#
Entropy Generation of Cu–Al_{2}O_{3}/Water Flow with Convective Boundary Conditions through a Porous Stretching Sheet with Slip Effect, Joule Heating and Chemical Reaction

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## Abstract

**:**

_{2}CO

_{3}/water nanofluidic flows on a porous stretched sheet of velocity slip, convective boundary conditions, Joule heating, and chemical reactions using an adapted Tiwari–Das model. Nonlinear fundamental equations such as continuity, momentum, energy, and concentration are transmuted into a non-dimensional ordinary nonlinear differential equation by similarity transformations. Numerical calculations are performed using HAM and the outcomes are traced on graphs such as velocity, temperature, and concentration. Temperature and concentration profiles are elevated as porosity is increased, whereas velocity is decreased. The Biot number increases the temperature profile. The rate of entropy is enhanced as the Brinkman number is raised. A decrease in the velocity is seen as the slip increases.

## 1. Introduction

## 2. Mathematical Formulation

**Continuity Equation**$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0,$$**Momentum Equation**$$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=\frac{{\mu}_{hnf}}{{\rho}_{hnf}}\left(\frac{{\partial}^{2}u}{\partial {y}^{2}}-\frac{u}{{k}^{*}}\right)-\frac{{\sigma}_{hnf}}{{\rho}_{hnf}}{B}_{o}^{2}u,$$**Temperature Equation**$$u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\frac{{k}_{hnf}}{{\left(\rho {C}_{p}\right)}_{hnf}}\left(\frac{{\partial}^{2}T}{\partial {y}^{2}}\right)+\frac{{\sigma}_{hnf}{B}_{o}^{2}}{{\left(\rho {C}_{p}\right)}_{hnf}}{u}^{2},$$**Concentration Equation**$$u\frac{\partial C}{\partial x}+v\frac{\partial C}{\partial y}={D}_{M}\left(\frac{{\partial}^{2}C}{\partial {y}^{2}}\right)-{k}_{r}\left(C-{C}_{\infty}\right),$$$$\left.\begin{array}{c}\mathrm{at}y=0:u={u}_{w}+{u}_{slip}=ax+L\frac{\partial u}{\partial y},v={v}_{w}=0,{k}_{hnf}\frac{\partial T}{\partial y}={h}_{s}\left(T-{T}_{w}\right),C={C}_{w}\\ asy\to \infty :u\to 0,T\to {T}_{\infty},C\to {C}_{\infty}\end{array}\right\}.$$

## 3. Entropy Generation

## 4. Numerical Evaluation Using HAM

- Choosing initial guesses as:

## 5. Results and Discussion

## 6. Conclusions

- An increase in porosity results in a decrease in velocity, whereas, the thermal and concentration profiles are increased. As a result, the rates of both heat and mass transfer increase.
- An increase in the slip parameter decreases the rate of flow of a hybrid nanofluid. The concentration increases as the slip increases.
- An increase in the magnetic parameter increases temperature and concentration but decreases velocity.
- As the Biot number increases, the temperature also increases.
- An increase in the Brinkman number improves the viscous force, which amplifies the collision of fluid particles. Hence, the rate of entropy generation is enhanced.
- When the magnetic parameter is increased there is a decrease in the Bejan number.
- An increase in the Brinkman number increases the rate of entropy generation. Thus, entropy is enhanced due to viscous dissipation.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

u,v | Velocity components taken along x- and y-axes (m/s) |

u_{w} | Velocity at the surface (m/s) |

T_{w} | Surface temperature (K) |

C_{w} | Surface concentration |

T_{∞} | Ambient temperature |

C_{∞} | Ambient concentration |

M | Magnetic parameter |

E | Porosity parameter |

D | Mass diffusivity (m^{2}s^{−1}) |

q_{r} | Heat flux (kg·m^{2}s^{−3}) |

C_{p} | Specific heat (J/kg) |

T | Temperature of the fluid (K) |

C | Concentration of the fluid |

k | Thermal conductivity |

Pr | Prandtl number |

Ec | Eckert number |

Sc | Schmidt number |

C_{f} | Skin friction coefficient |

Nu | Nusselt number |

${K}_{r}$ | Chemical reaction parameter |

B | Diffusion parameter |

$A$ | Temperature difference parameter |

${A}^{\prime}$ | Concentration difference parameter |

$Br$ | Brinkman number |

Be | Bejan number |

Greek symbols | |

${\nu}_{f}$ | Kinematic viscosity of the fluid (m^{2}/s) |

${\rho}_{f}$ | Density of the fluid (kgm^{−3}) |

${\mu}_{f}$ | Dynamic viscosity of the fluid (m^{2}/s) |

${\alpha}_{f}$ | Thermal diffusivity (m^{2}/s) |

$\sigma $ | Electrical conductivity (s/m) |

$\mathsf{\Omega}$ | Biot number |

$\mathsf{\Lambda}$ | Slip parameter |

Subscripts | |

∞ | Ambient |

f | base fluid |

nf | Nanofluid |

hnf | hybrid nanofluid |

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Property | Water | Copper | Alumina |
---|---|---|---|

Specific Heat (J/kgK) | 4180 | 385 | 765 |

Density (kg/m^{3}) | 997.0 | 8933 | 3970 |

Thermal Conductivity (W/mK) | 0.6071 | 400 | 40 |

Electrical Conductivity (s/m) | 5.5 × 10^{−6} | 59.6 × 10^{6} | 35 × 10^{6} |

Thermophysical Property | Nanofluid | Hybrid Nanofluid |
---|---|---|

Density | ${\rho}_{nf}$= $\left[\left(1-{\varphi}_{1}\right){\rho}_{f}+{\varphi}_{1}{\rho}_{n1}\right]$ | ${\rho}_{hnf}$= $\left(1-{\varphi}_{2}\right)\left[\left(1-{\varphi}_{1}\right){\rho}_{f}+{\varphi}_{1}{\rho}_{n1}\right]+{\varphi}_{2}{\rho}_{n2}$ |

