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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

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Abstract

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.

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Abbreviations

B 0 :

External magnetic field component (T)

c :

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

C f :

Skin friction factor

d f :

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)

d p :

Diameter of the nanoparticles (m)

D B :

Brownian diffusion coefficient (m2 s−1)

D T :

Thermophoresis diffusion coefficient (m2 s−1)

F :

Accelerating parameter

G rT :

Thermal Grashof number

G rϕ :

Concentration Grashof number

g :

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

h :

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

k :

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

k B :

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

L e :

Lewis number

M :

Magnetic parameter

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

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

n :

Velocity order

N AV :

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

N b :

Brownian motion parameter

N r :

Radiation parameter

N t :

Thermophoresis parameter

Nu x :

Local Nusselt number

P′ :

Pressure (Pa)

P r :

Prandtl number (P r  = ν/α)

q r :

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

Sh x :

Local Sherwood number

T′ :

Temperature (K)

T fr :

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

t′ :

Time (s)

x′, y′, z′ :

Cartesian coordinates (m)

U 0 :

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

\( \alpha \) :

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

\( \beta \) :

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

β ϕ :

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

β R :

Mean absorption coefficient (m−1)

λ :

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

μ :

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

ν :

Kinematic viscosity (m2 s−1)

ρ :

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

ρ f0 :

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

(ρc):

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

\( \sigma \) :

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

σ e :

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

\(\phi^{\prime}\) :

Volumetric fraction of nanoparticles

∂y,η :

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

′:

Dimensional variables

c :

Cold

f :

Base fluid

h :

Hot

nf :

Nanofluid

p :

Nanoparticle

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Acknowledgements

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.

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Authors and Affiliations

  1. Laboratory of Mechanics, Faculty of Sciences Aïn Chock, Hassan II University, B.P. 5366, Mâarif, Casablanca, Morocco

    Abderrahim Wakif, Zoubair Boulahia & Rachid Sehaqui

  2. Computational Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Farhad Ali

  3. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Farhad Ali

  4. Department of Mathematics, City University of Science and Information Technology, Peshawar, Khyber Pakhtunkhwa, Pakistan

    Farhad Ali

  5. Department of Mathematics, Faculty of Science, New Valley Branch, Assiut University, Al-Kharga, Al-Wadi Al-Jadid, 72511, Egypt

    Mohamed R. Eid

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  1. Abderrahim Wakif
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Correspondence to Abderrahim Wakif.

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Wakif, A., Boulahia, Z., Ali, F. et al. 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. Int. J. Appl. Comput. Math 4, 81 (2018). https://doi.org/10.1007/s40819-018-0513-y

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