# Quasi-Linearization Analysis for Entropy Generation in MHD Mixed-Convection Flow of Casson Nanofluid over Nonlinear Stretching Sheet with Arrhenius Activation Energy

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## Abstract

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## 1. Introduction

_{80}Ta

_{20}/Cu

_{20}Ta

_{80}) under the simultaneous effects of surface interface and layer thickness. The impacts of surface layer, loading velocity and indenter size on nanolaminates mechanism were also justified in this study. Ekiciler et al. [10] examined the effect of different shapes of Al

_{2}O

_{3}nanoparticles in the flow of water through a triangular duct with various volume fractions. Yadav et al. [11] conferred an inclusive review study on the nanoparticles of Carbon nanotubes (CNTs). They determined the physical and chemical stability of CNTs and claimed that Carbon nanotubes had better thermal properties and high tensile strength as compared to other nanomaterials. Arslan and Ekiciler [12] presented a numerical solution of the square duct flow problem of SiO

_{2}/water nanofluid with 1.0% to 4.0% different volume fractions of nanoparticles. Ahmad et al. [13,14] numerically explored nanofluid flows under different circumstances using successive over-relaxation (SOR) technique.

## 2. Formulation of Flow Model

## 3. Entropy Generation Analysis

## 4. Solution Procedure

## 5. Results and Discussion

#### 5.1. Velocity Profiles

#### 5.2. Thermal Profiles

#### 5.3. Concentration Profiles

#### 5.4. Entropy Generation Profiles

## 6. Physical Quantities of Engineering Interest

## 7. Conclusions

- Growing values of $M$, $\beta $ and $\delta $ cause a significant decline in the velocity while $N$ exhibits an opposite trend.
- Thermal profiles fall off with enhancing values of $\beta $ and $\mathrm{Pr}$, whereas increasing values of $M$, $\lambda $, $Ec$, ${R}_{d}$, ${N}_{t}$ and ${N}_{b}$ improve thermal profiles significantly.
- Improving the values of $\lambda $, ${N}_{t}$, $B{i}_{2}$ and $E$ upsurge the concentration profiles; however, the parameters $N$, ${N}_{b}$, $Le$ and ${k}_{1}$ cause a remarkable reduction in the concentration distributions.
- The linear stretching dominates the nonlinear stretching for all controlling parameters of the concentration profiles.
- Entropy generation is an incremental function of the parameters $Br$, $B{i}_{1}$, $B{i}_{2}$, $Rd$, ${N}_{t}$ and ${N}_{b}$ while a declining function of parameters $\beta $ and $\mathrm{Pr}$.
- Implications of Brownian motion and thermophoresis are to minimize the entropy generation near the surface of the stretching sheet while maximizing the sheet.
- Entropy generation against nonlinear stretching leads as compared to the linear stretching for all pertinent parameters of the MHD flow of Casson nanofluid.
- Nusselt number and Sherwood number decrease significantly with growing values of activation energy in the case of nonlinear stretching.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$g$ | acceleration due to gravity ($m{s}^{-2}$) | $\mathrm{Re}$ | Reynolds number |

${T}_{f}$ | temperature of fluid ($K$) | $Le$ | Lewis number |

${C}_{p}$ | specific heat at constant pressure ($J\text{}mo{l}^{-1}{\text{}\mathrm{k}}^{-1}$) | $\mathrm{Pr}$ | Prandtl number |

${B}_{0}$ | magnetic field strength ($kg\text{}{s}^{-2}\text{}{A}^{-1}$) | $Ec$ | Eckert number |

$k$ | thermal conductivity ($W\text{}{m}^{-1}{\text{}\mathrm{k}}^{-1}$) | $Gr$ | Grashof number |

${D}^{\ast}$ | molecular diffusivity (${m}^{2}\text{}{s}^{-1}$) | $Br$ | Brinkman number |

$C$ | fluid concentration ($mol\text{}{m}^{-3}$) | ||

$B{i}_{1}$ | thermal Biot number | Greek Symbols | |

$B{i}_{2}$ | solutalBiot number | $\mu $ | fluid viscosity ($\mathrm{kg}\text{}{m}^{-1}{s}^{-1}$) |

${q}_{r}$ | radiative heat flux ($W{m}^{-2}$) | $\rho $ | fluid density ($\mathrm{kg}\text{}{m}^{-3}$) |

$M$ | magnetic parameter | $\beta $ | Casson parameter |

$Q$ | heat generation coefficient ($W$) | $\sigma $ | electrical conductivity ($S{m}^{-1}$) |

${E}_{a}$ | activation energy factor ($J\xb7mo{l}^{-1}$) | $\psi $ | stream function (${m}^{2}\text{}{s}^{-1}$) |

${N}_{t}$ | thermophoresis parameter | $\theta $ | dimensionless temperature |

${N}_{b}$ | Brownian motion parameter | $\lambda $ | mixed convection parameter |

${k}_{r}$ | rate of reaction (${s}^{-1}$) | $\varphi $ | dimensionless concentration |

$m$ | index parameter | $\eta $ | similarity variable |

$u,v$ | components of velocity ($m{s}^{-1}$) | $\gamma $ | dimensionless reaction rate |

$x,y$ | Cartesian coordinates along the stretching sheet, respectively ($m$) | $\epsilon $ | heat generation parameter |

$\delta $ | slip parameter |

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**Figure 3.**Comparison of numerical scheme adopted in the present study over nonlinear stretching sheet with the classical analytical results.

