# Significance of Velocity Slip in Convective Flow of Carbon Nanotubes

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

## 3. Solutions by OHAM

## 4. Optimal Convergence-Control Parameters

## 5. Results and Discussion

## 6. Conclusions

- Both velocities ${f}^{\prime}\left(\zeta \right)$ and $g\left(\zeta \right)$ show reduction for higher values of velocity slip parameter $\alpha $.
- Larger stretching-strength parameter C presents an increase in radial velocity ${f}^{\prime}\left(\zeta \right)$ while opposite trend is noticed for azimuthal velocity $g\left(\zeta \right)$ and temperature $\theta \left(\zeta \right)$.
- For higher estimations of the volume fraction $\varphi ,$ both the velocity and temperatue field are enhanced.
- Temperature field $\theta \left(\zeta \right)$ is enhanced for larger values of the Biot number $Bi$.
- Nusselt number Re${}_{r}^{-1/2}N{u}_{r}$ is increased for larger values of volume fraction $\varphi $.
- Coefficient of skin-friction Re${}_{r}^{1/2}{C}_{f}$ increases for higher volume fraction $\varphi $ and velocity slip parameter $\alpha .$
- The used technique for the solution’s development has advantages over the other in the sense of the following points:
- It is independent of small/large physical parameters.
- It provides a simple way to ensure the convergence of series solutions.
- It provides a large freedom to choose the base functions and related auxiliary linear operators.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$r,$$\phi ,$z | space coordinates [m] | $u,$$v,$w | velocity components [m·s${}^{-1}$] |

${\rho}_{f}$ | fluid density [kg·m${}^{-3}$] | ${\mu}_{f}$ | fluid dynamic viscosity [Pa·s] |

${k}_{nf}$ | nanofluids themal | ${\nu}_{nf}$ | kinematic nanofluid |

conductivity [W·m${}^{-1}$·K${}^{-1}$] | viscosity [m${}^{2}$·s${}^{-1}$] | ||

${k}_{f}$ | basefluid themal | ${\nu}_{f}$ | kinematic fluid |

conductivity [W·m${}^{-1}$·K${}^{-1}$] | viscosity [m${}^{2}$·s${}^{-1}$] | ||

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

of base fluid [m${}^{2}$·s${}^{-1}$] | of nanofluid [m${}^{2}$·s${}^{-1}$] | ||

${T}_{f}$ | hot fluid temperature [K] | ${T}_{\infty}$ | ambient temperature [K] |

C | stretching-strength parameter | ${k}_{CNT}$ | CNTs thermal conductivity [W·m${}^{-1}$·K${}^{-1}$] |

$\alpha $ | velocity slip parameter | $\varphi $ | nanomaterial volume fraction |

