# A Numerical Exploration of Modified Second-Grade Nanofluid with Motile Microorganisms, Thermal Radiation, and Wu’s Slip

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

## 3. Numerical Procedure

## 4. Graphical Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**–

**d**): Variations of ${f}^{\prime},\theta ,\varphi ,\chi $ for various values of $Rb$ when $Nr=0.2,\alpha =0.2,\mathsf{\Gamma}=1.0,\beta =-1.0,$ ${\mathsf{\Lambda}}_{1}=0.2,Le=2.0,\mathrm{Pr}=1.0,{\theta}_{w}=1.5,Rd=0.4,Bi=2.0,Pr=0.7,E=0.2,Pe=0.1,Lb=2.0$.

**Figure 2.**(

**a**–

**d**): Variations of ${f}^{\prime},\theta ,\varphi ,\chi $ for various values of $Nr$ when $Rb=0.2,\alpha =0.2,\mathsf{\Gamma}=1.0,\beta =-1.0,$ ${\mathsf{\Lambda}}_{1}=0.2,Le=2.0,\mathrm{Pr}=1.0,{\theta}_{w}=1.5,Rd=0.4,Bi=2.0,Pr=0.7,E=0.2,Pe=0.1,Lb=2.0$.

**Figure 3.**(

**a**–

**d**): Variations of ${f}^{\prime},\theta ,\varphi ,\chi $ for various values of $\alpha $ when $Rb=0.2,Nr=0.2,\mathsf{\Gamma}=1.0,\beta =-1.0,$ ${\mathsf{\Lambda}}_{1}=0.2,Le=2.0,\mathrm{Pr}=1.0,{\theta}_{w}=1.5,Rd=0.4,Bi=2.0,Pr=0.7,E=0.2,Pe=0.1,Lb=2.0$.

**Figure 4.**(

**a**–

**c**): Variations of ${f}^{\prime}$ for various parameters $\mathsf{\Gamma},\beta ,{\mathsf{\Lambda}}_{1}$ when $Rb=0.2,Nr=0.2,\alpha =0.2,Le=2.0,$ $\mathrm{Pr}=1.0,{\theta}_{w}=1.5,Rd=0.4,Bi=2.0,Pr=0.7,E=0.2,Pe=0.1,Lb=2.0$.

**Figure 5.**(

**a**,

**b**): Variations of $\theta ,\varphi $ for various values of $Le$ when $Rb=0.2,Nr=0.2,\alpha =0.2,\mathsf{\Gamma}=1.0,\beta =-1.0,$ ${\mathsf{\Lambda}}_{1}=0.2,\mathrm{Pr}=1.0,{\theta}_{w}=1.5,Rd=0.4,Bi=2.0,Pr=0.7,E=0.2,Pe=0.1,Lb=2.0$.

**Figure 6.**(

**a**–

**c**): Variations of $\theta ,$ for various parametrs $\mathrm{Pr},{\theta}_{w},Rd,$ when $Rb=0.2,Nr=0.2,\alpha =0.2,\mathsf{\Gamma}=1.0,$ $\beta =-1.0,{\mathsf{\Lambda}}_{1}=0.2,Le=2.0,Bi=2.0,Pr=0.7,E=0.2,Pe=0.1,Lb=2.0$.

**Figure 7.**(

**a**,

**b**): Variations of $\theta ,\varphi ,$ for various values of $Bi$ when $Rb=0.2,Nr=0.2,\alpha =0.2,\mathsf{\Gamma}=1.0,$ $\beta =-1.0,{\mathsf{\Lambda}}_{1}=0.2,Le=2.0,\mathrm{Pr}=1.0,{\theta}_{w}=1.5,Rd=0.4,Pr=0.7,E=0.2,Pe=0.1,Lb=2.0$.

**Figure 8.**(

**a**,

**b**): Variations of $\varphi $ for various values of parameters $\mathrm{Pr},E$ when $Rb=0.2,Nr=0.2,\alpha =0.2,\mathsf{\Gamma}=1.0,$ $\beta =-1.0,{\mathsf{\Lambda}}_{1}=0.2,Le=2.0,\mathrm{Pr}=1.0,{\theta}_{w}=1.5,Rd=0.4,Bi=2,0,Pe=0.1,Lb=2.0$.

