# Modified MHD Radiative Mixed Convective Nanofluid Flow Model with Consideration of the Impact of Freezing Temperature and Molecular Diameter

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## Abstract

**:**

_{2}O

_{3}+ H

_{2}O nanofluid at 310 K and the freezing temperature of 273.15 K. Firstly, the model was reduced into a coupled set of ordinary differential equations using similarity transformations. The impact of the freezing temperature and the molecular diameter were incorporated in the energy equation. Then, the Runge–Kutta scheme, along with the shooting technique, was adopted for the mathematical computations and code was written in Mathematica 10.0. Further, a comprehensive discussion of the flow characteristics is provided. The results for the dynamic viscosity, heat capacity and effective density of the nanoparticles were examined for various nanoparticle diameters and volume fractions.

## 1. Introduction

_{2}O

_{3}/H

_{2}O and CuO/H

_{2}O nanofluids. In 2011, Corcione [11] developed a model for Al

_{2}O

_{3}+ H

_{2}O nanofluids by incorporating the effects of freezing temperatures. By incorporating thermal conductivity models, the above researchers presented various models and described the heat transfer enhancement due to thermal conductivity. Some useful studies for nanofluids are described in [12,13,14,15,16].

_{2}O

_{3}+ H

_{2}O were used to study the characteristics of the flow and other effective thermophysical properties, such as effective density, heat capacity and thermal conductivity. The results for shear stress and local heat transfer are also described and discussed comprehensively. Finally, major findings of the study is presented.

## 2. Model Formulation

_{2}O

_{3}+ H

_{2}O nanofluid by taking into account the influence of a nonlinear, radiative heat flux and the imposed variable magnetic field over an arc geometry situated in the curvilinear frame r and s. Further, the r-axis was perpendicular, and the arc was placed in the direction of s. The velocity and magnetic field were functions of s and mathematically described as below:

_{2}O

_{3}+ H

_{2}O nanofluid flow, incorporating the phenomena of Lorentz forces, viscous dissipation and nonlinear radiative heat flux, is described by the following system [21]:

^{−1}). ${d}_{p}$ represents the molecular diameter which is calculated by the expression [24]:

## 3. Mathematical Analysis

_{2}O

_{3}+ H

_{2}O is a highly nonlinear fourth-order model defined at a semi-infinite domain. Therefore, the model was tackled numerically using the Runge-Kutta scheme [13,16,25], as the RK scheme is used for the first order initial value problem (IVP). First the following substitutions were made and the model was reduced into first order IVP.

## 4. Graphical Results and Discussion

_{2}O

_{3}. The values for the thermophysical characteristics were calculated at 310 K [23]. The results for the shear stress and heat transfer rate are elaborated using bar charts and are discussed comprehensively.

#### 4.1. Velocity and Temperature Distribution

_{2}O

_{3}+ H

_{2}O nanofluids is elaborated in Figure 3a. It was shown that the applied magnetic field opposed the nanofluid motion, and the velocity of the Al

_{2}O

_{3}+ H

_{2}O nanofluid dropped. The velocity declined more slowly for a weaker magnetic field, and a rapid decrement in the nanofluid velocity was observed for a stronger magnetic field. Near the arched surface, variations in the velocity ($L\prime \left(\eta \right)$) were almost negligible. This behavior of the velocity distribution was due to the friction between the surface and the nanolayer of Al

_{2}O

_{3}+ H

_{2}O. In the successive nanolayers, the velocity field was altered significantly. These influences became negligible far from the curve and showed an asymptotic pattern of velocity distribution at the free surface. Figure 3b shows the velocity distribution of the parameter m. The velocity of the Al

_{2}O

_{3}+ H

_{2}O nanofluid dropped rapidly for $m$ in comparison with M. As the values for parameter m became larger, the velocity decreased promptly.

_{2}O

_{3}+ H

_{2}O nanofluid is depicted in Figure 4b. For $\alpha $, a prominent behavior of the velocity was noticed in $0.5\le \eta \le 1.5$. Besides this, the velocity ($L\prime \left(\eta \right)$) was almost inconsequential.

_{2}O

_{3}+ H

_{2}O nanofluid grew for the more dissipative nanofluid. For the larger Ec, the temperature distribution rose rapidly.

