Application of Atangana-Baleanu Fractional Derivative to ‎Carbon Nanotubes Based Non-Newtonian Nanofluid: ‎Applications in Nanotechnology

Document Type : Research Paper


1 Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Pakistan

2 Departamento de Ingeniería Electrónica, CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca Morelos, México‎

3 Universidad Virtual CNCI. Av. Ruiz Cortines No. 5901, Col. Valle Verde, 1er. Sector. Monterrey, Nuevo León, C.P. 64360, México‎


Single and multi-walled carbon nanotubes (SWCNTs & MWCNTs) comprise a large group of nanometer-thin hollow fibrous nanomaterials having physico-chemical characteristics like atomic configuration, length to diameter ratios, defects, impurities and functionalization. This manuscript is devoted for the analysis of carbon nanotubes based non-Newtonian nanofluid suspended in ethylene glycol taken as base fluid. The problem is modeled through modern method of fractional calculus namely Atangana-Baleanu fractional derivative and then solved analytically by invoking Laplace transform. The analytic solutions are established for the temperature and velocity distribution and expressed in terms of special function. The graphical results are depicted through computational software Mathcad and discussed for carbon nanotubes with various embedded parameters. An interesting comparison is explored graphically between single and multi-walled carbon nanotubes subject to the single and multi-walled carbon nanotubes are suspended in ethylene glycol. The several similarities and differences suggested that carbon nanotubes are accelerated and decelerated, while for unit time t = 1s, carbon nanotubes have identical velocities with and without fractional approach.


