rho $ (kg /m${}^{3}$) | 863 | 10,500 | 8993 | 4250 |

${C}_{p}$ (J/kg.K) | 2048 | 235 | 385 | 686.20 |

k (W/m.K) | 0.1404 | 429 | 401 | 8.9538 |

$\sigma $ (s/m) | $55\times {10}^{-6}$ | $3.6\times {10}^{7}$ | $5.96\times {10}^{7}$ | $1\times {10}^{-12}$ |

$\beta \times {10}^{-5}$ (1/K) | 70 | 1.89 | 1.67 | 0.90 |

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**MDPI and ACS Style**

Asjad, M.I.; Karim, R.; Hussanan, A.; Iqbal, A.; Eldin, S.M.
Applications of Fractional Partial Differential Equations for MHD Casson Fluid Flow with Innovative Ternary Nanoparticles. *Processes* **2023**, *11*, 218.
https://doi.org/10.3390/pr11010218

**AMA Style**

Asjad MI, Karim R, Hussanan A, Iqbal A, Eldin SM.
Applications of Fractional Partial Differential Equations for MHD Casson Fluid Flow with Innovative Ternary Nanoparticles. *Processes*. 2023; 11(1):218.
https://doi.org/10.3390/pr11010218

**Chicago/Turabian Style**

Asjad, Muhammad Imran, Rizwan Karim, Abid Hussanan, Azhar Iqbal, and Sayed M. Eldin.
2023. "Applications of Fractional Partial Differential Equations for MHD Casson Fluid Flow with Innovative Ternary Nanoparticles" *Processes* 11, no. 1: 218.
https://doi.org/10.3390/pr11010218