American Institute of Mathematical Sciences

August 2018, 11(4): 583-594. doi: 10.3934/dcdss.2018033

Numerical investigation of Cattanneo-Christov heat flux in CNT suspended nanofluid flow over a stretching porous surface with suction and injection

 a. DBS & H CEME, National University of Sciences and Technology, Islamabad, Pakistan b. Department of Mechanical Engineering, Manipal University Jaipur, Rajasthan-303007, India c. Department of Mathematics, University of Malakand, Dir (Lower), Khyber Pakhtunkhwa, Pakistan

* Corresponding author: Noreen Sher Akbar

Received  October 2016 Revised  May 2017 Published  November 2017

The present study analyzes the heat energy transfer in nano fluids flow through the porous stretching surface. Cattanneo-Christov heat flux model is employed to study the heat energy transfer. Darcy law is used to discuss the flow characteristics over the different types of permeable sheets with suction and injection. Nanofluids is considered as water based single-walled carbon nanotubes (SWNTs) and multi-walled carbon nanotubes (MWNTs) nanofluids. A comparative study for SWCNT and MWCNT is also made. Governing equations are transformed into set of ordinary differential equations using similarity transformations. The computational results are obtained by using Runge-Kutta fourth order method along with shooting technique. Numerical and graphical results are presented to discuss the effects of various physical parameters on velocity profile, temperature profile, Nusselt number, Sherwood number and skin friction coefficient for different type of nanoparticles for suction and injection cases. Stream lines and isotherms are also plotted for three different cases viz. permeable sheet with suction, impermeable sheet and permeable sheet with injection. A comparative analysis with existing results is tabulated which validate that the numerical results of present study have good correlation with existing results. The outcomes of the results show that skin friction coefficient is more for SWCNT in caparison of MWCNT and the boundary layer thickness is maximum for permeable stretching sheet with suction parameter.

Citation: Noreen Sher Akbar, Dharmendra Tripathi, Zafar Hayat Khan. Numerical investigation of Cattanneo-Christov heat flux in CNT suspended nanofluid flow over a stretching porous surface with suction and injection. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 583-594. doi: 10.3934/dcdss.2018033
References:
 [1] N. S. Akbar, N. Kazmi, D. Tripathi and N. A. Mir, Study of heat transfer on physiological driven movement with CNT nanofluids and variable viscosity, Computer Methods and Programs in Biomedicine, 136 (2016), 21-29. doi: 10.1016/j.cmpb.2016.08.001. [2] N. S. Akbar, Entropy generation analysis for a CNT suspension nanofluid in plumb ducts with peristalsis, Entropy, 17 (2015), 1411-1424. doi: 10.3390/e17031411. [3] N. S. Akbar, D. Tripathi, Z. H. Khan and O. A. Bég, A numerical study of magnetohydrodynamic transport of nanofluids over a vertical stretching sheet with exponential temperature-dependent viscosity and buoyancy effects, Chemical Physics Letters, 661 (2016), 20-30. doi: 10.1016/j.cplett.2016.08.043. [4] N. S. Akbar and Z. H. Khan, Effect of variable thermal conductivity and thermal radiation with CNTS suspended nanofluid over a stretching sheet with convective slip boundary conditions: Numerical study, Journal of Molecular Liquids, 222 (2016), 279-286. doi: 10.1016/j.molliq.2016.06.102. [5] N. S. Akbar, A. Ebaid and Z. H. Khan, Numerical analysis of magnetic field effects on Eyring-Powell fluid flow towards a stretching sheet, Journal of Magnetism and Magnetic Materials, 382 (2015), 355-358. [6] M. M. Bhatti and M. M. Rashidi, Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet, Journal