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Duan–Rach Approach to Study Al_{2}O_{3}-Ethylene Glycol C_{2}H_{6}O_{2} Nanofluid Flow Based upon KKL Model

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## Abstract

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_{2}O

_{3}) as nanoparticles. Because of its enhanced thermophysical properties, Nanofluids are used in many application areas of mechanical and engineering in the form of nanofluid coolants such as electronics and vehicle cooling, transformer, and computer cooling. Depending on the heating and cooling systems, it is also used as an anti-freezing agent, which lowers the freezing point but enhances boiling point and temperature coolant. After using appropriate similarity transformation, the present Koo–Kleinstreuer–Li model for solving the boundary value problem (BVP) is tackled analytically. A comparison is made with a purely analytical approach by a modified version of the semi-analytical Adomian Decomposition Method (ADM), which is introduced by Duan and Rach (Duan–Rach Approach) and shooting technique. Analytical and graphical treatment of the flow regime is carried out, and the behavior of the leading parameters on the velocity, temperature, concentration profile with the behavior of physical quantities i.e., skin friction coefficient, local Nusselt number, and local Sherwood number are illustrated. This study confirms that, due to extraction in width the flow moves away from the lower plate whereas it moves towards near the upper plate and a rapid decrease in temperature is marked when alumina–EG nanofluids are taken into account.

## 1. Introduction

_{2}OH

_{2}). So, in the production process of polyester fibers, EG can be used in the raw material. It can also be used as an anti-freezing agent to reduce the freezing point of a water-based nanofluid, which, in turn, enhances the boiling point and coolant temperature of the nanofluid, automobiles and liquid-cooled computers. It has no odor, no color, but is a sweet-tasting viscous liquid. Because of these above properties, it is applicable in internal combustion engines and solar water heaters and HVAC chillers. This is also used in commercial purposes both in the pure concentrate and the diluted solution form, depending on the context.

_{2}O

_{3}–Ethylene glycol nanofluid with the KKL model. A comparison is also presented between the Duan–Rach approach and numerical scheme. The effects of the magnetic field and porosity are also contemplated. The impact of various significant outcomes is discussed with tables and graphs. The inclusion of buoyancy parameters (both thermal and solutal) in the momentum equation, heat source, and radiation absorption parameters in the energy equation boost the physical phenomena of ethylene-glycol based Al

_{2}O

_{3}nanofluid.

## 2. Mathematical Formulation

## 3. Formulation of Physical Quantities

## 4. Formulation of Duan–Rach Approach

## 5. Implementation of the Method

## 6. Discussion

#### 6.1. Validation and Comparative Study of Various Profiles

#### 6.2. Variation of the Velocity Distribution

#### 6.3. Variation of Temperature Distributions

#### 6.4. Variation of Temperature Distributions

#### 6.5. Variation of Engineering Coefficients

## 7. Conclusions

_{2}O

_{3}) as nanoparticles with the KKL model using the Duan–Rach approach. A comparison is made with a purely analytical approach by a modified version of the semi-analytical Adomian Decomposition Method (ADM), which is introduced by Duan and Rach (Duan–Rach Approach) and fourth-order Runge–Kutta method with shooting technique. Analytical and graphical treatment of the entire flow regime is carried out, and the effects of the pertinent parameters on the velocity, temperature, concentration profile with the behavior of physical quantities such as skin friction coefficient, local Nusselt number, and local Sherwood number are illustrated in the present work. The significant outcomes are explained below:

- It is noticed that a rapid decrease in temperature profile is observed when Alumina–EG nanofluids are contemplated.
- Porosity effects enhance the flow, whereas the magnetic field opposes the flow.
- Increase in buoyancy forces from heating to cooling, the enhancing nature of the profiles exhibits to lower down the thickness at the lower plate in the half region. In contrast, from the point of contact at the central area, the opposite effect is rendered near the upper plate region.
- It is found that increasing volume fraction retards the thickness at the lower plate; however, the expansion at the upper plate is more.
- It is seen that the profile overshoots with increasing suction Reynolds number i.e., for positive values and reverse impact is rendered for the injection where it indicates the negative Reynolds number.
- It is concluded that the growing volume fraction will cause a development in the heat transfer properties.
- Prandtl number shows converse behavior in the upper region as compared with the low area.
- Reynolds number and Schmidt number show converse behavior on the concentration profile.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${A}_{1}$ | Permeability |

