# The Impact of Cattaneo–Christov Double Diffusion on Oldroyd-B Fluid Flow over a Stretching Sheet with Thermophoretic Particle Deposition and Relaxation Chemical Reaction

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

#### Numerical Procedure

## 3. Results and Discussion

## 4. Conclusions

- ❖
- The rise in values of $\lambda $ declines ${f}^{\prime}\left(\eta \right)$ and $g\left(\eta \right)$.
- ❖
- The increasing values of ${\beta}_{1}$ declines ${f}^{\prime}\left(\eta \right)$, but a converse trend is seen for enhanced ${\beta}_{2}$ values.
- ❖
- The rising values of $Q$ improve $\theta \left(\eta \right)$.
- ❖
- The rising values of ${\lambda}_{E}$ reduces $\theta \left(\eta \right)$.
- ❖
- The escalating values of ${\lambda}_{C}$ and $Sc$ declines $\chi \left(\eta \right)$.
- ❖
- The increasing values of $\sigma $ declines $\chi \left(\eta \right)$, but a reverse trend is seen for enhanced ${N}_{t}$ values.
- ❖
- The rise in values of $\lambda $ and ${\beta}_{2}$ declines ${\theta}^{\prime}$, but the opposite trend is detected for upward values of $Q$,${\beta}_{1}$, and ${\lambda}_{E}$.
- ❖
- The growth in values of $Sc$,$\sigma $, and ${N}_{t}$ declines ${\chi}^{\prime}$, but the conflicting trend is detected for upward values of ${\lambda}_{C}$.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$\left(u,v,w\right)$ | velocity components |

$\left(x,y,z\right)$ | directions |

$\rho $ | density |

$\mu $ | dynamic viscosity |

${\lambda}_{1}$ | relaxation time |

${T}_{\infty}$ | ambient temperature |

${\Gamma}_{c}$ | relaxation time for mass flux |

${k}_{r}$ | reaction rate |

$D$ | diffusion coefficient |

$\upsilon $ | kinematic viscosity |

$Q=\frac{{Q}_{0}}{\rho a{C}_{p}}$ | heat source/sink parameter |

${C}_{w}$ | wall concentration |

${V}_{T}$ | thermophoretic velocity |

$\eta $ | similarity variable |

$\chi \left(\eta \right)$ | dimensionless concentration profile. |

$\lambda =\frac{\Omega}{a}$ | rotation parameter |

${\lambda}_{E}=a{\Gamma}_{e}$ | relaxation time parameter of temperature |

$Sc=\frac{\nu}{D}$ | Schmidt number |

${N}_{t}=\frac{{k}^{*}\left({T}_{w}-{T}_{\infty}\right)}{{T}_{r}}$ | thermophoretic parameter |

$a$ | positive constant |

$\Omega $ | angular velocity |

${\lambda}_{2}$ | retardation time |

${C}_{p}$ | specific heat |

$k$ | thermal conductivity |

$C$ | concentration |

${\beta}_{2}={\lambda}_{2}a$ | Deborah number for retardation time |

${Q}_{0}$ | heat source/sink coefficient |

$T$ | temperature |

${T}_{w}$ | wall temperature |

${k}^{*}$ | thermophoretic coefficient |

${T}_{r}$ | reference temperature |

${C}_{\infty}$ | ambient concentration |

$f\left(\eta \right),{g}^{\prime}\left(\eta \right)$ | dimensionless velocity profiles |

$\theta \left(\eta \right)$ | dimensionless thermal profile |

${\Gamma}_{e}$ | relaxation time for heat flux |

${\beta}_{1}={\lambda}_{1}a$ | Deborah number for relaxation time |

${\lambda}_{C}=a{\Gamma}_{c}$ | relaxation time parameter of concentration |

$Pr=\frac{\nu}{\alpha}$ | Prandtl number |

$\sigma =\frac{{k}_{r}}{a}$ | chemical reaction rate parameter |

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${\mathit{\beta}}_{1}.$ | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.2 |
---|---|---|---|---|---|---|

Abel et al. [47] | 0.999996 | 1.051948 | 1.101850 | 1.150163 | 1.196692 | 1.285257 |

Megahed [48] | 0.999978 | 1.051945 | 1.101848 | 1.150160 | 1.196690 | 1.285253 |

Sadeghy et al. [49] | 1.00000 | 1.05490 | 1.10084 | 1.15016 | 1.19872 | ---------- |

Mustafa et al. [50] | 1.000000 | 1.051890 | 1.101903 | 1.150137 | 1.196711 | 1.285363 |

Khan et al. [42] | 1.000000 | 1.051889 | 1.101903 | 1.150137 | 1.196711 | 1.285363 |

Present results | 1.000000 | 1.051890 | 1.101903 | 1.150137 | 1.196711 | 1.285363 |

