# American Institute of Mathematical Sciences

## Channel flow of fractionalized H2O-based CNTs nanofluids with Newtonian heating

 1 Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam 2 Department of Mathematics, City University of Science and Information Technology, Peshawar, 25000, Pakistan 3 Department of Applied Mathematics, Princess Nourah bint Abdulrahman University Riyadh, Saudi Arabia

* Corresponding author: Ilyas Khan, ilyaskhan@tdt.edu.vn

Received  March 2018 Revised  May 2018 Published  March 2019

The present article deals to study heat transfer analysis due to convection occurs in a fractionalized H2O-based CNTs nanofluids flowing through a vertical channel. The problem is modeled in terms of fractional partial differential equations using a modern trend of the fractional derivative of Atangana and Baleanu. The governing equation (momentum and energy equations) are subjected to physical initial and boundary conditions. The fractional Laplace transformation is used to obtain solutions in the transform domain. To obtain semi-analytical solutions for velocity and temperature distributions, the Zakian's algorithm is utilized for the Laplace inversions. For validation, the obtained solutions are compared in tabular form using Tzou's and Stehfest's numerical methods for Laplace inversion. The influence of fractional parameter is studied and presented in graphs and discussed.

Citation: Ilyas Khan, Muhammad Saqib, Aisha M. Alqahtani. Channel flow of fractionalized H2O-based CNTs nanofluids with Newtonian heating. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020043
##### References:
 [1] A. A. Alrashed, O. A. Akbari, A. Heydari, D. Toghraie, M. Zarringhalam, G. A. S. Shabani, A. R. Seifi and M. Goodarzi, The numerical modeling of water/fmwcnt nanofluid flow and heat transfer in a backward-facing contracting channel, Physica B: Condensed Matter, 537 (2018), 176-183. doi: 10.1016/j.physb.2018.02.022. Google Scholar [2] S. Aman, I. Khan, Z. Ismail, M. Z. Salleh, A. S. Alshomrani and M. S. Alghamdi, Magnetic field effect on poiseuille flow and heat transfer of carbon nanotubes along a vertical channel filled with casson fluid, AIP Advances, 7 (2017), 015036. doi: 10.1063/1.4975219. Google Scholar [3] S. Aminossadati and B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure, European Journal of Mechanics-B/Fluids, 28 (2009), 630-640. doi: 10.1016/j.euromechflu.2009.05.006. Google Scholar [4] A. Arabpour, A. Karimipour and D. Toghraie, The study of heat transfer and laminar flow of kerosene/multi-walled carbon nanotubes (mwcnts) nanofluid in the microchannel heat sink with slip boundary condition, Journal of Thermal Analysis and Calorimetry, 131 (2018), 1553-1566. doi: 10.1007/s10973-017-6649-x. Google Scholar [5] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with markovian and non-markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056. Google Scholar [6] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Chaos, 28 (2018), 063109, 6 pp, arXiv:1602.03408. doi: 10.1063/1.5026284. Google Scholar [7] A. Atangana and J. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3. Google Scholar [8] W. A. Azhar, D. Vieru and C. Fetecau, Free convection flow of some fractional nanofluids over a moving vertical plate with uniform heat flux and heat source, Physics of Fluids, 29 (2017), 082001. doi: 10.1063/1.4996034. Google Scholar [9] H. Brinkman, The viscosity of concentrated suspensions and solutions, The Journal of Chemical Physics, 20 (1952), 571-571. doi: 10.1063/1.1700493. Google Scholar [10] S. Chol and J. Estman, "Enhancing thermal conductivity of fluids with nanoparticles," ASME-Publications-Fed, vol. 