# Mathematical Analysis of Two Phase Saturated Nanofluid Influenced by Magnetic Field Gradient

^{*}

## Abstract

**:**

_{2}O-Fe

_{3}O

_{4}and C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni magnetic nanofluids. Finally, the study’s results are compared with the previously available data and are found to be in good agreement.

## 1. Introduction

_{20}W

_{15}Ni) alloy and magnetite ferrite (Fe

_{3}O

_{4}) nano-sized particles were added into a base fluid of ethylene glycol (C

_{2}H

_{6}O

_{2}) and water (H

_{2}O). The graphical results are computed in the discussion section. The particles and the base fluid properties are taken under the assumption of isothermal equilibrium. The analysis is made in such a way that a single solid particle is suspended in the base fluid. The suspension is then computed for thermal energy transmission. The comparison is made for the ferrite (Fe

_{3}O

_{4}) and alloy (CoCr

_{20}W

_{15}Ni) two phase nanofluid flow. The results are discussed and presented for the base fluids ethylene glycol (C

_{2}H

_{6}O

_{2}) and water (H

_{2}O).

## 2. Mathematical Modeling

_{3}O

_{4}and alloy CoCr

_{20}W

_{15}Ni solid nanoparticles satisfy the Curie temperature ${T}_{c}$. By contrast, the ferrite Fe

_{3}O

_{4}and alloy CoCr

_{20}W

_{15}Ni along with the base fluids ethylene glycol C

_{2}H

_{6}O

_{2}and water H

_{2}O are taken under the assumption of isothermal equilibrium and no slip occurs between them.

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni are given in Table 2.

#### 2.1. Magnetic Dipole

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni nanofluids. The introduction of magnetic forces into the magnetizable liquid gives rise to the effect known as ferrohydrodynamics. Now, since the nanoparticles are mechanically free to align with the fields of lines, the expression of the body force could be a good approximation for the flows of nanofluids. The magnetic scalar potential $\mathsf{\Omega}$ is stated below:

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni nanofluids is evident in Figure 1.

#### 2.2. Similarity Analysis

## 3. Numerical Simulation

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni nanofluids was computed with the help of the shooting technique in Matlab. The analysis in the discussion section is based on the dimensionless parameters and numbers in the above equations. The results show the impact of the solid particles on the heat transfer rate and friction drag theoretically. The system of equations is transformed into the first order equations in order to solve via the shooting technique.

## 4. Discussion

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}O-CoCr

_{20}W

_{15}Ni magnetic nanofluids were computed via the shooting technique in Matlab. The influence of physical parameters and dimensionless numbers on the flowing magnetic nanofluid is discussed in this section. Two base fluids along with two distinct solid particles are examined. The ferrite and alloy nanoparticles are entertained in the analysis. The magnetic two phase C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni nanofluids are simulated. The flows of C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni nanofluids were computed and then compared with each other. The variation of velocity field was noticed to be in an order, H

_{2}O-Fe

_{3}O

_{4}, H

_{2}O-CoCr

_{20}W

_{15}Ni, C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, and C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni. Therefore, we conclude that the maximum velocity field is determined for the H

_{2}O-Fe

_{3}O

_{4}nanofluid and the minimum distribution of velocity is determined for the magnetic C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni nanofluid. This variation occurs due to the thermophysical values of the transport properties of the solid particles and base fluids. The comparison is shown in Figure 2. The relation is compared for the temperature field in Figure 3. The variation in temperature distribution is noticed in the order C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, H

_{2}O-CoCr

_{20}W

_{15}Ni, and H

_{2}O-Fe

_{3}O

_{4}. This means that the H

_{2}O-Fe

_{3}O

_{4}nanofluid leads to the lowest temperature, whereas the C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni magnetic nanofluid exhibits the higher temperature field. The ferrite nanoparticles considered in the problem have higher thermal properties, which help in the enhancement of heat transfer; as a result, the lower temperature field occurs for the magnetite ferrite base magnetic nanofluid. Cobalt particles, however, have the property of Curie temperature, but that is not sufficient for a higher temperature field. Thus, the lower temperature field is observed for the alloy-based magnetic nanofluid.

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni. The impact of $\beta $ on the axial velocity is determined in Figure 4, whereas its influence on the temperature field is evident in Figure 5. The interaction between the nanoparticles of the ferrite and alloys with the flowing fluid leads to resisting the flow and the internal energy of the fluid. The higher resistance to the flowing fluid is induced by the alloy as well as the ferrite nanoparticles; as a result, the velocity field declines, as presented in Figure 4. On the other hand, the internal energy in the presence of the alloy and ferrite nanoparticles arises when $\beta $ is enhanced; thus, the temperature field enhancement is evident, as shown in Figure 5.

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni magnetic nanofluids were incorporated in this work. A higher resistance induced by the surface to the flowing fluid was noticed for the C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni magnetic fluid, which indicates that the alloy particles can cause more resistance as compared to the ferrite nanoparticles, as evident in Figure 6. The higher resistance has a better impact on the corresponding velocity. This means that the corresponding velocity will be higher for the considered solid particle and the base fluid. Figure 7 demonstrates the Nusselt number for the C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni magnetic nanofluids. The maximum transmission of heat is observed for the H

_{2}O-Fe

_{3}O

_{4}magnetic nanofluid. The thermal properties are higher for the base fluid as well as for the solid ferrite nanoparticles. Thus, the maximum heat transfer is noticed for the H

_{2}O-Fe

_{3}O

_{4}magnetic nanofluid, while the temperature behavior is lowered. A comparison of the present results was made with the available results examined by Ishak et al. [30]. The comparison is made for the Nusselt number as presented in Table 3.

