Influence of Non-Linear Boussinesq Approximation and Convective Thermal Boundary Condition on MHD Natural Convection Flow of a Couple Stress-Nanofluid in a Porous Medium

Abstract:

Nonlinear density and temperature variation’s role (NDT) on the magnetohydrodynamic (MHD) natural convective flow of couple stress fluid with nanoparticles through a vertical porous channel modeled as Darcy-Forchheimer flow is the purpose of our work. The nanoparticles volume fraction is taken into consideration (Buongiorno model). The nonlinear partial differential equations governing this flow were transformed into ordinary differential equations via the similarity technique and simulated numerically using Matlab, following boundary value problem (BVP4c) code. Graphical illustrations, including non-dimensional velocity, temperature, concentration, nanoparticle’s concentration and numerical results containing Nusselt and Sherwood numbers were presented for different values of the non-linear part of the Boussinesq approximation; couple stress parameter, and the Biot number on the walls.

• Periodical: Nano Hybrids and Composites (Volume 26)
• Pages: 45-61
• DOI: https://doi.org/10.4028/www.scientific.net/NHC.26.45
• Online since: August 2019
• Authors: Mhamed Tayeb*, M.N. Bouaziz, Salah Hanini
• Keywords: Couple Stress Fluid, MHD, Nanoparticles, Natural Convection, Nonlinear Boussinesq Approximation, Porous Medium

References

[1] J. Fourier, Theorie analytique de la chaleur, par M. Fourier, Chez Firmin Didot, père et fils, 1822.

[2] A. Fick, Ueber diffusion, Ann. Phys. 170 (1855) 59‑86.

[3] H. Darcy, Les fontaines publiques de la ville de Dijon: exposition et application..., Victor Dalmont, 1856.

[4] J.R. Philip, D.A. De Vries, Moisture movement in porous materials under temperature gradients, Trans. Am. Geophys. Union. 38 (1957) 222.

DOI: 10.1029/TR038i002p00222

[5] O.A. Saunders, Natural Convection in Liquids, Proc. R. Soc. A Math. Phys. Eng. Sci. 172 (1939) 55‑71.

DOI: 10.1098/rspa.1939.0089

[6] W. Elenbaas, Heat dissipation of parallel plates by free convection, Physica. 9 (1942) 1‑28.

DOI: 10.1016/s0031-8914(42)90053-3

[7] H. Sithole, H. Mondal, S. Goqo, P. Sibanda, S. Motsa, Numerical simulation of couple stress nanofluid flow in magneto-porous medium with thermal radiation and a chemical reaction, Appl. Math. Comput. 339 (2018) 820‑836.

DOI: 10.1016/j.amc.2018.07.042

[8] C.O. Bennett, J.E. Myers, Momentum, heat, and mass transfer, McGraw-Hill, (1962).

[9] A. Bejan, K.R. Khair, Heat and mass transfer by natural convection in a porous medium, Int. J. Heat Mass Transf. 28 (1985) 909‑918.

DOI: 10.1016/0017-9310(85)90272-8

[10] R.G. Carbonell, S. Whitaker, Heat and mass transfer in porous media, in: Fundam. Transp. Phenom. porous media, Springer, 1984: p.121‑198.

DOI: 10.1007/978-94-009-6175-3_3

[11] T.L. Bergman, F.P. Incropera, D.P. DeWitt, A.S. Lavine, Fundamentals of heat and mass transfer, John Wiley & Sons, (2011).

[12] S.~Chandrasekhar, Hydrodynamic and hydromagnetic stability, (1961).

[13] T.G. Cowling, Magnetohydrodynamics Interscience Publishers^ Inc, New York. (1957).

[14] M.F. El-Amin, Magnetohydrodynamic free convection and mass transfer flow in micropolar fluid with constant suction, J. Magn. Magn. Mater. 234 (2001) 567‑574.

