Navier’s Slip Effect on Mixed Convection Flow of Non-Newtonian Nanofluid: Buongiorno’s Model with Passive Control Approach

  • M. K. Mishra
  • G. S. Seth
  • R. SharmaEmail author
Original Paper


This article is devoted to the study of Navier’s slip effect on MHD mixed convection boundary layer flow of non-Newtonian nanofluid over a stretching surface. The flow is induced due to the continuously stretching surface. The mathematical modeling of the flow situation, is governed by Buongiorno’s nanofluid model with passively controlled nanoparticle boundary condition. To analyze the flow field behavior, the similarity transformation approach is adopted. The approximate similar solution of the problem is obtained using finite element method. Effects of active flow parameters on the quantities of engineering interests viz. skin friction coefficient, Nusselt number and flow variables viz velocity, temperature and nanoparticle concentration, are presented graphically. The Numerical results of the present investigation are validated with earlier published results. The findings suggest that the velocity slip plays a vital role in skin friction coefficient and Nusselt number of non-Newtonian nanofluid. It has been observed that the Brownian diffusion does not have very promising impact on Nusselt number as compared to results obtained in previous studies with active control of wall nanoparticles. The potential application of present investigation are in various industrial manufacturing processes such as cooling and/or drying of textile and paper, rolling sheet drawn from a die, manufacturing of crystalline materials, polymeric sheets, glass sheets, etc.


MHD FEM Passive control Slip velocity Stretching sheet 



Constant parameter (\(\hbox {s}^{-1}\))


Rate of strain tensor (\(\hbox {s}^{-1}\))


\(=\left( b_{x} ,b_{y} ,0\right) \) body force vector (N)


Magnetic field (\(\hbox {kg}\,\hbox {s}^{-2}\,\hbox {A}^{-1}\))


Biot number


Local skin friction coefficient


Brownian diffusion coefficient (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))


Thermophoretic diffusion coefficient (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))


Dimensionless stream function


Gravitational acceleration (\(\hbox {m}\,\hbox {s}^{-2}\))


Grashof number


Element size (m)


Coefficient of convective heat transfer (\(\hbox {W}\,\hbox {m}^{-2}\,\hbox {K}^{-1}\))


Unit tensor


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


Rosseland mean absorption coefficient (\(\hbox {m}^{-1}\))


First order coefficient of short relaxation (\(\hbox {kg}\,\hbox {m}^{-1}\))


Velocity slip factor (m)


Magnetic parameter


Brownian motion parameter


Thermophoresis parameter


Pressure (Pa)


Local Nusselt number

\(\Pr \)

Prandtl number

\(\Pr _{eff}\)

Effective Prandtl number


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


Surface heat flux (\(\hbox {W}\,\hbox {m}^{-2}\))


Radiation parameter


Local Reynolds number


Dimensionless nanoparticle volume fraction


Temperature (K)


Temperature of the left side of surface (K)

\(T_{\infty }\)

Temperature in free stream (K)


Time (s)


Stretching sheet velocity (\(\hbox {m}\,\hbox {s}^{-1}\))


Slip velocity (\(\hbox {m}\,\hbox {s}^{-1}\))


\(=\left( u,v,0\right) \) Velocity vector (\(\hbox {m}\,\hbox {s}^{-1}\))

x, y

Cartesian coordinate axes (m)

Greek Symbols

\(\alpha \)

Viscoelasticity parameter

\(\alpha _{m}\)

Thermal diffusivity (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))

\(\beta \)

Thermal expansion coefficient (\(\hbox {K}^{-1}\))

\(\gamma \)

Velocity slip parameter

\(\Upsilon \)

Cauchy stress tensor (Pa)

\(\sigma \)

Electrical conductivity (\(\hbox {S}\,\hbox {m}^{-1}\))

\(\rho \)

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

\(\lambda \)

Thermal buoyancy parameter

\(\phi \)

Nanoparticle volume fraction

\(\lambda ^{*}\)

Nanoparticle buoyancy force

\(\left( \rho C_{p} \right) _{np}\)

Specific heat capacity of nanoparticles (\(\hbox {J}\,\hbox {K}^{-1}\))

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

Specific heat capacity of the nanofluid (\(\hbox {J}\,\hbox {K}^{-1}\))

\(\theta \)

Dimensionless temperature

\(\upsilon \)

Kinematic coefficient of viscosity (\(\hbox {m}^{2}\,\hbox {s}^{-1}\))

\(\psi \)

Stream function (\(\hbox {m}^{2}\,\hbox {s}\))

\(\eta \)

Similarity variable

\(\mu \)

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

\(\sigma ^{*}\)

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

\(\tau _{s}\)

Surface shear stress (\(\hbox {kg}\,\hbox {m}^{-1}\,\hbox {s}^{-2}\))



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Copyright information

© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsVIT-AP UniversityAmaravatiIndia
  2. 2.Department of Applied MathematicsIndian Institute of Technology (ISM)DhanbadIndia
  3. 3.Department of MathematicsGITAMBengaluruIndia

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