# Entropy Generation of Carbon Nanotubes Flow in a Rotating Channel with Hall and Ion-Slip Effect Using Effective Thermal Conductivity Model

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## Abstract

**:**

## 1. Introduction

## 2. Basic Thermal Conductivities Models for CNTs

## 3. Mathematical Modeling

#### 3.1. Physical Quantities of Interest

#### 3.2. Entropy Analysis and Bejan Numbers

## 4. Solution by HAM

## 5. Results and Discussion

#### 5.1. Velocity Profile

#### 5.2. Temperature Function $\theta \left(\xi \right)$

#### 5.3. Entropy Generation ($Ns$) and Bejan Number ($Be$)

#### 5.4. Tables Discussion

## 6. Conclusions

- (a)
- The velocity function ${f}^{\prime}\left(\xi \right)$ increased with the augmentation in $\phi $, positive $Q$, ${n}_{i}$, and ${n}_{e}$, while it reduced with higher values of $R$, $M$, $Kr$, and negative $Q$.
- (b)
- It is observed that the transverse velocity function $g\left(\xi \right)$ increased with greater value of $\phi ,{n}_{i},\mathrm{and}\text{}{n}_{e}$, while it showed a reducing behavior for higher values of $R$, $Kr$, and $M$.
- (c)
- The temperature function $\theta \left(\xi \right)$ was augmented with the augmentation in $\phi $, while it showed reducing behavior with the escalation in $R,Pr$.
- (d)
- For entropy profile, it was observed that entropy generation $Ns$ increased with higher value of $M,Re,\mathrm{and}\text{}Br,$ while it showed decreasing behavior with an increase in ${n}_{i}\mathrm{and}\text{}{n}_{e}$.
- (e)
- The Bejan number $Be$ showed increasing behavior with an increase in $M$, $Br$, while it showed decreasing behavior with an increase in ${n}_{i}\mathrm{and}\text{}{n}_{e}$.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Pr | Prandtl number $\left(=\upsilon /\alpha \right)$ | $\psi $ | similarity variables |

P | fluid pressure $\left(\mathrm{Pa}\right)$ | ${\tau}_{xy}$ | surface shear stress |

$Nu$ | Nusselt number | $\sigma $ | electrical conductivity |

$g$ | internal energy distribution functions | $\alpha $ | thermal diffusivity |

${B}_{0}$ | magnetic flux density $\left({\mathrm{NmA}}^{-1}\right)$ | τ | lattice relaxation time |

$Be$ | Bejan number. | $\phi $ | volume friction |

$m$ | Hall parameter | $k$ | thermal conductivity |

$M$ | magnetic parameter | $\alpha $ | thermal diffusivity $\left({\mathrm{m}}^{2}{\mathrm{s}}^{-1}\right)$ |

$J$ | current density | $\Omega $ | angular velocity $\left({\mathrm{ms}}^{-1}\right)$ |

$E$ | electric intensity | $\mu $ | dynamic viscosity $\left(\mathrm{mPa}\right)$ |

