Document Type: Research Paper


1 Department of Mechanical Engineering, Islamic Azad University, Gonabad Branch, Gonabad, Iran

2 Young Researchers and Elite Club, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran

3 Young Researchers and Elite Club, Gonabad Branch, Islamic Azad University, Gonabad, Iran

4 Faculty of Civil Engineering and Environment, Khavaran Institute of Higher Education, Mashhad, Iran



In the present work, the study of alumina-water nanofluid heat transfer between two concentric vertical cylinders has been done by modified Buongiorno’s model (BM) to examine the impacts of temperature jump and slip velocity boundary conditions for a wide range of Knudsen number. Runge-Kutta-Fehlberg method as a standard integration scheme along with a shooting method, has been chosen for solving nonlinear ordinary differential equations (ODEs) along with boundary conditions. The main concentration of this paper focuses on the temperature jump since the slip velocity has been extensively studied in many studies. The presence of temperature jump boundary condition by varying Knudsen number was considered to investigate the effects of the bulk mean nanoparticle volume fraction ϕB, mixed convection parameter Nr, buoyancy parameter Ng, and heat flux ratio ε on the total dimensionless heat transfer coefficient HTC and the dimensionless pressure gradient Ndp. The obtained results indicate that temperature jump boundary condition plays a pivotal role in temperature profile, heat transfer coefficient and pressure drop; for instance, the negligence of temperature jump near walls causes to undervalue heat transfer coefficient in continuum flow regime and overestimate it in slip flow regime.

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Main Subjects

[1] J. Maxwell, "Treatise on Magnetism and Electricity," Vol II, Art.  pp. 717-9, (1873).


[2] S. U. Choi, J. A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles. Argonne National Lab., IL (United States); 1995.


[3] H. Masuda, A. Ebata, K. Teramae, "Alteration of thermal conductivity and viscosity of liquid by dispersing ultra-fine particles. Dispersion of Al2O3, SiO2 and TiO2 ultra-fine particles," NETSU BUSSEI.  Vol. 7,  No. 4,  pp. 227-33, (1993).


[4] M. Sheikholeslami, "Influence of magnetic field on Al2O3-H2O nanofluid forced convection heat transfer in a porous lid driven cavity with hot sphere obstacle by means of LBM," Journal of Molecular Liquids.  Vol. 263,  pp. 472-88, (2018).


[5] M. Sheikholeslami, Z. Li, A. Shafee, "Lorentz forces effect on NEPCM heat transfer during solidification in a porous energy storage system," International Journal of Heat and Mass Transfer.  Vol. 127,  pp. 665-74, (2018).


[6] M. Sheikholeslami, "Finite element method for PCM solidification in existence of CuO nanoparticles," Journal of Molecular Liquids.  Vol. 265,  pp. 347-55, (2018).


[7] M. Sheikholeslami, M. Darzi, Z. Li, "Experimental investigation for entropy generation and exergy loss of nano-refrigerant condensation process," International Journal of Heat and Mass Transfer.  Vol. 125,  pp. 1087-95, (2018).


[8] M. Sheikholeslami, M. B. Gerdroodbary, S. V. Mousavi, D. Ganji, R. Moradi, "Heat transfer enhancement of ferrofluid inside an 90 elbow channel by non-uniform magnetic field," Journal of Magnetism and Magnetic Materials.  Vol. 460,  pp. 302-11, (2018).


[9] S. Jain, S. Bohra. Hall current and radiation effects on unsteady MHD squeezing nanofluid flow in a rotating channel with lower stretching permeable wall.  Applications of Fluid Dynamics: Springer. (2018)


[10] S. Jain, R. Choudhary, "Combined effects of sunction/injection on MHD boundary Layer flow of nanofluid over horizontal permeable cylinder with radiation," J Adv Res Dyn Control Syst.  Vol. 11,  pp. 88-98, (2017).


[11] A. Parmar, S. Jain, "Influence of Non-Linear Chemical Reaction on MHD Convective Flow for Maxwell Fluid Over Three Different Permeable Vertical Surfaces," Journal of Nanofluids.  Vol. 8,  No. 4,  pp. 671-82, (2019).


[12] M. Sheikholeslami, M. Jafaryar, S. Saleem, Z. Li, A. Shafee, Y. Jiang, "Nanofluid heat transfer augmentation and exergy loss inside a pipe equipped with innovative turbulators," International Journal of Heat and Mass Transfer.  Vol. 126,  pp. 156-63, (2018).


