Numerical Analysis of the Unsteady Natural Convection MHD Couette Nanofluid Flow in the Presence of Thermal Radiation Using Single and Two-Phase Nanofluid Models for Cu–Water Nanofluids

  • Abderrahim Wakif
  • Zoubair Boulahia
  • Farhad Ali
  • Mohamed R. Eid
  • Rachid Sehaqui
Original Paper


The unsteady Couette nanofluid flow with heat transfer is investigated numerically for copper–water nanofluids under the combined effects of thermal radiation and a uniform transverse magnetic field with variable thermo-physical properties, in the case where the flow is established vertically between two parallel plates, so that one of them has an accelerated motion. The homogeneous single-phase model (i.e., Tiwari and Das’s nanofluid model) and the two-phase mixture model (i.e., Buongiorno’s nanofluid model) are utilized in this study together with Corcione’s model to further investigate and clarify the differences between those models and evaluate the validity of the single-phase model for studying the unsteady natural convection MHD Couette nanofluid flow with thermal radiation. In this investigation, we assume that the studied nanofluid is electrically conducting and has a Newtonian rheological behavior. The nonlinear dynamical system of partial differential equations are solved numerically by means of the Gear–Chebyshev–Gauss–Lobatto collocation technique for zero nanoparticles mass flux and no-slip impermeable conditions at the isothermal vertical plates. In a special case, the present numerical solution is also validated analytically and numerically with the earlier available results. For both nanofluid models, the effects of major parameters on the dimensionless velocity, temperature and volumetric fraction of nanoparticles are analysed via representative profiles, whereas the skin friction factor and the heat transfer rate are estimated numerically and discussed through tabular illustrations.


Nanofluid MHD Couette flow Natural convection Thermal radiation Spectral method 

List of symbols


External magnetic field component (T)


Specific heat (\( {\text{J}}\,{\text{kg}}^{ - 1} \,{\text{K}}^{ - 1} \))


Skin friction factor


Diameter of water molecules \( \left( {d_{f}^{3} = 3 M_{{{\text{H}}_{2} {\text{O}}}} /\left( {500 \,\pi \, \rho_{f0} \,N_{AV} } \right)} \right) \) (m)


Diameter of the nanoparticles (m)


Brownian diffusion coefficient (m2 s−1)


Thermophoresis diffusion coefficient (m2 s−1)


Accelerating parameter


Thermal Grashof number


Concentration Grashof number


Gravitational acceleration, \( \left( {g = 9.8\,{\text{m}}\,{\text{s}}^{ - 2} } \right) \)


Layer thickness \( \left( {h = 0.01 \,{\text{m}}} \right) \)


Thermal conductivity (\( {\text{W}}\,{\text{K}}^{ - 1} \,{\text{m}}^{ - 1} \))


Boltzmann constant \( \left( { k_{B} = 1.38066 \times 10^{ - 23} \,{\text{J}}\,{\text{K}}^{ - 1} } \right) \)


Lewis number


Magnetic parameter

\( M_{{{\text{H}}_{2} {\text{O}}}} \)

Molecular mass weight of water \( \left( {M_{{{\text{H}}_{2} {\text{O}}}} = 18\,{\text{g}}\,{\text{mol}}^{ - 1} } \right) \)


Velocity order


Avogadro number \( \left( {N_{AV} = 6.022 \times 10^{23} \,{\text{mol}}^{ - 1} } \right) \)


Brownian motion parameter


Radiation parameter


Thermophoresis parameter


Local Nusselt number


Pressure (Pa)


Prandtl number (P r  = ν/α)


Radiative heat flux \( ( {\text{W}}\,{\text{m}}^{ - 2} ) \)


Local Sherwood number


Temperature (K)


Freezing point of water \( \left( {T_{fr} = 273.15 \,{\text{K}}} \right) \)


Time (s)

x′, y′, z′

Cartesian coordinates (m)


Velocity constant

u′, v′, w′

Velocity components \( ({\text{m}}\,{\text{s}}^{ - 1} ) \)

\( \vec{e}_{x} , \vec{e}_{y} ,\vec{e}_{z} \)

Unit vectors along the Cartesian axes

Greek symbols

\( \alpha \)