${n}_{e}$ | ion-slip parameter | $\omega $ | electron cyclotron |

a, b, c | constants | ||

$T$ | fluid temperature $K$ | Subscripts | |

${c}_{p}$ | specific heat $\left(\frac{J}{kgK}\right)$ | $nf$ | nanofluid |

$CNTs$ | carbon nanotubes | ||

${\tilde{C}}_{f}$ | skin friction coefficient | $h$ | hot |

$R$ | Reynolds number | $ave$ | average |

$Q$ | suction and injection | ||

$Kr$ | rotation parameter | ||

$h$ | distance between the plates $\left(\mathrm{m}\right)$ | ||

${Q}_{w}$ | surface heat flux | ||

${S}_{h},{S}_{R},{S}_{f}$ | Dimensional entropy generation | ||

$Ns$ | non-dimensional entropy generation | ||

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

$x,y,z$ | coordinates | ||

$O$ | origin | ||

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

Greek symbols | |||

$\upsilon $ | kinematic viscosity | ||

$\rho $ | fluid density |

## References

- Xiao, B.; Chen, H.; Xiao, S.; Cai, J. Research on Relative Permeability of Nanofibers with Capillary Pressure Effect by Means of Fractal-Monte Carlo Technique. J. Nanosci. Nanotechnol.
**2017**, 17, 6811–6817. [Google Scholar] [CrossRef] - Xiao, B.; Wang, W.; Fan, J.; Chen, H.; Hu, X.; Zhao, D.; Zhang, X.; Ren, W. Optimization of the Fractal-Like Architecture of Porous Fibrous Materials Related to Permeability, Diffusivity and Thermal Conductivity. Fractals
**2017**. [Google Scholar] [CrossRef] - Xiao, B.; Zhang, X.; Wang, W.; Long, G.; Chen, H.; Kang, H.; Ren, W. A fractal model for water flow through unsaturated porous rocks. Fractals
**2018**. [Google Scholar] [CrossRef] - Liang, M.; Liu, Y.; Xiao, B.; Yang, S.; Han, H. An analytical model for the transverse permeability of gas diffusion layer with electrical double layer effects in proton exchange membrane fuel cells. Int. J. Hydrog. Energy
**2018**. [Google Scholar] [CrossRef] - Long, G.; Xu, G. The Effects of Perforation Erosion on Practical Hydraulic-Fracturing Applications. SPE J.
**2017**, 22, 645–659. [Google Scholar] [CrossRef] - Long, G.; Liu, S.; Xu, G.; Wong, S.W.; Chen, H.; Xiao, B. A Perforation-Erosion Model for Hydraulic-Fracturing Applications. SPE Prod. Oper.
**2018**, 33, 770–783. [Google Scholar] [CrossRef] - Kroto, H.W.; Heath, J.R.; O’Brien, S.C.; Curl, R.F.; Smalley, R.E. C60: The Best Constant of Discrete Sobolev Inequality on a Weighted Truncated Tetrahedron. Buckminsterfullerene. Nature
**1985**, 318, 162–163. [Google Scholar] [CrossRef] - Iijima, S. Helical microtubules of graphitic carbon. Nature
**1991**, 354, 56–58. [Google Scholar] [CrossRef] - Muhammad, S.; Ali, G.; Shah, Z.; Islam, S.; Hussain, A. The Rotating Flow of Magneto Hydrodynamic Carbon Nanotubes over a Stretching Sheet with the Impact of Non-Linear Thermal Radiation and Heat Generation/Absorption. Appl. Sci.
**2018**, 8, 482. [Google Scholar] [CrossRef] - Novoselov, K.S.; Geim, A.K.; Morozov, S.V.; Jiang, D.; Zhang, Y.; Dubonos, S.V. Electric field effect in atomically thin carbon films. Science
**2004**, 306, 666–669. [Google Scholar] [CrossRef] [PubMed] - Casari, C.S.; Tommasini, M.; Tykwinski, R.R.; Milani, A. Carbon-atom wires 1-D systems with tunable properties. Nanoscale
**2016**, 8, 4414–4435. [Google Scholar] [CrossRef] [PubMed] - Choi, S.U.S. Enhancing thermal conductivity of fluids with nanoparticles. In Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition, San Francisco, CA, USA, 12–17 November 1995; pp. 99–105. [Google Scholar]
- Kang, H.U.; Kim, S.H.; Oh, J.M. Estimation of thermal conductivity of nanofluid using experimental effective particle volume. Exp. Heat Transf.