[13] M. Sheikholeslami, M. Jafaryar, Z. Li, "Second law analysis for nanofluid turbulent flow inside a circular duct in presence of twisted tape turbulators," Journal of Molecular Liquids.  Vol. 263,  pp. 489-500, (2018).


[14] M. Sheikholeslami, S. Shehzad, Z. Li, A. Shafee, "Numerical modeling for alumina nanofluid magnetohydrodynamic convective heat transfer in a permeable medium using Darcy law," International Journal of Heat and Mass Transfer.  Vol. 127,  pp. 614-22, (2018).


[15] M. Kerdarian, E. Kianpour, "Flow field, heat transfer and entropy generation of nanofluid in a microchannel using the finite volume method," Journal of Computational & Applied Research in Mechanical Engineering (JCARME).  Vol. 8,  No. 2,  pp. 211-22, (2019).


[16] J. Buongiorno, "Convective transport in nanofluids," Journal of heat transfer.  Vol. 128,  No. 3,  pp. 240-50, (2006).


[17] M. Sheikholeslami, "Numerical investigation of MHD nanofluid free convective heat transfer in a porous tilted enclosure," Engineering Computations.  Vol. 34,  No. 6,  pp. 1939-55, (2017).


[18] S. Moshizi, I. Pop, "Conjugated effect of joule heating and magnetohydrodynamic on laminar convective heat transfer of nanofluids inside a concentric annulus in the presence of slip condition," International Journal of Thermophysics.  Vol. 37,  No. 7,  pp. 72, (2016).


[19] T. Grosan, I. Pop, "Fully developed mixed convection in a vertical channel filled by a nanofluid," Journal of heat transfer.  Vol. 134,  No. 8,  pp. 082501, (2012).


[20] A. Aziz, W. Khan, I. Pop, "Free convection boundary layer flow past a horizontal flat plate embedded in porous medium filled by nanofluid containing gyrotactic microorganisms," International Journal of Thermal Sciences.  Vol. 56,  pp. 48-57, (2012).


[21] M. Bahiraei, S. Mostafa Hosseinalipour, M. Hangi, "Prediction of convective heat transfer of Al2O3-water nanofluid considering particle migration using neural network," Engineering Computations.  Vol. 31,  No. 5,  pp. 843-63, (2014).


[22] S. Moshizi, M. Zamani, S. Hosseini, A. Malvandi, "Mixed convection of magnetohydrodynamic nanofluids inside microtubes at constant wall temperature," Journal of Magnetism and Magnetic Materials.  Vol. 430,  pp. 36-46, (2017).


[23] A. Malvandi, S. Moshizi, D. Ganji, "Nanoparticle transport effect on magnetohydrodynamic mixed convection of electrically conductive nanofluids in micro-annuli with temperature-dependent thermophysical properties," Physica E: Low-dimensional Systems and Nanostructures.  Vol. 88,  pp. 35-49, (2017).


[24] S. Moshizi, A. Malvandi, "Magnetic field effects on nanoparticle migration at mixed convection of MHD nanofluids flow in microchannels with temperature-dependent thermophysical properties," Journal of the Taiwan Institute of Chemical Engineers.  Vol. 66,  pp. 269-82, (2016).


[25] H. Xu, T. Fan, I. Pop, "Analysis of mixed convection flow of a nanofluid in a vertical channel with the Buongiorno mathematical model," International Communications in Heat and Mass Transfer.  Vol. 44,  pp. 15-22, (2013).


[26] C. Yang, W. Li, A. Nakayama, "Convective heat transfer of nanofluids in a concentric annulus," International Journal of Thermal Sciences.  Vol. 71,  pp. 249-57, (2013).


[27] C. Yang, W. Li, Y. Sano, M. Mochizuki, A. Nakayama, "On the anomalous convective heat transfer enhancement in nanofluids: a theoretical answer to the nanofluids controversy," Journal of heat transfer.  Vol. 135,  No. 5,  pp. 054504, (2013).


[28] D. Ganji, A. Malvandi, "Natural convection of nanofluids inside a vertical enclosure in the presence of a uniform magnetic field," Powder technology.  Vol. 263,  pp. 50-7, (2014).


[29] A. Malvandi, D. Ganji, "Effects of nanoparticle migration on force convection of alumina/water nanofluid in a cooled parallel-plate channel," Advanced Powder Technology.  Vol. 25,  No. 4,  pp. 1369-75, (2014).


[30] S. Moshizi, A. Malvandi, D. Ganji, I. Pop, "A two-phase theoretical study of Al2O3–water nanofluid flow inside a concentric pipe with heat generation/absorption," International Journal of Thermal Sciences.  Vol. 84,  pp. 347-57, (2014).