**Figure 9.**(

**a**,

**b**): Variations of $\chi $ for various values of parameters $Pe,Lb$ when $Rb=0.2,Nr=0.2,\alpha =0.2,\mathsf{\Gamma}=1.0,$ $\beta =-1.0,{\mathsf{\Lambda}}_{1}=0.2,Le=2.0,\mathrm{Pr}=1.0,{\theta}_{w}=1.5,Rd=0.4,Bi=2,0,Pr=0.7,E=0.2$.

**Table 1.**Comparison for the results of $-{f}^{\u2033}\left(0\right)$ and $-{\theta}^{\prime}\left(0\right)$ in the case of second-grade fluid $m=0,\alpha =1$, $\lambda =Nr=Nc=,{\theta}_{w}=Rd=E=Pe=Lb=0$.

${\mathit{\alpha}}^{*}$ | $\mathbf{Pr}$ | Masood et al. [39] | Present Results | ||||||
---|---|---|---|---|---|---|---|---|---|

Exact Solution | Numerical Solution | Exact Solution | Numerical Solution | ||||||

$-{\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | $-{\mathit{f}}^{\u2033}\left(0\right)$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | ||

0.5 | 10 | 0.81649658 | 2.3478745 | 0.816451160 | 2.3478704 | 0.816451161 | 2.3478701 | 0.81649645 | 2.3478744 |

1.0 | 0.70710678 | 2.3715683 | 0.70716177 | 2.3715544 | 0.70716170 | 2.3715542 | 0.70710696 | 2.3715684 | |

1.5 | 0.63245553 | 2.3877034 | 0.63257670 | 2.3876736 | 0.63257672 | 2.3876730 | 0.6324557 | 2.3877036 | |

2.0 | 0.57735027 | 2.399595 | 0.57755726 | 2.3995450 | 0.5775572 | 2.3995452 | 0.5773501 | 2.3995950 | |

2.0 | 0.95141934 | 0.9514135 | 0.9514137 | 0.9514193 | |||||

5.0 | 1.6081636 | 1.6081591 | 1.6081599 | 1.6081636 | |||||

7.0 | 1.9354025 | 1.9353982 | 1.9353985 | 1.9354025 |

**Table 2.**Variations in $-{f}^{\u2033}\left(0\right)$ against $\alpha $, ${\mathsf{\Lambda}}_{1}$, $Rb$, $Nr$, $\mathsf{\Gamma}$, and $\beta $.

$\mathit{\alpha}$ | ${\mathbf{\Lambda}}_{1}$ | $\mathit{R}\mathit{b}$ | $\mathit{N}\mathit{r}$ | $\mathbf{\Gamma}$ | $\mathit{\beta}$ | $-{\mathit{f}}^{\u2033}\left(0\right)$ | ||
---|---|---|---|---|---|---|---|---|