Main Subjects

[1] Choi, S.U.S., Zhang, Z.G., Yu, W., Lockwood, F.E., Grulke, E.A., Anomalous Thermal Conductivity Enhancement in Nanotube ‎Suspensions. Applied Physics Letters, 79(14), 2001, 2252-2254.‎
‎[2] Ramasubramaniam, R., Chen, J., Liu, H., Homogeneous Carbon Nanotube/Polymer Composites for Electrical Applications. Applied Physics Letters, ‎‎83, 2003, 2928-2930.‎
‎[3] Xue, Q.Z., Model for Thermal Conductivity of Carbon Nanotube-Based Composites. Physica B, 7, 2005, 302-307.‎
‎[4] Berber, S., Kwon, Y.K., Tomanek, D., Unusually High Thermal Conductivity of Carbon Nanotubes, Physical Review Letters, 84(20), 2000, ‎‎4613-4616.‎
‎[5] Hone, J., Llaguno, M.C., Biercuk, M.J., Johnson, A.T., Batlogg, B., Benes, Z., Fischer, J.E., Thermal Properties of Carbon Nanotubes and ‎Nanotube-Based Material. Applied Physics A, 74, 2002, 339–343.‎
‎[6] Jiang, W., Ding, G., Pang, H., Measurement and Model on Thermal Conductivities of Carbon Nanotube Nano-Refrigerants. International Journal of Thermal Sciences, 48, 2009, 1108–1115.‎
‎[7] Tan, G., Mieno, T., Experimental and Numerical Studies of Heat Convection in the Synthesis of Single-Walled Carbon Nanotubes by Arc ‎Vaporization. Japanese Journal of Applied Physics, 49(4), 2010, 1-9.‎
‎[8] Mayer, J., Mckrell, T., Grote, K., The Influence of Multi-Walled Carbon Nanotubes on Single-Phase Heat Transfer and Pressure Drop ‎Characteristics in the Transitional Flow Regime of Smooth Tubes. International Journal of Heat and Mass Transfer, 58(1-2), 2013, 597–609.‎
‎[9] Haq, R.U., Hammouch, Z., Khan, W.A., Water-Based Squeezing Flow in the Presence of Carbon Nanotubes Between Two Parallel Disks, ‎Thermal Sciences, 1, 2014, 148–158.‎
‎[10] Haq, R.U., Khan, Z.H., Khan, W.A., Thermophysical Effects of Carbon Nanotubes on MHD Flow Over a Stretching Surface. Physica E, 63, 2014, 215–222.‎
‎[11] Ellahi, R., Hassan, M., Zeeshan, A., Study of Natural Convection MHD Nanofluid by Means of Single and Multi Walled Carbon ‎Nanotubes Suspended in a Salt Water Solutions. IEEE Transactions on Nanotechnology, 14, 2015, 726–734.‎
‎[12]‎ Khan, U., Ahmed, N., Tauseef Mohyud-Din, S., Numerical Investigation for Three Dimensional Squeezing Flow of Nanofluid in a ‎Rotating Channel with Lower Stretching Wall Suspended by Carbon Nanotubes. Applied Thermal Engineering, 1, 2016, 1-11.‎
‎[13] Hayat, T., Khan, M.I., Farooq, M., Alsaedi, A., Yasmeen, T., Impact of Marangoni Convection in the Flow of Carbon-Water Nanofluid ‎with Thermal Radiation. International Journal of Heat and Mass Transfer, 106, 2017, 810-815.‎
‎[14] Khan, U., Ahmed, N., Mohyud-Din, S.T., Stoke's First Problem for Carbon Nanotubes Suspended Nanofluid Flow Under the Effect ‎of Slip Boundary Condition. Journal of Nanofluids, 5, 2016, 239-244.‎
‎[15] Khan, W.A., Khan, Z.H., Rahi. M., Fluid Flow and Heat Transfer of Carbon Nanotubes Along a Flat Plate with Navier Slip ‎Boundary. Applied Nanoscience, 4(5), 2014, 633-641.‎
‎[16] Fleming, E., Du, F., Ou, E., Dai, L., Shi. L., Thermal Conductivity of Carbon Nanotubes Grown by Catalyst-Free Chemical Vapor ‎Deposition in Nanopores. Carbon, 145, 2019, 195-200.‎
‎[17]‎ Ghazanfari, S.A., Wahid, M.A., HEAT TRANSFER ENHANCEMENT AND PRESSURE DROP FOR FIN-AND-TUBE COMPACT HEAT EXCHANGERS WITH DELTA WINGLET-TYPE VORTEX GENERATORS, Facta Universitatis, Series: Mechanical Engineering, 16(2), 2018, 233-247.‎
‎[18]‎ Bhojraj, L., Kashif, A.A., Abdul, W.S., Thermodynamical Analysis of Heat Transfer of Gravity‑Driven Fluid Flow via Fractional ‎Treatment: An Analytical Study. Journal of Thermal Analysis and Calorimetry, (2020)‎
‎[19]‎ Acar, B., Laminar Forced Convection of Various Nanofluids in Sudden Expansion Channels Under Constant Heat Flux: A CFD Study, International Journal of Applied Mechanics, 11(5), 2019, 1950049.
‎[20]‎ Farahi Shahri, m., Hossein Nezhad, A., Application of Various Electromagnetic Coupling Modes for the Better MHD Flow Distribution and Thermal Management Within a Liquid Metal Manifold, International Journal of Applied Mechanics, 10(5), 2018, 1850052.‎
‎[21]‎ Aziz, U.A., Mukarram, A., Kashif, A.A., Electroosmotic Slip Flow of Oldroyd-B Fluid Between Two Plates with Non-Singular Kernel. Journal of Computational and Applied Mathematics, 376, 2020, 112885.‎