Thermal diffusivity (α = k/(ρc)) (m2 s−1)

\( \beta \)

Coefficient of volume expansion for heat transfer (K−1)


Coefficient of volume expansion for mass transfer (β ϕ  = (ρ p  − ρ f )/ρ nf )


Mean absorption coefficient (m−1)


Ratio between the electrical conductivity of Cu-nanoparticles and water (λ = σ p /σ f )


Dynamic viscosity (μ = ρν) \( ({\text{Pa}} \;{\text{s}}) \)


Kinematic viscosity (m2 s−1)


Density \( ({\text{kg}}\,{\text{m}}^{ - 3} ) \)


Water density at 293 K \( \left( {\rho_{f0} = 998\,{\text{Kg}}\,{\text{m}}^{ - 3} } \right) \)


Heat capacity (\( {\text{J}}\,{\text{m}}^{ - 3} \,{\text{K}}^{ - 1} \))

\( \sigma \)

Electrical conductivity \( {(\Omega }^{ - 1} \,{\text{m}}^{ - 1} ) \)


Stefan-Boltzmann constant \( \left( {\sigma_{e} = 5.67 \times 10^{ - 8} \,{\text{W}}\,{\text{K}}^{ - 4} \,{\text{m}}^{ - 2} } \right) \)


Volumetric fraction of nanoparticles


Partial derivative with respect to \( y \) or η


Dimensional variables





Base fluid









The authors wish to express their very sincerely thanks to the peer reviewers, for their helpful suggestions and valuable comments, which have improved the paper appreciably. The corresponding author is also thankful to Dr. C. H. Amanulla from madanapalle institute of technology and Science in India, for his technical support.