$\mathit{P}\mathit{r}.$ | 0.7 | 2.0 | 7.0 |
---|---|---|---|

Khan and Pop [51] | 0.4539 | 0.9113 | 1.8954 |

Wang [52] | 0.4539 | 0.9114 | 1.8954 |

Gorla and Sidawi [53] | 0.4539 | 0.9114 | 1.8954 |

Khan et al. [42] | 0.454374 | 0.911155 | 1.822020 |

Present results | 0.454369 | 0.911148 | 1.822015 |

$\mathit{\lambda}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{f}}^{\u2033}$ |
---|---|---|---|

$0.2$ | $0.8$ | $1.1$ | $-0.8484$ |

$0$ | $-0.8395$ | ||

$0.1$ | $-0.8418$ | ||

$0.11$ | $-0.8422$ | ||

$0.12$ | $-0.8427$ | ||

$0.1$ | $-0.7431$ | ||

$0.15$ | $-0.7508$ | ||

$0.2$ | $-0.7585$ | ||

$0.1$ | $-1.1745$ | ||

$0.13$ | $-1.1578$ | ||

$0.15$ | $-1.1593$ | ||

$0.18$ | $-1.6316$ |

$\mathit{\lambda}$ | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | $\mathit{Q}$ | ${\mathit{\lambda}}_{\mathit{E}}$ | ${\mathit{\theta}}^{\prime}$ |
---|---|---|---|---|---|

$0.2$ | $0.8$ | $1.1$ | $0.5$ | $0.7$ | $-0.1544$ |

$0$ | $-0.1652$ | ||||

$0.1$ | $-0.1625$ | ||||

$0.11$ | $-0.1619$ | ||||

$0.12$ | $-0.1615$ | ||||

$0.1$ | $-0.2587$ | ||||

$0.13$ | $-0.2537$ | ||||

$0.15$ | $-0.2504$ | ||||

$0.18$ | $-0.2455$ | ||||

$0.1$ | $0.0022$ | ||||

$0.13$ | $-0.0045$ | ||||

$0.15$ | $-0.0089$ | ||||

$0.18$ | $-0.0153$ | ||||

$0.1$ | $-0.5847$ | ||||

$0.2$ | $-0.4995$ | ||||

$0.3$ | $-0.4028$ | ||||

$0.4$ | $-0.2902$ | ||||

$0.1$ | $-0.1371$ | ||||

$0.2$ | $-0.1331$ | ||||

$0.3$ | $-0.1307$ | ||||

$0.4$ | $-0.1306$ |

$\mathit{S}\mathit{c}$ | ${\mathit{\lambda}}_{\mathit{C}}$ | $\mathit{\sigma}$ | ${\mathit{N}}_{\mathit{t}}$ | ${\mathit{\chi}}^{\prime}$ |
---|---|---|---|---|

$1.2$ | $0.2$ | $0.01$ | $0.01$ | $-0.6562$ |

$0.8$ | $-0.5464$ | |||

$0.9$ | $-0.574$ | |||

$1$ | $-0.6016$ | |||

$1.1$ | $-0.6276$ | |||

$0.1$ | $-0.671$ | |||

$0.13$ | $-0.6665$ | |||

$0.15$ | $-0.6636$ | |||

$0.18$ | $-0.6592$ | |||

$0.1$ | $-0.569$ | |||

$0.13$ | $-0.5379$ | |||

$0.15$ | $-0.5166$ | |||

$0.18$ | $-0.4834$ | |||

$0.1$ | $-0.6498$ | |||

$0.13$ | $-0.6477$ | |||

$0.15$ | $-0.6464$ | |||

$0.18$ | $-0.6445$ |

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**MDPI and ACS Style**

Shankaralingappa, B.M.; Prasannakumara, B.C.; Gireesha, B.J.; Sarris, I.E.
The Impact of Cattaneo–Christov Double Diffusion on Oldroyd-B Fluid Flow over a Stretching Sheet with Thermophoretic Particle Deposition and Relaxation Chemical Reaction. *Inventions* **2021**, *6*, 95.
https://doi.org/10.3390/inventions6040095

**AMA Style**

Shankaralingappa BM, Prasannakumara BC, Gireesha BJ, Sarris IE.
The Impact of Cattaneo–Christov Double Diffusion on Oldroyd-B Fluid Flow over a Stretching Sheet with Thermophoretic Particle Deposition and Relaxation Chemical Reaction. *Inventions*. 2021; 6(4):95.
https://doi.org/10.3390/inventions6040095

**Chicago/Turabian Style**

Shankaralingappa, Bheemasandra M., Ballajja C. Prasannakumara, Bijjanal J. Gireesha, and Ioannis E. Sarris.
2021. "The Impact of Cattaneo–Christov Double Diffusion on Oldroyd-B Fluid Flow over a Stretching Sheet with Thermophoretic Particle Deposition and Relaxation Chemical Reaction" *Inventions* 6, no. 4: 95.
https://doi.org/10.3390/inventions6040095