#### 4.2. Streamlines and Isotherms

#### 4.3. Thermophysical Characteristics

_{2}O

_{3}+ H

_{2}O. The volume fraction of Al

_{2}O

_{3}showed a vibrant role in enhancing the dynamic viscosity of the nanofluid. The observed high dynamic viscosity corresponded to a greater volume fraction. On the other hand, the nanoparticle diameter (${d}_{p}$) induced inverse variations in the dynamic viscosity. Increasing the diameter of the nanoparticles caused the dynamic viscosity to drop. This means that nanoparticles with a smaller diameter are important to enhance the dynamic viscosity of nanofluids.

_{2}O

_{3}+ H

_{2}O became denser, which enhanced the effective density (${\rho}_{nf}$). Figure 14 highlights that the volume fraction ($\varphi $) and the effective electrical conductivity were in inverse proportion to each other.

#### 4.4. Skin Fraction and Heat Transfer Rate

## 5. Conclusions

_{2}O

_{3}nanofluids in the presence of Lorentz forces and nonlinear radiative heat flux, was examined over an arc-shaped geometry. To enhance the heat transfer rate, a thermal conductivity model that considered the impact of freezing temperature and molecular diameter was used. It was found that the nanofluid velocity ($L\prime \left(\eta \right)$) dropped for a stronger magnetic parameter (M), which is very significant from an industrial point of view. Further, the velocity of the nanofluid increased due to the mixed convection and larger curvature of the surface. The temperature $N\left(\eta \right)$ intensified for more dissipative fluid, and an inverse relationship between the temperature and the thermal radiation parameter was found. The dynamic viscosity of the nanofluid increased with the volume fraction, and the diameter of the nanoparticles showed reverse alterations for dynamic viscosity. The nanofluid became denser for a high volume fraction, and the electrical conductivity dropped. It was found that the heat transfer reduced the surface of a smaller curvature and intensified for larger curvatures. A more convective fluid and a larger curvature better opposed the shear stress. Finally, when considering the influence of the freezing temperature and the molecular diameter, the nanofluid flow model is very useful for heat transfer in comparison with existing studies.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$u$ | Component of the velocity |

