# Numerical Investigation of Copper-Water (Cu-Water) Nanofluid with Different Shapes of Nanoparticles in a Channel with Stretching Wall: Slip Effects

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Numerical Solution

## 4. Results and Discussion

## 5. Conclusions

- The heat transfer rate ${\theta}^{\prime}\left(-1\right)$ increases by enhancing in the strength of solid volume fraction $\phi =1\%$ to $10\%$
- The stretching Reynolds number $R$ decreases the velocity profile ${f}^{\prime}\left(\eta \right)$ near the lower wall of the channel $\eta \approx -1$ for spherical and cylindrical shape of nanoparticles.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

${B}_{\xb0}$ | External uniform magnetic field |

$p$ | Pressure (Pa) |

${k}_{s}$ | Thermal conductivity of the solid fraction (W/m·K) |

${k}_{nf}$ | Thermal conductivity of the nanofluid (W/m·K) |

${\rho}_{s}$ | Density of the solid fraction (Kg/m^{3}) |

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

$T$ | Fluid temperature (K or °C) |

${v}_{\xb0}$ | Injection/suction |

${k}_{f}$ | Thermal conductivity of the fluid (W/m·K) |

${k}_{nf}$ | Thermal conductivity of the nanofluid (W/m·K) |

$n$ | Shape factor through H-C Model |

${c}_{p}$ | Specific heat at constant pressure (J/kg·K) |

$\left(u,v\right)$ | Velocity component in Cartesian coordinate |

Greek symbols | |

$\eta $ | Scaled boundary layer coordinate |

${\sigma}_{nf}$ | Effective electrical conductivity of nanofluid (S.m^{-1}) |

$\mu $ | Dynamic viscosity |

$\theta $ | Self-similar temperature |

$\phi $ | Nanoparticle volume fraction parameter |

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

$\rho $ | Density (kg/m^{3}) |

Dimensionless numbers | |

$R=\frac{{a}^{2}b}{{\upsilon}_{f}}$ | Stretching Reynolds number |

$Pr=\text{}\frac{{a}^{2}b{\left(\rho {C}_{p}\right)}_{f}}{{\kappa}_{f}}$ | Prandtl number |

${\rho}_{nf}=\text{}{\rho}_{f}\left(1-\phi \right)+{\rho}_{s}$ | Density of the nanofluid |

${M}^{2}=\frac{\sigma {B}_{\xb0}^{2}{a}^{2}}{{\mu}_{f}}$ | Magnetic parameter |

${\mu}_{nf}=\text{}\frac{{\mu}_{f}}{{\left(1-\phi \right)}^{2.5}}$ | Dynamic viscosity of the nanofluid (Pa·s) |

$\frac{{\sigma}_{nf}}{{\sigma}_{f}}=1+\text{}\frac{3\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\phi}{\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}+2\right)-\left(\frac{{\sigma}_{s}}{{\sigma}_{f}}-1\right)\phi \text{}}$ | Ratio of effective electrical conductivity of nanofluid to the base fluid |

Subscripts | |

$nf$ | Nanofluid |

$s$ | Solid phase |

$2$ | Upper wall |

$f$ | Fluid phase |

$1$ | Lower wall |

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**Figure 2.**Effect of the variation of the solid volume fraction (ϕ) on the velocity of spherical nanoparticles

**Figure 3.**Effect of the variation of the solid volume fraction on velocity of cylindrical nanoparticles.

**Figure 4.**Effect of the variation of the Reynolds number (R) on the velocity of spherical nanoparticles.

**Figure 5.**Effect of the variation of the Reynolds number (R) on the velocity for cylindrical nanoparticles.

**Figure 6.**Effect of the variation of the magnetic field (M) on the velocity of spherical nanoparticles.

**Figure 7.**Effect of the variation of the magnetic field (M) on the velocity of cylindrical nanoparticles.

**Figure 8.**Effect of the variation of the solid volume fraction (ϕ) on the heat transfer for spherical nanoparticles.

**Figure 9.**Effect of the variation of the solid volume fraction (ϕ) on the heat transfer for cylindrical nanoparticles.

