## THERMAL SCIENCE

International Scientific Journal

### EFFECTS OF CARBON NANOTUBES ON MAGNETOHYDRODYNAMIC FLOW OF METHANOL BASED NANOFLUIDS VIA ATANGANA-BALEANU AND CAPUTO-FABRIZIO FRACTIONAL DERIVATIVES

**ABSTRACT**

This paper dedicatedly reports the heat transfer analysis of single and multi-walls carbon nanotubes (SWCNTs and MWCNTs) for electrically conducting flow of Casson fluid. Both types of carbon nanotubes are suspended in methanol that is considered as a conventional base fluid. The governing partial differential equations of nanofluids have been modeled by employing newly defined fractional approaches (derivatives) namely Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional derivatives. The comparison of analytical solutions for temperature distribution and velocity field has been established via both approaches i-e (AB) and (CF) fractional operators. The general analytical solutions are expressed in the layout of Mittage-Leffler function Myε,δ(T)and generalized M-function Mpq (F) satisfying initial and boundary conditions. In order to have vivid rheological effects, the general analytical solutions in both cases (and CF fractional derivatives) are depicted for graphical illustrations. The comparison of three types of fluids (i) pure methanol (ii) methanol with single walls carbon nanotubes and (iii) methanol with multi-walls carbon nanotubes is portrayed via Atangana-Baleanu (AB) and Caputo-Fabrizio (CF) fractional derivatives. Finally, the results indicate that, pure methanol moves quicker in comparison with methanol-SWCNTs via Caputo-Fabrizio (CF)and methanol-MWCNTs, while for larger time, methanol-MWCNTs moves more rapidly in comparison with pure methanol and methanol-SWCNTs via Atangana-Baleanu (AB).

**KEYWORDS**

PAPER SUBMITTED: 2018-01-16

PAPER REVISED: 2018-05-14

PAPER ACCEPTED: 2018-05-16

PUBLISHED ONLINE: 2018-06-03

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**2**, PAGES [883 - 898]