of Molecular Liquids, 221 (2016), 567-573. doi: 10.1016/j.molliq.2016.05.049. [7] M. M. Bhatti, T. Abbas, M. M. Rashidi and M. E. S. Ali, Numerical simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy, 18 (2016), Paper No. 200, 14 pp. doi: 10.3390/e18060200. [8] M. M. Bhatti, T. Abbas, M. M. Rashidi, M. E. S. Ali and Z. Yang, Entropy generation on MHD eyring-powell nanofluid through a permeable stretching surface Entropy, 18 (2016), Paper No. 224, 14 pp. doi: 10.3390/e18060224. [9] D. M. Camarano, F. A. Mansur, T. L. C. F. Araújo, G. C. Salles and A. P. Santos, Thermophysical properties of ethylene glycol mixture based CNT nanofluids Journal of Physics: Conference Series, 733 (2016), 012015. doi: 10.1088/1742-6596/733/1/012015. [10] C. I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mechanics Research Communications, 36 (2009), 481-486. doi: 10.1016/j.mechrescom.2008.11.003. [11] M. Ciarletta and B. Straughan, Uniqueness and structural stability for the Cattaneo-Christov equations, Mechanics Research Communications, 37 (2010), 445-447. doi: 10.1016/j.mechrescom.2010.06.002. [12] M. Ciarletta, B. Straughan and V. Tibullo, Christov-Morro theory for non-isothermal diffusion, Nonlinear Analysis: Real World Applications, 13 (2012), 1224-1228. doi: 10.1016/j.nonrwa.2011.10.014. [13] R. Ellahi, M. Hassan and A. Zeeshan, Study of natural convection MHD nanofluid by means of single and multi-walled carbon nanotubes suspended in a salt-water solution, IEEE Transactions on Nanotechnology, 14 (2015), 726-734. doi: 10.1109/TNANO.2015.2435899. [14] P. Estellé, S. Halelfadl and T. Maré, Thermal conductivity of CNT water based nanofluids: Experimental trends and models overview, Journal of Thermal Enginnering, 1 (2015), 381-390. [15] S. A. M. Haddad, Thermal instability in Brinkman porous media with Cattaneo-Christov heat flux, International Journal of Heat and Mass Transfer, 68 (2014), 659-668. doi: 10.1016/j.ijheatmasstransfer.2013.09.039. [16] S. Halelfadl, T. Maré and P. Estellé, Efficiency of carbon nanotubes water based nanofluids as coolants, Experimental Thermal and Fluid Science, 53 (2014), 104-110. doi: 10.1016/j.expthermflusci.2013.11.010. [17] S. Han, L. Zheng, C. Li and X. Zhang, Coupled flow and heat transfer in viscoelastic fluid with Cattaneo-Christov heat flux model, Applied Mathematics Letters, 38 (2014), 87-93. doi: 10.1016/j.aml.2014.07.013. [18] R. U. Haq, S. Nadeem, Z. H. Khan and N. F. M. Noor, Convective heat transfer in MHD slip flow over a stretching surface in the presence of carbon nanotubes, Physica B: Condensed Matter, 457 (2015), 40-47. [19] T. Hayat, Z. Hussain, T. Muhammad and A. Alsaedi, Effects of homogeneous and heterogeneous reactions in flow of nanofluids over a nonlinear stretching surface with variable surface thickness, Journal of Molecular Liquids, 221 (2016), 1121-1127. doi: 10.1016/j.molliq.2016.06.083. [20] T. Hayat, M. Imtiaz, A. Alsaedi and S. Almezal, On Cattaneo-Christov heat flux in MHD flow of Oldroyd-B fluid with homogeneous-heterogeneous reactions, Journal of Magnetism and Magnetic Materials, 401 (2016), 296-303. doi: 10.1016/j.jmmm.2015.10.039. [21] N. Ishfaq, Z. H. Khan, W. A. Khan and R. J. Culham, Estimation of boundary-layer flow of a nanofluid past a stretching sheet: A revised model, Journal of Hydrodynamics, Ser. B, 28 (2016), 596-602. doi: 10.1016/S1001-6058(16)60663-7. [22] R. Kandasamy, P. Loganathan and P. P. Arasu, Scaling group transformation for MHD boundary-layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection, Nuclear Engineering and Design, 241 (2011), 2053-2059. [23] W. A. Khan and I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, International Journal of Heat and Mass Transfer, 53 (2010), 2477-2483. doi: 