**Figure 10.**Effect of the variation of the Reynolds number (R) on the heat transfer for spherical nanoparticles.

**Figure 11.**Effect of the variation of the Reynolds number (R) on the heat transfer for cylindrical nanoparticles.

**Figure 12.**Effect of the variation of the magnetic field (M) on the heat transfer for spherical nanoparticles.

**Figure 13.**Effect of the variation of the magnetic (M) field on the heat transfer for cylindrical nanoparticles.

**Table 1.**Thermophysical properties of water and nanoparticles [20].

Density ρ (kg/m ^{3}) | Specific heat at constant pressure ${\mathit{c}}_{\mathit{p}}$ (J/kg·K) | Thermal conductivity $\mathit{k}$ (W/m·K) | Electrical conductivity $\mathit{\sigma}$ (S·m^{–1}) | |
---|---|---|---|---|

Pure water | 991.1 | 4179 | 0.613 | 0.05 |

Copper (Cu) | 8933 | 385 | 401 | $5.96\times {10}^{7}$ |

Alumina ($A{l}_{2}{O}_{3}$) | 3970 | 765 | 40 | $3.69\times {10}^{7}$ |

Silver (Ag) | 10500 | 235 | 429 | $6.3\times {10}^{7}$ |

Titanium Oxide $\left(Ti{O}_{2}\right)$ | 4250 | 686.2 | 8.9538 | $0.24\times {10}^{7}$ |

**Table 2.**Effect of the solid volume fraction on ${\theta}^{\prime}\left(-1\right)$ with spherical shape nanoparticles.

Solid Volume Fraction $\mathit{\phi}$ | Magnetic Parameter $\mathit{M}$ | Reynolds Number $\mathit{R}$ | Suction Parameter $\mathit{S}$ | Velocity Slip Condition ${\mathit{S}}_{1}$ | Thermal Slip Condition ${\mathit{S}}_{2}$ | Heat transfer Rate ${\mathit{\theta}}^{\prime}\left(-1\right)$ |
---|---|---|---|---|---|---|

0.01 | 0.5 | 4 | 1 | 0.1 | 0.1 | -4.964682800390394 |

0.03 | -4.795613667915525 | |||||

0.05 | -4.629249877087038 | |||||

0.1 | -4.225974478882998 |

**Table 3.**Effect of solid volume fraction on ${\theta}^{\prime}\left(-1\right)$ with cylindrical shape nanoparticles.

Solid Volume Fraction $\mathit{\phi}$ | Magnetic Parameter $\mathit{M}$ | Reynolds Number $\mathit{R}$ | Suction Parameter $\mathit{S}$ | Velocity Slip Condition ${\mathit{S}}_{1}$ | Thermal Slip Condition ${\mathit{S}}_{2}$ | Heat transfer Rate ${\mathit{\theta}}^{\prime}\left(-1\right)$ |
---|---|---|---|---|---|---|

0.01 | 0.5 | 4 | 1 | 0.3 | 0.2 | -3.2951491963380715 |

0.03 | -3.218220931297281 | |||||

0.05 | -3.141039169488954 | |||||

0.1 | -2.947344452998095 |

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**MDPI and ACS Style**

Raza, J.; Rohni, A.M.; Omar, Z.
Numerical Investigation of Copper-Water (Cu-Water) Nanofluid with Different Shapes of Nanoparticles in a Channel with Stretching Wall: Slip Effects. *Math. Comput. Appl.* **2016**, *21*, 43.
https://doi.org/10.3390/mca21040043

**AMA Style**

Raza J, Rohni AM, Omar Z.
Numerical Investigation of Copper-Water (Cu-Water) Nanofluid with Different Shapes of Nanoparticles in a Channel with Stretching Wall: Slip Effects. *Mathematical and Computational Applications*. 2016; 21(4):43.
https://doi.org/10.3390/mca21040043

**Chicago/Turabian Style**

Raza, Jawad, Azizah Mohd Rohni, and Zurni Omar.
2016. "Numerical Investigation of Copper-Water (Cu-Water) Nanofluid with Different Shapes of Nanoparticles in a Channel with Stretching Wall: Slip Effects" *Mathematical and Computational Applications* 21, no. 4: 43.
https://doi.org/10.3390/mca21040043