- Hameed, M. Nadeem, S., Unsteady MHD flow of a non-Newtonian fluid on a porous plate, J. Math. Anal. Appl, 325 (2007), pp 724-733.
- Turkyilmazoglu, M., The analytical solution of mixed convection heat transfer and fluid flow of a MHD viscoelastic fluid over a permeable stretching surface, Int. J. Mech. Sci, 77 (2013), pp. 263-268.
- Rashid, M. M., et al., A study on heat transfer in a second grade fluid through a porous medium with the modified differential transform method, Heat Transfer Asian Res, 42 (2013), pp. 31-45.
- Hayat, T., et al., Flow of a second grade fluid with convective boundary conditions, Therm. Sci, 15 (2011), pp. 253-261.
- Kashif A. A., Porous Effects on Second Grade Fluid in Oscillating Plate, J. Appl. Environ. Biol. Sci, 6(5) (2016), pp. 71-82.
- Casson N., A flow equation for pigment oil suspensions of the printing ink type. In: Rheology of disperse systems, Mill CC (Ed.) Pergamon Press, Oxford, (1959), pp. 84-102.
- Mustafa M, et al., Unsteady boundary layer flow of a Casson fluid due to an impulsively started moving flat plate, Heat Transfer-Asian Research, 40 (2011), pp. 553-576.
- S.U.S. Choi., Enhancing thermal conductivity of fluids with nanoparticles, USA, ASME, FED 231/MD, 66 (1995), pp. 99-105.
- Buongiorno, J., Convective tranort in nanofluids, ASME J. Heat Transfer, 128 (2006), pp. 240-250.
- Tiwari, R. K., and Das, M. K., Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, International Journal of Heat and Mass Transfer, 50 (2007), pp. 9-10.
- Sheikhzadeh, G. A., et al., Laminar mixed convection of Cu-water nano-fluid in two sided lid-driven enclosures, Journal of Nanostructures, 1 (2012), pp. 44-53.
- Kashif, A. A., et al., An analytic study of molybdenum disulfide nanofluids using modern approach of Atangana-Baleanu fractional derivatives, European Physical Journal Plus, Eur. Phys. J. Plus (2017) 132: 439, DOI 10.1140/epjp/i2017-11689-y (2017).
- Sheikholeslami, M., et al., Effects of MHD on Cu-water nanofluid flow and heat transfer by means of CVFEM, J. Magn. Magn. Mater, 349 (2014), pp. 188-200.
- Aaiza, G., et al., Heat transfer in MHD mixed convection flow of a ferrofluid along a vertical channel, PLoS ONE, 10(11), e0141213 (2015).
- Khalid, A., et al., Exact solutions for free convection flow of nanofluids with ramped wall temperature, The European Physical Journal Plus, 130 (2015), pp. 57-71.
- Hayat, T., et al., On magnetohydrodynamic three dimensional flow of nanofluid over a convectively heated nonlinear stretching surface, Int. J. Heat Mass Transfer, 100 (2016), pp. 566-572.
- Hayat, T., et al., On magnetohydrodynamic flow of nanofluid due to a rotating disk with slip e¤ect: A numerical study, Comp. Methods Appl. Mech. Eng, 315 (2017), pp. 467-477.
- Aaiza, G., et al., Energy transfer in mixed convection MHD flow of nanofluid containing different shapes of nanoparticles in a channel filled with saturated porous medium, Nanoscale Research Letters, 10(1) (2015), doi: 10.1186/s11671-015-1144-4
- Liu, M.S., et al., Enhancement of thermal conductivity with carbon nanotube for nanofluids, Int. Comm. in Heat and Mass Transfer, 32 (2005), pp. 1202-1210.
- Marquis, F. D. S. and Chibnate, L. P. F., Improving the heat transfer of nanofluids and nanolubricants with Carbon nanotubes, JOM, (2005), pp. 32-43.
- Xie, H., et al., Nanofluids containing multiwall carbon nanotubes and their enhanced thermal conductivities, J. Appl. Phys, 94 (2003), pp. 4967-4971.
- Khan, W. A., et al., Fluid flow and heat transfer of carbon nanotubes along a flat plate with Navier slip boundary, Appl Nanosci., 4, (2014), pp. 633-641.
- Haq, R.U., et al., Convective heat transfer in MHD slip flow over a stretching surface in the presence of carbon nanotubes, Phys. B Condens. Matter, 457 (2015), pp. 40-47.
- Haq, R. U., et al., Thermophysical effects of carbon nanotubes on MHD flow over a stretching surface, Phys. E: Low Dimens. Sys. Nanostruct, 63 (2014), pp. 215-222.
- Camilli, L., et al., A three-dimensional carbon nanotube network for water treatment, Nanotechnology, 25 (2014).
- Kamli, R., and Binesh, A., Numerical investigation of heat transfer enhancement using carbon nanotube-based non-newtonian nanofluids, Int. Commun. Heat Mass Transfer, 37(8), pp. 1153-1157.