10.1016/j.ijheatmasstransfer.2010.01.032. [24] M. Khan and W. A. Khan, Three-dimensional flow and heat transfer to burgers fluid using Cattaneo-Christov heat flux model, Journal of Molecular Liquids, 221 (2016), 651-657. doi: 10.1016/j.molliq.2016.06.041. [25] W. A. Khan, M. Khan and A. S. Alshomrani, Impact of chemical processes on 3D Burgers fluid utilizing Cattaneo-Christov double-diffusion: Applications of non-Fourier's heat and non-Fick's mass flux models, Journal of Molecular Liquids, 223 (2016), 1039-1047. doi: 10.1016/j.molliq.2016.09.027. [26] L. Liu, L. Zheng, F. Liu and X. Zhang, Anomalous convection diffusion and wave coupling transport of cells on comb frame with fractional Cattaneo-Christov flux, Communications in Nonlinear Science and Numerical Simulation, 38 (2016), 45-58. doi: 10.1016/j.cnsns.2016.02.009. [27] M. Mustafa, Cattaneo-Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid AIP Advances, 5 (2015), 047109. doi: 10.1063/1.4917306. [28] S. Nadeem, R. Mehmood and N. S. Akbar, Oblique Stagnation Point Flow of Carbon Nano Tube Based Fluid Over a Convective Surface, Journal of Computational and Theoretical Nanoscience, 12 (2015), 605-612. doi: 10.1166/jctn.2015.3774. [29] S. Naramgari and C. Sulochana, MHD flow over a permeable stretching/shrinking sheet of a nanofluid with suction/injection, Alexandria Engineering Journal, 55 (2016), 819-827. doi: 10.1016/j.aej.2016.02.001. [30] J. Qing, M. M. Bhatti, M. A. Abbas, M. M. Rashidi and M. E. S. Ali, Entropy generation on MHD Casson nanofluid flow over a porous stretching/shrinking surface, Entropy, 18 (2016), 123. [31] W. Rashmi, M. Khalid, A. F. Ismail, R. Saidur and A. K. Rashid, Experimental and numerical investigation of heat transfer in CNT nanofluids, Journal of Experimental Nanoscience, 10 (2013), 545-563. doi: 10.1080/17458080.2013.848296. [32] T. Salahuddin, M. Y. Malik, A. Hussain, S. Bilal and M. Awais, MHD flow of Cattanneo-Christov heat flux model for Williamson fluid over a stretching sheet with variable thickness: Using numerical approach, Journal of Magnetism and Magnetic Materials, 401 (2016), 991-997. doi: 10.1016/j.jmmm.2015.11.022. [33] M. Sheikholeslami, Effect of uniform suction on nanofluid flow and heat transfer over a cylinder, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 37 (2015), 1623-1633. doi: 10.1007/s40430-014-0242-z. [34] S. M. Sohel Murshed and C. A. Nieto de Castro, Superior thermal features of carbon nanotubes-based nanofluids-A review, Renewable and Sustainable Energy Reviews, 37 (2014), 155-167. doi: 10.1016/j.rser.2014.05.017. [35] B. Straughan, Thermal convection with the Cattaneo-Christov model, International Journal of Heat and Mass Transfer, 53 (2010), 95-98. doi: 10.1016/j.ijheatmasstransfer.2009.10.001. [36] C. Sulochana and N. Sandeep, Stagnation point flow and heat transfer behavior of Cu-water nanofluid towards horizontal and exponentially stretching/shrinking cylinders, Applied Nanoscience, 6 (2016), 451-459. doi: 10.1007/s13204-015-0451-5. [37] B. H. Thang, P. H. Khoi and P. N. Minh, A modified model for thermal conductivity of carbon nanotube-nanofluids Physics of Fluids, 27 (2015), 032002. doi: 10.1063/1.4914405. [38] B. H. Thang, P. H. Khoi and P. N. Minh, A modified model for thermal conductivity of carbon nanotube-nanofluids Physics of Fluids, 27 (2015), 032002. doi: 10.1063/1.4914405. [39] V. Tibullo and V. Zampoli, A uniqueness result for the Cattaneo-Christov heat conduction model applied to incompressible fluids, Mechanics Research Communications, 38 (2011), 77-79. doi: 10.1016/j.mechrescom.2010.10.008. [40] C. Y. Wang, Free convection on a vertical stretching surface, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 69 (1989), 418-420. doi: 10.1002/zamm.19890691115.

