# The Impact of Viscous Dissipation on the Thin Film Unsteady Flow of GO-EG/GO-W Nanofluids

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}, SP

^{3}. Carbon is an arresting component, graphene is defined as the single coated 2D sheets of graphite. Graphene is mostly used as a nanofluid as the organized potential wall of the solubility is very poor. Thus, we used graphene in the form of graphene oxide (GO), as it changes as a result and forms indisputable dispersion in water due to its high oxidize structure. GO is solvable in oils, the dynamic research area of GO is practically in the rotating disk model of nanofluids. Graphene is a single layer of carbon atom in a hexagonal lattice. The researchers take more interest in graphene oxide as compared to graphene. Graphene oxide (GO) is a two-dimensional material, and it is the oxidized form of graphene with O functional groups. Graphene oxide (GO) was first synthesized in 1859 by Sir Second Baronent Benjamin Colline Brodie via oxidation of bulk graphite with potassium chlorate and nitric acid by chemical method. The natural structure has been studied by Lerf and Klinowsky, at present the most well-known method to synthesize graphene oxide (GO) is the modified hummers method.

_{2}O nanofluid by using two rotating discs. Gul et al. [14] examined the water and ethylene glycol-based graphene oxide nanofluid flow under the influence of Marangoni convection. They analyzed the impact of the physical constraints and compared the effect of the embedded parameters using the GO-W and GO-EG nanofluids. The thin film is a thin layer of the fluid having finite domain. Qasim et al. [15] studied heat and mass transfer in nanofluid thin film over an unsteady stretching sheet using Buongiorno’s model.

## 2. Mathematical Formulation

## 3. Method of Solution

## 4. Table Discussion

## 5. Result and Discussion

## 6. Conclusions

- By increasing the unsteadiness parameter $S$ the velocity profile decreases and this effect is comparatively strong in the GO-EG nanofluid.
- The increasing value of the Prandtl number reduces the temperature profile. The effect is comparatively strong using the GO-EG nanofluid.
- Increasing the Eckert number increases the kinetic energy to enhance the temperature field.
- The increasing thickness of the thin film reduces the fluid motion.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclatures

$g$ | Acceleration due to gravity $\left({\mathrm{ms}}^{-2}\right)$ |

${\mu}_{nf}$ | Dynamic viscosity of the nanofluids axial directions $\left({\mathrm{kgms}}^{-1}\right)$ |

${\rho}_{nf}$ | Density of the nanofluids $\left({\mathrm{kgm}}^{-3}\right)$ |

$Ec$ | Eckert number $\left(\frac{\mu {c}_{p}}{{k}_{0}}\right)$ |

$Q$ | Heat generation/absorption parameter $\left(\mathrm{deg}\right)$ |

$T$ | Local temperature $\left(\mathrm{K}\right)$ |

$\beta $ | Non-dimensional thickness of the liquid film |

$Nu$ | Nusselt number $\left(r{q}_{w}|\theta =\alpha \backslash {k}_{0}{T}_{w}\right)$ |

$Pr$ | Prandtl number $\left(\frac{{U}^{2}}{{C}_{p}{T}_{w}}\right)$ |

${C}_{p}$ | Specific heat $\left({\mathrm{Jkg}}^{-1}{\mathrm{K}}^{-1}\right)$ |

${W}_{w}$ | Stretching velocity $\left({\mathrm{ms}}^{-1}\right)$ |

$\eta $ | Similarity variable $\left(\frac{\theta}{\alpha}\right)$ |

${C}_{f}$ | Skin friction $\left(\frac{{\tau}_{w}}{\rho {U}^{2}}\right)$ |

$\phi $ | Solid particle volume fraction |

$\delta $ | Thickness of the liquid film |

${T}_{\delta}$ | Temperature at the free surface $\left(\mathrm{K}\right)$ |

${k}_{nf}$ | Thermal conductivity of the nanoparticles $\left({\mathrm{WM}}^{-1}{\mathrm{K}}^{-1}\right)$ |