**Table 1.**Numerical values of skin friction coefficient ${C}_{fx}$, Nusselt number $N{u}_{x}$ and Sherwood number $S{h}_{x}$ for fixed values of $Le=1$, $\lambda =5$, $N=0.5$, $\epsilon =0.5$, ${N}_{t}=0.1$, ${N}_{b}=0.2$, ${k}_{1}=0.5$ and $E=5$.

$\mathbf{Pr}$ | $\mathit{M}$ | $\mathit{R}\mathit{d}$ | $\mathit{\beta}$ | $\mathit{E}\mathit{c}$ | $\mathit{\delta}$ | $\mathit{B}{\mathit{i}}_{1}$ | $\mathit{B}{\mathit{i}}_{2}$ | Linear Stretching (m = 1.0) | Nonlinear Stretching (m = 10) | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{C}{\mathit{f}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{\frac{1}{2}}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{\frac{-1}{2}}$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{\frac{-1}{2}}$ | $\mathit{C}{\mathit{f}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{\frac{1}{2}}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{\frac{-1}{2}}$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{\frac{-1}{2}}$ | ||||||||

7.38 | 5 | 1 | 0.3 | 0.1 | 0.5 | 3 | 2 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 |

6.50 | −2.0127 | 2.0215 | 0.6359 | −3.4058 | 6.3840 | 2.6385 | |||||||

7.38 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 | |||||||

8.00 | −2.2307 | 2.6936 | 0.3959 | −4.6823 | 7.9332 | 2.3796 | |||||||

2 | −1.2522 | 2.3938 | 0.5796 | −4.1249 | 8.6631 | 2.3350 | |||||||

3 | −1.5627 | 2.2560 | 0.5846 | −4.0620 | 7.9686 | 2.3984 | |||||||

5 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 | |||||||

1 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 | |||||||

2 | −1.8494 | 2.4286 | 0.6672 | −3.0301 | 8.2835 | 2.6653 | |||||||

3 | −1.6812 | 2.6148 | 0.7170 | −2.2438 | 9.3106 | 2.7633 | |||||||

0.3 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 | |||||||

0.4 | −1.7368 | 2.1591 | 0.5699 | −3.3178 | 6.9686 | 2.4786 | |||||||

0.5 | −1.5323 | 2.2149 | 0.5582 | −2.9036 | 7.2003 | 2.4498 | |||||||

0.1 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 | |||||||

0.2 | −1.9087 | 1.5545 | 0.6638 | −2.1724 | 3.1598 | 2.8806 | |||||||

0.3 | −1.7386 | 0.9730 | 0.7487 | 0.4449 | −2.3094 | 3.4186 | |||||||

1 | −1.4760 | 2.2093 | 0.5532 | −2.6048 | 6.9133 | 2.4712 | |||||||

2 | −0.9426 | 2.3126 | 0.5238 | −1.5491 | 7.1415 | 2.4314 | |||||||

3 | −0.6922 | 2.3526 | 0.5110 | −1.1027 | 7.2341 | 2.4147 | |||||||

2 | −2.1431 | 1.7297 | 0.6219 | −4.3114 | 5.0947 | 2.6169 | |||||||

4 | −1.9997 | 2.2955 | 0.5658 | −3.6926 | 7.7490 | 2.4516 | |||||||

6 | −1.9288 | 2.5737 | 0.5388 | −3.3207 | 9.3719 | 2.3522 | |||||||

1 | −2.1399 | 2.1294 | 0.4108 | −4.3904 | 6.8786 | 1.5333 | |||||||

3 | −2.0109 | 2.0375 | 0.6871 | −3.6629 | 6.4114 | 3.2167 | |||||||

5 | −1.9613 | 2.0017 | 0.7946 | −3.2843 | 6.1605 | 4.1298 |

**Table 2.**Numerical values of skin friction coefficient ${C}_{fx}$, Nusselt number $N{u}_{x}$ and Sherwood number $S{h}_{x}$ for fixed values of $\mathrm{Pr}=6450$, $\beta =0.3$, $\delta =0.5$, $M=5$, $B{i}_{1}=3$, $B{i}_{2}=2$, $Rd=1$ and $Ec=0.1$.