show all references

References:
 [1] N. S. Akbar, N. Kazmi, D. Tripathi and N. A. Mir, Study of heat transfer on physiological driven movement with CNT nanofluids and variable viscosity, Computer Methods and Programs in Biomedicine, 136 (2016), 21-29. doi: 10.1016/j.cmpb.2016.08.001. [2] N. S. Akbar, Entropy generation analysis for a CNT suspension nanofluid in plumb ducts with peristalsis, Entropy, 17 (2015), 1411-1424. doi: 10.3390/e17031411. [3] N. S. Akbar, D. Tripathi, Z. H. Khan and O. A. Bég, A numerical study of magnetohydrodynamic transport of nanofluids over a vertical stretching sheet with exponential temperature-dependent viscosity and buoyancy effects, Chemical Physics Letters, 661 (2016), 20-30. doi: 10.1016/j.cplett.2016.08.043. [4] N. S. Akbar and Z. H. Khan, Effect of variable thermal conductivity and thermal radiation with CNTS suspended nanofluid over a stretching sheet with convective slip boundary conditions: Numerical study, Journal of Molecular Liquids, 222 (2016), 279-286. doi: 10.1016/j.molliq.2016.06.102. [5] N. S. Akbar, A. Ebaid and Z. H. Khan, Numerical analysis of magnetic field effects on Eyring-Powell fluid flow towards a stretching sheet, Journal of Magnetism and Magnetic Materials, 382 (2015), 355-358. [6] M. M. Bhatti and M. M. Rashidi, Effects of thermo-diffusion and thermal radiation on Williamson nanofluid over a porous shrinking/stretching sheet, Journal of Molecular Liquids, 221 (2016), 567-573. doi: 10.1016/j.molliq.2016.05.049. [7] M. M. Bhatti, T. Abbas, M. M. Rashidi and M. E. S. Ali, Numerical simulation of entropy generation with thermal radiation on MHD Carreau nanofluid towards a shrinking sheet Entropy, 18 (2016), Paper No. 200, 14 pp. doi: 10.3390/e18060200. [8] M. M. Bhatti, T. Abbas, M. M. Rashidi, M. E. S. Ali and Z. Yang, Entropy generation on MHD eyring-powell nanofluid through a permeable stretching surface Entropy, 18 (2016), Paper No. 224, 14 pp. doi: 10.3390/e18060224. [9] D. M. Camarano, F. A. Mansur, T. L. C. F. Araújo, G. C. Salles and A. P. Santos, Thermophysical properties of ethylene glycol mixture based CNT nanofluids Journal of Physics: Conference Series, 733 (2016), 012015. doi: 10.1088/1742-6596/733/1/012015. [10] C. I. Christov, On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction, Mechanics Research Communications, 36 (2009), 481-486. doi: 10.1016/j.mechrescom.2008.11.003. [11] M. Ciarletta and B. Straughan, Uniqueness and structural stability for the Cattaneo-Christov equations, Mechanics Research Communications, 37 (2010), 445-447. doi: 10.1016/j.mechrescom.2010.06.002. [12] M. Ciarletta, B. Straughan and V. Tibullo, Christov-Morro theory for non-isothermal diffusion, Nonlinear Analysis: Real World Applications, 13 (2012), 1224-1228. doi: 10.1016/j.nonrwa.2011.10.014. [13] R. Ellahi, M. Hassan and A. Zeeshan, Study of natural convection MHD nanofluid by means of single and multi-walled carbon nanotubes suspended in a salt-water solution, IEEE Transactions on Nanotechnology, 14 (2015), 