Arabian Journal for Science and Engineering

, Volume 44, Issue 2, pp 1269–1282 | Cite as

3D MHD Free Convective Stretched Flow of a Radiative Nanofluid Inspired by Variable Magnetic Field

  • M. K. NayakEmail author
  • Sachin Shaw
  • Ali J. Chamkha
Research Article - Mechanical Engineering


This paper carries on investigation to study the effects of variable magnetic field and thermal radiation on free convective flow of an electrically conducting incompressible nanofluid over an exponential stretching sheet. The model implemented in the present study significantly enriches the thermal conductivity and hence more heat transfer capability of nanofluids. The transformed governing equations have been solved numerically using fourth-order Runge–Kutta method along with shooting technique. The influence of variable magnetic field and thermal radiation associated with thermal buoyancy on the dimensionless velocity, temperature, skin friction and Nusselt number have been analyzed. The obtained numerical results in the present study are validated and found to be in excellent agreement with some previous results seen in the literature. The present study contributes to the result that augmented Hartmann number belittles the fluid flow and enhances the fluid temperature and the related thermal boundary layer thickness.


3D flow Variable magnetic field Nanofluid Exponential stretching sheet Free convection Thermal radiation 

List of symbols


Velocity components in (xyz) directions (\(\hbox {m}\,\hbox {s}^{-1}\))

\(\left( {k_\mathrm{nf}, k_\mathrm{f}, k_\mathrm{s} } \right) \)

Thermal conductivities of (nanofluid, base fluid, nanoparticle) (\(\hbox {W}\,\hbox {m}^{-1}\,\hbox {K}^{-1}\))

\(\left( {\mu _\mathrm{nf}, \mu _\mathrm{f} } \right) \)

Effective viscosity of (nanofluid, base fluid)

\(\left( {\sigma _\mathrm{nf}, \sigma _\mathrm{f} } \right) \)

Effective viscosity of (nanofluid, base fluid)

\(\left( {{F}'(\eta ), {G}'(\eta ), \theta \left( \eta \right) } \right) \)

Dimensionless (axial velocity, transverse velocity and temperature)

\(\left( {\beta _\mathrm{f} , \beta _\mathrm{s} } \right) \)

Thermal expansion of (base fluid and nanoparticles)

\((U_0 ,V_0 ,U_\mathrm{w} ,V_\mathrm{w} ,c)\)


\(\left( {\rho C_\mathrm{p} } \right) _\mathrm{f} \)

Heat capacitance of base fluid (\(\hbox {J}\,\hbox {m}^{-3}\,\hbox {K}^{-1}\))

\(\left( {A_\mathrm{s} , A_\mathrm{f} } \right) \)

Heat transfer area corresponding to (particles and fluid)

\(\upsilon _\mathrm{f}\)

Dynamic viscosity \((\hbox {m}^{2}\,\hbox {s}^{-1}\))


Temperature of fluid (\(\hbox {K}\))


Surface temperature (\(\hbox {K}\))


Reference length (\(\hbox {m}\))

\(B_0 \)

Uniform magnetic field strength


Mean absorption coefficient

\(\left( {\rho C_\mathrm{p} } \right) _\mathrm{nf}\)

Heat capacitance of nanofluid (\(\hbox {J}\,\hbox {m}^{-3}\,\hbox {K}^{-1}\))

\(\left( {\tau _{wx},\tau _{wy} } \right) \)

Wall shear stresses

\(\rho _\mathrm{s} \)

Density of nanoparticles (\(\hbox {kg}\,\hbox {m}^{-3}\))

\(\mu _\mathrm{f} \)

Dynamic viscosity of base fluid (\(\hbox {NS}\,\hbox {m}^{-2}\))


Hartmann number


Prandtl number

\(q_\mathrm{w} \)

Wall heat flux

\(u_\mathrm{s} \)

Brownian motion velocity of nanoparticles

\(\left( {C_{fx} , C_{fy} } \right) \)

Skin frictions

\(\phi \)

Solid volume fraction

\(\left( {\textit{Re}_x , \textit{Re}_y } \right) \)

Local Reynolds number

\(T_\infty \)

Ambient fluid temperature (\(\hbox {K}\))

\(\alpha _\mathrm{f} \)

Thermal diffusivity of the fluid


Peclet number

\(\sigma ^{*}\)

Stefan–Boltzmann constant(\(\hbox {W}\,\hbox {m}^{-2}\,\hbox {K}^{-4}\))

\(\rho _\mathrm{nf}\)

Effective density of nanofluid (\(\hbox {kg}\,\hbox {m}^{-3}\))

\(\rho _\mathrm{f} \)

Density of base fluid (\(\hbox {kg}\,\hbox {m}^{-3}\))


Acceleration due to gravity

\(\mu _\mathrm{nf}\)

Dynamic viscosity (\(\hbox {N}\,\hbox {S}\,\hbox {m}^{-2}\))

\(\eta \)

Non-dimensional vertical distance

\(d_\mathrm{s} \)

Nanoparticle diameter


Radiation parameter

\(d_\mathrm{f} \)

Molecular size of the fluid

\(\left( {\rho C_\mathrm{p} } \right) _\mathrm{s} \)

Heat capacitance (\(\hbox {J}\,\hbox {m}^{-3}\hbox {K}^{-1}\))







Quantities at wall





\(\infty \)

Quantities at free stream

Greek symbols

\(\phi \)

Solid volume fraction

\(\eta \)

Distance from the leading edge of the plate

\(\rho _{f}\)

Density of the fluid

\(\sigma \)

Electrically conductivity


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Copyright information

© King Fahd University of Petroleum & Minerals 2018

Authors and Affiliations

  1. 1.Department of PhysicsRadhakrishna Institute of Technology and EngineeringBhubaneswarIndia
  2. 2.Department of Mathematics and Statistical SciencesBotswana International University of Science and TechnologyPalapyeBotswana
  3. 3.Mechanical Engineering Department, Prince Sultan Endowment for Energy and EnvironmentPrince Mohammad Bin Fahd UniversityAl-KhobarSaudi Arabia
  4. 4.RAK Research and Innovation CenterAmerican University of Ras Al KhaimahRas Al KhaimahUnited Arab Emirates

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