  1. 1.
    Ali, F., Sheikh, N.A., Khan, I., Saqib, M.: Magnetic field effect on blood flow of Casson fluid in axisymmetric cylindrical tube: a fractional model. J. Magn. Magn. Mater. 423, 327–336 (2017)CrossRefGoogle Scholar
  2. 2.
    Seth, G.S., Ansari, M.S., Nandkeolyar, R.: Unsteady hydromagnetic Couette flow within porous plates in a rotating system. Adv. Appl. Math. Mech. 2, 286–302 (2010)MathSciNetGoogle Scholar
  3. 3.
    Seth, G.S., Ansari, M.S., Nandkeolyar, R.: Unsteady hydromagnetic Couette flow induced due to accelerated movement of one of the porous plates of the channel in a rotating system. Int. J. Appl. Math. Mech. 6, 24–42 (2010)Google Scholar
  4. 4.
    Seth, G.S., Ansari, M.S., Nandkeolyar, R.: Unsteady hydromagnetic couette flow within a porous channel. Tamkang J. Sci. Eng. 14, 7–14 (2011)Google Scholar
  5. 5.
    Seth, G.S., Singh, J.K.: Effects of Hall current and rotation on unsteady MHD Couette flow within a porous channel in the presence of a moving magnetic field. J. Nat. Sci. Sustain. Tech. 5, 263–283 (2011)Google Scholar
  6. 6.
    Seth, G.S., Singh, J.K., Mahato, G.K.: Effects of Hall current and rotation on unsteady hydromagnetic Couette flow within a porous channel. Int. J. Appl. Mech. 4, 1–25 (2012). CrossRefGoogle Scholar
  7. 7.
    Seth, G.S., Kumbhakar, B., Sharma, R.: Unsteady hydromagnetic natural convection flow of a heat absorbing fluid within a rotating vertical channel in porous medium with Hall effects. J. Appl. Fluid Mech. 8, 767–779 (2015)CrossRefGoogle Scholar
  8. 8.
    Seth, G.S., Sharma, R., Kumbhakar, B.: Effects of Hall current on unsteady MHD convective Couette flow of heat absorbing fluid due to accelerated movement of one of the plates of the channel in a porous medium. J. Porous Media 19, 13–20 (2016)CrossRefGoogle Scholar
  9. 9.
    Singh, A.K.: Natural convection in unsteady Couette motion. Def. Sci. J. 38, 35–41 (1988)CrossRefzbMATHGoogle Scholar
  10. 10.
    Jha, B.K.: Natural convection in unsteady MHD couette flow. Heat Mass Transf. 37, 329–331 (2001)CrossRefGoogle Scholar
  11. 11.
    Singh, R.K., Singh, A.K., Sacheti, N.C., Chandran, P.: On hydromagnetic free convection in the presence of induced magnetic field. Heat Mass Transf. 46, 523–529 (2010)CrossRefGoogle Scholar
  12. 12.
    Raju, R.S., Reddy, G.J., Rao, J.A., Rashidi, M.M.: Thermal diffusion and diffusion thermo effects on an unsteady heat and mass transfer magnetohydrodynamic natural convection Couette flow using FEM. J. Comput. Des. Eng. 3, 349–362 (2016)Google Scholar
  13. 13.
    Seth, G., Mandal, P., Chamkha, A.: MHD free convective flow past an impulsively moving vertical plate with ramped heat flux through porous medium in the presence of inclined magnetic field. Front. Heat Mass Transf. 7, 1–12 (2016)CrossRefGoogle Scholar
  14. 14.
    Ali, F., Gohar, M., Khan, I.: MHD flow of water-based Brinkman type nanofluid over a vertical plate embedded in a porous medium with variable surface velocity, temperature and concentration. J. Mol. Liq. 223, 412–419 (2016)CrossRefGoogle Scholar
  15. 15.
    Choi, S.U.S.: Enhancing thermal conductivity of fluids with nanoparticles. ASME Publ. Fed. 231, 99–106 (1995)Google Scholar
  16. 16.
    Corcione, M.: Empirical correlating equations for predicting the effective thermal conductivity and dynamic viscosity of nanofluids. Energy Convers. Manag. 52, 789–793 (2011)CrossRefGoogle Scholar
  17. 17.
    Tiwari, R.K., Das, M.K.: Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids. Int. J. Heat Mass Transf. 50, 2002–2018 (2007)CrossRefzbMATHGoogle Scholar
  18. 18.
    Buongiorno, J.: Convective transport in nanofluids. J. Heat Transf. 128, 240–250 (2006)CrossRefGoogle Scholar
  19. 19.
    Baag, S., Mishra, S.R.: Heat and mass transfer analysis on MHD 3-D water-based nanofluid. J. Nanofluids 4, 352–361 (2015)CrossRefGoogle Scholar
  20. 20.
    Bhatti, M.M., Mishra, S.R., Abbas, T., Rashidi, M.M.: A mathematical model of MHD nanofluid flow having gyrotactic microorganisms with thermal radiation and chemical reaction effects. Neural Comput. Appl. 1–13 (2016).
  21. 21.
    Boulahia, Z., Wakif, A., Sehaqui, R.: Numerical study of mixed convection of the nanofluids in two-sided lid-driven square cavity with a pair of triangular heating cylinders. J. Eng. 2016, 1–8 (2016). CrossRefGoogle Scholar
  22. 22.
    Boulahia, Z., Wakif, A., Chamkha, A.J., Sehaqui, R.: Numerical study of natural and mixed convection in a square cavity filled by a Cu–water nanofluid with circular heating and cooling cylinders. Mech. Ind. 18, 1–21 (2017). Google Scholar