## 5. Concluding Remarks

_{2}O-Fe

_{3}O

_{4}was found to exhibit a maximum heat transfer rate and C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni with higher resistance as compared to the rest of the nanofluid combinations considered.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$u,v$ | Velocity components |

${\left(\rho {c}_{p}\right)}_{nf}$ | Specific heat of hybrid nanofluid |

${\mu}_{nf}$ | Dynamic viscosity |

${k}_{nf}$ | Thermal conductivity ofhybrid nanofluid |

${\rho}_{nf}$ | Density of hybrid nanofluid |

T | Temperature |

P | Pressure |

M | Magnetization |

S | Stretching rate |

${T}_{c}$ | Curie temperature |

$R{e}_{x}$ | Reynolds number |

${K}_{c}$ | Pyromagnetic coefficient |

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

${\rho}_{nf}$ | density of hybrid nanofluid |

$N{u}_{x}$ | Nusselt number |

${\gamma}_{1}$ | Magnetic field induction |

H | Magnetic field |

$\mathsf{\Omega}$ | Magnetic scalar potential function |

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**Figure 2.**Velocity field comparison of magnetic nanofluids C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni.

**Figure 3.**Temperature field comparison of magnetic nanofluids C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni.

**Figure 4.**Influence of $\beta $ on the velocity of magnetic nanofluids C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni.

**Figure 5.**Influence of $\beta $ on the temperature of magnetic nanofluids C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni.

**Figure 6.**Comparison of wall shear stress in the flow of magnetic nanofluids C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni.

**Figure 7.**Comparison of Nusselt number in magnetic nanofluids C

_{2}H

_{6}O

_{2}-Fe

_{3}O

_{4}, C

_{2}H

_{6}O

_{2}-CoCr

_{20}W

_{15}Ni, H

_{2}O-Fe

_{3}O

_{4}, and H

_{2}O-CoCr

_{20}W

_{15}Ni.

Properties | Islam et al. [29] |
---|---|

Thermal Conductivity k | $\frac{{k}_{nf}}{{k}_{f}}=\frac{2{k}_{f}+{k}_{s}-2\mathsf{\Phi}({k}_{f}-{k}_{s})}{2{k}_{f}+{k}_{s}+\mathsf{\Phi}({k}_{f}-{k}_{s})}$ |

Viscosity $\mu $ | ${\mu}_{nf}={\mu}_{f}{(1-\mathsf{\Phi})}^{-25/10}$ |

Heat Capacity $\rho {C}_{p}$ | ${\left(\rho {c}_{p}\right)}_{nf}=(1-\mathsf{\Phi}){\left(\rho {c}_{p}\right)}_{f}+\mathsf{\Phi}{\left(\rho {c}_{p}\right)}_{s}$ |

Density $\rho $ | ${\rho}_{nf}=(1-\mathsf{\Phi}){\rho}_{f}+\mathsf{\Phi}{\rho}_{s}$ |

k (W/mK) | $\mathit{\rho}$ (kg/m^{3}) | ${\mathit{C}}_{\mathit{p}}$ (J/kgK) | |
---|---|---|---|

CoCr_{20}W_{15}Ni | 15 | 9.60 | 400.0 |

Fe_{3}O_{4} | 9.7 | 5180 | 670 |

C_{2}H_{6}O_{2} | 0.249 | 1116.6 | 2382 |

H_{2}O | 0.60 | 998.3 | 4182 |

**Table 3.**Comparison of Nusselt number for various values of Pr (Prandtl number) in the flow of C

_{2}H

_{6}O

_{2}, H

_{2}O, CoCr

_{20}W

_{15}Ni, and Fe

_{3}O

_{4}.

Pr | Ishak et al. [30] | Present Results |
---|---|---|

0.01 | 0.0197 | 0.0155 |

0.72 | 0.8086 | 0.8088 |

1.0 | 1.0000 | 1.0000 |

3.0 | 1.9237 | 1.9298 |

7.0 | 3.0723 | 3.0765 |

10.0 | 3.7207 | 3.7200 |

100.0 | 5.2941 | 5.2965 |

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**MDPI and ACS Style**

Khan, F.; Yang, X.
Mathematical Analysis of Two Phase Saturated Nanofluid Influenced by Magnetic Field Gradient. *Inventions* **2021**, *6*, 26.
https://doi.org/10.3390/inventions6020026

**AMA Style**

Khan F, Yang X.
Mathematical Analysis of Two Phase Saturated Nanofluid Influenced by Magnetic Field Gradient. *Inventions*. 2021; 6(2):26.
https://doi.org/10.3390/inventions6020026

**Chicago/Turabian Style**

Khan, Farhan, and Xiaodong Yang.
2021. "Mathematical Analysis of Two Phase Saturated Nanofluid Influenced by Magnetic Field Gradient" *Inventions* 6, no. 2: 26.
https://doi.org/10.3390/inventions6020026