**2006**, 19, 181–191. [Google Scholar] [CrossRef] - Haq, R.U.; Nadeem, S.; Khan, Z.H.; Noor, N.F.M. Convective heat transfer in MHD slips flow over a stretching surface in the presence of carbon nanotubes. Phys. B Condens. Matter
**2015**, 457, 40–47. [Google Scholar] [CrossRef] - Liu, M.S.; Lin, M.C.C.; Te, H.I.; Wang, C.C. Enhancement of thermal conductivity with carbon nanotube for nanofluids. Int. Commun. Heat Mass Transf.
**2005**, 32, 1202–1210. [Google Scholar] [CrossRef] - Shah, Z.; Dawar, A.; Islam, S.; Khan, I.; Ching, D.L.C. Darcy-Forchheimer Flow of Radiative Carbon Nanotubes with Microstructure and Inertial Characteristics in the Rotating Frame. Case Stud. Eng.
**2018**. [Google Scholar] [CrossRef] - Alrashed, A.A.A.A.; Gharibdousti, M.S.; Goodarzi, M.; Oliveira, L.R.; Filho, E.P. Effects on thermophysical properties of carbon based nanofluids: Experimental data, modelling using regression, ANFIS and ANN. Int. J. Heat Mass Transf.
**2018**, 23, 920–932. [Google Scholar] [CrossRef] - Safaei, M.R.; Togun, K.H.; Vafai, S.; Kazi, N.; Badarudin, A. Investigation of Heat Transfer Enhancement in a Forward-Facing Contracting Channel Using FMWCNT Nanofluids. Int. J. Comput. Methodol.
**2014**. [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] - Sheikholeslami, M.; Hatami, D.; Ganji, D.D. Nanofluid flow and heat transfer in a rotating system in the presence of a magnetic field. J. Mol. Liq.
**2014**, 190, 112–120. [Google Scholar] [CrossRef] - Sheikholeslami, M. Lattice Boltzmann Method simulation of MHD non-Darcy nanofluid free convection. Physica B
**2017**, 516, 55–71. [Google Scholar] [CrossRef] - Sheikholeslami, M. Influence of magnetic field on nanofluid free convection in an open porous cavity by means of Lattice Boltzmann Method. J. Mol. Liq.
**2017**, 234, 364–374. [Google Scholar] [CrossRef] - Sheikholeslami, M. Magnetohydrodynamic nanofluid forced convection in a porous lid driven cubic cavity using Lattice Boltzmann Method. J. Mol. Liq.
**2017**, 231, 555–565. [Google Scholar] [CrossRef] - Jawad, M.; Shah, Z.; Islam, S.; Islam, S.; Bonyah, E.; Khan, Z.A. Darcy-Forchheimer flow of MHD nanofluid thin film flow with Joule dissipation and Navier’s partial slip. J. Phys. Commun.
**2018**. [Google Scholar] [CrossRef] - Khan, N.; Zuhra, S.; Shah, Z.; Bonyah, E.; Khan, W.; Islam, S. Slip flow of Eyring-Powell nanoliquid film containing graphene nanoparticles. AIP Adv.
**2018**, 8, 115302. [Google Scholar] [CrossRef] - Khan, A.S.; Nie, Y.; Shah, Z.; Dawar, A.; Khan, W.; Islam, S. Three-Dimensional Nanofluid Flow with Heat and Mass Transfer Analysis over a Linear Stretching Surface with Convective Boundary Conditions. Appl. Sci.
**2018**, 8, 2244. [Google Scholar] [CrossRef] - Mendoza, E. Reflections on the Motive Power of Fire and other Papers on the Second Law of Thermodynamics; Clapeyron, E., Clausius, R., Eds.; Dover Publications: New York, NY, USA, 1988; ISBN 0-486-44641-7. [Google Scholar]
- Clausius, R. Mechanical Theory of Heat; Institute of Human Thermodynamics Publishing Ltd.: Chicago, IL, USA, 2006; pp. 1850–1865. [Google Scholar]
- Bejan, A. Second law analysis in heat transfer. Energy