DOI: 10.1016/S0304-8853(01)00374-2

[15] K. Vajravelu, Combined free and forced convection in hydromagnetic flows, in vertical wavy channels, with travelling thermal waves, Int. J. Eng. Sci. 27 (1989) 289‑300.

DOI: 10.1016/0020-7225(89)90117-1

[16] M.A. Seddeek, Heat and mass transfer on a stretching sheet with a magnetic field in a visco-elastic fluid flow through a porous medium with heat source or sink, Comput. Mater. Sci. 38 (2007) 781‑787.

DOI: 10.1016/J.COMMATSCI.2006.05.015

[17] M.F. El-Amin, Magnetohydrodynamic free convection and mass transfer flow in micropolar fluid with constant suction, J. Magn. Magn. Mater. 234 (2001) 567‑574.

DOI: 10.1016/S0304-8853(01)00374-2

[18] O.A. Bég, A.Y. Bakier, V.R. Prasad, Numerical study of free convection magnetohydrodynamic heat and mass transfer from a stretching surface to a saturated porous medium with Soret and Dufour effects, Comput. Mater. Sci. 46 (2009) 57‑65.

DOI: 10.1016/j.commatsci.2009.02.004

[19] A. Raptis, C. Massalas, G. Tzivanidis, Hydromagnetic free convection flow through a porous medium between two parallel plates, Phys. Lett. A. 90 (1982) 288‑289.

DOI: 10.1016/0375-9601(82)90118-9

[20] G.K. Singha, P.N. Deka, Skin-friction for unsteady free convection MHD flow between two heated vertical parallel plates, Theor. Appl. Mech. 33 (2006) 259‑280.

DOI: 10.2298/tam0604259s

[21] A.K. Singh, MHD free convective flow through a porous medium between two vertical parallel plates, (2002).

[22] V.K. Stokes, Couple Stresses in Fluids, Phys. Fluids. 9 (1966) 1709.

DOI: 10.1063/1.1761925

[23] D. M., S. D., S. B., Analytical solutions of couple stress fluid flows with slip boundary conditions, Alexandria Eng. J. 53 (2014) 723‑730.

DOI: 10.1016/J.AEJ.2014.06.005

[24] S. Akhtar, N.A. Shah, Exact solutions for some unsteady flows of a couple stress fluid between parallel plates, Ain Shams Eng. J. (2016).

DOI: 10.1016/J.ASEJ.2016.05.008

[25] V.M. Soundalgekar, Effects of Couple Stresses in Fluids on Dispersion of a Solute in a Channel Flow, Phys. Fluids. 14 (1971) 19.

DOI: 10.1063/1.1693276

[26] J.C. Umavathi, M.S. Malashetty, Oberbeck convection flow of a couple stress fluid through a vertical porous stratum, Int. J. Non. Linear. Mech. 34 (1999) 1037‑1045.

DOI: 10.1016/S0020-7462(98)00074-2

[27] R.C. Sharma, K.D. Thakur, On couple-stress fluid heated from below in porous medium in hydromagnetics, Czechoslov. J. Phys. 50 (2000) 753‑758.

DOI: 10.1023/A:1022886903213

[28] J.-R. Lin, Effects of couple stresses on the lubrication of finite journal bearings, Wear. 206 (1997) 171‑178.

DOI: 10.1016/s0043-1648(96)07357-7

[29] U.M. Mokhiamer, W.A. Crosby, H.A. El-Gamal, A study of a journal bearing lubricated by fluids with couple stress considering the elasticity of the liner, Wear. 224 (1999) 194‑201.

DOI: 10.1016/s0043-1648(98)00320-2

[30] D. Srinivasacharya, K. Kaladhar, Soret and dufour effects on free convection flow of a couple stress fluid in a vertical channel with chemical reaction, Chem. Ind. Chem. Eng. Q. 19 (2013) 45‑55.