$\mathit{m}=-0.5$ | $\mathit{m}=0$ | $\mathit{m}=0.5$ | ||||||

0.1 | 0.2 | 0.2 | 0.2 | 1.0 | −1.0 | 0.3415 | 0.3559 | 0.3691 |

0.4 | 0.3277 | 0.3424 | 0.3546 | |||||

0.8 | 0.3158 | 0.3300 | 0.3407 | |||||

0.5 | 0.2 | 0.2 | 0.2 | 1.0 | −1.0 | 0.3388 | 0.3547 | 0.3684 |

0.4 | 0.3354 | 0.3542 | 0.3680 | |||||

0.6 | 0.3339 | 0.3559 | 0.3674 | |||||

0.5 | 0.2 | 0.1 | 0.2 | 1.0 | −1.0 | 0.3442 | 0.3571 | 0.3699 |

0.5 | 0.3440 | 0.3569 | 0.3697 | |||||

1.0 | 0.3438 | 0.3567 | 0.3695 | |||||

0.5 | 0.2 | 0.2 | 0.1 | 1.0 | −1.0 | 0.3417 | 0.3560 | 0.3692 |

0.5 | 0.3415 | 0.3558 | 0.3690 | |||||

1.0 | 0.3413 | 0.3556 | 0.3688 | |||||

0.5 | 0.2 | 0.2 | 0.2 | 2.0 | −1.0 | 0.2487 | 0.2577 | 0.2643 |

3.0 | 0.1965 | 0.2026 | 0.2067 | |||||

4.0 | 0.1624 | 0.1670 | 0.1699 | |||||

0.5 | 0.2 | 0.2 | 0.2 | 1.0 | −2.0 | 0.2732 | 0.2829 | 0.2941 |

−3.0 | 0.2337 | 0.2392 | 0.2474 | |||||

−4.0 | 0.2071 | 0.2098 | 0.2155 |

**Table 3.**Variations in $-{\theta}^{\prime}\left(0\right)$ against $\mathrm{Pr}$, $\epsilon ,{\theta}_{w}$, $Nt,$ $Nb$, $Le$, ${\mathsf{\Lambda}}_{1}$, $Rb$, and $Nr$.

$\mathbf{Pr}$ | $\mathit{\epsilon}$ | ${\mathit{\theta}}_{\mathit{w}}$ | $\mathit{N}\mathit{t}$ | $\mathit{N}\mathit{b}$ | $\mathit{L}\mathit{e}$ | ${\mathbf{\Lambda}}_{1}$ | $\mathit{R}\mathit{d}$ | $\mathit{R}\mathit{b}$ | $\mathit{N}\mathit{r}$ | $-{\mathit{\theta}}^{\prime}\left(0\right)$ | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{m}=-0.5$ | $\mathit{m}=0$ | $\mathit{m}=0.5$ | ||||||||||

1 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 0.2787 | 0.2926 | 0.3024 |

2 | 0.3683 | 0.3919 | 0.4077 | |||||||||

3 | 0.4336 | 0.4636 | 0.4824 | |||||||||

10 | 0.1 | 0.5 | 0.3 | 0.2 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 0.3833 | 0.3563 | 0.3489 |

0.4 | 0.3687 | 0.3424 | 0.3355 | |||||||||

0.8 | 0.3514 | 0.3260 | 0.3193 | |||||||||

10 | 1 | 0.1 | 0.3 | 0.2 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 0.3533 | 0.3276 | 0.3209 |

0.4 | 0.3332 | 0.3083 | 0.3019 | |||||||||

0.8 | 0.3133 | 0.2892 | 0.2832 | |||||||||

10 | 1 | 0.5 | 0.1 | 0.2 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 0.5577 | 0.5986 | 0.6238 |

0.4 | 0.4895 | 0.5196 | 0.5394 | |||||||||

0.5 | 0.4213 | 0.4381 | 0.4506 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.1 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 0.5228 | 0.5584 | 0.5810 |

0.5 | 0.5223 | 0.5580 | 0.5807 | |||||||||

1.0 | 0.5221 | 0.5578 | 0.5805 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 1 | 0.2 | 0.5 | 0.2 | 0.2 | 0.5532 | 0.5915 | 0.6151 |

1.5 | 0.5404 | 0.5773 | 0.6005 | |||||||||

1.8 | 0.5324 | 0.5687 | 0.5916 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.5236 | 0.5589 | 0.5812 |

0.3 | 0.5264 | 0.5607 | 0.5818 | |||||||||

0.4 | 0.5277 | 0.5613 | 0.5816 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.2 | 0.1 | 0.2 | 0.2 | 0.5668 | 0.6046 | 0.6286 |

0.4 | 0.5100 | 0.5450 | 0.5672 | |||||||||

0.5 | 0.4680 | 0.5003 | 0.5211 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.2 | 0.5 | 0.1 | 0.2 | 0.5166 | 0.5526 | 0.5761 |