**1980**, 5, 720–732. [Google Scholar] [CrossRef] - Rashidi, M.M.; Kavyani, N.; Abelman, S. Investigation of entropy generation in MHD and slip flow over a rotating porous disk with variable. Int. J. Heat Mass Transf.
**2014**, 70, 892–917. [Google Scholar] [CrossRef] - Soomro, F.A.; Rizwan-ul-Haq, K.Z.H.; Zhang, Q. Numerical study of entropy generation in MHD water-based carbon nanotubes along an inclined permeable surface. Eur. Phys. J. Plus
**2017**, 132, 412. [Google Scholar] [CrossRef] - Mohammad, I.; Gohar, A.; Shah, Z.; Islam, S.; Muhammad, S. Entropy Generation on Nanofluid Thin Film Flow of Eyring–Powell Fluid with Thermal Radiation and MHD Effect on an Unsteady Porous Stretching Sheet. Entropy
**2018**, 20, 412. [Google Scholar] [CrossRef] - Darbari, B.; Rashidi, S.; Esfahani, J.A. Sensitivity analysis of entropy generation in nanofluid flow inside a channel by response surface methodology. Entropy
**2016**, 18, 52. [Google Scholar] [CrossRef] - Bhatti, M.M.; Abbas, T.; Mehdi, M.; Rashidi, M.; Mohamed, S.; Ali, E. Numerical simulation of Entropy Generation with thermal radiation on MHD Carreau Nanofluid towards a Shrinking Sheet. Entropy
**2016**, 18, 200. [Google Scholar] [CrossRef] - Mohammad, Y.A.J.; Mohammad, R.S.; Abdullah, A.; Truong, K.N.; Enio, P.B.F. Entropy Generation in Thermal Radiative Loading of Structures with Distinct Heaters. Entropy
**2017**, 19, 506. [Google Scholar] [CrossRef] - Mohammad, M.R.; Mohammad, N.; Mustafa, S.S.; Zhighang, Y. Entropy Generation in a Circular Tube Heat Exchanger Using Nanofluids: Effects of Different Modeling Approaches. J. Heat Transf. Eng.
**2017**. [Google Scholar] [CrossRef] - Cramer, K.; Pai, S. Magnetofluid Dynamics for Engineers and Applied Physicists; McGraw-Hill: New York, NY, USA, 1973. [Google Scholar]
- Attia, H.A. Effect of the ion slip on the MHD flow of a dusty fluid with heat transfer under exponential decaying pressure gradient. Cent. Eur. J. Phys.
**2005**, 3, 484–507. [Google Scholar] [CrossRef] [Green Version] - Motsa, S.S.; Shatery, S. The effects of chemical reaction, Hall and ion-slip currents on MHD micropolar fluid flow with thermal diffusivity using a noval numerical technique. J. Appl. Math.
**2012**. [Google Scholar] [CrossRef] - Shah, Z.; Islam, S.; Ayaz, H.; Khan, S. Radiative Heat and Mass Transfer Analysis of Micropolar Nanofluid Flow of Casson Fluid between Two Rotating Parallel Plates with Effects of Hall Current. ASME J. Heat Transf.
**2018**. [Google Scholar] [CrossRef] - Shah, Z.; Islam, S.; Gul, T.; Bonyah, E.; Altaf Khan, M. The Elcerical MHD And Hall Current Impact On Micropolar Nanofluid Flow Between Rotating Parallel Plates. Results Phys.
**2018**. [Google Scholar] [CrossRef] - Greenspan, H.P.; Howard, L.N. On a time-dependent motion of a rotating fluid. J. Fluid Mech.
**1963**, 17, 385–404. [Google Scholar] [CrossRef] - Nazar, R.; Amin, N.; Pop, I. Unsteady boundary layer flow due to a stretching surface in a rotating fluid. Mech. Res. Commun.
**2004**, 31, 121–128. [Google Scholar] [CrossRef] - Mustafa, M.; Wasim, M.; Hayat, T.; Alsaedi, A. A revised model to study the rotating flow of nanofluid over an exponentially deforming sheet: Numerical solutions. J. Mol. Liq.
**2017**, 225, 320–327. [Google Scholar] [CrossRef] - Khan, A.; Shah, Z.; Islam, S.; Khan, S.; Khan, W.; Khan, Z.A. Darcy–Forchheimer flow of micropolar nanofluid between two plates in the rotating frame with non-uniform heat generation/absorption. Adv. Mech. Eng.
**2018**. [Google Scholar] [CrossRef] - Nor AthirahMohd, Z.; Khan, I.; Sharidan, S.; Alshomrani, A.S. Analysis of heat transfer for unsteady MHD free convection flow of rotating Jeffrey nanofluid saturated in a porous medium. Results Phys.
