# MHD Bioconvection Flow and Heat Transfer of Nanofluid through an Exponentially Stretchable Sheet

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## Abstract

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## 1. Introduction

## 2. Mathematical Modeling

_{0}, was acted in the y-trend. Note that the flow encouraged by bioconvection occupies a place in elevated nanoparticles suspension, with a view to obtain the release of inhibiting bioconvection. According to these assumptions, the governing equations concerning the present investigation can be addressed as (see Rashad et al. [23], Babu and Sandeep [25], and Khan and Pop [26]);

## 3. Spectral Relaxation Method

Algorithm 1: Spectral Relaxation Method |

Step 1: Introducing the transformation: ${f}^{\prime}\left(\eta \right)=F\left(\eta \right)$.Step 2: Depict the original equation in terms of $F\left(\eta \right)$ to decrease the order of the momentum equation for $f\left(\eta \right)$ $F\left(\eta \right)$.Step 3: Assume that $F\left(\eta \right)$, $\theta \left(\eta \right)$, $\varphi \left(\eta \right)$, $\chi \left(\eta \right)$ are similar to the former iteration (indicated by ${f}_{r}\left(\eta \right)$, ${\theta}_{r}\left(\eta \right)$, ${\varphi}_{r}\left(\eta \right)$, and ${\chi}_{r}\left(\eta \right)$).Step 4: Construct an iteration scheme for $F\left(\eta \right)$, $\theta \left(\eta \right)$, $\varphi \left(\eta \right)$, and $\chi \left(\eta \right)$.Step 5: Assume that only linear terms in $F\left(\eta \right)$, $\theta \left(\eta \right)$, $\varphi \left(\eta \right)$, and $\chi \left(\eta \right)$ are to be estimated at the present iteration scale (indicated by ${F}_{r+1}\left(\eta \right)$, ${\theta}_{r+1}\left(\eta \right)$, ${\varphi}_{r+1}\left(\eta \right)$, ${\chi}_{r+1}\left(\eta \right)$), and all other terms are presumed to be similar to the former iteration. |

## 4. Results and Discussion

## 5. Conclusions

- The comparison values of heat rate transfer were in good agreement with the former study, and hence led to the confidence of the present results to be reported further.
- The resultant velocity diminished with the increments in the magnetic parameter.
- Fluid temperature increased as the magnetic parameter, thermophoresis, and Brownian motion parameters increased.
- The concentration was reduced with the boost in the Lewis number and Brownian motion parameter.
- The concentration was increased with the increment in the Prandtl number, thermophoresis, and magnetic parameters.
- The density of the motile microorganism is a decreasing function of the Prandtl number, Lewis number, Peclet number, bioconvection Lewis number, and bioconvection parameter.
- The residual errors of $f\left(\eta \right)$, $\theta \left(\eta \right)$, $\varphi \left(\eta \right)$, and $\chi \left(\eta \right)$ were iteration dependent.
- For future research, it is suggested for the present study to consider all possible multiple solutions or dual solutions. This is driven by the fact that the multiple solutions cannot be seen experimentally and can only be obtained by using numerical simulation.
- It was also proposed for the stability of multiple solutions to be included as one of the main objective studies for future work. Stability analysis is important for identifying the reliability of the multiple solutions, which depend on the assumptions of the physical model.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**The velocity profile (${f}^{\prime}\left(\eta \right)$) for different values of $M$ when $Nb=0.5,Nt=0.1,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,\mathrm{Pr}=0.71,Ec=0.01$.

**Figure 3.**The temperature profile ($\theta \left(\eta \right)$) for different values of $M$ when $Nb=0.5,Nt=0.1,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,\mathrm{Pr}=0.71,Ec=0.01$.

**Figure 4.**The nanoparticle volume fraction profile ($\varphi \left(\eta \right)$) for different values of $M$ when $Nb=0.5,Nt=0.1,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,\mathrm{Pr}=0.71,Ec=0.01$.

**Figure 5.**The density of the motile microorganism profile ($\chi \left(\eta \right)$) for different values of $M$ when $Nb=0.5,Nt=0.1,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,\mathrm{Pr}=0.71,Ec=0.01$.

**Figure 6.**The temperature profile ($\theta \left(\eta \right)$) for different values of $\mathrm{Pr}$ when $M=0.5,Nb=0.5,Nt=0.1,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,Ec=0.01$.

**Figure 7.**The nanoparticle volume fraction profile ($\varphi \left(\eta \right)$) for different values of $\mathrm{Pr}$ when $M=0.5,Nb=0.5,Nt=0.1,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,Ec=0.01$.

**Figure 8.**The density of the motile microorganism profile ($\chi \left(\eta \right)$) for different values of $\mathrm{Pr}$ when $M=0.5,Nb=0.5,Nt=0.1,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,Ec=0.01$.