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

$C$ | concentration fluid |

${c}_{p}$ | specific heat |

$D$ | solutal diffusivity |

$f$ | velocity profiles |

$F$ | self-similar velocity |

$g$ | gravity |

$n$ | shape factor for nanoparticle |

$M$ | magnetic parameter |

$k$ | thermal conductivity of water |

$Kp$ | porosity parameter |

$(x,y)$ | horizontal and vertical co-ordinate axes |

$p$ | pressure |

${\mathrm{Pr}}_{f}$ | Prandtl number |

${Q}_{1}$ | heat source parameter |

${Q}_{2}$ | coefficient of radiation absorption |

$Ra$ | non-dimensional radiation absorption |

$\mathrm{Re}$ | Reynolds number |

$S$ | non-dimensional source parameter |

$Sc$ | Schmidt number |

$(u,v)$ | Horizontal and vertical velocity |

$T$ | Temperature |

Greek Symbols | |

$\varphi $ | nanoparticle volume fraction |

$\psi $ | sphericity of the nanoparticles |

$\upsilon $ | kinematic viscosity |

$\rho $ | fluid density |

$\sigma $ | electrical conductivity |

${\beta}_{T}$ | volumetric coefficient of expansion for heat transfer |

${\beta}_{C}$ | volumetric coefficient of expansion for mass transfer |

$\mu $ | dynamic viscosity |

$\alpha $ | wall expansion/contraction parameter |

${\alpha}_{nf}$ | thermal diffusivity |

$\eta $ | scaled boundary layer coordinate |

$\theta $ | dimensionless temperature |

$\chi $ | dimensionless concentration |

${\lambda}_{1}$ | thermal buoyancy parameter |

${\lambda}_{2}$ | solutal buoyancy parameter |

$(\rho {c}_{p})$ | heat capacity |

Subscripts | |

nf | nanofluid |

f | base fluid |

p | solid particle |

l | lower walls |

u | upper walls |

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**Figure 8.**Consequences of velocity profile with (

**a**) $M$ and $Kp$ (

**b**) ${\lambda}_{1}$ and ${\lambda}_{2}$.

**Figure 15.**Consequences of skin friction coefficient with $(a)\hspace{0.17em}\mathrm{Re}\hspace{0.17em}\hspace{0.17em}(b)\hspace{0.17em}\alpha \hspace{0.17em}\hspace{0.17em}(c)\hspace{0.17em}{\lambda}_{1}\hspace{0.17em}\hspace{0.17em}(d)\hspace{0.17em}{\lambda}_{2}$.

**Figure 16.**Consequences of Nusselt number with $(a)\hspace{0.17em}\mathrm{Re}\hspace{0.17em}\hspace{0.17em}(b)\hspace{0.17em}\alpha \hspace{0.17em}\hspace{0.17em}(c)\hspace{0.17em}S\hspace{0.17em}\hspace{0.17em}(d)\hspace{0.17em}Ra$.

**Figure 17.**Consequences of Sherwood number with $(a)\hspace{0.17em}\mathrm{Re}\hspace{0.17em}\hspace{0.17em}(b)\hspace{0.17em}\alpha \hspace{0.17em}\hspace{0.17em}(c)\hspace{0.17em}Sc$.

**Table 1.**Thermophysical properties of alumina ${(\mathrm{Al}}_{2}{\mathrm{O}}_{3})$ –EG ${(\mathrm{C}}_{2}{\mathrm{H}}_{6}{\mathrm{O}}_{2})$.

$\mathit{\rho}$${(\mathbf{Kg}/\mathbf{m}}^{3})$ | ${\mathit{c}}_{\mathit{p}}$$(\mathbf{J}/\mathbf{Kg}/\mathbf{K})$ | $\mathit{k}$$(\mathbf{W}/\mathbf{m}/\mathbf{K})$ | |
---|---|---|---|

Ethylene-Glycol | 1114 | 2415 | 0.252 |

Alumina | 3970 | 765 | 40 |

**Table 2.**Coefficient values of alumina–ethylene–glycol-based nanofluids [34].

Coefficient Values | Alumina-Ethylene-Glycol |
---|---|

${a}_{1}$ | 52.813488759 |

${a}_{2}$ | 6.115637295 |

${a}_{3}$ | 0.6955745084 |

${a}_{4}$ | 4.17455552786 × 10^{−2} |

${a}_{5}$ | 0.176919300241 |

${a}_{6}$ | −298.19819084 |

${a}_{7}$ | −34.532716906 |

${a}_{8}$ | −3.9225289283 |

${a}_{9}$ | −0.2354329626 |

${a}_{10}$ | −0.999063481 |

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**MDPI and ACS Style**

Pattnaik, P.K.; Mishra, S.; Bhatti, M.M.
Duan–Rach Approach to Study Al_{2}O_{3}-Ethylene Glycol C_{2}H_{6}O_{2} Nanofluid Flow Based upon KKL Model. *Inventions* **2020**, *5*, 45.
https://doi.org/10.3390/inventions5030045

**AMA Style**

Pattnaik PK, Mishra S, Bhatti MM.
Duan–Rach Approach to Study Al_{2}O_{3}-Ethylene Glycol C_{2}H_{6}O_{2} Nanofluid Flow Based upon KKL Model. *Inventions*. 2020; 5(3):45.
https://doi.org/10.3390/inventions5030045

**Chicago/Turabian Style**

Pattnaik, Pradyumna Kumar, Satyaranjan Mishra, and Muhammad Mubashir Bhatti.
2020. "Duan–Rach Approach to Study Al_{2}O_{3}-Ethylene Glycol C_{2}H_{6}O_{2} Nanofluid Flow Based upon KKL Model" *Inventions* 5, no. 3: 45.
https://doi.org/10.3390/inventions5030045