$a$ | Radius |

$T$ | Temperature |

${T}_{\infty}$ | Temperature far from the surface |

${k}_{f}$ | Thermal conductivity of the host fluid |

${k}_{s}$ | Thermal conductivity of the nanoparticles |

${k}_{nf}$ | Effective thermal conductivity of the nanofluid |

${\left({C}_{p}\right)}_{f}$ | Heat capacity of the host fluid |

${\sigma}_{s}$ | Electrical conductivity of the nanoparticles |

${\sigma}_{nf}$ | Electrical conductivity of the nanofluid |

$\mu )\_f$ | Dynamic viscosity of the fluid |

${M}^{*}$ | Molecular weight |

${d}_{f}$ | Molecular diameter |

${\beta}_{w}$ | Temperature ratio parameter |

$Pr$ | Prandtl number |

$M$ | Hartmann number |

$L\left(\eta \right)$ | Dimensionless velocity |

$Nu$ | Nusselt number |

$v$ | Component of the velocity |

$p$ | Pressure |

${T}_{w}$ | Temperature at the surface |

${\rho}_{f}$ | Density of the host fluid |

${\rho}_{s}$ | Density of the nanoparticles |

${\rho}_{nf}$ | Effective density of the nanofluid |

$({C}_{p})\_s$ | Heat capacity of the nanoparticles |

${\left({C}_{p}\right)}_{nf}$ | Heat capacity of the nanofluid |

${\sigma}_{f}$ | Electrical conductivity of the host fluid |

${\mu}_{nf}$ | Effective dynamic viscosity |

$\varphi $ | Volume fraction of the nanoparticles |

${N}^{*}$ | Avogadro number |

${k}_{b}$ | Stefan Boltzmann constant |

$Rd$ | Radiation parameter |

$Ec$ | Eckert number |

$K$ | Curvature parameter |

$N\left(\eta \right)$ | Dimensionless temperature |

${C}_{f}$ | Skin fraction coefficient |

## References

- Choi, S. Enhancing thermal conductivity of fluids with nanoparticles in developments and applications of non-newtonians flows. ASME J. Heat Transf.
**1995**, 66, 99–105. [Google Scholar] - Clerk, M.J. Treatise on Electricity and Magnetism; Oxford University Press: Oxford, UK, 1873. [Google Scholar]
- Bruggeman, D.A.G. Berechnung verschiedener physikalischer konstanten von heterogenen substanzen, I—dielektrizitatskonstanten und leitfahigkeiten der mischkorper aus isotropen substanzen. Ann. Phys. Leipz.
**1935**, 24, 636–679. [Google Scholar] [CrossRef] - Hamilton, H.L.; Crosser, O.K. Thermal conductivity of heterogeneous two-component systems. Ind. Eng. Chem. Fundam.
**1962**, 1, 187–191. [Google Scholar] [CrossRef] - Lu, S.; Lin, H. Effective conductivity of composites containing aligned spherical inclusions of finite conductivity. J. Appl. Phys.
**1996**, 79, 6761–6769. [Google Scholar] [CrossRef] - Koo, J.; Kleinstreuer, C. A new thermal conductivity model for nanofluids. J. Nanopart. Res.
**2004**, 6, 577–588. [Google Scholar] [CrossRef] - Koo, J.; Kleinstreuer, C. Laminar nanofluid flow in micro-heat sinks. Int. J. Heat Mass Transf.
**2005**, 48, 2652–2661. [Google Scholar] [CrossRef] - Xue, Q.Z. Model for thermal conductivity of carbon nanotube-based composites. Phys. B Phys. Condens. Matter.
**2005**, 368, 302–307. [Google Scholar] [CrossRef] - Prasher, R.; Bhattacharya, P.; Phelan, P.E. Thermal conductivity of nanoscale colloidal solutions (nanofluids). Phys. Rev. Lett.
**2005**, 92, 25901. [Google Scholar] [CrossRef] - Li, C.H.; Peterson, G.P. Experimental investigation of temperature and volume fraction variations on the effective thermal conductivity of nanoparticle suspensions (nanofluids). J. Appl. Phys.
**2006**, 99. [Google Scholar] [CrossRef] - Corcione, M. Rayleigh–Be´nard convection heat transfer in nanoparticle suspensions. Int. J. Heat Fluid Flow
**2011**, 32, 65–77. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Li, Z.; Shamlooei, M. Nanofluid MHD natural convection through a porous complex shaped cavity considering thermal radiation. Phys. Lett. A
**2018**, 382, 1615–1632. [Google Scholar] [CrossRef] - Ahmed, N.; Khan, A.U.; Mohyud-Din, S.T. Influence of an effective prandtl number model on squeezed flow of γAl_2 O_3-H_2 O and γAl_2 O_3-C_2 H_6 O_2 nanofluids. J. Mol. Liq.
**2017**, 238, 447–454. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Zia, Q.M.Z.; Ellahi, R. Influence of induced magnetic field on free convection of nanofluid considering Koo-Kleinstreuer-Li (KKL) correlation. Appl. Sci.
**2016**, 6, 324. [Google Scholar] [CrossRef] - Asadullah, A.M.; Khan, U.; Naveed, A.; Mohyud-Din, S.T. Analytical and numerical investigation of thermal radiation effects on flow of viscous incompressible fluid with stretchable convergent/divergent channels. J. Mol. Liq.
**2016**, 224, 768–775. [Google Scholar] - Khan, U.; Naveed, A.A.; Mohyud-Din, S.T. 3D squeezed flow of γAl_2 O_3-H_2 O and γAl_2 O_3-C_2 H_6 O_2 nanofluids: A numerical study. Int. J. Hydrog. Energy
**2017**, 42, 24620–24633. [Google Scholar] [CrossRef] - Iijima, S. Helical microtubules of graphitic carbon. Nature
**1991**, 354, 56–58. [Google Scholar] [CrossRef] - Naveed, A.A.; Khan, U.; Mohyud-Din, S.T. Influence of thermal radiation and viscous dissipation on squeezed flow of water between two riga plates saturated with carbon nanotubes. Colloids Surf. A Physciochem. Eng. Asp.
**2017**, 522, 389–398. [Google Scholar] - Saba, F.; Naveed, A.; Hussain, S.; Khan, U.; Mohyud-Din, S.T.; Darus, M. Thermal analysis of nanofluid flow over a curved stretching surface suspended by carbon nanotubes with internal heat generation. Appl. Sci.
**2018**, 8, 395. [Google Scholar] [CrossRef] - Khan, U.; Naveed, A.; Mohyud-Din, S.T. Heat transfer effects on carbon nanotubes suspended nanofluid flow in a channel with non-parallel walls under the effect of velocity slip boundary condition: A numerical study. Neural Comput. Appl.
**2017**, 28, 37–46. [Google Scholar] [CrossRef] - Reddy, J.V.R.; Sugunamma, V.; Sandeep, N. Dual solutions for nanofluid flow past a curved surface with nonlinear radiation, soret and dufour effects. J. Phys. Conf. Ser.
**2018**, 1000, 12152. [Google Scholar] [CrossRef] - Hayat, T.; Qayyum, S.; Imtiaz, M.; Alsaedi, A. Double stratification in flow by curved stretching sheet with thermal radiation and joule heating. J. Therm. Sci. Eng. Appl.
**2017**, 10. [Google Scholar] [CrossRef] - Alsabery, A.I.; Sheremet, M.A.; Chamkha, A.J.; Hashim, I. MHD convective heat transfer in a discretely heated square cavity with conductive inner block using two-phase nanofuid model. Sci. Rep.
**2018**, 8, 1–23. [Google Scholar] [CrossRef] - Corcione, M. Empirical correlating equations for predicting the efective thermal conductivity and dynamic viscosity of nanofuids. Energy Convers. Manag.
**2011**, 52, 789–793. [Google Scholar] [CrossRef] - Naveed, A.A.; Khan, U.; Mohyud-Din, S.T. Unsteady radiative flow of chemically reacting fluid over a convectively heated stretchable surface with cross-diffusion gradients. Int. J. Therm. Sci.
**2017**, 121, 182–191. [Google Scholar]