‎[22] ‎Dogonchi, A.S., Chamkha, A.J., Seyyedi, S.M., Hashemi-Tilehnoee, M., Ganji, D.D., Viscous Dissipation Impact on Free ‎Convection Flow of Cu-Water Nanofluid in a Circular Enclosure with Porosity Considering Internal Heat Source. Journal of Applied and Computational Mechanics, 5(4), 2019, 717-726.‎
‎[23] Koca, I., Atangana, A., Solutions of Cattaneo-Hristov Model of Elastic Heat Diffusion with Caputo-Fabrizio and Atangana-Baleanu ‎Fractional Derivatives. Thermal Science, 21(6A), 2017, 2299-2305.‎
‎[24]‎ Abro, K.A., Abro, I.A., Almani, S.M., Khan, I., On the Thermal Analysis of Magnetohydrodynamic Jeffery Fluid via Modern Non Integer ‎Order derivative. Journal of King Saud University – Science, 31(4), 2019, 973-979.‎
‎[25]‎ Sheikholeslami, M., Ellahi, R., Three Dimensional Mesoscopic Simulation of Magnetic Field Effect on Natural Convection of Nanofluid. ‎International Journal of Heat and Mass Transfer, 89, 2015, 799–808.‎
‎[26]‎ Abro, K.A., Rashidi, M.M., Khan, I., Abro, I.A., Tassadiq, A., Analysis of Stokes’ Second Problem for Nanofluids Using Modern Fractional ‎Derivatives. Journal of Nanofluids, 7, 2018, 738–747.‎
‎[27]‎ Sheikholeslami, M., Ellahi, R., Simulation of Ferrofluid Flow for Magnetic Drug Targeting Using Lattice Boltzmann Method. Zeitschrift Fur Naturforschung A, 70, 2015, 115–124.‎
‎[28]‎ Khan, I., Abro, K.A., Thermal Analysis in Stokes’ Second Problem of Nanofluid: Applications in Thermal Engineering, Case Studies in Thermal Engineering, 12, 2018, 271-275.‎
‎[29]‎ Zeehan, A., Ellahi, R., Hassan, M., Magnetohydrodynamic Flow of Water/Ethylene Glycol Based Nanofluids with Natural Convection ‎Through Porous Medium. European Physical Journal Plus, 129, 2014, 1-12.‎
‎[30]‎ Ellahi, R., Aziz, S., Zeeshan, A., Non-Newtonian Nanofluids Flow Through a Porous Medium Between Two Coaxial Cylinders with Heat ‎Transfer and Variable Viscosity. Journal of Porous Media, 16(3), 2013, 205–216.‎
‎[31]‎ Abro, K.A., Chandio, A.D., Abro, I.A., Khan, I., 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, 135, 2019, 2197–2207. ‎
‎[32]‎ Rehman, S.U., Haq, R.U., Khan, Z.H., Lee, C., Entropy Generation Analysis for Non-Newtonian Nanofluid with Zero Normal Flux of ‎Nanoparticles at the Stretching Surface. Journal of the Taiwan Institute of Chemical Engineers, 63, 2016, 226-235.‎
‎[33] Khan, Z.H., Hussain, S.T., Hammouch, Z., Flow and Heat Transfer Analysis of Water and Ethylene Glycol Based Cu nanoparticles ‎between two parallel disks with suction/injection effects. Journal of Molecular Liquids, 221, 2016, 298-304.‎
‎[34]‎ Kashif, A.A, Abdon, A., Role of Non-integer and Integer Order Differentiations on the Relaxation Phenomena of Viscoelastic Fluid, ‎Physica Scripta, 2020, doi: 10.1088/1402-4896/ab560c‎
‎[35] Atangana, A., Koca, I., On the New Fractional Derivative and Application to Nonlinear Baggs and Freedman Model. Journal of Nonlinear Sciences and Applications, 9, 2016, 2467-2480.‎
‎[36]‎ Abro, K.A., Gomez-Aguilar, J.F., Khan, I., Nisar, K.S., Role of Modern Fractional Derivatives in an Armature-Controlled DC Servomotor. ‎European Physical Journal Plus, 134(553), 2019, 1-16.‎
‎[37] Yavuz, M., Özdemir, N., Comparing the New Fractional Derivative Operators Involving Exponential and Mittag-Leffler ‎Kernel. Discrete & Continuous Dynamical Systems-S, 2, 2019, 1098-1107.‎
‎[38] Atangana, A., Baleanu, D., New Fractional Derivatives with Nonlocal and Nonsingular Kernel: Theory and Application to Heat ‎Transfer Model. Thermal Science, 20(2), 2016, 763-769.‎
‎[39]‎ Abro, K.A., Pervaiz, H.S., Jose, F.G.A, Ilyas, K., Analysis of De-Levie’s Model via Modern Fractional Differentiations: An Application to ‎Supercapacitor. Alexandria Engineering Journal, 58(4), 2019 58, 1375–1384.‎
‎[40]‎ Abro, K.A., Hussain, M., Baig, M.M., An Analytic Study of Molybdenum Disulfide Nanofluids Using Modern Approach of Atangana-‎Baleanu Fractional Derivatives. European Physical Journal Plus, 132, 2017, 1-14.‎
‎[41]‎ Atangana, A., Baleanu, D., Caputo–Fabrizio Derivative Applied to Groundwater Flow Within Confined Aquifer. Journal of Engineering Mechanics, 142, 2016, 1-‎‎8.‎