$Bi$ | Biot number | Pr | Prandtl number |

${f}^{\prime}$ | dimensionless velocity | $N{u}_{r}$ | Nusselt number |

${C}_{f}$ | skin friction coefficient | $\zeta $ | dimensionless variable |

Re${}_{r}$ | local Reynolds number | $\theta $ | dimensionless temperature |

CNTs | carbon nanotubes | ${F}_{i}^{****}$ | arbitrary constants |

## References

- Von Karman, T. Uber laminare and turbulente Reibung. ZAMM Z. Angew. Math. Mech.
**1921**, 1, 233–252. [Google Scholar] [CrossRef] - Turkyilmazoglu, M.; Senel, P. Heat and mass transfer of the flow due to a rotating rough and porous disk. Int. J. Therm. Sci.
**2013**, 63, 146–158. [Google Scholar] [CrossRef] - Rashidi, M.M.; Kavyani, N.; Abelman, S. Investigation of entropy generation in MHD and slip flow over rotating porous disk with variable properties. Int. J. Heat Mass Transf.
**2014**, 70, 892–917. [Google Scholar] [CrossRef] - Turkyilmazoglu, M. Nanofluid flow and heat transfer due to a rotating disk. Comput. Fluids
**2014**, 94, 139–146. [Google Scholar] [CrossRef] - Hatami, M.; Sheikholeslami, M.; Gangi, D.D. Laminar flow and heat transfer of nanofluids between contracting and rotating disks by least square method. Power Technol.
**2014**, 253, 769–779. [Google Scholar] [CrossRef] - Mustafa, M.; Khan, J.A.; Hayat, T.; Alsaedi, A. On Bödewadt flow and heat transfer of nanofluids over a stretching stationary disk. J. Mol. Liq.
**2015**, 211, 119–125. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Hatami, M.; Ganji, D.D. Numerical investigation of nanofluid spraying on an inclined rotating disk for cooling process. J. Mol. Liq.
**2015**, 211, 577–583. [Google Scholar] [CrossRef] - Hayat, T.; Muhammad, T.; Shehzad, S.A.; Alsaedi, A. On magnetohydrodynamic flow of nanofluid due to a rotating disk with slip effect: A numerical study. Comput. Methods Appl. Mech. Eng.
**2017**, 315, 467–477. [Google Scholar] [CrossRef] - Choi, S.U.S.; Zhang, Z.G.; Yu, W.; Lockwood, F.E.; Grulke, E.A. Anomalous thermal conductivity enhancement in nanotube suspensions. Appl. Phys. Lett.
**2001**, 79, 2252. [Google Scholar] [CrossRef] - Ramasubramaniam, R.; Chen, J.; Liu, H. Homogeneous carbon nanotube/polymer composites for electrical applications. Appl. Phys. Lett.
**2003**, 83, 2928. [Google Scholar] [CrossRef] - Xue, Q.Z. Model for thermal conductivity of carbon nanotube-based composites. Phys. B Condens. Matter
**2005**, 368, 302–307. [Google Scholar] [CrossRef] - Kamali, R.; Binesh, A. Numerical investigation of heat transfer enhancement using carbon nanotube-based non-Newtonian nanofluids. Int. Commun. Heat Mass Transf.
**2010**, 37, 1153–1157. [Google Scholar] [CrossRef] - Wang, J.; Zhu, J.; Zhang, X.; Chen, Y. Heat transfer and pressure drop of nanofluids containing carbon nanotubes in laminar flows. Exp. Therm. Fluid Sci.
**2013**, 44, 716–721. [Google Scholar] [CrossRef] - Haq, R.U.; Hammouch, Z.; Khan, W.A. Water-based squeezing flow in the presence of carbon nanotubes between two parallel disks. Therm. Sci.
**2014**, 20, 148. [Google Scholar] [CrossRef] - Safaei, M.R.; Togun, H.; Vafai, K.; Kazi, S.N.; Badarudin, A. Investigation of heat transfer enhancement in a forward-facing contracting channel using FMWCNT nanofluids. Numer. Heat Transf. Part A
**2014**, 66, 1321–1340. [Google Scholar] [CrossRef] - Ellahi, R.; Hassan, M.; Zeeshan, A. Study of natural convection MHD nanofluid by means of single and multi walled carbon nanotubes suspended in a salt water solutions. IEEE Trans. Nanotechnol.
**2015**, 14, 726–734. [Google Scholar] [CrossRef] - Karimipour, A.; Taghipour, A.; Malvandi, A. Developing the laminar MHD forced convection flow of water/FMWNT carbon nanotubes in a microchannel imposed the uniform heat flux. J. Magn. Magn. Mater.
**2016**, 419, 420–428. [Google Scholar] [CrossRef] - Hayat, T.; Hussain, Z.; Muhammad, T.; Alsaedi, A. Effects of homogeneous and heterogeneous reactions in flow of nanofluids over a nonlinear stretching surface with variable surface thickness. J. Mol. Liq.