231, pp. 99-106, 1995.Google Scholar [11] C. Fetecau, D. Vieru and W. A. Azhar, Natural convection flow of fractional nanofluids over an isothermal vertical plate with thermal radiation, Applied Sciences, 7 (2017), 247. doi: 10.3390/app7030247. Google Scholar [12] R. Ganvir, P. Walke and V. Kriplani, Heat transfer characteristics in nanofluid-a review, Renewable and Sustainable Energy Reviews, 75 (2017), 451-460. doi: 10.1016/j.rser.2016.11.010. Google Scholar [13] D. Halsted and D. Brown, Zakian's technique for inverting laplace transforms, The Chemical Engineering Journal, 3 (1972), 312-313. doi: 10.1016/0300-9467(72)85037-8. Google Scholar [14] R. U. Haq, F. Shahzad and Q. M. Al-Mdallal, Mhd pulsatile flow of engine oil based carbon nanotubes between two concentric cylinders, Results in Physics, 7 (2017), 57-68. doi: 10.1016/j.rinp.2016.11.057. Google Scholar [15] M. Hassan, A. Faisal and M. M. Bhatti, Interaction of aluminum oxide nanoparticles with flow of polyvinyl alcohol solutions base nanofluids over a wedge, Applied Nanoscience, 8 (2018), 53-60. doi: 10.1007/s13204-018-0651-x. Google Scholar [16] S. Iijima, Helical microtubules of graphitic carbon, nature, 354 (1991), 56-58. doi: 10.1038/354056a0. Google Scholar [17] S. A. A. Jan, F. Ali, N. A. Sheikh, I. Khan, M. Saqib and M. Gohar, Engine oil based generalized brinkman-type nano-liquid with molybdenum disulphide nanoparticles of spherical shape: Atangana-baleanu fractional model, Numerical Methods for Partial Differential Equations, 34 (2018), 1472-1488. doi: 10.1002/num.22200. Google Scholar [18] M. H. Matin and I. Pop, Forced convection heat and mass transfer flow of a nanofluid through a porous channel with a first order chemical reaction on the wall, International Communications in Heat and Mass Transfer, 46 (2013), 134-141. doi: 10.1016/j.icheatmasstransfer.2013.05.001. Google Scholar [19] S. S. Murshed, C. N. De Castro, M. Lourenço, M. Lopes and F. Santos, A review of boiling and convective heat transfer with nanofluids, Renewable and Sustainable Energy Reviews, 15 (2011), 2342-2354. doi: 10.1016/j.rser.2011.02.016. Google Scholar [20] M. R. Safaei, G. Ahmadi, M. S. Goodarzi, A. Kamyar and S. Kazi, Boundary layer flow and heat transfer of fmwcnt/water nanofluids over a flat plate, Fluids, 1 (2016), 31. doi: 10.3390/fluids1040031. Google Scholar [21] M. Saqib, F. Ali, I. Khan, N. A. Sheikh and S. B. Shafie, Convection in ethylene glycol-based molybdenum disulfide nanofluid, Journal of Thermal Analysis and Calorimetry, (2018), 1-10. doi: 10.1007/s10973-018-7054-9. Google Scholar [22] N. A. Sheikh, F. Ali, I. Khan, M. Gohar and M. Saqib, On the applications of nanofluids to enhance the performance of solar collectors: A comparative analysis of atangana-baleanu and caputo-fabrizio fractional models, The European Physical Journal Plus, 132 (2017), 540.Google Scholar [23] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 52. Google Scholar [24] D. Y. Tzou, Macro-to Microscale Heat Transfer: The Lagging Behavior, John Wiley & Sons, 2014. doi: 10.1002/9781118818275. Google Scholar [25] Q. Wang and H. Zhan, On different numerical inverse laplace methods for solute transport problems, Advances in Water Resources, 75 (2015), 80-92. doi: 10.1016/j.advwatres.2014.11.001. Google Scholar [26] Q. Xue, Model for thermal conductivity of carbon nanotube-based composites, Physica B: Condensed Matter, 368 (2005), 302-307. doi: 10.1016/j.physb.2005.07.024. Google Scholar [27] V. Zakian and R. Littlewood, Numerical inversion of laplace transforms by weighted least-squares approximation, The Computer Journal, 16 (1973), 66-68. doi: 10.1093/comjnl/16.1.66. Google Scholar