$\left(u,w\right)$ | Velocity components $\left({\mathrm{ms}}^{-1}\right)$ |

## References

- Choi, S.U.S.; Estman, J.A. Enhancing thermal conductivity of fluids with nanoparticles. ASME Publ. Fed
**1995**, 231, 99–106. [Google Scholar] - Choi, S.U.; 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] - Haq, R.U.; Kazmi, S.N.; Mekkaoui, T. Thermal management of water based SWCNTs enclosed in a partially heated trapezoidal cavity via FEM. Int. J. Heat Mass Transf.
**2017**, 112, 972–982. [Google Scholar] [CrossRef] - Soomro, F.A.; Haq, R.U.; Khan, Z.H.; Zhang, Q. Passive control of nanoparticle due to convective heat transfer of Prandtl fluid model at the stretching surface. Chin. J. Phys.
**2017**, 55, 1561–1568. [Google Scholar] [CrossRef] - Yu, W.; Xie, H.Q.; Chen, L.F.; Li, Y. Investigation on the thermal transport properties of ethylene glycol-based nanofluids containing copper nanoparticles. Powder Technol.
**2010**, 197, 218–221. [Google Scholar] [CrossRef] - Xie, H.Q.; Chen, L.F.J. Review on the preparation and thermal performances of carbon nanotube contained nanofluids. Chem. Eng. Data
**2011**, 56, 1030–1041. [Google Scholar] [CrossRef] - Xiao, B.; Yang, Y.; Chen, L. Developing a novel form of thermal conductivity of nanofluids with Brownian motion effect by means of fractal geometry. Powder Technol.
**2013**, 239, 409–414. [Google Scholar] [CrossRef] - Buongiorno, J. Convective transport in nanofluids. ASME J. Heat Transf.
**2006**, 128, 240–250. [Google Scholar] [CrossRef] - Khan, W.A.; Pop, I. Boundary layer flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transf.
**2010**, 53, 2477–2483. [Google Scholar] [CrossRef] - Shah, Z.; Gul, T.; Khan, M.A.; Ali, I.; Islam, S.; Hussain, F. Effects of hall current on steady three dimensional non-Newtonian nanofluid in a rotating frame with Brownian motion and thermophoresis effects. J. Eng. Technol.
**2017**, 6, 280–296. [Google Scholar] - Balandin, A.A.; Ghosh, S.; Bao, W. Superior thermal conductivity of single-layer graphene. Nano Lett.
**2008**, 8, 902–907. [Google Scholar] [CrossRef] [PubMed] - Wei, Y.; Huaqing, X.; Dan, B. Enhanced thermal conductivities of nanofluids containing graphene oxide nanosheets. Nanotechnology
**2010**, 21, 055705. [Google Scholar] - Gul, T.; Firdous, K. The experimental study to examine the stable dispersion of the graphene nanoparticles and to look at the GO-H
_{2}O nanofluid flow between two rotating disks. Appl. Nanosci.**2018**, 8, 1711–1727. [Google Scholar] [CrossRef] - Gul, T.; Noman, W.; Sohail, M.; Khan, M.A. Impact of the Marangoni and thermal radiation convection on the graphene-oxide-water-based and ethylene-glycol-based nanofluids. Adv. Mech. Eng.