**2016**, 221, 1121–1127. [Google Scholar] [CrossRef] - Hayat, T.; Muhammad, K.; Farooq, M.; Alsaedi, A. Unsteady squeezing flow of carbon nanotubes with convective boundary conditions. PLoS ONE
**2016**, 11, e0152923. [Google Scholar] [CrossRef] - Hayat, T.; Haider, F.; Muhammad, T.; Alsaedi, A. On Darcy-Forchheimer flow of carbon nanotubes due to a rotating disk. Int. J. Heat Mass Transf.
**2017**, 112, 248–254. [Google Scholar] [CrossRef] - Akbar, N.S.; Khan, Z.H.; Nadeem, S. The combined effects of slip and convective boundary conditions on stagnation-point flow of CNT suspended nanofluid over a stretching sheet. J. Mol. Liq.
**2014**, 196, 21–25. [Google Scholar] [CrossRef] - Arani, A.A.A.; Akbari, O.A.; Safaei, M.R.; Marzban, A.; Alrashed, A.A.A.A.; Ahmadi, G.R.; Nguyen, T.K. Heat transfer improvement of water/single-wall carbon nanotubes (SWCNT) nanofluid in a novel design of a truncated double-layered microchannel heat sink. Int. J. Heat Mass Transf.
**2017**, 113, 780–795. [Google Scholar] [CrossRef] - Goodarzi, M.; Javid, S.; Sajadifar, A.; Nojoomizadeh, M.; Motaharipour, S.H.; Bach, Q.V.; Karimipour, A. Slip velocity and temperature jump of a non-Newtonian nanofluid, aqueous solution of carboxy-methyl cellulose/aluminum oxide nanoparticles, through a microtube. Int. J. Numer. Methods Heat Fluid Flow
**2018**. [Google Scholar] [CrossRef] - Ellahi, R.; Zeeshan, A.; Hussain, F.; Asadollahi, A. Peristaltic blood flow of couple stress fluid suspended with nanoparticles under the influence of chemical reaction and activation energy. Symmetry
**2019**, 11, 276. [Google Scholar] [CrossRef] - Suleman, M.; Ramzan, M.; Ahmad, S.; Lu, D.; Muhammad, T.; Chung, J.D. A numerical simulation of silver-water nanofluid flow with impacts of Newtonian heating and homogeneous-heterogeneous reactions past a nonlinear stretched cylinder. Symmetry
**2019**, 11, 295. [Google Scholar] [CrossRef] - Liao, S.J. An optimal homotopy-analysis approach for strongly nonlinear differential equations. Commun. Nonlinear. Sci. Numer. Simul.
**2010**, 15, 2003–2016. [Google Scholar] [CrossRef] - Dehghan, M.; Manafian, J.; Saadatmandi, A. Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Meth. Part. Diff. Equ.
**2010**, 26, 448–479. [Google Scholar] [CrossRef] - Malvandi, A.; Hedayati, F.; Domairry, G. Stagnation point flow of a nanofluid toward an exponentially stretching sheet with nonuniform heat generation/absorption. J. Thermodyn.
**2013**, 2013, 764827. [Google Scholar] [CrossRef] - Abbasbandy, S.; Hayat, T.; Alsaedi, A.; Rashidi, M.M. Numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid. Int. J. Numer. Methods Heat Fluid Flow
**2014**, 24, 390–401. [Google Scholar] [CrossRef] - Sheikholeslami, M.; Hatami, M.; Ganji, D.D. Micropolar fluid flow and heat transfer in a permeable channel using analytic method. J. Mol. Liq.
**2014**, 194, 30–36. [Google Scholar] [CrossRef] - Hayat, T.; Muhammad, T.; Alsaedi, A.; Alhuthali, M.S. Magnetohydrodynamic three-dimensional flow of viscoelastic nanofluid in the presence of nonlinear thermal radiation. J. Magn. Magn. Mater.
**2015**, 385, 222–229. [Google Scholar] [CrossRef] - Turkyilmazoglu, M. An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method. Filomat
**2016**, 30, 1633–1650. [Google Scholar] [CrossRef] - Zeeshan, A.; Majeed, A.; Ellahi, R. Effect of magnetic dipole on viscous ferro-fluid past a stretching surface with thermal radiation. J. Mol. Liq.
**2016**, 215, 549–554. [Google Scholar] [CrossRef] - Hayat, T.; Abbas, T.; Ayub, M.; Muhammad, T.; Alsaedi, A. On squeezed flow of Jeffrey nanofluid between two parallel disks. Appl. Sci.
**2016**, 6, 346. [Google Scholar] [CrossRef] - Muhammad, T.; Alsaedi, A.; Shehzad, S.A.; Hayat, T. A revised model for Darcy-Forchheimer flow of Maxwell nanofluid subject to convective boundary condition. Chin. J. Phys.
**2017**, 55, 963–976. [Google Scholar] [CrossRef]