[31] A. Malvandi, S. Moshizi, E. G. Soltani, D. Ganji, "Modified Buongiorno’s model for fully developed mixed convection flow of nanofluids in a vertical annular pipe," Computers & Fluids.  Vol. 89,  pp. 124-32, (2014).


[32] S. Kandlikar, S. Garimella, D. Li, S. Colin, M. R. King, Heat transfer and fluid flow in minichannels and microchannels: elsevier, (2005).


[33] A. Sohankar, M. Riahi, E. Shirani, "Numerical investigation of heat transfer and pressure drop in a rotating U-shaped hydrophobic microchannel with slip flow and temperature jump boundary conditions," Applied Thermal Engineering.  Vol. 117,  pp. 308-21, (2017).


[34] M. Smoluchowski von Smolan, "Ueber wärmeleitung in verdünnten gasen," Annalen der Physik.  Vol. 300,  No. 1,  pp. 101-30, (1898).


[35] M. Knudsen, "Die molekulare Wärmeleitung der Gase und der Akkommodationskoeffizient," Annalen der Physik.  Vol. 339,  No. 4,  pp. 593-656, (1911).


[36] A. Akbarinia, M. Abdolzadeh, R. Laur, "Critical investigation of heat transfer enhancement using nanofluids in microchannels with slip and non-slip flow regimes," Applied Thermal Engineering.  Vol. 31,  No. 4,  pp. 556-65, (2011).


[37] C. Yang, Q. Wang, A. Nakayama, T. Qiu, "Effect of temperature jump on forced convective transport of nanofluids in the continuum flow and slip flow regimes," Chemical Engineering Science.  Vol. 137,  pp. 730-9, (2015).


[38] H. Shokouhmand, A. M. Isfahani, E. Shirani, "Friction and heat transfer coefficient in micro and nano channels filled with porous media for wide range of Knudsen number," International Communications in Heat and Mass Transfer.  Vol. 37,  No. 7,  pp. 890-4, (2010).


[39] J. Shalini, S. BOHRA, "Soret/Dufour Effects on Radiative Free Convection Flow and Mass Transfer over a Sphere with Velocity Slip and Thermal Jump," Walailak Journal of Science and Technology (WJST).  Vol. 16,  No. 9,  pp. 701-21, (2019).


[40] A. Karimipour, "New correlation for Nusselt number of nanofluid with Ag/Al2O3/Cu nanoparticles in a microchannel considering slip velocity and temperature jump by using lattice Boltzmann method," International Journal of Thermal Sciences.  Vol. 91,  pp. 146-56, (2015).


[41] S. A. Sajadifar, A. Karimipour, D. Toghraie, "Fluid flow and heat transfer of non-Newtonian nanofluid in a microtube considering slip velocity and temperature jump boundary conditions," European Journal of Mechanics-B/Fluids.  Vol. 61,  pp. 25-32, (2017).


[42] M. Shojaeian, M. Yildiz, A. Koşar, "Heat transfer characteristics of plug flows with temperature-jump boundary conditions in parallel-plate channels and concentric annuli," International Journal of Thermal Sciences.  Vol. 84,  pp. 252-9, (2014).


[43] H. Brinkman, "The viscosity of concentrated suspensions and solutions," The Journal of Chemical Physics.  Vol. 20,  No. 4,  pp. 571-, (1952).


[44] B. C. Pak, Y. I. Cho, "Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles," Experimental Heat Transfer an International Journal.  Vol. 11,  No. 2,  pp. 151-70, (1998).


[45] M. Gad-el-Hak, MEMS: introduction and fundamentals: CRC press, (2005).


[46] M. Hatami, M. Sheikholeslami, D. Ganji, "Laminar flow and heat transfer of nanofluid between contracting and rotating disks by least square method," Powder technology.  Vol. 253,  pp. 769-79, (2014).


[47] N. A. Yacob, A. Ishak, I. Pop, K. Vajravelu, "Boundary layer flow past a stretching/shrinking surface beneath an external uniform shear flow with a convective surface boundary condition in a nanofluid," Nanoscale research letters.  Vol. 6,  No. 1,  pp. 314, (2011).


[48] T. Motsumi, O. Makinde, "Effects of thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a permeable moving flat plate," Physica Scripta.  Vol. 86,  No. 4,  pp. 045003, (2012).


[49] W. M. Kays, M. E. Crawford, B. Weigand, Convective heat and mass transfer: McGraw-Hill Higher Education Boston, (2005).


[50] W. Kays, M. Crawford, Convective Heat and Mass Transfer. New York: McGraw-Hill, (1980).