# Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

## 3. Preliminaries

## 4. Solution of the Problem

#### 4.1. Exact Solution of Heat Profile

#### Nusselt Number

#### 4.2. Exact Solution of Velocity Profile

## 5. Limiting Cases

## 6. Results and Discussion

## 7. Conclusions

- The temperature field decline with the larger values of ${P}_{r}$;
- It is examined that the impacts of ${\lambda}_{1}$ and ${\lambda}_{2}$ on velocity profile are quite opposite;
- The accumulative values of the parameters M and ${P}_{r}$ decrease in the velocity distribution noticed;
- The increasing values of the grashof number ${G}_{r}$ stimulates the velocity distribution;
- It is analyzed that the effect of fractional parameters $\alpha $ and $\beta $ on velocity contour are quite converse;
- Caputo Fabrizio fractional model approaches to classical model when $\alpha ,\beta \to 1$;
- It is noted that for two different functions $f\left(t\right)={e}^{at}$ and $f\left(t\right)=sint$, velocity profile shows same behavior.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Quantity | Units |

$\alpha ,\beta $ | Fractional parameters | $\left(-\right)$ |

$\mu $ | Dynamic viscosity | (Kgm${}^{-1}$s${}^{-1}$) |

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

g | Acceleration due to gravity | (ms${}^{-2}$) |

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

$\rho $ | Fluid density | (Kgm${}^{-3}$) |

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

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

s | Laplace parameter | $\left(-\right)$ |

Q | Heat generation/absorption | (JK${}^{-1}$m${}^{-3}$s${}^{-1}$) |

$\omega $ | Non-dimensional velocity | $\left(-\right)$ |

$\theta $ | Dimensionless temperature | $\left(-\right)$ |

${G}_{r}$ | Thermal Grashof number | $\left(-\right)$ |

${T}_{w}$ | Temperature of the plate | $\left(K\right)$ |

${T}_{\infty}$ | Temperature of fluid far away from the plat | $\left(K\right)$ |

${\lambda}_{1}$ | Relaxation time | $\left(-\right)$ |

${\lambda}_{2}$ | Retardation time | $\left(-\right)$ |

${P}_{r}$ | Prandtl number | $\left(-\right)$ |

${M}_{0}$ | Imposed Magnetic field | (Wm${}^{-2}$) |

M | Total Magnetic field | $\left(-\right)$ |

k | Thermal conductivity of the fluid | (Wm${}^{-2}$K${}^{-1}$) |

t | Time | $\left(s\right)$ |

P | Pressure | (N m${}^{-2}$) |

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**Figure 2.**Temperature profile for varied values of ${P}_{r}$ for two different time levels $t=0.9$ and $t=1.5$.

**Figure 3.**Velocity profile for varied values of ${P}_{r}$ with and without slip effect $\gamma $ when $f\left(t\right)={e}^{at}$, $a=0.25$, $t=1.5$, $\alpha =0.5$, $\beta =0.3$, $Gr=3.5$, ${\lambda}_{1}=0.6$, ${\lambda}_{2}=0.2$, $M=2$.

**Figure 4.**Velocity for varied values of ${G}_{r}$ with and without slip effect $\gamma $ when $f\left(t\right)={e}^{at}$, $a=0.25$, $t=1.5$, $\alpha =0.5$, $\beta =0.3$, ${P}_{r}=0.71$, ${\lambda}_{1}=0.6$, ${\lambda}_{2}=0.2$, $M=2$.

**Figure 5.**Velocity for varied values of M with and without slip effect $\gamma $ when $f\left(t\right)={e}^{at}$, $a=0.25$, $t=1.5$, $\alpha =0.5$, $\beta =0.3$, $Gr=3.5$, ${\lambda}_{1}=0.6$, ${\lambda}_{2}=0.2$, ${P}_{r}=0.71$.

**Figure 6.**Velocity for varied values of ${\lambda}_{1}$ with and without slip effect $\gamma $ when $f\left(t\right)={e}^{at}$, $a=0.25$, $t=1.5$, $\alpha =0.5$, $\beta =0.3$, $Gr=3.5$, ${P}_{r}=0.71$, ${\lambda}_{2}=0.2$, $M=2$.

**Figure 7.**Velocity for varied values of ${\lambda}_{2}$ with and without slip effect $\gamma $ when $f\left(t\right)={e}^{at}$, $a=0.25$, $t=1.5$, $\alpha =0.5$, $\beta =0.3$, $Gr=3.5$, ${\lambda}_{1}=0.6$, ${P}_{r}=0.71$, $M=2$.