Physical Features | Water | CNT | |
---|---|---|---|

SWCNTs | MWCNTs | ||

$\rho $ (kg/m${}^{3})$ | $997.1$ | 2600 | 1600 |

k (W/mK) | $0.613$ | 6600 | 3000 |

${c}_{p}$ (J/kgK) | 4179 | 425 | 796 |

**Table 2.**Individual averaged squared residuals errors for single-walled carbon nanotubes (SWCNTs)–water.

m | ${\mathit{\epsilon}}_{\mathit{m}}^{\mathit{f}}$ | ${\mathit{\epsilon}}_{\mathit{m}}^{\mathit{g}}$ | ${\mathit{\epsilon}}_{\mathit{m}}^{\mathit{\theta}}$ |
---|---|---|---|

2 | $9.95225\times {10}^{-5}$ | $2.35341\times {10}^{-2}$ | $7.29447\times {10}^{-7}$ |

6 | $4.17686\times {10}^{-5}$ | $1.03083\times {10}^{-2}$ | $6.04738\times {10}^{-7}$ |

10 | $2.95796\times {10}^{-5}$ | $7.24672\times {10}^{-3}$ | $5.69806\times {10}^{-7}$ |

14 | $2.37325\times {10}^{-5}$ | $5.77429\times {10}^{-3}$ | $5.53122\times {10}^{-7}$ |

18 | $2.01939\times {10}^{-5}$ | $4.88653\times {10}^{-3}$ | $5.43213\times {10}^{-7}$ |

20 | $1.88867\times {10}^{-5}$ | $4.55942\times {10}^{-3}$ | $5.39608\times {10}^{-7}$ |

**Table 3.**Individual averaged squared residuals errors for single-walled carbon nanotubes (MWCNTs)–water.

m | ${\mathit{\epsilon}}_{\mathit{m}}^{\mathit{f}}$ | ${\mathit{\epsilon}}_{\mathit{m}}^{\mathit{g}}$ | ${\mathit{\epsilon}}_{\mathit{m}}^{\mathit{\theta}}$ |
---|---|---|---|

2 | $1.0164\times {10}^{-4}$ | $2.40503\times {10}^{-2}$ | $7.29447\times {10}^{-7}$ |

6 | $4.27165\times {10}^{-5}$ | $1.05447\times {10}^{-2}$ | $6.04522\times {10}^{-7}$ |

10 | $3.02678\times {10}^{-5}$ | $7.41739\times {10}^{-3}$ | $5.69547\times {10}^{-7}$ |

14 | $2.42942\times {10}^{-5}$ | $5.91293\times {10}^{-3}$ | $5.52829\times {10}^{-7}$ |

18 | $2.06785\times {10}^{-5}$ | $5.00567\times {10}^{-3}$ | $5.42892\times {10}^{-7}$ |

20 | $1.93426\times {10}^{-5}$ | $4.67132\times {10}^{-3}$ | $5.39275\times {10}^{-7}$ |

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**MDPI and ACS Style**

Alshomrani, A.S.; Ullah, M.Z.
Significance of Velocity Slip in Convective Flow of Carbon Nanotubes. *Symmetry* **2019**, *11*, 679.
https://doi.org/10.3390/sym11050679

**AMA Style**

Alshomrani AS, Ullah MZ.
Significance of Velocity Slip in Convective Flow of Carbon Nanotubes. *Symmetry*. 2019; 11(5):679.
https://doi.org/10.3390/sym11050679

**Chicago/Turabian Style**

Alshomrani, Ali Saleh, and Malik Zaka Ullah.
2019. "Significance of Velocity Slip in Convective Flow of Carbon Nanotubes" *Symmetry* 11, no. 5: 679.
https://doi.org/10.3390/sym11050679