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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

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

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.

Keywords

Nanofluid MHD Couette flow Natural convection Thermal radiation Spectral method 

List of symbols

B0

External magnetic field component (T)

c

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

Cf

Skin friction factor

df

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)

dp

Diameter of the nanoparticles (m)

DB

Brownian diffusion coefficient (m2 s−1)

DT

Thermophoresis diffusion coefficient (m2 s−1)

F

Accelerating parameter

GrT

Thermal Grashof number

G

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

kB

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

Le

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

NAV

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

Nb

Brownian motion parameter

Nr

Radiation parameter

Nt

Thermophoresis parameter

Nux

Local Nusselt number

P′

Pressure (Pa)

Pr

Prandtl number (P r  = ν/α)

qr

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

Shx

Local Sherwood number

T′

Temperature (K)

Tfr

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

t′

Time (s)

x′, y′, z′

Cartesian coordinates (m)

U0

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)

βϕ

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

βR

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

ρf0

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

σe

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

Superscript

Dimensional variables

Subscripts

c

Cold

f

Base fluid

h

Hot

nf

Nanofluid

p

Nanoparticle

Notes

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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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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