726-734. doi: 10.1109/TNANO.2015.2435899. [14] P. Estellé, S. Halelfadl and T. Maré, Thermal conductivity of CNT water based nanofluids: Experimental trends and models overview, Journal of Thermal Enginnering, 1 (2015), 381-390. [15] S. A. M. Haddad, Thermal instability in Brinkman porous media with Cattaneo-Christov heat flux, International Journal of Heat and Mass Transfer, 68 (2014), 659-668. doi: 10.1016/j.ijheatmasstransfer.2013.09.039. [16] S. Halelfadl, T. Maré and P. Estellé, Efficiency of carbon nanotubes water based nanofluids as coolants, Experimental Thermal and Fluid Science, 53 (2014), 104-110. doi: 10.1016/j.expthermflusci.2013.11.010. [17] S. Han, L. Zheng, C. Li and X. Zhang, Coupled flow and heat transfer in viscoelastic fluid with Cattaneo-Christov heat flux model, Applied Mathematics Letters, 38 (2014), 87-93. doi: 10.1016/j.aml.2014.07.013. [18] R. U. Haq, S. Nadeem, Z. H. Khan and N. F. M. Noor, Convective heat transfer in MHD slip flow over a stretching surface in the presence of carbon nanotubes, Physica B: Condensed Matter, 457 (2015), 40-47. [19] T. Hayat, Z. Hussain, T. Muhammad and A. Alsaedi, Effects of homogeneous and heterogeneous reactions in flow of nanofluids over a nonlinear stretching surface with variable surface thickness, Journal of Molecular Liquids, 221 (2016), 1121-1127. doi: 10.1016/j.molliq.2016.06.083. [20] T. Hayat, M. Imtiaz, A. Alsaedi and S. Almezal, On Cattaneo-Christov heat flux in MHD flow of Oldroyd-B fluid with homogeneous-heterogeneous reactions, Journal of Magnetism and Magnetic Materials, 401 (2016), 296-303. doi: 10.1016/j.jmmm.2015.10.039. [21] N. Ishfaq, Z. H. Khan, W. A. Khan and R. J. Culham, Estimation of boundary-layer flow of a nanofluid past a stretching sheet: A revised model, Journal of Hydrodynamics, Ser. B, 28 (2016), 596-602. doi: 10.1016/S1001-6058(16)60663-7. [22] R. Kandasamy, P. Loganathan and P. P. Arasu, Scaling group transformation for MHD boundary-layer flow of a nanofluid past a vertical stretching surface in the presence of suction/injection, Nuclear Engineering and Design, 241 (2011), 2053-2059. [23] W. A. Khan and I. Pop, Boundary-layer flow of a nanofluid past a stretching sheet, International Journal of Heat and Mass Transfer, 53 (2010), 2477-2483. doi: 10.1016/j.ijheatmasstransfer.2010.01.032. [24] M. Khan and W. A. Khan, Three-dimensional flow and heat transfer to burgers fluid using Cattaneo-Christov heat flux model, Journal of Molecular Liquids, 221 (2016), 651-657. doi: 10.1016/j.molliq.2016.06.041. [25] W. A. Khan, M. Khan and A. S. Alshomrani, Impact of chemical processes on 3D Burgers fluid utilizing Cattaneo-Christov double-diffusion: Applications of non-Fourier's heat and non-Fick's mass flux models, Journal of Molecular Liquids, 223 (2016), 1039-1047. doi: 10.1016/j.molliq.2016.09.027. [26] L. Liu, L. Zheng, F. Liu and X. Zhang, Anomalous convection diffusion and wave coupling transport of cells on comb frame with fractional Cattaneo-Christov flux, Communications in Nonlinear