- Mohsen Sheikhholeslami, Houman B Rokni, Simulation of nanofluid heat transfer in presence of magnetic field:A review, International Journal of Heat and Mass Transfer 115 (2017) 1203-1233
- Mohsen Sheikholeslami, M.K. Sadoughi, Simulation of CuO-water nanofluid heat transfer enhancement in presence of melting surface, International Journal of Heat and Mass Transfer 116 (2018) 909-919.
- Mohsen Sheikholeslami, Numerical investigation of nanofluid free convection under the influence of electric field in a porous enclosure, Journal of Molecular Liquids 249 (2018) 1212-1221.
- Kashif Ali Abro, Ali Dad Chandio, Irfan Ali Abro, Ilyas Khan, Dual thermal analysis of magnetohydrodynamic flow of nanofluids via modern approaches of Caputo-Fabrizio and Atangana-Baleanu fractional derivatives embedded in porous medium, Journal of Thermal Analysis and Calorimetry, (2018) 1-11. doi.org/10.1007/s10973-018-7302-z
- M. Sheikholeslami, S.A. Shehzad, Simulation of water based nanofluid convective flow inside a porous enclosure via non-equilibrium model, International Journal of Heat and Mass Transfer 120 (2018) 1200-1212
- Ilyas Khan, Kashif Ali Abro, Thermal analysis in Stokes' second problem of nanofluid: Applications in thermal engineering, Case Studies in Thermal Engineering, (2018), Available online 10 April 2018, DOI: doi.org/10.1016/j.csite.2018.04.005
- Mohsen Sheikholeslami, Houman B. Rokni, Simulation of nanofluid heat transfer in presence of magnetic field: A review, International Journal of Heat and Mass Transfer 115 (2017) 1203-1233
- Mohsen Sheikholeslami, Mohadeseh Seyednezhad, Simulation of nanofluid flow and natural convection in a porous media under the influence of electric field using CVFEM, International Journal of Heat and Mass Transfer 120 (2018) 772-781.
- Kandasamy, R., et al., Single walled carbon nanotubes on MHD unsteady flow over a porous wedge with thermal radiation with variable stream conditions, Alexandria Engineering Journal, 55 (2016), pp. 275-285.
- Rahmat, E., et al., A study of heat transfer in power law nanofluid, Thermal Science, 20(6) (2016) 2015-2026.
- Kamel, M. S., et al., Effect of Wavy Surface Characteristics on Heat Transfer in a Wavy Square Cavity Filled with Nanofluid, International Journal of Heat and Mass Transfer, 107 (2017) 1110-1118.
- Rahmat, E., et al., On boundary layer magnetic flow of nano-Ferroliquid under the influence of low oscillating over stretchable rotating disk, Journal of Molecular Liquids, Vol. 229 (2017) 339-345
- Kamel, M. S., et al., Numerical Investigation of Heat Exchanger Effectiveness in a Double Pipe Heat Exchanger Filled With Nanofluid: A Sensitivity Analysis by Response Surface Methodology, Power Technology, 313 (2017) 99-111
- Esfahani, J.A., et al., Influences of wavy wall and nanoparticles on entropy generation in a plate heat exchanger, International Journal of Heat and Mass Transfer, 109 (2017) 1162-1171.
- Saman, R., et al., Convective heat transfer and particle motion in an obstructed duct with two side-by-side obstacles by means of DPM model, Applied Sciences, 7 (2017) 431-442
- Mohsan, H., et al., Particle shape effects on ferrofuids flow and heat transfer under influence of low oscillating magnetic field, Journal of Magnetism and Magnetic Materials 443 (2017) 36-44.
- Saman, R., et al., Volume of fluid model to simulate the nanofluid flow and entropy generation in a single slope solar still, Renewable Energy, 115 (2018) 400-410.
- N. Ijaz, A. Zeeshan, M. M. Bhatti, R. Ellahi, Analytical study on liquid-solid particles interaction in the presence of heat and mass transfer through a wavy channel, Journal of Molecular Liquids, 250 (2018) 80-87.
- A. Zeeshan, N. Shehzad, R. Ellahi, Analysis of activation energy in Couette-Poiseuille flow of nanofluid in the presence of chemical reaction and convective boundary conditions, Results in Physics, 8 (2018) 502-512.
- Kamel Milani Shirvan, Mojtaba Mamourian, R. Ellahi, Numerical Investigation and Optimization of Mixed Convection in Ventilated Square Cavity Filled with Nanofluid of different Inlet and Outlet port, International Journal of Numerical Methods for Heat and Fluid Flow, 27(9) 2053-2069 (2017).