DOI: 10.2298/CICEQ111231041S

[31] S.O. Adesanya, C.R. Makhalemele, L. Rundora, Natural convection flow of heat generating hydromagnetic couple stress fluid with time periodic boundary conditions, Alexandria Eng. J. (2017).

DOI: 10.1016/J.AEJ.2017.04.006

[32] M. Devakar, A. Raje, S. Hande, Unsteady Flow of Couple Stress Fluid Sandwiched Between Newtonian Fluids Through a Channel, Zeitschrift für Naturforsch. A. 73 (2018) 629‑637.

DOI: 10.1515/zna-2017-0434

[33] T. Hayat, S. Asghar, A. Tanveer, A. Alsaedi, Chemical reaction in peristaltic motion of MHD couple stress fluid in channel with Soret and Dufour effects, Results Phys. 10 (2018) 69‑80.

DOI: 10.1016/j.rinp.2018.04.040

[34] N.T. Eldabe, S.M. Elshaboury, A.A. Hasan, M.A. Elogail, MHD Peristaltic Flow of a Couple Stress Fluids with Heat and Mass Transfer through a Porous Medium, Innov. Syst. Des. Eng. 3 (2012) 51‑67. https://www.iiste.org/Journals/index.php/ISDE/article/view/1841 (Consulted on October 21st, 2018).

DOI: 10.1016/j.apm.2015.04.043

[35] D. Srinivasacharya, K. Kaladhar, Natural convection flow of a couple stress fluid between two vertical parallel plates with Hall and ion-slip effects, Acta Mech. Sin. 28 (2012) 41‑50.

DOI: 10.1007/s10409-011-0523-z

[36] N.T. Eldabe, S.M. Elshaboury, A.A. Hasan, M.A. Elogail, MHD peristaltic flow of a couple stress fluids with heat and mass transfer through a porous medium, Innov. Syst. Des. Eng. 3 (2012).

DOI: 10.1016/j.apm.2015.04.043

[37] R. Muthuraj, S. Srinivas, R.K. Selvi, Heat and mass transfer effects on MHD flow of a couple-stress fluid in a horizontal wavy channel with viscous dissipation and porous medium, Heat Transf. Res. 42 (2013) 403‑421.

DOI: 10.1002/htj.21040

[38] H.P. Rani, G.J. Reddy, C.N. Kim, The effect of the couple stress parameter and Prandtl number on the transient natural convection flow over a vertical cylinder, Acta Mech. Sin. 29 (2013) 649‑656.

DOI: 10.1007/s10409-013-0079-1

[39] K. Kaladhar, Natural Convection Flow of Couple Stress Fluid in a Vertical Channel With Hall and Joule Heating Effects, Procedia Eng. 127 (2015) 1071‑1078.

DOI: 10.1016/J.PROENG.2015.11.465

[40] S.U.S. Choi, J.A. Eastman, Enhancing thermal conductivity of fluids with nanoparticles, in: United States, 1995. https://www.osti.gov/servlets/purl/196525.

[41] J. Buongiorno, Convective transport in nanofluids, J. Heat Transfer. 128 (2006) 240‑250. doi:.

DOI: 10.1115/1.2150834

[42] N.A. Khan, F. Sultan, F. Riaz, M. Jamil, Investigation of combined heat and mass transfer between vertical parallel plates in a two-layer flow of couple stress nanofluid, Open Eng. 6 (2016).

DOI: 10.1515/eng-2016-0004

[43] A.S. Eegunjobi, O.D. Makinde, Irreversibility analysis of hydromagnetic flow of couple stress fluid with radiative heat in a channel filled with a porous medium, Results Phys. 7 (2017) 459‑469.

DOI: 10.1016/J.RINP.2017.01.002

[44] S. Adesanya, H. Ogunseye, J. Falade, R.S. Lebelo, S.O. Adesanya, H.A. Ogunseye, J.A. Falade, R.S. Lebelo, Thermodynamic Analysis for Buoyancy-Induced Couple Stress Nanofluid Flow with Constant Heat Flux, Entropy. 19 (2017) 580.