‎[42] Abro, K.A., Irfan, A.A, Ahmed, Y., A Comparative Analysis of Sulfate Ion Concentration via Modern Fractional Derivatives: An ‎Industrial Application to Cooling System of Power Plant. Physica A: Statistical Mechanics and its Applications, 541, 2020, 123306.‎
‎[43]‎ Abro, K.A., Memon, A.A., Uqaili, M.A., A Comparative Mathematical Analysis of RL and RC Electrical Circuits via Atangana-Baleanu ‎and Caputo-Fabrizio Fractional Derivatives. European Physical Journal Plus, 133, 2018, 1-8.‎
‎[44] Kashif, A.A., Atangana, A., Mathematical Analysis of Memristor Through Fractal‐Fractional Differential Operators: A Numerical ‎Study, Mathematical Methods in the Applied Sciences, 2020,‎
‎[45] Casson, N., A Flow Equation for the Pigment Oil Suspensions of the Printing Ink Type in Rheology of Disperse Systems. ‎Pergamon, New York, NY, USA, 1, 1959.‎
‎[46] Wang, J., Zhu, J., Zhang, X., Chen, Y., Heat Transfer and Pressure Drop of Nanofluids Containing Carbon Nanotubes in Laminar ‎Flows. Experimental Thermal and Fluid Science, 44, 2013, 716-721.‎
‎[47]‎ Mustafa, M., Khan, J.A., Model for Flow of Casson Nanofluid Past a Nonlinearly Stretching Sheet Considering Magnetic Field Effects. ‎AIP Advances, 5, 2015, 1-9.‎
‎[48]‎ Abro, K.A., Khan, I., Analysis of Heat and Mass Transfer in MHD Flow of Generalized Casson Fluid in a Porous Space Via Non-Integer ‎Order Derivative without Singular Kernel. Chinese Journal of Physics, 55(4), 2017, 1583-1595.‎
‎[49] Chaudhary, R.C., Jain, P., Unsteady Free Convection Boundary Layer Flow Past an Impulsively Started Vertical Surface with ‎Newtonian Heating. Romanian Journal of Physics, 51, 2006, 911–925.‎
‎[50]‎ Al-Mdallal, Q., Abro, K.A., Khan, I., Analytical Solutions of Fractional Walter's-B Fluid with Applications. Complexity, 1, 2018, 1-8.‎
‎[51]‎ Abro, K.A., Solangi, M.A., Heat Transfer in Magnetohydrodynamic Second Grade Fluid with Porous Impacts using Caputo-Fabrizio ‎Fractional Derivatives. Punjab University Journal of Mathematics, 49(2), 2017, 113-125. ‎
‎[52]‎ Ouakad, H.M., Sedighi, H.M., Rippling Effect on the Structural Response of Electrostatically Actuated Single-Walled Carbon Nanotube ‎Based NEMS Actuators. International Journal of Non-Linear Mechanics, 87, 2016, 97-108.‎
‎[53]‎ Sedighi, H.M., Daneshmand, F., Static and Dynamic Pull-in Instability of Multi-Walled Carbon Nanotube Probes by He’s Iteration ‎Perturbation Method. Journal of Mechanical Science and Technology, 28(9), 2014, 3459-3469.‎
‎[54]‎ Abro, K.A., Atangana, A., A Comparative Study of Convective Fluid Motion in Rotating Cavity via Atangana–Baleanu and Caputo–‎Fabrizio Fractal-Fractional Differentiations. European Physical Journal Plus, 135(226), 2020, 1-12.‎
‎[55]‎ Kashif, A.A., Siyal, A., Atangana, A., Thermal Stratification of Rotational Second-Grade Fluid through Fractional Differential Operators. Journal of Thermal Analysis and Calorimetry, 2020,‎
‎[56] Eltaher, M., Agwa, M., Kabeel, A., Vibration Analysis of Material Size-Dependent CNTs Using Energy Equivalent Model, Journal of Applied and Computational Mechanics, 4(2), 2018, 75-86.‎
‎[57]‎ Sedighi, H.M., Yaghootian, A., Dynamic Instability of Vibrating Carbon Nanotubes near Small Layers of Graphite Sheets Based on ‎Nonlocal Continuum Elasticity. Journal of Applied Mechanics and Technical Physics, 57(1), 2016, 90-100.‎
‎[58] Sedighi, H.M., Farjam, N., A Modified Model for Dynamic Instability of CNT Based Actuators by Considering Rippling ‎Deformation Tip-Charge Concentration and Casimir Attraction. Microsystem Technologies, 23(6), 2016, 2175-2191.‎
‎[59]‎ Kashif, A.A., A Fractional and Analytic Investigation of Thermo-Diffusion Process on Free Convection Flow: An Application to Surface ‎Modification Technology. European Physical Journal Plus, 1, 2020, 1-16.‎
‎[60]‎ Sedighi, H.M., Malikan, M., Stress-Driven Nonlocal Elasticity for Nonlinear Vibration Characteristics of Carbon/Boron-Nitride Hetero-‎Nanotube Subject to Magneto-Thermal Environment. Physica Scripta, 95(5), 2020, 055218.‎
‎[61]‎ Sedighi, H.M., Divergence and Flutter Instability of Magneto-Thermo-Elastic C-BN Hetero-Nanotubes Conveying Fluid. Acta Mechanica Sinica, 36, 2020, 381–396.‎