**2017**, 7, 288–309. [Google Scholar] [CrossRef] - Mohammadreza, H.; Rad, S.; Goodarz, A.; Mahidzal, B.D.; Salim, N.K.; Mohammad, R.S.; Emad, S. Numerical Study of Entropy Generation in a Flowing Nanofluid Used in Micro- and Minichannels. Entropy
**2013**, 15, 144–155. [Google Scholar] [CrossRef] [Green Version] - Nasiri, H.; Jamalabadi, M.Y.A.; Safaei, M.R.; Nguyen, T.K.; Shadlo, M.S. A smoothed particle hydrodynamics approach for numerical simulation of nano-fluid flows. J. Therm. Anal.
**2018**. [Google Scholar] [CrossRef] - Bhatti, M.M.; Sheikhulislami, M.; Zeeshan, A. Entropy Analysis on Electro-Kinetically Modulated Peristaltic Propulsion of Magnetized Nanofluid Flow through a Microchannel. Entropy
**2017**, 19, 481. [Google Scholar] [CrossRef] - Yarmand, H.; Ahmadi, G.; Gharehkhani, G.; Kazi, S.N.; Safei, M.R.; Alehshem, M.S.; Mahat, A.B. Entropy Generation during Turbulent Flow of Zirconia-water and Other Nanofluids in a Square Cross Section Tube with a Constant Heat Flux. Entropy
**2014**, 16, 6116–6132. [Google Scholar] [CrossRef] [Green Version] - Cho, C.C.; Yau, H.T.; Chiu, C.H.; Chiu, K.C. Numerical Investigation into Natural Convection and Entropy Generation in a Nanofluid-Filled U-Shaped Cavity. Entropy
**2015**, 17, 5980–5994. [Google Scholar] [CrossRef] [Green Version] - Shadlo, M.S.; kimiaeifar, A.; Bagheri, D. Series solution for heat transfer of continuous stretching sheet immersed in a micropolar fluid in the existence of radiation. Int. J. Numer. Methods Heat Fluid Flow
**2013**, 23, 289–304. [Google Scholar] [CrossRef] - Aghaei, A.; Sheikhzadeh, G.A.; Goodarzi, M.; Hasani, H.; Damirchi, H.; Afrand, M. Effect of horizontal and vertical elliptic baffles inside an enclosure on the mixed convection of a MWCNTs-water nanofluid and its entropy generation. Eur. Phys. J. Plus
**2018**, 133, 486. [Google Scholar] [CrossRef] - Liao, S.J. On the Analytic Solution of Magnetohydrodynamic Flows of Non-Newtonian Fluids over a Stretching Sheet. J. Fluid Mech.
**2013**, 488, 189–212. [Google Scholar] [CrossRef] - Liao, S.J. On Homotopy Analysis Method for Nonlinear Problems. Appl. Math. Comput.
**2004**, 147, 499–513. [Google Scholar] [CrossRef] - Shah, Z.; Bonyah, E.; Islam, S.; Khan, W.; Ishaq, M. Radiative MHD thin film flow of Williamson fluid over an unsteady permeable stretching. Heliyon
**2018**, 4, e00825. [Google Scholar] [CrossRef] - Hammed, K.; Haneef, M.; Shah, Z.; Islam, I.; Khan, W.; Asif, S.M. The Combined Magneto hydrodynamic and electric field effect on an unsteady Maxwell nanofluid Flow over a Stretching Surface under the Influence of Variable Heat and Thermal Radiation. Appl. Sci.
**2018**, 8, 160. [Google Scholar] [CrossRef] - Shadlo, M.S.; Kimiaeifar, A. Application of homotopy perturbation method to find an analytical solution for magnetohydrodynamic flows of viscoelastic fluids in converging/diverging channels. J. Mech. Eng. Sci. Part C
**2010**. [Google Scholar] [CrossRef] - Maxwell, J.C. Electricity and Magnetism, 3rd ed.; Clarendon: Oxford, UK, 1904. [Google Scholar]
- Jaffery, D.J. Conduction through a random suspension of spheres. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci.
**1973**, 335, 335–336. [Google Scholar] [CrossRef] - Davis, R. The effective thermal conductivity of a composite material with spherical inclusions. Int. J.
**1986**, 7, 609–620. [Google Scholar] [CrossRef] - Hamilton, R.L.; Crosser, O.K. Thermal conductivity of heterogenous two-component systems. Ind. Eng. Chem. Fund.
**1962**, 3, 187–191. [Google Scholar] [CrossRef] - Xue, Q. Model for thermal conductivity of carbon nanotube-based composites. Phys. B Condens. Matter
**2005**, 368, 302–307. [Google Scholar] [CrossRef]