show all references

##### References:
 [1] A. A. Alrashed, O. A. Akbari, A. Heydari, D. Toghraie, M. Zarringhalam, G. A. S. Shabani, A. R. Seifi and M. Goodarzi, The numerical modeling of water/fmwcnt nanofluid flow and heat transfer in a backward-facing contracting channel, Physica B: Condensed Matter, 537 (2018), 176-183. doi: 10.1016/j.physb.2018.02.022. Google Scholar [2] S. Aman, I. Khan, Z. Ismail, M. Z. Salleh, A. S. Alshomrani and M. S. Alghamdi, Magnetic field effect on poiseuille flow and heat transfer of carbon nanotubes along a vertical channel filled with casson fluid, AIP Advances, 7 (2017), 015036. doi: 10.1063/1.4975219. Google Scholar [3] S. Aminossadati and B. Ghasemi, Natural convection cooling of a localised heat source at the bottom of a nanofluid-filled enclosure, European Journal of Mechanics-B/Fluids, 28 (2009), 630-640. doi: 10.1016/j.euromechflu.2009.05.006. Google Scholar [4] A. Arabpour, A. Karimipour and D. Toghraie, The study of heat transfer and laminar flow of kerosene/multi-walled carbon nanotubes (mwcnts) nanofluid in the microchannel heat sink with slip boundary condition, Journal of Thermal Analysis and Calorimetry, 131 (2018), 1553-1566. doi: 10.1007/s10973-017-6649-x. Google Scholar [5] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with markovian and non-markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056. Google Scholar [6] A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Chaos, 28 (2018), 063109, 6 pp, arXiv:1602.03408. doi: 10.1063/1.5026284. Google Scholar [7] A. Atangana and J. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3. Google Scholar [8] W. A. Azhar, D. Vieru and C. Fetecau, Free convection flow of some fractional nanofluids over a moving vertical plate with uniform heat flux and heat source, Physics of Fluids, 29 (2017), 082001. doi: 10.1063/1.4996034. Google Scholar [9] H. Brinkman, The viscosity of concentrated suspensions and solutions, The Journal of Chemical Physics, 20 (1952), 571-571. doi: 10.1063/1.1700493. Google Scholar [10] S. Chol and J. Estman, "Enhancing thermal conductivity of fluids with nanoparticles," ASME-Publications-Fed, vol. 231, pp. 99-106, 1995.Google Scholar [11] C. Fetecau, D. Vieru and W. A. Azhar, Natural convection flow of fractional nanofluids over an isothermal vertical plate with thermal radiation, Applied Sciences, 7 (2017), 247. doi: 10.3390/app7030247. Google Scholar [12] R. Ganvir, P. Walke and V. Kriplani, Heat transfer characteristics in nanofluid-a review, Renewable and Sustainable Energy Reviews, 75 (2017), 451-460. doi: 10.1016/j.rser.2016.11.010. Google Scholar [13] D. Halsted and D. Brown, Zakian's technique for inverting laplace transforms, The Chemical Engineering Journal, 3 (1972), 312-313. doi: 10.1016/0300-9467(72)85037-8. Google Scholar [14] R. U. Haq, F. Shahzad and Q. M. Al-Mdallal, Mhd pulsatile flow of engine oil based carbon nanotubes between two concentric cylinders, Results in Physics, 7 (2017), 57-68. doi: 10.1016/j.rinp.2016.11.057. Google Scholar [15] M. Hassan, A. Faisal and M. M. Bhatti, Interaction of aluminum oxide nanoparticles with flow of polyvinyl alcohol solutions base nanofluids over a wedge, Applied Nanoscience, 8 (2018), 53-60. doi: 10.1007/s13204-018-0651-x. Google Scholar [16] S. Iijima, Helical microtubules of graphitic carbon, nature, 354 (1991), 56-58. doi: 10.1038/354056a0. Google Scholar [17] S. A. A. Jan, F. Ali, N. A. Sheikh, I. Khan, M. Saqib and M. Gohar, Engine oil based generalized brinkman-type nano-liquid with molybdenum disulphide nanoparticles of spherical shape: Atangana-baleanu fractional model, Numerical Methods for Partial Differential Equations, 34 (2018), 