# Applications of Prabhakar-like Fractional Derivative for the Solution of Viscous Type Fluid with Newtonian Heating Effect

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description

**Basic preliminaries:**

**(Prabhakar kernel)**The function

**(Prabhakar Integral)**The Prabhakar integral can be defined as [47,48]

**(The regularized Prabhakar derivative)**In [47,48], the regularized Prabhakar derivative is distinct as

- When $\beta \ge 0,\gamma =0$, the Prabhakar derivative will transform into the Caputo derivative ${}_{}{}^{C}\mathfrak{D}{}_{\alpha ,\beta ,\alpha}^{0}g\left(t\right)={}_{}{}^{C}\mathfrak{D}{}_{t}^{\beta}g\left(t\right)$.
- When $\alpha =\beta =1,\gamma =-1$, then the relation between Prabhakar and CF derivative will become ${}_{}{}^{C}\mathfrak{D}{}_{1,1,\frac{-\sigma}{1-\sigma}}^{}g\left(t\right)={g}^{\prime}\left(t\right)-\sigma {}_{}{}^{CF}\mathfrak{D}{}_{t}^{\sigma}g\left(t\right)$.
- When $\beta =1,\gamma =-1,0\alpha 1$, then the connection between the AB derivative and Prabhakar derivative will develop as ${}_{}{}^{C}\mathfrak{D}{}_{\alpha ,1,\frac{-\sigma}{1-\sigma}}^{-1}g\left(t\right)=\left(1-\alpha \right)\frac{d}{dt}{}_{}{}^{ABC}\mathfrak{D}{}_{t}^{\alpha}g\left(t\right)$.
- When $\beta =\gamma =0,m=0$, the Prabhakar derivative will be ${}_{}{}^{C}\mathfrak{D}{}_{\alpha ,0,\alpha}^{0}g\left(t\right)=g\left(t\right)$ with its kernel ${\overline{h}}_{p}\left(\alpha ,0,0,\alpha ,q\right)=1$.
- When $\beta =1,\gamma =0,m=1$, the Prabhakar derivative ${}_{}{}^{C}\mathfrak{D}{}_{\alpha ,1,\alpha}^{0}g\left(t\right)={g}^{\prime}\left(t\right)$ with its kernel ${\overline{h}}_{p}\left(\alpha ,1,0,\alpha ,q\right)=\frac{d}{dt}\left(g\left(t\right)\right)$. As the LT of Prabhakar, fractional operator ${}_{}{}^{C}\mathfrak{D}{}_{\alpha ,\beta ,\alpha}^{\gamma}$ is, consequently,

## 3. Solution of the Problem

#### 3.1. Solution of the Energy Profile

#### Classical Solution of the Energy Field $\left(\beta =\gamma =0\right)$

#### 3.2. Solution of the Concentration Profile

#### Classical Solution of the Concentration Profile $\left(\beta =\gamma =0\right)$

#### 3.3. Solution of Momentum Field

**Nusselt number, Sherwood number, and skin friction**are as follows:

## 4. Results and Discussion

## 5. Validation of Attained Results

## 6. Conclusions

- The impact of larger values of fractional parameter and an adequate Prandtl number declines the profiles of temperature distributions.
- The boundary layer concentration also decays with the enhancement in fractional parameter and Schmidt number.
- The momentum profile is an increasing function for $Re,Gr,Gm$,while it decreases with the variation in $P{r}_{eff},S{c}_{eff},M,{\theta}_{1}$ and Prabhakar fractional parameters.
- Thermal profile, concentration, and momentum profiles asymptotically increase with time.
- The overlapping of both numerical schemes validates the attained solution of all governed equations.
- The momentum profile is maximal near the plate. It approaches its distinctive peak values in the stream region and then decreases away along the y-axis.
- The rate of heat transfer, mass transfer, and skin friction varies with the increment in time values.
- In the comparison of numerical techniques and with the attained results of Imran et al. [54], the overlapping of both curves validates the attained results of this study.
- As the Prabhakar fractional derivative is the more recent definition of the fractional derivatives technique, it has more efficient and accurate results as compared to other fractional operators as depicted in the comparison of Imran et al. [54].

