# Computational Examination of Heat and Mass Transfer Induced by Ternary Nanofluid Flow across Convergent/Divergent Channels with Pollutant Concentration

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

${\upsilon}_{thnf}$ | : | Kinematic viscosity $\left({\upsilon}_{thnf}={\mu}_{thnf}/{\rho}_{thnf}\right)$ |

${D}_{f}$ | : | Coefficient of mass diffusivity |

${m}_{3}$ | : | External pollutant source variation parameter |

${L}^{*}$ | : | Temperature dependent heat source/sink |

$P$ | : | Pressure |

${G}^{*}$ | : | The space dependent heat source/sink |

${\left({C}_{p}\right)}_{thnf}$ | : | Specific heat |

${k}_{thnf}$ | : | Thermal conductivity |

${\left(\rho {C}_{p}\right)}_{thnf}$ | : | Specific heat capacitance |

${\mu}_{thnf}$ | : | Absolute viscosity |

${\rho}_{thnf}$ | : | Density |

## 3. Numerical Procedure

## 4. Result and Discussion

## 5. Conclusions

- For the convergent channel situation, the inclusion of ${\varphi}_{3}$ values will improve the velocity profile, but the divergent channel exhibits the opposite characteristic.
- The concentration profile in both channels will be improved by both ${\lambda}_{1}$ and ${\delta}_{2}$.
- In the presence of the Eckert number, the heat spreading is more concentrated in divergent channels than convergent channels.
- Convergent channels display a lower surface drag than divergent channels due to an increase in the solid volume percentage and Reynolds number.
- When an external pollutant source parameter is present, the rate of mass transfer increases.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbols | |

$P$ | Pressure |

${T}_{1}$ | Temperature |

$k$ | Thermal conductivity |

${S}_{1}$ | The external pollutant concentration |

${D}_{f}$ | Coefficient of mass diffusivity |

${C}_{1}$ | Concentration |

${q}^{\prime \prime \prime}$ | Non-uniform heat source/sink |

${G}^{*}$ | Space dependent heat source/sink |

${L}^{*}$ | Temperature dependent heat source/sink |

${C}_{P}$ | Specific heat |

$\mathrm{Pr}$ | Prandtl number |

$\mathrm{Re}$ | Reynolds number |

$S{c}_{1}$ | Schmidt number |

${m}_{3}$ | External pollutant source variation parameter |

$E{c}_{1}$ | Eckert number |

${u}_{\stackrel{\u2322}{r}}$ | Uniform velocity |

${C}_{f}$ | Skin friction |

$Sh$ | Sherwood number |

$Nu$ | Nusselt number |

Greek symbols | |

$\upsilon $ | Kinematic viscosity |

$\mu $ | Dynamic viscosity |

$\rho $ | Density |

${\delta}_{2}$ | Local pollutant external source parameter |

${\lambda}_{1}$ | External pollutant source variation parameter |

$\varphi $ | Solid volume fraction |

${\beta}^{*}\left(\eta \right)$ | Temperature profile |

${\phi}^{*}\left(\eta \right)$ | Concentration profile |

Subscripts | |

$f$ | Fluid |

$nf$ | Nanofluid |

$hnf$ | Hybrid nanofluid |

$thnf$ | Ternary hybrid nanofluid |

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**Figure 6.**(

**a**) Variation of ${C}_{f}$ on $\mathrm{Re}$ for variation in ${\varphi}_{3}$ (

**b**) variation of $Nu$ on $L*$ for variation in $G*$ (

**c**) variation of $Sh$ on $S{c}_{1}$ for variation in ${\lambda}_{1}$.

**Table 1.**The effective thermophysical characteristics of chosen nanoparticles and carrier fluid, are given by, (see [22]).

Properties | Unit | ${\mathit{H}}_{2}\mathit{O}$ | $\mathit{A}{\mathit{l}}_{2}{\mathit{O}}_{3}$ | $\mathit{A}\mathit{g}$ | $\mathit{C}\mathit{u}$ |
---|---|---|---|---|---|

$\rho $ | $\left({\mathrm{kg}/\mathrm{m}}^{3}\right)$ | 997.1 | 3970 | 10,500 | 8933 |

${C}_{p}$ | $\left({\mathrm{m}}^{2}{\mathrm{s}}^{-2}{\mathrm{K}}^{-1}\right)$ | 4179 | 765.0 | 235 | 385.0 |

$k$ | $\left({\mathrm{kgms}}^{-3}{\mathrm{K}}^{-1}\right)$ | 0.613 | 40 | 429 | 401 |

