# Entropy Minimization for Generalized Newtonian Fluid Flow between Converging and Diverging Channels

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## Abstract

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## 1. Introduction

## 2. Description and Formulation of the Problem

#### 2.1. Physical Configuration

#### 2.2. Similarity Solutions

#### 2.3. Entropy Generation within the System

#### 2.4. Irreversibility Distribution Ratio

#### 2.5. Curiosity in Physical Measurements

## 3. Numerical Scheme for the Solution

## 4. Results and Discussion

#### 4.1. Consequences of the Reynolds Number

#### 4.2. Consequences of the Weissenberg Number

#### 4.3. Consequences of Indexed Power

#### 4.4. Consequences of Magnetic Parameter

#### 4.5. Effect of the Eckert Number

#### 4.6. Effect of the Brinkman Number on Entropy Generation Rate and the Bejan Number

#### 4.7. Influence of Various Physical Parameters on the Bejan Number

#### 4.8. Influence of Physical Parameters on Skin-Drag Force and Heat-Transfer Rate

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Impacts of the Reynolds number $Re$ on (

**a**) velocity $f\left(\eta \right)$, (

**b**) temperature $\beta \left(\eta \right)$, (

**c**) concentration $\gamma \left(\eta \right)$, and (

**d**) entropy production $Ns$.

**Figure 3.**Impacts of the Weissenberg number $We$ on (

**a**) velocity f(η), (

**b**) temperature β(η), (

**c**) concentration γ(η), and (

**d**) entropy production $Ns$.

**Figure 4.**Impact of power-indexed parameter $n$ on (

**a**) velocity $f\left(\eta \right)$, (

**b**) temperature $\beta \left(\eta \right)$, (

**c**) concentration $\gamma \left(\eta \right)$, and (

**d**) entropy production $Ns$.

**Figure 5.**Impact of magnetic parameter $M$ on (

**a**) velocity $f\left(\eta \right)$, (

**b**) temperature $\beta \left(\eta \right)$, (

**c**) concentration $\gamma \left(\eta \right)$, and (

**d**) entropy production $Ns$.

**Figure 6.**Impact of the Eckert number $Ec$ on (

**a**) velocity $f\left(\eta \right)$, (

**b**) temperature $\beta \left(\eta \right)$, (

**c**) concentration $\gamma \left(\eta \right)$, and (

**d**) entropy production $Ns$.

**Figure 7.**Impact of the Brinkmann number $Br$ on (

**a**) entropy production $Ns$ and (

**b**) the Bejan profile $Be$.

**Figure 8.**(

**a**) Impacts of the Weissenberg number $We$, (

**b**) Reynolds number $Re$, (

**c**) power-indexed parameter $n$, (

**d**) magnetic parameter $M$, and (

**e**) Eckert number $Ec$ on the Bejan profile $Be$.

**Figure 9.**Impact of (

**a**) the Weissenberg number $We$ and (

**b**) the power-indexed parameter $n$ on skin friction ${C}_{f}$.

**Figure 10.**Impact of (

**a**) the Weissenberg number $We$ and (

**b**) the power-indexed parameter $n$ on the Nusselt number $Nu$.

**Table 1.**Comparison of numerical values of f(η) against multiples values of an opening angle ψ = 3^0, when Re = 4, We = 0, Γ = 0 or n = 1, M = 0.

$\mathit{\eta}$ | Al-Saif and Jasim [34] | Ghagha et al. [35] | Present Study |
---|---|---|---|

0.0 | 1 | 1 | 1 |

0.1 | 0.98901 | 0.98953 | 0.98953 |

0.3 | 0.95626 | 0.95819 | 0.95991 |

0.4 | 0.90917 | 0.90619 | 0.90998 |

0.5 | 0.84124 | 0.83386 | 0.84001 |

0.6 | 0.75123 | 0.741635 | 0.74887 |

0.7 | 0.64012 | 0.630019 | 0.63981 |

0.8 | 0.51324 | 0.499554 | 0.51018 |

0.9 | 0.36129 | 0.350769 | 0.35918 |

1.0 | 0.19913 | 0.184134 | 0.19023 |

0.3 | 0 | 0 | 0 |

**Table 2.**Comparison of numerical values of skin friction ${f}_{\eta}\left(1\right)$ against multiples values of the parameters and an opening angle $\psi ,$ when $Re=50,We=1.0,\mathsf{\Gamma}=0\mathrm{or}n=1$.

$\mathit{\alpha}$ | ${\mathit{f}}_{\mathit{\eta}}\left(1\right)$ | ||
---|---|---|---|

Alam et al. [21] | Rehman et al. [36] | Present Study | |

$-{5}^{0}$ | $-$5.13092 | −5.13092 | $-$5.13094 |

$-$4.65216 | −4.65215 | $-$4.65216 | |

$-$2.83395 | −2.83391 | $-$2.83393 | |

0 | 0 | 0 | |

${5}^{0}$ | 3.66971 | 3.66971 | 3.66963 |

$-$3.50810 | −3.50810 | $-$3.50831 | |

$-$1.10933 | −1.10932 | $-$1.10941 | |

0 | 0 | 0 |

**Table 3.**Comparison of numerical values of −1/α β_η (1) against multiples values of an opening angle ψ, when Re = 50, Pr = 3.0, Nb = 0.4, Nt = 0.2, Γ = 0, or n = 1, M = 0.

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**MDPI and ACS Style**

Rehman, S.; Hashim; Nasr, A.; Eldin, S.M.; Malik, M.Y.
Entropy Minimization for Generalized Newtonian Fluid Flow between Converging and Diverging Channels. *Micromachines* **2022**, *13*, 1755.
https://doi.org/10.3390/mi13101755

**AMA Style**

Rehman S, Hashim, Nasr A, Eldin SM, Malik MY.
Entropy Minimization for Generalized Newtonian Fluid Flow between Converging and Diverging Channels. *Micromachines*. 2022; 13(10):1755.
https://doi.org/10.3390/mi13101755

**Chicago/Turabian Style**

Rehman, Sohail, Hashim, Abdelaziz Nasr, Sayed M. Eldin, and Muhammad Y. Malik.
2022. "Entropy Minimization for Generalized Newtonian Fluid Flow between Converging and Diverging Channels" *Micromachines* 13, no. 10: 1755.
https://doi.org/10.3390/mi13101755