0.5 | 0.5243 | 0.5599 | 0.5823 | |||||||||

1.0 | 0.5307 | 0.5662 | 0.5879 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.2 | 0.5 | 0.2 | 0.1 | 0.5221 | 0.5579 | 0.5806 |

0.5 | 0.5226 | 0.5582 | 0.5809 | |||||||||

1.0 | 0.5231 | 0.5586 | 0.5811 |

**Table 4.**Variations in ${\varphi}^{\prime}\left(0\right)$ against $\mathrm{Pr}$, $E$, $\sigma ,$ $Nt,$ $Nb$, $Le$, ${\mathsf{\Lambda}}_{1}$, $Rd$, $Rb$, and $Nr$.

$\mathbf{Pr}$ | $\mathit{E}$ | $\mathit{\sigma}$ | $\mathit{N}\mathit{t}$ | $\mathit{N}\mathit{b}$ | $\mathit{L}\mathit{e}$ | ${\mathbf{\Lambda}}_{1}$ | $\mathit{R}\mathit{d}$ | $\mathit{R}\mathit{b}$ | $\mathit{N}\mathit{r}$ | ${\mathit{\varphi}}^{\prime}\left(0\right)$ | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{m}=-0.5$ | $\mathit{m}=0$ | $\mathit{m}=0.5$ | ||||||||||

1 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 0.4181 | 0.4388 | 0.4536 |

2 | 0.5525 | 0.5878 | 0.6115 | |||||||||

3 | 0.6504 | 0.6948 | 0.7236 | |||||||||

10 | 0.1 | 0.5 | 0.3 | 0.2 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 0.5609 | 0.5207 | 0.5100 |

0.5 | 0.5633 | 0.5320 | 0.5110 | |||||||||

1.0 | 0.5650 | 0.5230 | 0.5120 | |||||||||

10 | 1 | 0.1 | 0.3 | 0.2 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 0.5528 | 0.5158 | 0.5062 |

0.5 | 0.5371 | 0.5045 | 0.4968 | |||||||||

1.0 | 0.5305 | 0.4992 | 0.4920 | |||||||||

10 | 1 | 0.5 | 0.1 | 0.2 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 0.2789 | 0.2993 | 0.3119 |

0.4 | 1.2238 | 1.2990 | 1.3485 | |||||||||

0.5 | 2.1065 | 2.1907 | 2.2531 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.1 | 2 | 0.1 | 0.5 | 0.2 | 0.2 | 1.5685 | 1.6752 | 1.7429 |

0.5 | 0.3134 | 0.3348 | 0.3484 | |||||||||

1.0 | 0.1567 | 0.1674 | 0.1742 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 1.0 | 0.2 | 0.5 | 0.2 | 0.2 | 0.8997 | 0.8872 | 0.9226 |

1.4 | 0.8105 | 0.8660 | 0.9007 | |||||||||

1.7 | 0.7986 | 0.8531 | 0.8875 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.2 | 0.5 | 0.2 | 0.2 | 0.7854 | 0.8383 | 0.8718 |

0.3 | 0.7896 | 0.8410 | 0.8724 | |||||||||

0.4 | 0.7916 | 0.8420 | 0.8727 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.2 | 0.1 | 0.2 | 0.2 | 0.8501 | 0.9069 | 0.9428 |

0.4 | 0.7650 | 0.8174 | 0.8509 | |||||||||

0.5 | 0.7020 | 0.7505 | 0.7817 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.2 | 0.5 | 0.1 | 0.2 | 0.7749 | 0.8289 | 0.8642 |

0.5 | 0.7864 | 0.8398 | 0.8735 | |||||||||

1.0 | 0.7960 | 0.8493 | 0.8818 | |||||||||

10 | 1 | 0.5 | 0.3 | 0.2 | 2 | 0.2 | 0.5 | 0.2 | 0.1 | 0.7832 | 0.8368 | 0.8710 |

0.5 | 0.7839 | 0.8374 | 0.8713 | |||||||||

1.0 | 0.7846 | 0.8379 | 0.8717 |

**Table 5.**Variations in $-{\chi}^{\prime}\left(0\right)$ against $\mathrm{P}e,$ $\mathsf{\Gamma},$ $\beta ,$ ${\mathsf{\Lambda}}_{1}$, $Lb,$ $Rb$, and $Nr$.