- Ramana, J.V., et al., Impact of nonlinear radiation on 3D magnetohydrodynamic flow of methanol and kerosene based ferrofluids with temperature dependent viscosity, Journal of Molecular Liquids, 236 (2017), pp. 93-100.
- Muhammad Saqib, Farhad Ali, Ilyas Khan, Nadeem Ahmad Sheikh,Syed Aftab Alam Jan, Samiulhaq, Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo-Fabrizio fractional model, DOI: 10.1016/j.aej.2017.03.017
- Nadeem Ahmad Sheikh, Farhad Ali, Muhammad Saqib, Ilyas Khan, and Syed Aftab Alam Jan. A comparative study of Atangana-Baleanu and Caputo-Fabrizio fractional derivatives to the convective flow of a generalized Casson fluid, Eur. Phys. J. Plus (2017) 132 :54, DOI 10.1140/epjp/i2017-11326-y
- Nadeem Ahmad Sheikh, Farhad Ali, Ilyas Khan, Madeha Gohar, and Muhammad Saqib, On the applications of nanofluids to enhance the performance of solar collectors: A comparative analysis of Atangana-Baleanu and Caputo-Fabrizio fractional models, Eur. Phys. J. Plus (2017) 132: 540, DOI 10.1140/epjp/i2017-11809-9.
- Syed Aftab Alam Jan, Farhad Ali, Nadeem Ahmad Sheikh, Ilyas Khan, Muhammad Saqib, Madeha Gohar, Engine oil based generalized brinkman-type nano-liquid with molybdenum disulphide nanoparticles of spherical shape: Atangana-Baleanu fractional model, Numer Methods Partial Differential Eq. 2017;1-17. 10.1002/num.22200.
- Hone, J., et al., Thermal properties of carbon nanotubes and nanotube-based material, Appl. Phys. A, 74 (2002), pp. 339-343.
- Antar, Z., et al., Thermophysical and radiative properties of conductive biopolymer composite, Matter Sci Forum, 714 (2012), pp. 115-122.
- Maxwell, J. C., Electricity and magnetism, 3rd ed. (clarendon, Oxford, 1904).
- Jaffery, D. J., conduction through a random suspension of spheres, Proc Roy soc Lond ser A Math Phys Sci, 335 (1973), pp. 335-367.
- Davis, R., The effective thermal conductivity of a composite material with spherical inclusions, Int J Thermophys, 7 (1986), pp. 609-620.
- Hamilton, R. L., and Crosser, O. K., Thermal conductivity of heterogenous two-component systems, Ind Eng Chem Fund, 1(3) (1962), pp. 187-191.
- Xue, Q., Model for thermal conductivity of carbon nanotube-based composites, Phys B condens Matter, 368 (2005), pp. 302-307.
- Mustafa, M., and Junaid, K. A, Model for flow of Casson nanofluid past a non-linearly stretching sheet considering magnetic field effects, AIP Advances, 5 (2015), 077148.
- Kandasamy, R., et al., Impact of chemical reaction on Cu, Al2O3, and SWCNTs-nanofluid flow under slip conditions, Engineering Science and Technology, an International Journal, (2015).
- Abdon, A., Baleanu D., New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Sci, (2016),18, doi: 10.2298/TSCI160111018A .
- Kashif, A. A., et al., Analysis of Stokes' Second Problem for Nanofluids Using Modern Fractional Derivatives, Journal of Nanofluids, 7 (2018), pp. 738-747.
- Kashif, A. A., et al., Application of Atangana-Baleanu fractional derivative to convection flow of MHD Maxwell fluid in a porous medium over a vertical plate, Mathematical Modelling of Natural Phenomena, 13 (2018) 1, doi.org/10.1051/mmnp/2018007.
- Kashif, A., et al., Slippage of Magnetohydrodynamic Fractionalized Oldroyd-B Fluid in Porous Medium Progress in Fractional Differentiation and Applications, An International Journal, Progr. Fract. Differ. Appl, 3(1) (2017), pp. 69-80.
- Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl, 1(2) (2015), pp. 73-85.
- Qasem, A.M., et al., Analytical Solutions of Fractional Walter's-B Fluid with Applications, Complexity, (2018), Article ID 8918541.
- Kashif, A. A., et al., A comparative mathematical analysis of RL and RC electrical circuits via Atangana-Baleanu and Caputo-Fabrizio fractional derivatives, Eur. Phys. J. Plus, (2018) 133: 113, DOI 10.1140/epjp/i2018-11953-8.
- Sidra, A., Ilyas, K., Zulkhibri, I., Mohd, Z. S., Ali, S. A., Metib, S. A., Magnetic field effect on Poiseuille flow and heat transfer of carbon nanotubes along a vertical channel filled with Casson fluid, AIP ADVANCES 7, 015036 (2017).