**Figure 8.**Velocity for varied values of $\alpha $ with and without slip effect $\gamma $ when $f\left(t\right)={e}^{at}$, $t=1.5$, $\beta =0.3$, $Gr=3.5$, $M=2$, ${\lambda}_{1}=0.6$, ${\lambda}_{2}=0.2$, ${P}_{r}=0.71$.

**Figure 9.**Velocity for varied values of $\beta $ with and without slip effect $\gamma $ when $f\left(t\right)={e}^{at}$, $t=1.5$, $Gr=3.5$, $\alpha =0.5$, ${\lambda}_{1}=0.6$, ${P}_{r}=0.71$, ${\lambda}_{2}=0.2$, $M=2$.

**Figure 10.**Velocity comparison for varied values of $\alpha $ with and without slip effect $\gamma $ when $f\left(t\right)={e}^{at}$, $t=1.5$, $\beta =0.6$, $Gr=3.5$, ${\lambda}_{1}=0.6$, ${P}_{r}=0.71$, ${\lambda}_{2}=0.2$, $M=2$.

**Figure 11.**Velocity profile for varied values of ${P}_{r}$ with and without slip effect $\gamma $ when $f\left(t\right)=sint$, $t=1.5$, $\alpha =0.5$, $\beta =0.3$, $Gr=3.5$, ${\lambda}_{1}=0.6$, ${\lambda}_{2}=0.2$, $M=2$.

**Figure 12.**Velocity for varied values of ${G}_{r}$ with and without slip effect $\gamma $ when $f\left(t\right)=sint$, $t=1.5$, $\alpha =0.5$, $\beta =0.3$, ${P}_{r}=0.71$, ${\lambda}_{1}=0.6$, ${\lambda}_{2}=0.2$, $M=2$.

**Figure 13.**Velocity for varied values of M with and without slip effect $\gamma $ when $f\left(t\right)=sint$, $t=1.5$, $\alpha =0.5$, $\beta =0.3$, $Gr=3.5$, ${\lambda}_{1}=0.6$, ${\lambda}_{2}=0.2$, ${P}_{r}=0.71$.

**Figure 14.**Velocity for varied values of ${\lambda}_{1}$ with and without slip effect $\gamma $ when $f\left(t\right)=sint$, $t=1.5$, $Gr=3.5$, $\alpha =0.5$, $\beta =0.3$, ${P}_{r}=0.71$, ${\lambda}_{2}=0.2$, $M=2$.

**Figure 15.**Velocity for varied values of ${\lambda}_{2}$ with and without slip effect $\gamma $ when $f\left(t\right)=sint$, $t=1.5$, $\alpha =0.5$, $\beta =0.3$, $Gr=3.5$, ${\lambda}_{1}=0.6$, ${P}_{r}=0.71$, $M=2$.

**Figure 16.**Velocity for varied values of $\alpha $ with and without slip effect $\gamma $ when $f\left(t\right)=sint$, $t=1.5$, $\beta =0.3$, $Gr=3.5$, ${\lambda}_{1}=0.6$, ${\lambda}_{2}=0.2$, $M=2$, ${P}_{r}=0.71$.

**Figure 17.**Velocity for varied values of $\beta $ with and without slip effect $\gamma $ when $f\left(t\right)=sint$, $t=1.5$, $\alpha =0.5$, ${\lambda}_{2}=0.2$, $Gr=3.5$, ${P}_{r}=0.71$, ${\lambda}_{2}=0.2$, $M=2$.

**Figure 18.**Velocity comparison for varied values of $\alpha $ with and without slip effect $\gamma $ when $f\left(t\right)=sint$, $t=1.5$, $\beta =0.3$, $Gr=3.5$, ${\lambda}_{1}=0.6$, ${P}_{r}=0.71$, ${\lambda}_{2}=0.2$, $M=2$.

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**MDPI and ACS Style**

Riaz, M.B.; Awrejcewicz, J.; Rehman, A.-U.; Akgül, A.
Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach. *Fractal Fract.* **2021**, *5*, 124.
https://doi.org/10.3390/fractalfract5030124

**AMA Style**

Riaz MB, Awrejcewicz J, Rehman A-U, Akgül A.
Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach. *Fractal and Fractional*. 2021; 5(3):124.
https://doi.org/10.3390/fractalfract5030124

**Chicago/Turabian Style**

Riaz, Muhammad Bilal, Jan Awrejcewicz, Aziz-Ur Rehman, and Ali Akgül.
2021. "Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach" *Fractal and Fractional* 5, no. 3: 124.
https://doi.org/10.3390/fractalfract5030124