$\mathit{P}\mathit{e}$ | $\mathbf{\Gamma}$ | $\mathit{\beta}$ | ${\mathbf{\Lambda}}_{1}$ | $\mathit{L}\mathit{b}$ | $\mathit{R}\mathit{b}$ | $\mathit{N}\mathit{r}$ | $-{\mathit{\chi}}^{\prime}\left(0\right)$ | ||
---|---|---|---|---|---|---|---|---|---|

$\mathit{m}=-0.5$ | $\mathit{m}=0$ | $\mathit{m}=0.5$ | |||||||

0.3 | 1 | 0.5 | 0.1 | 2 | 0.2 | 0.2 | 0.5502 | 0.6417 | 0.6981 |

0.5 | 0.6270 | 0.7235 | 0.7831 | ||||||

0.7 | 0.7039 | 0.8635 | 0.9632 | ||||||

0.5 | 1.2 | 0.5 | 0.1 | 2 | 0.2 | 0.2 | 0.4321 | 0.5000 | 0.5367 |

1.6 | 0.4060 | 0.4631 | 0.4915 | ||||||

2.0 | 0.3879 | 0.4379 | 0.4613 | ||||||

0.5 | 1 | 0.5 | 0.2 | 2 | 0.2 | 0.2 | 0.4389 | 0.5105 | 0.5534 |

0.4160 | 0.4778 | 0.5131 | |||||||

0.3898 | 0.4363 | 0.4620 | |||||||

0.5 | 1 | 0.5 | 0.2 | 2 | 0.2 | 0.2 | 0.4781 | 0.5635 | 0.6155 |

0.3 | 0.4903 | 0.5729 | 0.6215 | ||||||

0.4 | 0.4974 | 0.5784 | 0.6247 | ||||||

0.5 | 1 | 0.5 | 0.2 | 1.0 | 0.2 | 0.2 | 0.3846 | 0.4403 | 0.4741 |

1.4 | 0.4297 | 0.5013 | 0.5452 | ||||||

1.8 | 0.4737 | 0.5602 | 0.6133 | ||||||

0.5 | 1 | 0.5 | 0.2 | 2 | 0.1 | 0.2 | 0.4568 | 0.5447 | 0.6003 |

0.5 | 0.4787 | 0.5649 | 0.6173 | ||||||

1.0 | 0.4966 | 0.5819 | 0.6322 | ||||||

0.5 | 1 | 0.5 | 0.2 | 2 | 0.2 | 0.1 | 0.4726 | 0.5594 | 0.6127 |

0.5 | 0.4740 | 0.5604 | 0.6135 | ||||||

1.0 | 0.4753 | 0.5615 | 0.6142 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Y.; Waqas, H.; Imran, M.; Farooq, U.; Mallawi, F.; Tlili, I.
A Numerical Exploration of Modified Second-Grade Nanofluid with Motile Microorganisms, Thermal Radiation, and Wu’s Slip. *Symmetry* **2020**, *12*, 393.
https://doi.org/10.3390/sym12030393

**AMA Style**

Li Y, Waqas H, Imran M, Farooq U, Mallawi F, Tlili I.
A Numerical Exploration of Modified Second-Grade Nanofluid with Motile Microorganisms, Thermal Radiation, and Wu’s Slip. *Symmetry*. 2020; 12(3):393.
https://doi.org/10.3390/sym12030393

**Chicago/Turabian Style**

Li, Yurong, Hassan Waqas, Muhammad Imran, Umar Farooq, Fouad Mallawi, and Iskander Tlili.
2020. "A Numerical Exploration of Modified Second-Grade Nanofluid with Motile Microorganisms, Thermal Radiation, and Wu’s Slip" *Symmetry* 12, no. 3: 393.
https://doi.org/10.3390/sym12030393