  23. 23.
    Boulahia, Z., Wakif, A., Sehaqui, R.: Modeling of free convection heat transfer utilizing nanofluid inside a wavy enclosure with a pair of hot and cold cylinders. Front. Heat Mass Transf. 8(14), 1–10 (2017)Google Scholar
  24. 24.
    Makinde, O.D., Mishra, S.R.: On stagnation point flow of variable viscosity nanofluids past a stretching surface with radiative heat. Int. J. Appl. Comput. Math. 3, 561–578 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Thumma, T., Mishra, S.R.: Effect of viscous dissipation and Joule heating on magnetohydrodynamic Jeffery nanofluid flow with and without multi slip boundary conditions. J. Nanofluids. 7, 516–526 (2018)CrossRefGoogle Scholar
  26. 26.
    Rashidi, M.M., Ganesh, N.V., Hakeem, A.K.A., Ganga, B.: Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation. J. Mol. Liq. 198, 234–238 (2014)CrossRefGoogle Scholar
  27. 27.
    Aman, S., Khan, I., Ismail, Z., Salleh, M.Z.: Impacts of gold nanoparticles on MHD mixed convection Poiseuille flow of nanofluid passing through a porous medium in the presence of thermal radiation, thermal diffusion and chemical reaction. Neural Comput. Appl. 1–9 (2016).
  28. 28.
    Mosayebidorcheh, S., Sheikholeslami, M., Hatami, M., Ganji, D.D.: Analysis of turbulent MHD Couette nanofluid flow and heat transfer using hybrid DTM–FDM. Particuology 26, 95–101 (2016)CrossRefGoogle Scholar
  29. 29.
    Zaraki, A., Ghalambaz, M., Chamkha, A.J., Ghalambaz, M., De Rossi, D.: Theoretical analysis of natural convection boundary layer heat and mass transfer of nanofluids: effects of size, shape and type of nanoparticles, type of base fluid and working temperature. Adv. Powder Technol. 26, 935–946 (2015)CrossRefGoogle Scholar
  30. 30.
    Tripathi, R., Seth, G.S., Mishra, M.K.: Double diffusive flow of a hydromagnetic nanofluid in a rotating channel with Hall effect and viscous dissipation: active and passive control of nanoparticles. Adv. Powder Technol. 28, 2630–2641 (2017)CrossRefGoogle Scholar
  31. 31.
    Seth, G.S., Mishra, M.K., Tripathi, R.: Modeling and analysis of mixed convection stagnation point flow of nanofluid towards a stretching surface: OHAM and FEM approach. Comput. Appl. Math. (2017). Google Scholar
  32. 32.
    Wakif, A., Boulahia, Z., Sehaqui, R.: A semi-analytical analysis of electro-thermo-hydrodynamic stability in dielectric nanofluids using Buongiorno’s mathematical model together with more realistic boundary conditions. Results Phys. (2018). Google Scholar
  33. 33.
    Viskanta, R., Grosh, R.J.: Boundary layer in thermal radiation absorbing and emitting media. Int. J. Heat Mass Transf. 5, 795–806 (1962)CrossRefGoogle Scholar
  34. 34.
    McNab, G.S., Meisen, A.: Thermophoresis in liquids. J. Colloid Interface Sci. 44, 339–346 (1973)CrossRefGoogle Scholar
  35. 35.
    Garnett, J.C.M.: Colours in metal glasses and in metallic films. Philos. Trans. R. Soc. Lond. Ser. A. 203, 385–420 (1904)CrossRefzbMATHGoogle Scholar
  36. 36.
    Siddheshwar, P.G., Kanchana, C., Kakimoto, Y., Nakayama, A.: Steady finite-amplitude Rayleigh–Bénard convection in nanoliquids using a two-phase model: theoretical answer to the phenomenon of enhanced heat transfer. J. Heat Transf. 139, 1–8 (2017)Google Scholar
  37. 37.
    Hayat, T., Ahmed, B., Abbasi, F.M., Alsaedi, A.: Hydromagnetic peristalsis of water based nanofluids with temperature dependent viscosity: a comparative study. J. Mol. Liq. 234, 324–329 (2017)CrossRefGoogle Scholar
  38. 38.
    Canuto, C., Hussaini, M.Y., Quarteroni, A.M., Thomas Jr., A.: Spectral Methods in Fluid Dynamics. Springer, Berlin (2012)zbMATHGoogle Scholar
  39. 39.
    Kumar, A., Singh, A.: Transient magnetohydrodynamic Couette flow with ramped velocity. Int. J. Fluid Mech. Res. 37, 435–446 (2010)CrossRefGoogle Scholar

Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratory of Mechanics, Faculty of Sciences Aïn ChockHassan II UniversityMâarif, CasablancaMorocco
  2. 2.Computational Analysis Research GroupTon Duc Thang UniversityHo Chi Minh CityVietnam
  3. 3.Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam
  4. 4.Department of MathematicsCity University of Science and Information TechnologyPeshawarPakistan
  5. 5.Department of Mathematics, Faculty of Science, New Valley BranchAssiut UniversityAl-Kharga, Al-Wadi Al-JadidEgypt

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