# An Analysis for Variable Physical Properties Involved in the Nano-Biofilm Transportation of Sutterby Fluid across Shrinking/Stretching Surface

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## Abstract

**:**

## 1. Introduction

## 2. Physical Model and Mathematical Formulation

## 3. Solution Evaluation

## 4. Physical Quantities

#### 4.1. Skin Friction Coefficient

#### 4.2. Nusselt Number

#### 4.3. Sherwood Number

#### 4.4. Density of Micro-Organisms

## 5. Solution Procedure

## 6. Analysis of Results

#### 6.1. Impacts of Distinct Parameters on Physical Quantities

#### 6.2. Influence of Power Law Index $\mathit{m}$

#### 6.3. Effects of Concentration-Dependent Parameters

#### 6.4. Velocity Profiles

#### 6.5. Temperature Distributions

#### 6.6. Concentration Profiles

#### 6.7. Motile Density Distributions

## 7. Conclusions

- The velocity profile accelerates as the power law coefficient m improves, while the heat, density as well as motile concentration trends deteriorate. Variable transiting parameters ${c}_{2}$, ${c}_{4}$, ${c}_{6}$ and ${c}_{8}$ optimize velocity, heat, density and motile concentration distribution.
- When the factors ${S}_{b}$, M, ${R}_{es}$, $Kp$, ${D}_{es}$, $Nr$ as well as $Rb$ updated, a declining velocity trend is viewed, which is dramatically exaggerated when the $\omega $ is inspected.
- Brownian motion parameter $Nb$, heat conduction factor Q as well as thermophoresis parameter $Nt$ all assist towards a relatively consistent temperature variation.
- The nano-particle density pattern strengthens while the activating energy E and thermophoresis factor $Nt$ expand, and it swiftly declines as the Lewis number $Le$ as well as the Brownian motion factor $Nb$ develop. item The density of microbes grows as the Schmidt coefficient $Sc$ as well as the Peclet number $Pe$ rise.
- As the amounts of the parameters ${c}_{4}$, ${c}_{6}$, $Nb$, $Nt$ and Q expanded, the heat transport capacity declined.
- The Sherwood number drops as the factors E and ${c}_{6}$ expand, but it grows with the variables $Le$, $Nb$ and $Nt$ boost.
- With growing species dispersion parameter ${c}_{8}$, microorganism concentration drops swiftly, which is accentuated by raising Schmidt quantity $Sc$, Peclet number $Pe$ and bioconvection variable $\sigma $.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