**Figure 2.**Impact of $\psi $ on ${f}^{\prime}\left(\xi \right)$, when $R=0.1,Kr=0.5,{n}_{e}=0.6,Q=0.7,{m}_{e}=0.8,\text{}M=0.9$.

**Figure 3.**Impact of $R$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,\text{}Kr=0.5,\text{}{n}_{e}=0.6,\text{}Q=0.7,\text{}{m}_{e}=0.8,\text{}M=0.9$.

**Figure 4.**Impact of $Kr$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Q=0.7,\text{}{m}_{e}=0.8,\text{}M=0.9$.

**Figure 5.**Impact of $M$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Kr=0.7,{m}_{e}=0.8,\text{}Q=0.9$.

**Figure 6.**Impact of $\left(Q>0\right)$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Kr=0.7,{m}_{e}=0.8,\text{}M=0.9$.

**Figure 7.**Impact of $\left(Q<0\right)$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Kr=0.7,{m}_{e}=0.8,\text{}M=0.9$.

**Figure 8.**Impact of ${n}_{i}$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{e}=0.6,Kr=0.7,M=0.8,\text{}Q=0.9$.

**Figure 9.**Impact of ${n}_{e}$ on ${f}^{\prime}\left(\xi \right)$, when $\psi =0.1,R=0.5,{n}_{i}=0.6,Kr=0.7,M=0.8,\text{}Q=0.9$.

**Figure 10.**Impact of $\psi $ on $g\left(\xi \right)$, when $Pr=0.4,Kr=0.2,R=0.5,{m}_{e}=0.6,{n}_{e}=0.7,\text{}M=0.8$.

**Figure 11.**Impact of $R$ on $g\left(\xi \right)$, when $Pr=0.4,Kr=0.2,\psi =0.5,{m}_{e}=0.6,{n}_{e}=0.7,M=0.8$.

**Figure 12.**Impact of $Kr$ on $g\left(\xi \right)$, when $Pr=0.4,R=0.2,\psi =0.5,{m}_{e}=0.6,{n}_{e}=0.7,\text{}M=0.8$.

**Figure 13.**Impact of $M$ on $g\left(\xi \right)$, when $Pr=0.4,R=0.2,\psi =0.5,{m}_{e}=0.6,{n}_{e}=0.7,\text{}R=0.8$.

**Figure 14.**Impact of ${n}_{i}$ on $g\left(\xi \right)$, when $Pr=0.4,R=0.2,\psi =0.5,M=0.6,{n}_{e}=0.7,\text{}R=0.8$.

**Figure 15.**Impact of ${n}_{e}$ on $g\left(\xi \right)$, when $Pr=0.4,R=0.2,\psi =0.5,M=0.6,{n}_{i}=0.7,\text{}R=0.8$.

**Table 1.**The numerical values of skin friction ${\tilde{C}}_{f}=\frac{{\mu}_{nf}}{{\mu}_{f}}{f}^{\u2033}\left(0\right),$ when $\psi =0.01$.

$\mathit{R}$ | $\mathit{K}\mathit{r}$ | $\mathit{M}$ | $\mathit{Q}$ | ${\mathit{n}}_{\mathit{e}}$ | ${\mathit{n}}_{\mathit{i}}$ | ${\tilde{\mathit{C}}}_{\mathit{f}}$ at $\mathit{\xi}=0$ | ${\tilde{\mathit{C}}}_{\mathit{f}}$ at $\mathit{\xi}=1$ |
---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.4 | 0.5 | 0.3 | 0.2 | −0.484638 | −0.515879 |

