# First Solution of Fractional Bioconvection with Power Law Kernel for a Vertical Surface

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## Abstract

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## 1. Introduction

## 2. Mathematical Formulation

## 3. The Solution of the Problem with Classical Time Derivative

#### 3.1. The Solution of Bioconvection

#### 3.2. The Solution of Temperature Field

#### 3.3. The Solution of the Velocity Field

#### 3.4. Fractional Modeling

#### 3.5. Solution of the Fractional Model Using Generalized Constitutive Relations

## 4. Results and Discussion

## 5. Conclusions

- The fractional parameter can be used to control the boundary layers of the fluid properties like bioconvection, temperature and velocity.
- The fractional approach can be the best fit in some experimental work, where the needed and desired results can be achieved for different values of fractional parameters and time.
- The fractional model obtained with generalized constitutive laws gives better and more accurate results in terms of memory than the fractional approach with artificial replacement.
- The present results are compared with the existing literature in the absence of bioconvection and they are in good agreement.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Name |

$\rho $ | Fluid density |

s | Laplace transform |

$\mu $ | Viscosity |

$\mathrm{Pr}$ | Prandtl number |

g | Gravitational acceleration |

$\mathrm{Lb}$ | Bioconvection Lewis number |

T | Fluid temperature |

$\mathrm{Gr}$ | Grashof number |

${T}_{w}$ | Temperature at wall |

$\mathrm{Ra}$ | Bioconvection Rayleigh number |

${T}_{\infty}$ | Ambient temperature of the fluid |

$\mathrm{erfc}(.)$ | Gauss’s error function of complimentary |

${\beta}_{T}$ | Volumetric coefficient of thermal expansion |

$\varphi $ | The dimensionless nanoparticle volume fraction |

${\rho}_{m}$ | Mass density |

t | Time (s) |

${\rho}_{\infty}$ | Fluid density |

${N}^{*}$ | Concentration of microorganisms |

N | The dimensionless concentration of microorganisms |

${N}_{\infty}$ | The density of motile microorganisms |

${C}_{p}$ | Specific heat at constant pressure |

k | Thermal conductivity |

${D}_{N}$ | Diffusivity of microorganisms |

$\theta $ | Dimensionless temperature |

u | Velocity |

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**Figure 2.**Comparative analysis of velocity distribution for different $\beta $ and $\gamma $ values.

**Figure 9.**Comparison between present Results in the absence of bioconvection and Nehad el al. [25].

**Figure 10.**Comparison of temperature between classical time derivative, Caputo fractional model with artificial replacement and Caputo model with generalized constitutive law.

**Figure 11.**Comparison of bioconvection between classical time derivative, Caputo fractional model with artificial replacement and Caputo model with generalized constitutive law.

**Figure 12.**Comparison of velocity between classical time derivative, Caputo fractional model with artificial replacement and Caputo model with generalized constitutive law.

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**MDPI and ACS Style**

Asjad, M.I.; Ur Rehman, S.; Ahmadian, A.; Salahshour, S.; Salimi, M.
First Solution of Fractional Bioconvection with Power Law Kernel for a Vertical Surface. *Mathematics* **2021**, *9*, 1366.
https://doi.org/10.3390/math9121366

**AMA Style**

Asjad MI, Ur Rehman S, Ahmadian A, Salahshour S, Salimi M.
First Solution of Fractional Bioconvection with Power Law Kernel for a Vertical Surface. *Mathematics*. 2021; 9(12):1366.
https://doi.org/10.3390/math9121366

**Chicago/Turabian Style**

Asjad, Muhammad Imran, Saif Ur Rehman, Ali Ahmadian, Soheil Salahshour, and Mehdi Salimi.
2021. "First Solution of Fractional Bioconvection with Power Law Kernel for a Vertical Surface" *Mathematics* 9, no. 12: 1366.
https://doi.org/10.3390/math9121366