Science and Numerical Simulation, 38 (2016), 45-58. doi: 10.1016/j.cnsns.2016.02.009. [27] M. Mustafa, Cattaneo-Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid AIP Advances, 5 (2015), 047109. doi: 10.1063/1.4917306. [28] S. Nadeem, R. Mehmood and N. S. Akbar, Oblique Stagnation Point Flow of Carbon Nano Tube Based Fluid Over a Convective Surface, Journal of Computational and Theoretical Nanoscience, 12 (2015), 605-612. doi: 10.1166/jctn.2015.3774. [29] S. Naramgari and C. Sulochana, MHD flow over a permeable stretching/shrinking sheet of a nanofluid with suction/injection, Alexandria Engineering Journal, 55 (2016), 819-827. doi: 10.1016/j.aej.2016.02.001. [30] J. Qing, M. M. Bhatti, M. A. Abbas, M. M. Rashidi and M. E. S. Ali, Entropy generation on MHD Casson nanofluid flow over a porous stretching/shrinking surface, Entropy, 18 (2016), 123. [31] W. Rashmi, M. Khalid, A. F. Ismail, R. Saidur and A. K. Rashid, Experimental and numerical investigation of heat transfer in CNT nanofluids, Journal of Experimental Nanoscience, 10 (2013), 545-563. doi: 10.1080/17458080.2013.848296. [32] T. Salahuddin, M. Y. Malik, A. Hussain, S. Bilal and M. Awais, MHD flow of Cattanneo-Christov heat flux model for Williamson fluid over a stretching sheet with variable thickness: Using numerical approach, Journal of Magnetism and Magnetic Materials, 401 (2016), 991-997. doi: 10.1016/j.jmmm.2015.11.022. [33] M. Sheikholeslami, Effect of uniform suction on nanofluid flow and heat transfer over a cylinder, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 37 (2015), 1623-1633. doi: 10.1007/s40430-014-0242-z. [34] S. M. Sohel Murshed and C. A. Nieto de Castro, Superior thermal features of carbon nanotubes-based nanofluids-A review, Renewable and Sustainable Energy Reviews, 37 (2014), 155-167. doi: 10.1016/j.rser.2014.05.017. [35] B. Straughan, Thermal convection with the Cattaneo-Christov model, International Journal of Heat and Mass Transfer, 53 (2010), 95-98. doi: 10.1016/j.ijheatmasstransfer.2009.10.001. [36] C. Sulochana and N. Sandeep, Stagnation point flow and heat transfer behavior of Cu-water nanofluid towards horizontal and exponentially stretching/shrinking cylinders, Applied Nanoscience, 6 (2016), 451-459. doi: 10.1007/s13204-015-0451-5. [37] B. H. Thang, P. H. Khoi and P. N. Minh, A modified model for thermal conductivity of carbon nanotube-nanofluids Physics of Fluids, 27 (2015), 032002. doi: 10.1063/1.4914405. [38] B. H. Thang, P. H. Khoi and P. N. Minh, A modified model for thermal conductivity of carbon nanotube-nanofluids Physics of Fluids, 27 (2015), 032002. doi: 10.1063/1.4914405. [39] V. Tibullo and V. Zampoli, A uniqueness result for the Cattaneo-Christov heat conduction model applied to incompressible fluids, Mechanics Research Communications, 38 (2011), 77-79. doi: 10.1016/j.mechrescom.2010.10.008. [40] C. Y. Wang, Free convection on a vertical stretching surface, ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 69 (1989), 418-420. doi: 10.1002/zamm.19890691115.