0.3 | −0.470976 | −0.529161 | |||||

0.5 | 0.2 | −0.458474 | −0.541466 | ||||

0.4 | −0.461839 | −0.538401 | |||||

0.6 | 0.4 | −0.465886 | −0.534719 | ||||

0.7 | −0.472578 | −0.529764 | |||||

1.0 | −1.5 | −0.483196 | −0.521086 | ||||

−0.1 | −2.187070 | −1.202060 | |||||

0.1 | −1.006670 | −0.971885 | |||||

1.5 | 0.3 | −0.835078 | 0.523539 | ||||

0.4 | 0.491335 | 0.511560 | |||||

0.5 | 0.2 | 0.498996 | −0.503849 | ||||

0.6 | 0.490763 | 0.511484 | |||||

1.0 | 0.495080 | 0.507058 |

**Table 2.**The numerical values of Nusselt number $N{u}_{x}=-\left(\frac{{k}_{nf}}{{k}_{f}}\right){\phi}^{\prime}\left(0\right),$ when $\psi =0.01$.

$\mathit{R}$ | $\mathit{K}\mathit{r}$ | $\mathit{M}$ | $\mathit{Q}$ | ${\mathit{n}}_{\mathit{e}}$ | ${\mathit{n}}_{\mathit{i}}$ | $\mathit{P}\mathit{r}$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}$ at $\mathit{\xi}=0$ | $\mathit{N}{\mathit{u}}_{\mathit{x}}$ at $\mathit{\xi}=1$ |
---|---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.4 | 0.5 | 0.3 | 0.2 | 7.2 | −0.001105 | 0.000884 |

0.3 | −0.001107 | 0.000886 | ||||||

0.5 | 0.2 | −0.001110 | 0.000889 | |||||

0.4 | −0.001445 | 0.001157 | ||||||

0.6 | 0.4 | −0.001779 | 0.001429 | |||||

0.7 | −0.002337 | 0.001887 | ||||||

1.0 | −1.5 | −0.002932 | 0.002347 | |||||

−0.1 | −0.000808 | 0.000995 | ||||||

0.1 | −0.002280 | 0.001443 | ||||||

1.5 | 0.3 | −0.002596 | 0.004649 | |||||

0.4 | −0.003721 | 0.004249 | ||||||

0.5 | 0.2 | −0.003404 | 0.003887 | |||||

0.6 | −0.002348 | 0.002682 | ||||||

1.0 | 7.2 | −0.001862 | 0.002126 | |||||

7.3 | −0.001862 | 0.002126 | ||||||

7.5 | −0.001862 | 0.002126 |

**Table 3.**Physical properties of carbon nanotubes (CNTs) (Xie et al. [50]).

Materials | SWCNTs | MWCNTs |
---|---|---|

Thermal Conductivity ${\mathit{k}}_{\mathit{n}\mathit{f}}$ ($W/mK$) | 3000 | 3000 |

Specific gravity (g/cm^{3}) | $0.8$ | $1.8$ |

Strength $\mathbf{\left(}GPa\mathbf{\right)}$ | 50–500 | 10–60 |

Elastic Modulus $\mathbf{\left(}TPa\mathbf{\right)}$ | 1 | 0.3–1 |

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**MDPI and ACS Style**

Feroz, N.; Shah, Z.; Islam, S.; Alzahrani, E.O.; Khan, W.
Entropy Generation of Carbon Nanotubes Flow in a Rotating Channel with Hall and Ion-Slip Effect Using Effective Thermal Conductivity Model. *Entropy* **2019**, *21*, 52.
https://doi.org/10.3390/e21010052

**AMA Style**

Feroz N, Shah Z, Islam S, Alzahrani EO, Khan W.
Entropy Generation of Carbon Nanotubes Flow in a Rotating Channel with Hall and Ion-Slip Effect Using Effective Thermal Conductivity Model. *Entropy*. 2019; 21(1):52.
https://doi.org/10.3390/e21010052

**Chicago/Turabian Style**

Feroz, Nosheen, Zahir Shah, Saeed Islam, Ebraheem O. Alzahrani, and Waris Khan.
2019. "Entropy Generation of Carbon Nanotubes Flow in a Rotating Channel with Hall and Ion-Slip Effect Using Effective Thermal Conductivity Model" *Entropy* 21, no. 1: 52.
https://doi.org/10.3390/e21010052