1472-1488. doi: 10.1002/num.22200. Google Scholar [18] M. H. Matin and I. Pop, Forced convection heat and mass transfer flow of a nanofluid through a porous channel with a first order chemical reaction on the wall, International Communications in Heat and Mass Transfer, 46 (2013), 134-141. doi: 10.1016/j.icheatmasstransfer.2013.05.001. Google Scholar [19] S. S. Murshed, C. N. De Castro, M. Lourenço, M. Lopes and F. Santos, A review of boiling and convective heat transfer with nanofluids, Renewable and Sustainable Energy Reviews, 15 (2011), 2342-2354. doi: 10.1016/j.rser.2011.02.016. Google Scholar [20] M. R. Safaei, G. Ahmadi, M. S. Goodarzi, A. Kamyar and S. Kazi, Boundary layer flow and heat transfer of fmwcnt/water nanofluids over a flat plate, Fluids, 1 (2016), 31. doi: 10.3390/fluids1040031. Google Scholar [21] M. Saqib, F. Ali, I. Khan, N. A. Sheikh and S. B. Shafie, Convection in ethylene glycol-based molybdenum disulfide nanofluid, Journal of Thermal Analysis and Calorimetry, (2018), 1-10. doi: 10.1007/s10973-018-7054-9. Google Scholar [22] N. A. Sheikh, F. Ali, I. Khan, M. Gohar and M. Saqib, On the applications of nanofluids to enhance the performance of solar collectors: A comparative analysis of atangana-baleanu and caputo-fabrizio fractional models, The European Physical Journal Plus, 132 (2017), 540.Google Scholar [23] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 52. Google Scholar [24] D. Y. Tzou, Macro-to Microscale Heat Transfer: The Lagging Behavior, John Wiley & Sons, 2014. doi: 10.1002/9781118818275. Google Scholar [25] Q. Wang and H. Zhan, On different numerical inverse laplace methods for solute transport problems, Advances in Water Resources, 75 (2015), 80-92. doi: 10.1016/j.advwatres.2014.11.001. Google Scholar [26] Q. Xue, Model for thermal conductivity of carbon nanotube-based composites, Physica B: Condensed Matter, 368 (2005), 302-307. doi: 10.1016/j.physb.2005.07.024. Google Scholar [27] V. Zakian and R. Littlewood, Numerical inversion of laplace transforms by weighted least-squares approximation, The Computer Journal, 16 (1973), 66-68. doi: 10.1093/comjnl/16.1.66. Google Scholar
Flow configuration and coordinate system
Variation of the velocity profile for water-based MWCNT nanofluid due to $\alpha$when $\phi = 0.03\, \mathit{\rm{and}}\, Gr = 5$
Variation of the temperature profile for water-based MWCNT nanofluid due to $\alpha$when $\phi = 0.03.$
Variation of the velocity profile for water based MWCNT nanofluid due to $\phi$when $\phi = 0.5\, \mathit{\rm{and}}\, Gr = 5$
Variation of the temperature profile for water-based MWCNT nanofluid due to $\phi$when $\phi = 0.5\, \mathit{\rm{and}}\, Gr = 5$
Comparison of velocity profiles for water-based MWCNT and SWCNT nanofluids when $\alpha = 0.5,\, \phi = 0.3\, \mathit{\rm{and}}\, Gr = 5$
Comparison of temperature profiles for water-based MWCNT and SWCNT nanofluids when $\alpha = 0.5,\, \phi = 0.3\, \mathit{\rm{and}}\, Gr = 5$
Comparison of velocity profile using different algorithms
Comparison of temperature distribution using different algorithms
Thermophysical properties of water and CNTs nanoparticles
 Material Base fluid Nanoparticles Water MWCNT SWCNT ${\rho \left( {{\rm{kg}}/{{\rm{m}}^3}} \right)}$ 997 1600 2600 ${C_p}\left( {{\rm{J}}/{\rm{kg}}\;{\rm{K}}} \right)$ 4179 796 425 ${K\left( {{\rm{W}}/{\rm{mK}}} \right)}$ 0.613 6600 Pr 6.2 - -
 Material Base fluid Nanoparticles Water MWCNT SWCNT ${\rho \left( {{\rm{kg}}/{{\rm{m}}^3}} \right)}$ 997 1600 2600 ${C_p}\left( {{\rm{J}}/{\rm{kg}}\;{\rm{K}}} \right)$ 4179 796 425 ${K\left( {{\rm{W}}/{\rm{mK}}} \right)}$ 0.613 6600 Pr 6.2 - -
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