Dynamic Viscosity | ${\mu}_{nf}=\frac{{\mu}_{f}}{{\left(1-{\varphi}_{1}\right)}^{2.5}}$ | ${\mu}_{hnf}=\frac{{\mu}_{f}}{{\left(1-{\varphi}_{1}\right)}^{2.5}{\left(1-{\varphi}_{2}\right)}^{2.5}}$ |

Thermal Conductivity | k_{nf} = $\frac{{k}_{n1}+2{k}_{f}-2{\varphi}_{1}\left({k}_{f}-{k}_{n1}\right)}{{k}_{n1}+2{k}_{f}+{\varphi}_{1}\left({k}_{f}-{k}_{n1}\right)}$k_{f} | k_{hnf} = $\frac{{k}_{n2}+2{k}_{nf}-2{\varphi}_{2}\left({k}_{nf}-{k}_{n2}\right)}{{k}_{n2}+2{k}_{nf}+{\varphi}_{2}\left({k}_{nf}-{k}_{n2}\right)}$k_{nf}, where k_{nf} = $\frac{{k}_{n1}+2{k}_{f}-2{\varphi}_{1}\left({k}_{f}-{k}_{n1}\right)}{{k}_{n1}+2{k}_{f}+{\varphi}_{1}\left({k}_{f}-{k}_{n1}\right)}$k_{f} |

Heat Capacity | ${\left(\rho {C}_{p}\right)}_{nf}=\left(1-{\varphi}_{1}\right){\left(\rho {C}_{p}\right)}_{f}+{\varphi}_{1}{\left(\rho {C}_{p}\right)}_{n1}$ | ${\left(\rho C\right)}_{hnf}=\left(1-{\varphi}_{2}\right)\left[\left(1-{\varphi}_{1}\right){\left(\rho {C}_{p}\right)}_{f}+{\varphi}_{1}{\left(\rho {C}_{p}\right)}_{n1}\right]{\varphi}_{2}{\left(\rho {C}_{p}\right)}_{n2}$ |

Electrical Conductivity | ${\sigma}_{nf}$=$\frac{{\sigma}_{n1}+2{\sigma}_{f}-2{\varphi}_{1}\left({\sigma}_{f}-{\sigma}_{n1}\right)}{{\sigma}_{n1}+2{\sigma}_{f}+{\varphi}_{1}\left({\sigma}_{f}-{\sigma}_{n1}\right)}$${\sigma}_{f}$ | ${\sigma}_{hnf}$=$\frac{{\sigma}_{n2}+2{\sigma}_{f}-2{\varphi}_{2}\left({\sigma}_{f}-{\sigma}_{n2}\right)}{{\sigma}_{n2}+2{\sigma}_{f}+{\varphi}_{2}\left({\sigma}_{f}-{\sigma}_{n2}\right)}$${\sigma}_{nf}$ |

**Table 3.**Convergences of series ${\varphi}_{1}=0.01$, ${\varphi}_{2}=0.01$, $\mathsf{\Lambda}=0.2$, Bi = 0.1, Pr = 3.97, Ec = 0.3, M = 0.5, E = 0.3, Sc = 0.6, ${K}_{r}=0.2$.

Order | $-{\mathit{f}}^{\prime \prime}(0)$ | $-{\mathit{\theta}}^{\prime}(0)$ | $-{\mathit{\varphi}}^{\prime}(0)$ |
---|---|---|---|

1 | 0.9431 | 0.0821 | 0.7028 |

5 | 1.0069 | 0.0773 | 0.5170 |

10 | 1.0089 | 0.0770 | 0.4924 |

15 | 1.0089 | 0.0771 | 0.4873 |

20 | 1.0089 | 0.0771 | 0.4858 |

25 | 1.0089 | 0.0771 | 0.4853 |

30 | 1.0089 | 0.0771 | 0.4851 |

35 | 1.0089 | 0.0771 | 0.4850 |

40 | 1.0089 | 0.0771 | 0.4850 |

45 | 1.0089 | 0.0771 | 0.4850 |

50 | 1.0089 | 0.0771 | 0.4850 |

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**MDPI and ACS Style**

Joyce, M.I.; Kandasamy, J.; Sivanandam, S.
Entropy Generation of Cu–Al_{2}O_{3}/Water Flow with Convective Boundary Conditions through a Porous Stretching Sheet with Slip Effect, Joule Heating and Chemical Reaction. *Math. Comput. Appl.* **2023**, *28*, 18.
https://doi.org/10.3390/mca28010018

**AMA Style**

Joyce MI, Kandasamy J, Sivanandam S.
Entropy Generation of Cu–Al_{2}O_{3}/Water Flow with Convective Boundary Conditions through a Porous Stretching Sheet with Slip Effect, Joule Heating and Chemical Reaction. *Mathematical and Computational Applications*. 2023; 28(1):18.
https://doi.org/10.3390/mca28010018

**Chicago/Turabian Style**

Joyce, Maria Immaculate, Jagan Kandasamy, and Sivasankaran Sivanandam.
2023. "Entropy Generation of Cu–Al_{2}O_{3}/Water Flow with Convective Boundary Conditions through a Porous Stretching Sheet with Slip Effect, Joule Heating and Chemical Reaction" *Mathematical and Computational Applications* 28, no. 1: 18.
https://doi.org/10.3390/mca28010018