DOI: 10.3390/e19110580

[45] E.P. Siva, A.G.S. Srinivas, M. Vidhya, Thermal radiation effect on mhd peristaltic transport of a couple stress nanofluid in a tapered asymmetric channel, Int. J. Pure Appl. Math. 113 (2017) 106‑114. https://acadpubl.eu/jsi/2017-113-pp/articles/13/12.pdf.

DOI: 10.17654/am096060307

[46] R. Van der Borght, A numerical study of non-linear convection in a compressible medium, J. Comput. Appl. Math. 6 (1980) 283‑294.

DOI: 10.1016/0771-050x(80)90038-8

[47] J.K. Kamel, S. Paolucci, Heat transfer and fluid flow in a furnace using the non-Boussinesq approximation, in: ASME 2004 Heat Transf. Eng. Summer Conf., American Society of Mechanical Engineers, 2004: p.933‑938.

DOI: 10.1115/ht-fed2004-56466

[48] K. Vajravelu, K.S. Sastri, Fully developed laminar free convection flow between two parallel vertical walls—I, Int. J. Heat Mass Transf. 20 (1977) 655‑660.

DOI: 10.1016/0017-9310(77)90052-7

[49] H. Barrow, T.L. Sitharamarao, Effect of variation in volumetric expansion coefficient on free convection heat transfer, Br. Chem. Eng. 16 (1971) 704‑+.

[50] M.K. Partha, Nonlinear convection in a non-Darcy porous medium, Appl. Math. Mech. 31 (2010) 565‑574.

DOI: 10.1007/s10483-010-0504-6

[51] S.O. Adesanya, H.A. Ogunseye, R.S. Lebelo, K.C. Moloi, O.G. Adeyemi, Second law analysis for nonlinear convective flow of a reactive couple stress fluid through a vertical channel., Heliyon. 4 (2018) e00907.

DOI: 10.1016/j.heliyon.2018.e00907

[52] O.D. Makinde, A.S. Eegunjobi, MHD couple stress nanofluid flow in a permeable wall channel with entropy generation and nonlinear radiative heat, J. Therm. Sci. Technol. 12 (2017) JTST0033-JTST0033. doi:https://doi.org/10.1299/jtst.2017jtst0033.

DOI: 10.1299/jtst.2017jtst0033

[53] S.O. Adesanya, J.A. Falade, R. Rach, Effect of couple stresses on hydromagnetic Eyring-Powell fluid flow through a porous channel, Theor. Appl. Mech. 42 (2015) 135‑150.

DOI: 10.2298/tam1502135a

[54] P.K. Kameswaran, B. Vasu, P. Murthy, R.S.R. Gorla, Mixed convection from a wavy surface embedded in a thermally stratified nanofluid saturated porous medium with non-linear Boussinesq approximation, Int. Commun. Heat Mass Transf. 77 (2016) 78‑86.

DOI: 10.1016/j.icheatmasstransfer.2016.07.006

[55] K. Ramesh, Effects of slip and convective conditions on the peristaltic flow of couple stress fluid in an asymmetric channel through porous medium, Comput. Methods Programs Biomed. 135 (2016) 1‑14.

DOI: 10.1016/j.cmpb.2016.07.001

[56] T. Hayat, R. Sajjad, A. Alsaedi, T. Muhammad, R. Ellahi, On squeezed flow of couple stress nanofluid between two parallel plates, Results Phys. 7 (2017) 553‑561.

DOI: 10.1016/J.RINP.2016.12.038

[57] J. Umavathi, J.P. Kumar, I. Pop, M. Shekar, Flow and heat transfer of couple stress fluid in a vertical channel in the presence of heat source/sink, Int. J. Numer. Methods Heat Fluid Flow. 27 (2017) 795‑819.

DOI: 10.1108/HFF-12-2015-0540