**2019**, 11, 1–9. [Google Scholar] [CrossRef] - Qasim, M.; Khan, Z.H.; Lopez, R.J.; Khan, W.A. Heat and mass transfer in nanofluid thin film over an unsteady stretching sheet using Buongiorno’s model. Eur. Phys. J. Plus
**2016**, 131, 16. [Google Scholar] [CrossRef] - Alshomrani, A.S.; Gul, T. The convective study of the Al
_{2}O_{3}-H_{2}O and Cu-H_{2}O nano-liquid film sprayed over a stretching cylinder with viscous dissipation. Eur. Phys. J. Plus**2017**, 132, 495. [Google Scholar] [CrossRef] - Gul, T.; Nasir, S.; Islam, S.; Shah, Z.; Khan, M.A. Effective Prandtl number model influences on the γAl
_{2}O_{3}-C_{2}H_{6}O_{2}and γAl_{2}O_{3}-H_{2}O nanofluids spray along a stretching cylinder. Arab. J. Sci. Eng.**2018**, 44, 1601–1616. [Google Scholar] [CrossRef] - Aziz, R.C.; Hashim, I.; Alomari, A.K. Thin film flow and heat transfer on an unsteady stretching sheet with internal heating. Meccanica
**2011**, 46, 349–357. [Google Scholar] [CrossRef] - Pour, M.S.; Nassab, S.G. Numerical investigation of forced laminar convection flow of nanofluids over a backward facing step under bleeding condition. J. Mech.
**2012**, 28, 7–12. [Google Scholar] [CrossRef] - Abu-Nada, E. Numerical prediction of entropy generation in separated flows. Entropy
**2005**, 7, 234–252. [Google Scholar] [CrossRef] - Abu-Nada, E. Application of nanofluids for heat transfer enhancement of separated flows encountered in a backward facing step. Int. J. Heat Fluid Flow
**2008**, 29, 242–249. [Google Scholar] [CrossRef] - Liao, S.J. The Proposed Homotopy Analysis Method for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, China, June 1992. [Google Scholar]
- Liao, S.J. Advances in the Homotopy Analysis Method; World Scientific Press: Singapore, 2013. [Google Scholar]
- Rehman, A.; Gul, T.; Salleh, Z.; Mukhtar, S.; Hussain, F.; Nisar, K.S.; Kumam, P. Effect of the Marangoni convection in the unsteady thin film spray of CNT nanofluids. Processes
**2019**, 7, 392. [Google Scholar] [CrossRef] - Gul, T.; Sohail, M. Marangoni liquid film scattering over an extending cylinder. Theor. Appl. Mech. Lett.
**2019**, 9, 106–112. [Google Scholar] [CrossRef] - Khan, N.S.; Gul, T.; Kumam, P. Influence of inclined magnetic field on Carreau nanoliquid thin film flow and heat transfer with graphene nanoparticles. Energies
**2019**, 12, 1459. [Google Scholar] [CrossRef] - Ali, V.; Gul, T.; Afridi, S.; Ali, F.; Alharbi, S.O.; Khan, I. Thin film flow of micropolar fluid in a permeable medium. Coatings
**2019**, 9, 98. [Google Scholar] [CrossRef] - Gul, T.; Khan, W.A.; Tahir, M.; Bilal, R.; Khan, I.; Nisar, K.S. Unsteady Nano–Liquid spray with thermal radiation comprising CNTs. Processes
**2019**, 7, 181. [Google Scholar] [CrossRef] - Gul, T.; Khan, A.S.; Islam, S. Heat transfer investigation of the unsteady thin film flow of Williamson fluid past an inclined and oscillating moving plate. Appl. Sci.
**2017**, 7, 369. [Google Scholar] [CrossRef] - Khan, W.; Gul, T.; Idrees, M.; Islam, S.; Khan, I.; Dennis, L.C.C. Thin film Williamson nanofluid flow with varying viscosity and thermal conductivity on a time-dependent stretching sheet. Appl. Sci.
**2016**, 6, 334. [Google Scholar] [CrossRef]