List of Symbols | |

$u,v$ | Nanofluid velocity components |

$(x,y)$ | Cartesian Coordinates |

${u}_{w}\left(x\right)$ | Stretching/shrinking velocity |

${u}_{e}\left(x\right)$ | Ambient fluid velocity |

${v}_{w}\left(x\right)$ | Wall transpiration velocity |

${B}_{0}$ | Magnetic field strength |

C | Concentration of nanoparticles |

T | Temperature of nanoparticles |

n | Density of micro-organisms |

${k}_{1}$ | Permeability of porous medium |

${k}_{s}\left(C\right)$ | Variable thermal conductivity |

${k}_{\infty}$ | Constant thermal conductivity |

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

g | Gravitational accerlation |

c | Constant in stretching/shrinking velocity |

${c}_{2}$ | Viscosity parameter |

${c}_{4}$ | Thermal conductivity parameter |

${c}_{6}$ | Nanoparticle mass diffusivity |

${c}_{8}$ | Micro-organisms species diffusivity |

${D}_{B}\left(C\right)$ | Variable Brownian diffusion coefficient |

${D}_{T}$ | Thermophoresis diffusion coefficient |

${D}_{n}\left(C\right)$ | Variable diffusivity of micro-organisms |

${D}_{B,\infty}$ | Constant nano-particle mass diffusivity |

${D}_{n,\infty}$ | Constant micro-organisms diffusivity |

${T}_{\infty}$ | Uniform temperature in the free stream |

${C}_{\infty}$ | Uniform nanofluid concentration in free stream |

${n}_{\infty}$ | Uniform density of microorganisms in free stream |

${Q}_{1}$ | Heat source/sink coefficient |

${E}_{a}$ | Activation energy |

M | Magnetic field parameter |

$Kp$ | Porosity parameter |

$Pr$ | Prandtl number |

$Nb$ | Brownian motion parameter |

$Nt$ | Thermophoresis parameter |

$Sc$ | Schmidt number |

Q | Heat source parameter |

$Le$ | Lewis number |

$Pe$ | Peclet number |

$Nr$ | Buoyancy ratio parameter |

$Rb$ | Rayleigh number |

${D}_{es}$ | Sutterby Deborah number |

${R}_{es}$ | Sutterby Reynolds number |

${W}_{c}$ | Maximum cell swimming speed |

${T}_{w}$ | Uniform temperature at the sheet surface |

${C}_{w}$ | Uniform nanoparticles concentration at sheet surface |

${n}_{w}$ | Uniform density of micro-organisms at sheet surface |

f | Dimensionless stream function |

a | Constant in the ambient fluid velocity |

b | Chemotaxis constant |

${m}^{*}$ | Fitted rate parameter |

R | Wall transpiration parameter |

${S}_{b}$ | Deportment index flow |

${b}^{2}$ | Consistency index |

Greek Symbols | |

${\alpha}_{\infty}$ | Uniform thermal diffusivity |

${\mu}_{\infty}$ | Constant dynamic viscosity |

${\mu}_{s}\left(C\right)$ | Variable dynamic viscosity |

$\nu $ | Kinematic viscosity |

$\sigma $ | Bioconvection constant |

${\sigma}^{*}$ | Electrical conductivity |

$\rho $ | Density of fluid |

${\rho}_{\infty}$ | Constant fluid density |

${\gamma}^{*}$ | Average volume of micro-organisms |

$\omega $ | Mixed convection parameter |

$\lambda $ | Stretching/shrinking parameter |

$\psi $ | Dimensionless stream function |

$\eta $ | Dimensionless transverse coordinate |

$\theta $ | Dimensionless temperature function |

$\varphi $ | Dimensionless density of nanoparticles |

$\chi $ | Dimensionless density of micro-organisms |

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**Figure 2.**Fluctuation in profiles (velocity, temperature, concentration, and motile density) with m.

**Figure 3.**Fluctuation in profiles (velocity, temperature, concentration and motile density) with concentration dependent parameters.

Alsenafi et al. [40] | Zaimi et al. [41] | Present Results | |
---|---|---|---|

${f}^{\u2033}\left(0\right)$ | 0 | 0 | 0 |

$-{\theta}^{\prime}\left(0\right)$ | 0.476745 | 0.476737 | 0.476744 |

$-{\varphi}^{\prime}\left(0\right)$ | 1.045230 | 1.045154 | 1.04513 |

${\mathit{c}}_{2}$ | ${\mathit{S}}_{\mathit{b}}$ | ${\mathit{R}}_{\mathit{e}\mathit{s}}$ | ${\mathit{D}}_{\mathit{e}\mathit{s}}$ | m | M | $\mathit{\omega}$ | $\mathit{N}\mathit{r}$ | $\mathit{R}\mathit{b}$ | $-{\mathit{f}}^{\u2033}\left(0\right)$ |
---|---|---|---|---|---|---|---|---|---|

0.2 | 0.5 | 0.5 | 0.5 | 2.0 | 0.5 | 0.1 | 1.0 | 1.0 | 0.2399 |

0.4 | 0.1351 | ||||||||

0.6 | 0.0472 | ||||||||

0.4 | 0.1 | 0.1322 | |||||||

0.3 | 0.1336 | ||||||||

0.5 | 0.1351 | ||||||||

0.5 | 0.5 | 0.1351 | |||||||

1.0 | 0.1387 | ||||||||

1.5 | 0.1423 | ||||||||

0.5 | 0.5 | 0.1351 | |||||||

0.9 | 0.1380 | ||||||||

1.3 | 0.1409 | ||||||||

0.5 | 1.0 | 0.1003 | |||||||

2.0 | 0.1351 | ||||||||

3.0 | 0.1355 | ||||||||

2.0 | 0.1 | 0.1240 | |||||||

0.3 | 0.1296 | ||||||||

0.5 | 0.1351 | ||||||||

0.5 | 0.1 | 0.1351 | |||||||

0.3 | 0.1282 | ||||||||

0.5 | 0.1213 | ||||||||

0.1 | 0.1 | 0.1411 | |||||||

0.5 | 0.1384 | ||||||||

1.0 | 0.1351 | ||||||||

1.0 | 0.1 | 0.1412 | |||||||

0.5 | 0.1385 | ||||||||

1.0 | 0.1351 |

${\mathit{c}}_{4}$ | ${\mathit{c}}_{6}$ | $\mathit{N}\mathit{b}$ | $\mathit{N}\mathit{t}$ | Q | $-{\mathit{\theta}}^{\prime}\left(0\right)$ |
---|---|---|---|---|---|