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Quantity | Unit |

$\alpha ,\beta ,\gamma $ | Prabhakar fractional constraints | $(-)$ |

$\mu $ | Dynamic Viscosity | $\left({\mathrm{Kgm}}^{-1}{\mathrm{s}}^{-1}\right)$ |

$\upsilon $ | Kinematic viscosity coefficient | $\left({\mathrm{m}}^{2}{\mathrm{s}}^{-1}\right)$ |

$g$ | Gravitational acceleration | $\left({\mathrm{ms}}^{-2}\right)$ |

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

$\rho $ | Density | $\left({\mathrm{Kgm}}^{-3}\right)$ |

${C}_{p}$ | Specific heat at constant pressure | $\left({\mathrm{JKg}}^{-1}{\mathrm{K}}^{-1}\right)$ |

$s$ | Laplace-transformed parameter | $(-)$ |

$\sigma $ | Electrical conductivity | $\left({\mathrm{sm}}^{-1}\right)$ |

$k$ | Thermal conductivity | $\left({\mathrm{Wm}}^{-2}{\mathrm{K}}^{-1}\right)$ |

${T}_{\left(y,t\right)}$ | Dimensionless temperature profile | $(-)$ |

${v}_{\left(y,t\right)}$ | Dimensionless momentum field | $(-)$ |

${C}_{\left(y,t\right)}$ | Dimensionless concentration profile | $(-)$ |

$Gr$ | Heat Grashof number | $(-)$ |

$Gm$ | Mass Grashof number | $(-)$ |

$P{r}_{eff}$ | Effective Prandtl number | $(-)$ |

$Sc$ | Schmidt number | $(-)$ |

${B}_{o}$ | Magnetic field strength | $\left({\mathrm{NsC}}^{-1}\right)$ |

$M$ | Magnetic field | $(-)$ |

$LT$ | Laplace transformation | $(-)$ |

$Nu$ | Nusselt number | $(-)$ |

${C}_{f}$ | Skin friction | $(-)$ |

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**Figure 2.**Variation in temperature due to fractional constraints at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 3.**Variation in temperature due to effective Prandtl number at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 4.**Variation in concentration due to fractional constraints at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 6.**Variation in momentum field due to fractional constraints with $P{r}_{eff}=1.4,S{c}_{eff}=1.1,M=1.75,Gr=0.75,Gm=0.5,Re=1.3,{\theta}_{1}={\theta}_{2}=\frac{\pi}{4},K=1.4,h=0.5$ at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 7.**Variation in momentum field due to effective Prandtl number with $\alpha =\beta =\gamma =0.7,S{c}_{eff}=1.1,M=1.75,Gr=0.75,Gm=0.5,Re=1.3,{\theta}_{1}={\theta}_{2}=\frac{\pi}{4},K=1.4,h=0.5$ at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 8.**Variation in momentum field due to heat Grashof number with $\alpha =\beta =\gamma =0.7,P{r}_{eff}=1.4,S{c}_{eff}=1.1,M=1.75,Gm=0.5,Re=1.3,{\theta}_{1}={\theta}_{2}=\frac{\pi}{4},K=1.4,h=0.5$ at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 9.**Variation in momentum field due to mass Grashof number with $\alpha =\beta =\gamma =0.7,P{r}_{eff}=1.4,S{c}_{eff}=1.1,M=1.75,Gr=0.75,Re=1.3,{\theta}_{1}={\theta}_{2}=\frac{\pi}{4},K=1.4,h=0.5$ at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 10.**Variation in momentum field due to Schmidt number with $\alpha =\beta =\gamma =0.7,P{r}_{eff}=1.4,S{c}_{eff}=1.1,M=1.75,Gr=0.75,Gm=0.5,Re=1.3,{\theta}_{1}={\theta}_{2}=\frac{\pi}{4},K=1.4,h=0.5$ at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 11.**Variation in momentum field due to $Re$ parameter with $\alpha =\beta =\gamma =0.7,P{r}_{eff}=1.4,S{c}_{eff}=1.1,M=1.75,Gr=0.75,Gm=0.5,{\theta}_{1}={\theta}_{2}=\frac{\pi}{4},K=1.4,h=0.5$ at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 12.**Variation in momentum field due to magnetic field with $\alpha =\beta =\gamma =0.7,P{r}_{eff}=1.4,S{c}_{eff}=1.1,Gr=0.75,Gm=0.5,Re=1.3,{\theta}_{1}={\theta}_{2}=\frac{\pi}{4},K=1.4,h=0.5$ at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 13.**Variation in momentum field due to inclined of the magnetic field with $\alpha =\beta =\gamma =0.7,P{r}_{eff}=1.4,S{c}_{eff}=1.1,M=1.75,Gr=0.75,Gm=0.5,Re=1.3,{\theta}_{2}=\frac{\pi}{4},K=1.4,h=0.5$ at (