Sl. No | Name and Expression for the Constraint | Fixed Value |
---|---|---|

1 | Prandtl number $\left(\mathrm{Pr}=\frac{{\mu}_{f}{\left({C}_{p}\right)}_{f}}{{k}_{f}}\right)$ | 6.3 |

2 | Reynolds number $\left(\mathrm{Re}=\frac{{f}_{\mathrm{m}\text{}x}\gamma}{{\upsilon}_{f}}\right)$ | 5 |

3 | Schmidt number $\left(S{c}_{1}=\frac{{\upsilon}_{f}}{{D}_{f}}\right)$ | 0.8 |

4 | Eckert number $\left(E{c}_{1}=\frac{{\left(U*\right)}^{2}}{{\left({C}_{p}\right)}_{f}{T}_{{w}_{1}}}\right)$ | 3 |

5 | Local pollutant external source parameter $\left({\delta}_{2}=\frac{{H}_{1}l}{{C}_{{w}_{1}}U*}\right)$ | 0.1 |

6 | External pollutant source variation parameter $\left({\lambda}_{1}={C}_{{w}_{1}}{m}_{3}\right)$ | 0.1 |

7 | Internal heat absorption $\left({G}^{*}{L}^{*}0\right)$ | $\left(G*=L*=-0.1\right)$ |

8 | Internal heat generation $\left({G}^{*}{L}^{*}0\right)$ | $\left(G*=L*=0.1\right)$ |

Special cases: | ||

1 | Divergent channel scenario if $\gamma >0$ | |

2 | Convergent channel scenario if $\gamma <0$ |

**Table 3.**The verification of the study’s solutions using work by (see [22]) without the presence of nanoparticles.

Convergent Channel Case | ||||
---|---|---|---|---|

[22] | Current Work | |||

$h\left(\eta \right)$ | $h\left(\eta \right)$ | |||

$\eta $ | ADM | RK-4 | HAM | RKF-45 |

1 | $0$ | $0$ | $0$ | $0$ |

0.8 | $0.423183$ | $0.423183$ | $0.423183$ | $0.423187$ |

0.6 | $0.705698$ | $0.705698$ | $0.705698$ | $0.705701$ |

0.4 | $0.879028$ | $0.879028$ | $0.879028$ | $0.879031$ |

0.2 | $0.971234$ | $0.971234$ | $0.971234$ | $0.971235$ |

$0$ | $1.0$ | $1.0$ | $1.0$ | $1.000000$ |

Divergent Channel Case | ||||

$\eta $ | ADM | RK-4 | HAM | RKF-45 |

1 | $0$ | $0$ | $0$ | $0$ |

0.8 | $0.288378$ | $0.288378$ | $0.288378$ | $0.288381$ |

0.6 | $0.559036$ | $0.559036$ | $0.559036$ | $0.559039$ |

0.4 | $0.788205$ | $0.788205$ | $0.788205$ | $0.788207$ |

0.2 | $0.944324$ | $0.944324$ | $0.944324$ | $0.944326$ |

$0$ | $1.0$ | $1.0$ | $1.0$ | $1.000000$ |

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## Share and Cite

**MDPI and ACS Style**

K, V.; Sunitha, M.; Madhukesh, J.K.; Khan, U.; Zaib, A.; Sherif, E.-S.M.; Hassan, A.M.; Pop, I.
Computational Examination of Heat and Mass Transfer Induced by Ternary Nanofluid Flow across Convergent/Divergent Channels with Pollutant Concentration. *Water* **2023**, *15*, 2955.
https://doi.org/10.3390/w15162955

**AMA Style**

K V, Sunitha M, Madhukesh JK, Khan U, Zaib A, Sherif E-SM, Hassan AM, Pop I.
Computational Examination of Heat and Mass Transfer Induced by Ternary Nanofluid Flow across Convergent/Divergent Channels with Pollutant Concentration. *Water*. 2023; 15(16):2955.
https://doi.org/10.3390/w15162955

**Chicago/Turabian Style**

K, Vinutha, M Sunitha, J. K. Madhukesh, Umair Khan, Aurang Zaib, El-Sayed M. Sherif, Ahmed M. Hassan, and Ioan Pop.
2023. "Computational Examination of Heat and Mass Transfer Induced by Ternary Nanofluid Flow across Convergent/Divergent Channels with Pollutant Concentration" *Water* 15, no. 16: 2955.
https://doi.org/10.3390/w15162955