**Table 1.**The numerical values for the skin friction coefficient for different physical parameters when $h=-0.1,Pr=0.7,Ec=0.1,\beta =0.1$.

$\mathit{\beta}$ | $\mathit{S}$ | ${\mathit{f}}^{\u2033}\left(0\right)$ GO-W $\mathit{\varphi}=0.01$ | ${\mathit{f}}^{\u2033}\left(0\right)$ GO-W $\mathit{\varphi}=0.02$ | ${\mathit{f}}^{\u2033}\left(0\right)$ GO-EG $\mathit{\varphi}=0.01$ | ${\mathit{f}}^{\u2033}\left(0\right)$ GO-EG $\mathit{\varphi}=0.02$ |
---|---|---|---|---|---|

0.1 | 0.1 | 30.3264 | 30.3335 | 28.7696 | 30.4545 |

0.2 | - | 30.3140 | 30.3412 | 30.3040 | 30.3414 |

- | 0.2 | 30.2883 | 30.3261 | 30.2830 | 30.3264 |

- | 0.3 | 30.7430 | 28.7777 | 28.1609 | 28.7778 |

**Table 2.**The numerical values of local Nusselt numbers of different physical parameters, when $\beta =0.1,h=-0.1,S=0.1$.

$\mathit{E}\mathit{c}$ | $\mathit{P}\mathit{r}$ | ${\mathit{\theta}}^{\prime}\left(0\right)$ GO-W $\mathit{\varphi}=0.01$ | ${\mathit{\theta}}^{\prime}\left(0\right)$ GO-W $\mathit{\varphi}=0.02$ | ${\mathit{\theta}}^{\prime}\left(0\right)$ GO-EG $\mathit{\varphi}=0.01$ | ${\mathit{\theta}}^{\prime}\left(0\right)$ GO-EG $\mathit{\varphi}=0.02$ |
---|---|---|---|---|---|

0.1 | 0.5 | −0.1437 | −0.4409 | −0.62116 | −0.4411 |

0.2 | - | −1.2456 | −1.2571 | −1.2458 | −1.2576 |

0.3 | - | −1.8697 | −1.8869 | −1.8700 | −1.8875 |

- | 0.6 | 0.6026 | −1.8337 | −0.6027 | −0.6098 |

- | 0.7 | 0.5870 | −1.8875 | −0.5871 | −0.5952 |

**Table 3.**Individual averaged squared residual errors for graphene oxide water based nanofluid (GO-W) when $Pr=6.5,Ec=0.5,Nu=0.1,S=0.4,\beta =0.3,\varphi =0.1$.

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

6 | $1.7784\times {10}^{-2}$ | $1.91773\times {10}^{-1}$ |

12 | $9.14122\times {10}^{-4}$ | $8.77139\times {10}^{-2}$ |

18 | $6.97126\times {10}^{-5}$ | $5.4508\times {10}^{-4}$ |

24 | $6.44141\times {10}^{-6}$ | $3.79449\times {10}^{-5}$ |

30 | $6.50496\times {10}^{-7}$ | $2.7782\times {10}^{-6}$ |

**Table 4.**Individual averaged squared residual errors for graphene oxide ethylene glycol based nanofluid (GO-EG) when $Pr=6.5,Ec=0.5,Nu=0.1,S=0.4,\beta =0.3,\varphi =0.1$.

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

6 | $1.82761\times {10}^{-2}$ | $1.91773\times {10}^{-1}$ |

12 | $9.55814\times {10}^{-4}$ | $8.77139\times {10}^{-2}$ |

18 | $7.39644\times {10}^{-5}$ | $5.4508\times {10}^{-4}$ |

24 | $6.93791\times {10}^{-6}$ | $3.79449\times {10}^{-5}$ |

30 | $7.1187\times {10}^{-7}$ | $2.7782\times {10}^{-6}$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rehman, A.; Salleh, Z.; Gul, T.; Zaheer, Z.
The Impact of Viscous Dissipation on the Thin Film Unsteady Flow of GO-EG/GO-W Nanofluids. *Mathematics* **2019**, *7*, 653.
https://doi.org/10.3390/math7070653

**AMA Style**

Rehman A, Salleh Z, Gul T, Zaheer Z.
The Impact of Viscous Dissipation on the Thin Film Unsteady Flow of GO-EG/GO-W Nanofluids. *Mathematics*. 2019; 7(7):653.
https://doi.org/10.3390/math7070653

**Chicago/Turabian Style**

Rehman, Ali, Zabidin Salleh, Taza Gul, and Zafar Zaheer.
2019. "The Impact of Viscous Dissipation on the Thin Film Unsteady Flow of GO-EG/GO-W Nanofluids" *Mathematics* 7, no. 7: 653.
https://doi.org/10.3390/math7070653