**Figure 4.**Impacts of (

**a**) K and (

**b**) $\alpha $ on the velocity distribution ($L\prime \left(\eta \right)$).

**Figure 12.**The impact of the volume fraction ($\varphi $) and the nanoparticle diameter (${d}_{p}$) on the dynamic viscosity.

**Table 1.**Thermal and physical characteristics of the fluid phase and nanoparticles at T = 310 K [23].

Properties | ${\mathit{d}}_{\mathit{p}}\left(\mathbf{nm}\right)$ | $\mathit{\rho}$ (kg/m^{3}) | $\mathit{\beta}$ (1/k) | ${\mathit{c}}_{\mathit{p}}$ (J/Kg K) | ${\mathit{\mu}}_{\mathit{f}}$ (kg/ms) | $\mathit{k}$ (W/mk) | $\mathit{\sigma}$ (S/m) |
---|---|---|---|---|---|---|---|

H_{2}O | $0.385$ | $993$ | $36.2\times {10}^{5}$ | $4178$ | $695\times {10}^{6}$ | $0.628$ | $0.005$ |

Al_{2}O_{3} | 33 | $3970$ | $0.85\times {10}^{5}$ | $765$ | - | $40$ | $0.05\times {10}^{6}$ |

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**MDPI and ACS Style**

Khan, U.; Abbasi, A.; Ahmed, N.; Alharbi, S.O.; Noor, S.; Khan, I.; Mohyud-Din, S.T.; Khan, W.A.
Modified MHD Radiative Mixed Convective Nanofluid Flow Model with Consideration of the Impact of Freezing Temperature and Molecular Diameter. *Symmetry* **2019**, *11*, 833.
https://doi.org/10.3390/sym11060833

**AMA Style**

Khan U, Abbasi A, Ahmed N, Alharbi SO, Noor S, Khan I, Mohyud-Din ST, Khan WA.
Modified MHD Radiative Mixed Convective Nanofluid Flow Model with Consideration of the Impact of Freezing Temperature and Molecular Diameter. *Symmetry*. 2019; 11(6):833.
https://doi.org/10.3390/sym11060833

**Chicago/Turabian Style**

Khan, Umar, Adnan Abbasi, Naveed Ahmed, Sayer Obaid Alharbi, Saima Noor, Ilyas Khan, Syed Tauseef Mohyud-Din, and Waqar A. Khan.
2019. "Modified MHD Radiative Mixed Convective Nanofluid Flow Model with Consideration of the Impact of Freezing Temperature and Molecular Diameter" *Symmetry* 11, no. 6: 833.
https://doi.org/10.3390/sym11060833