**Figure 9.**The temperature profile ($\theta \left(\eta \right)$) for different values of $Nt$ when $M=0.5,Nb=0.5,\mathrm{Pr}=0.7,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,Ec=0.01$.

**Figure 10.**The nanoparticle volume fraction profile ($\varphi \left(\eta \right)$) for different values of $Nt$ when $M=0.5,Nb=0.5,\mathrm{Pr}=0.7,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,Ec=0.01$.

**Figure 11.**The temperature profile ($\theta \left(\eta \right)$) for different values of $Nb$ when $M=0.5,Nt=0.1,\mathrm{Pr}=0.7,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,Ec=0.01$.

**Figure 12.**The nanoparticle volume fraction profile ($\varphi \left(\eta \right)$) for different values of $Nb$ when $M=0.5,Nt=0.1,\mathrm{Pr}=0.7,Le=5,Lb=0.5,Pe=0.3,\sigma =0.2,Ec=0.01$.

**Figure 13.**The nanoparticle volume fraction profile ($\varphi \left(\eta \right)$) for different values of $Le$ when $M=0.5,Nb=0.5,Nt=0.1,\mathrm{Pr}=0.7,Lb=0.5,Pe=0.3,\sigma =0.2,Ec=0.01$.

**Figure 14.**The density of the motile microorganism profile ($\chi \left(\eta \right)$) for different values of $Le$ when $M=0.5,Nb=0.5,Nt=0.1,\mathrm{Pr}=0.7,Lb=0.5,Pe=0.3,\sigma =0.2,Ec=0.01$.

**Figure 15.**The density of the motile microorganism profile ($\chi \left(\eta \right)$) for different values of $\sigma $ when $M=0.5,Nb=0.5,Nt=0.1,\mathrm{Pr}=0.7,Lb=0.5,Pe=0.3,Le=5,Ec=0.01$.

**Figure 16.**The density of the motile microorganism profile ($\chi \left(\eta \right)$) for different values of $Pe$ when $M=0.5,Nb=0.5,Nt=0.1,\mathrm{Pr}=0.7,Lb=0.5,\sigma =0.2,Le=5,Ec=0.01$.

**Figure 17.**The density of the motile microorganism profile ($\chi \left(\eta \right)$) for different values of $Lb$ when $M=0.5,Nb=0.5,Nt=0.1,\mathrm{Pr}=0.7,Pe=0.3,\sigma =0.2,Le=5,Ec=0.01$.

**Figure 18.**Residual error of ${f}^{\prime}\left(\eta \right),\theta \left(\eta \right),\varphi \left(\eta \right),\chi \left(\eta \right)$ versus iterations when $M=0.5,Nt=0.1,\mathrm{Pr}=0.7,Nb=1.5,Pe=0.3,\sigma =0.2,Le=5,Lb=0.5,Ec=0.01$.

**Table 1.**Comparison of of local Nusselt number $-{\theta}^{\prime}\left(0\right)$ for various values of $\mathrm{Pr}$ for $Nt=Nb=Le=Lb=Pe=M=Ec=\sigma =0$.

Pr | Bidin and Nazar [30] | Magyari and Keller [31] | El-Aziz [32] | Loganthan and Vimala [33] | Present Study | |
---|---|---|---|---|---|---|

SRM | Bvp4c | |||||

1 | 0.9547 | 0.954782 | 0.954785 | 0.954955 | 0.9548 | 0.954782 |

1.5 | 1.2348 | 1.234755 | ||||

2 | 1.4714 | 1.4715 | 1.471460 | |||

2.5 | 1.6802 | 1.680229 | ||||

3 | 1.8691 | 1.869075 | 1.869074 | 1.869074 | 1.8691 | 1.869073 |

5 | 2.500135 | 2.500132 | 2.500184 | 2.5001 | 2.500131 | |

7 | 3.0133 | 3.013277 | ||||

10 | 3.660379 | 3.660372 | 3.660379 | 3.6604 | 3.660372 | |

20 | 5.3016 | 5.301625 |

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**MDPI and ACS Style**

Ferdows, M.; Zaimi, K.; Rashad, A.M.; Nabwey, H.A.
MHD Bioconvection Flow and Heat Transfer of Nanofluid through an Exponentially Stretchable Sheet. *Symmetry* **2020**, *12*, 692.
https://doi.org/10.3390/sym12050692

**AMA Style**

Ferdows M, Zaimi K, Rashad AM, Nabwey HA.
MHD Bioconvection Flow and Heat Transfer of Nanofluid through an Exponentially Stretchable Sheet. *Symmetry*. 2020; 12(5):692.
https://doi.org/10.3390/sym12050692

**Chicago/Turabian Style**

Ferdows, Mohammad, Khairy Zaimi, Ahmed M. Rashad, and Hossam A. Nabwey.
2020. "MHD Bioconvection Flow and Heat Transfer of Nanofluid through an Exponentially Stretchable Sheet" *Symmetry* 12, no. 5: 692.
https://doi.org/10.3390/sym12050692