0.2 | 0.4 | 0.1 | 0.1 | 0.3 | 0.5670 |

0.4 | 0.5074 | ||||

0.6 | 0.4605 | ||||

0.4 | 0.2 | 0.5130 | |||

0.4 | 0.5074 | ||||

0.6 | 0.5018 | ||||

0.4 | 0.1 | 0.5074 | |||

0.2 | 0.4573 | ||||

0.3 | 0.4122 | ||||

0.1 | 0.1 | 0.5074 | |||

0.2 | 0.4879 | ||||

0.3 | 0.4693 | ||||

0.1 | 0.1 | 0.5903 | |||

0.2 | 0.5496 | ||||

0.3 | 0.5074 |

${\mathit{c}}_{6}$ | $\mathit{L}\mathit{e}$ | $\mathit{N}\mathit{t}$ | $\mathit{N}\mathit{b}$ | E | $-{\mathit{\varphi}}^{\prime}\left(0\right)$ |
---|---|---|---|---|---|

0.2 | 4.0 | 0.1 | 0.1 | 0.3 | 1.8349 |

0.4 | 1.6460 | ||||

0.6 | 1.5017 | ||||

0.4 | 3.0 | 1.3848 | |||

4.0 | 1.6460 | ||||

5.0 | 1.8812 | ||||

4.0 | 0.1 | 1.6460 | |||

0.2 | 1.6511 | ||||

0.3 | 1.6686 | ||||

0.1 | 0.1 | 1.6460 | |||

0.2 | 1.6646 | ||||

0.3 | 1.6695 | ||||

0.1 | 0.1 | 1.6737 | |||

0.2 | 1.6593 | ||||

0.3 | 1.6460 |

${\mathit{c}}_{8}$ | $\mathit{S}\mathit{c}$ | $\mathit{P}\mathit{e}$ | $\mathit{\sigma}$ | $-{\mathit{\chi}}^{\prime}\left(0\right)$ |
---|---|---|---|---|

0.2 | 3.0 | 0.1 | 0.1 | 1.5608 |

0.4 | 1.3925 | |||

0.6 | 1.2618 | |||

0.4 | 3.0 | 1.3925 | ||

4.0 | 1.6249 | |||

5.0 | 1.8386 | |||

3.0 | 0.1 | 1.3925 | ||

0.2 | 1.4895 | |||

0.3 | 1.5875 | |||

0.1 | 0.1 | 1.3925 | ||

0.3 | 1.4068 | |||

0.5 | 1.4211 |

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**MDPI and ACS Style**

Abdal, S.; Siddique, I.; Afzal, S.; Sharifi, S.; Salimi, M.; Ahmadian, A.
An Analysis for Variable Physical Properties Involved in the Nano-Biofilm Transportation of Sutterby Fluid across Shrinking/Stretching Surface. *Nanomaterials* **2022**, *12*, 599.
https://doi.org/10.3390/nano12040599

**AMA Style**

Abdal S, Siddique I, Afzal S, Sharifi S, Salimi M, Ahmadian A.
An Analysis for Variable Physical Properties Involved in the Nano-Biofilm Transportation of Sutterby Fluid across Shrinking/Stretching Surface. *Nanomaterials*. 2022; 12(4):599.
https://doi.org/10.3390/nano12040599

**Chicago/Turabian Style**

Abdal, Sohaib, Imran Siddique, Saima Afzal, Somayeh Sharifi, Mehdi Salimi, and Ali Ahmadian.
2022. "An Analysis for Variable Physical Properties Involved in the Nano-Biofilm Transportation of Sutterby Fluid across Shrinking/Stretching Surface" *Nanomaterials* 12, no. 4: 599.
https://doi.org/10.3390/nano12040599