Schematic representation of SWCNT and MWCNT nanofluids flow over porous stretching Surface
Velocity profile for different values of solid nanoparticles volume fraction with porosity parameter. (a) Suction parameter. (b) Impermeable sheet. (c) Injection parameter
Temperature profile for different values of solid nanoparticles volume fraction with porosity parameter. (a) Suction parameter. (b) Impermeable sheet. (c) Injection parameter
Temperature profile for different values of solid nanoparticles volume fraction with thermal relaxation time. (a) Suction parameter. (b) Impermeable sheet. (c) Injection parameter
Skin friction coefficient for SWCNT and MWCNT with porosity parameter. (a) Suction parameter (b) Impermeable sheet. (c) Injection parameter
Nusselt number for SWCNT and MWCNT with porosity parameter (a) Suction parameter. (b) Impermeable sheet. (c) Injection parameter
Nusselt number for SWCNT and MWCNT with thermal relaxation time (a) Suction parameter. (b) Impermeable sheet. (c) Injection parameter
Streamlines for (a) Suction parameter (b) Impermeable sheet (c) Injection parameter at $\phi =0.2, \gamma=1,N=0.5$
Isotherms for (a) Suction parameter (b) Impermeable sheet (c) Injection parameter at $\phi =0.2, \gamma=1,N=0.5$
Thermophysical properties of different base fluid and CNT's
 Physical properties Base fluid Nanoparticles Water SWCNT MWCNT $\rho$ (kg/m$^3$) 997 2,600 1,600 $c_p$(J/kg K) 4,179 425 796 $k$(W/m K) 0.613 6,600 3,000 r (nm) 0.1 10 10
 Physical properties Base fluid Nanoparticles Water SWCNT MWCNT $\rho$ (kg/m$^3$) 997 2,600 1,600 $c_p$(J/kg K) 4,179 425 796 $k$(W/m K) 0.613 6,600 3,000 r (nm) 0.1 10 10
Comparison of results for the skin friction for pure fluid $(\phi = 0)$
 N Present results Salahuddin et al. [32] Noreen et al. [5] 0.0 1 1 1 0.5 -1.11703 -1.11701 -1.11703 1 -1.41321 -1.41318 -1.41321 5 -2.44849 -2.44842 -2.44849 10 -3.31653 -3.31656 -3.31653 100 -10.04978 -10.04971 -10.04978 500 -22.38313 -22.38383 -22.38313 1000 -31.63849 -31.63856 -31.63869
 N Present results Salahuddin et al. [32] Noreen et al. [5] 0.0 1 1 1 0.5 -1.11703 -1.11701 -1.11703 1 -1.41321 -1.41318 -1.41321 5 -2.44849 -2.44842 -2.44849 10 -3.31653 -3.31656 -3.31653 100 -10.04978 -10.04971 -10.04978 500 -22.38313 -22.38383 -22.38313 1000 -31.63849 -31.63856 -31.63869
Comparison of results for the Nusselt number for pure fluid $(\phi = 0)$ with $N=0$ and $\gamma=0$
 Pr Present results Khan et al. [21] Khan Pop. [23] Wang [40] Kandasamy et al. [22] 0.07 0.0664 0.0664 0.0664 0.0655 0.0662 0.20 0.1692 0.1692 0.1692 0.1692 0.1692 0.70 0.4538 0.4538 0.4538 0.4538 0.4543 2 0.9113 0.9113 0.9114 0.9115 0.9115 7 1.8953 1.8953 1.8953 1.8953 1.8952 20 3.3538 3.3538 3.3538 3.3538 - 70 6.4621 6.4623 6.4622 6.4623 -
 Pr Present results Khan et al. [21] Khan Pop. [23] Wang [40] Kandasamy et al. [22] 0.07 0.0664 0.0664 0.0664 0.0655 0.0662 0.20 0.1692 0.1692 0.1692 0.1692 0.1692 0.70 0.4538 0.4538 0.4538 0.4538 0.4543 2 0.9113 0.9113 0.9114 0.9115 0.9115 7 1.8953 1.8953 1.8953 1.8953 1.8952 20 3.3538 3.3538 3.3538 3.3538 - 70 6.4621 6.4623 6.4622 6.4623 -
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