$\mathit{L}\mathit{e}$ | $\mathit{\lambda}$ | $\mathit{\epsilon}$ | ${\mathit{N}}_{\mathit{t}}$ | ${\mathit{N}}_{\mathit{b}}$ | $\mathit{N}$ | ${\mathit{k}}_{1}$ | $\mathit{E}$ | Linear Stretching (m = 1.0) | Nonlinear Stretching (m = 10) | ||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{C}{\mathit{f}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{\frac{1}{2}}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-\frac{1}{2}}$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-\frac{1}{2}}$ | $\mathit{C}{\mathit{f}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{\frac{1}{2}}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-\frac{1}{2}}$ | $\mathit{S}{\mathit{h}}_{\mathit{x}}{\mathbf{Re}}_{\mathit{x}}^{-\frac{1}{2}}$ | ||||||||

1 | 5 | 0.5 | 0.1 | 0.2 | 0.5 | 0.5 | 5 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 |

3 | −2.2710 | 2.1076 | 0.8862 | −4.9178 | 7.1047 | 3.1427 | |||||||

5 | −2.3261 | 2.1325 | 1.0030 | −5.1124 | 7.2187 | 3.3437 | |||||||

3 | −2.5725 | 2.1562 | 0.5556 | −6.3210 | 7.5287 | 2.3616 | |||||||

5 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 | |||||||

7 | −1.5259 | 1.9566 | 0.6226 | −1.2865 | 5.4147 | 2.7014 | |||||||

0.1 | −2.1670 | 2.5039 | 0.5244 | −4.7874 | 8.3329 | 2.3381 | |||||||

0.3 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 | |||||||

0.7 | −1.8019 | 1.1148 | 0.7273 | −1.9781 | 2.2189 | 2.9742 | |||||||

0.1 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.6033 | 2.5225 | |||||||

0.2 | −1.8532 | 1.9129 | 0.4212 | −3.0659 | 5.9165 | 2.2900 | |||||||

0.3 | −1.6681 | 1.7556 | 0.3002 | −2.2718 | 5.2373 | 2.1905 | |||||||

0.2 | −1.9023 | 2.1395 | 0.3580 | −3.2647 | 6.4004 | 2.1942 | |||||||

0.4 | −2.0569 | 2.0705 | 0.5880 | −3.9580 | 6.5409 | 2.5225 | |||||||

0.6 | −2.0902 | 1.9707 | 0.6694 | −4.1510 | 6.6033 | 2.6524 | |||||||

1 | −1.5483 | 1.9574 | 0.6233 | −2.1734 | 5.7906 | 2.6467 | |||||||

2 | −0.6506 | 1.6999 | 0.6878 | 0.6823 | 4.3577 | 2.8435 | |||||||

3 | 0.1247 | 1.4286 | 0.7456 | 2.9520 | 3.1026 | 3.0005 | |||||||

2 | −2.0762 | 2.0765 | 0.6203 | −3.9735 | 6.6115 | 2.5343 | |||||||

4 | −2.1000 | 2.0837 | 0.6601 | −3.9938 | 6.6327 | 2.5498 | |||||||

6 | −2.1217 | 2.0902 | 0.6966 | −4.0136 | 6.6425 | 2.5650 | |||||||

1 | −2.1188 | 2.0818 | 0.6730 | −4.0139 | 6.6320 | 2.5568 | |||||||

3 | −2.0714 | 2.0739 | 0.6094 | −3.9700 | 6.6096 | 2.5306 | |||||||

5 | −2.0569 | 2.0693 | 0.5880 | −3.9580 | 6.6033 | 2.5225 |

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**MDPI and ACS Style**

Ali, K.; Faridi, A.A.; Ahmad, S.; Jamshed, W.; Hussain, S.M.; Tag-Eldin, E.S.M.
Quasi-Linearization Analysis for Entropy Generation in MHD Mixed-Convection Flow of Casson Nanofluid over Nonlinear Stretching Sheet with Arrhenius Activation Energy. *Symmetry* **2022**, *14*, 1940.
https://doi.org/10.3390/sym14091940

**AMA Style**

Ali K, Faridi AA, Ahmad S, Jamshed W, Hussain SM, Tag-Eldin ESM.
Quasi-Linearization Analysis for Entropy Generation in MHD Mixed-Convection Flow of Casson Nanofluid over Nonlinear Stretching Sheet with Arrhenius Activation Energy. *Symmetry*. 2022; 14(9):1940.
https://doi.org/10.3390/sym14091940

**Chicago/Turabian Style**

Ali, Kashif, Aftab Ahmed Faridi, Sohail Ahmad, Wasim Jamshed, Syed M. Hussain, and El Sayed M. Tag-Eldin.
2022. "Quasi-Linearization Analysis for Entropy Generation in MHD Mixed-Convection Flow of Casson Nanofluid over Nonlinear Stretching Sheet with Arrhenius Activation Energy" *Symmetry* 14, no. 9: 1940.
https://doi.org/10.3390/sym14091940