**a**) $t=0.5$ (

**b**) $t=0.7$ (

**c**) $t=0.9$.

**Figure 14.**Validation of all governed profiles by different numerical schemes. (

**a**) Temperature (

**b**) Concentration (

**c**) Velocity.

**Figure 15.**Validation attained results with the results of Imran et al. [54]. (

**a**) Temperature. (

**b**) Velocity.

y | T_{(y,t)} byStehfest | T_{(y,t)} byTzou’s | C_{(y,t)} byStehfest | C_{(y,t)} byTzou’s | v_{(y,t)} byStehfest | v_{(y,t)} byTzou’s |
---|---|---|---|---|---|---|

0.1 | 0.6524 | 0.6396 | 0.8453 | 0.8460 | 0.8039 | 0.7981 |

0.3 | 0.4169 | 0.4068 | 0.6029 | 0.6046 | 0.7247 | 0.7209 |

0.5 | 0.2658 | 0.2579 | 0.4291 | 0.4311 | 0.5643 | 0.5615 |

0.7 | 0.1690 | 0.1630 | 0.3046 | 0.3067 | 0.4118 | 0.4095 |

0.9 | 0.1071 | 0.1026 | 0.2156 | 0.2176 | 0.3007 | 0.2988 |

1.1 | 0.0677 | 0.0644 | 0.1521 | 0.1539 | 0.2196 | 0.2181 |

1.3 | 0.0426 | 0.0402 | 0.1069 | 0.1085 | 0.1605 | 0.1593 |

1.5 | 0.0267 | 0.0249 | 0.0749 | 0.0763 | 0.1173 | 0.1163 |

1.7 | 0.0167 | 0.0154 | 0.0522 | 0.0534 | 0.0858 | 0.0850 |

1.9 | 0.0104 | 0.0094 | 0.0362 | 0.0372 | 0.0628 | 0.0621 |

α | Nu at t = 0.5 | Nu at t = 0.7 | Sh at t = 0.5 | Sh at t = 0.7 | C_{f} at t = 0.5 | C_{f} at t = 0.7 |
---|---|---|---|---|---|---|

0.1 | 2.4196 | 2.5332 | 1.2886 | 1.3048 | 0.9056 | 1.2143 |

0.2 | 2.3024 | 2.3874 | 1.3457 | 1.3687 | 0.7385 | 0.9344 |

0.3 | 2.2067 | 2.2599 | 1.4021 | 1.4384 | 0.6290 | 0.7390 |

0.4 | 2.1308 | 2.1502 | 1.4546 | 1.5088 | 0.5583 | 0.5994 |

0.5 | 2.0734 | 2.0518 | 1.4973 | 1.5739 | 0.5158 | 0.4996 |

0.6 | 2.0330 | 1.9831 | 1.5286 | 1.6290 | 0.4952 | 0.4305 |

0.7 | 2.0080 | 1.9250 | 1.5481 | 1.6711 | 0.4918 | 0.3867 |

0.8 | 1.9966 | 1.8831 | 1.5569 | 1.6996 | 0.5027 | 0.3648 |

0.9 | 1.9971 | 1.8564 | 1.5568 | 1.7154 | 0.5249 | 0.3623 |

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**MDPI and ACS Style**

Raza, A.; Khan, U.; Zaib, A.; Mahmoud, E.E.; Weera, W.; Yahia, I.S.; Galal, A.M.
Applications of Prabhakar-like Fractional Derivative for the Solution of Viscous Type Fluid with Newtonian Heating Effect. *Fractal Fract.* **2022**, *6*, 265.
https://doi.org/10.3390/fractalfract6050265

**AMA Style**

Raza A, Khan U, Zaib A, Mahmoud EE, Weera W, Yahia IS, Galal AM.
Applications of Prabhakar-like Fractional Derivative for the Solution of Viscous Type Fluid with Newtonian Heating Effect. *Fractal and Fractional*. 2022; 6(5):265.
https://doi.org/10.3390/fractalfract6050265

**Chicago/Turabian Style**

Raza, Ali, Umair Khan, Aurang Zaib, Emad E. Mahmoud, Wajaree Weera, Ibrahim S. Yahia, and Ahmed M. Galal.
2022. "Applications of Prabhakar-like Fractional Derivative for the Solution of Viscous Type Fluid with Newtonian Heating Effect" *Fractal and Fractional* 6, no. 5: 265.
https://doi.org/10.3390/fractalfract6050265