Journal of Thermal Analysis and Calorimetry

, Volume 135, Issue 1, pp 207–222 | Cite as

Brownian motion effect on heat transfer of a three-dimensional nanofluid flow over a stretched sheet with velocity slip

  • N. Freidoonimehr
  • Asghar B. RahimiEmail author


Present article provides an analytical investigation of the fluid flow and heat and mass transfer for the steady laminar MHD three-dimensional nanofluid flow over a bidirectional stretching sheet with convective surface boundary condition using optimal homotopy analysis method (OHAM) via Mathematica package BVPh2.0. In contrast to the conventional no-slip condition at the surface, Navier’s slip condition has been applied. The governing partial differential equations are transformed into a highly nonlinear coupled ordinary differential equations consisting the momentum, energy and concentration equations via appropriate similarity transformations. The current OHAM solution demonstrates very good correlation with those of the previously published studies in the especial cases. The influence of different physical parameters such as magnetic parameter (M), Prandtl number (\( \Pr \)), Brownian motion parameter (\( {\text{Nb}} \)), thermophoresis parameter (\( {\text{Nt}} \)), Lewis number (Le), velocity slip parameter (γ), stretching rate ratio parameter (λ), and Biot number (Bi) on all fluid velocity components \( \left( {f^{\prime}(\eta ),\,\,g^{\prime}(\eta )} \right) \), temperature distribution \( \left( {\theta \,(\eta )} \right) \) and concentration \( \left( {\phi \,(\eta )} \right) \) as well as the skin friction coefficients in x and y directions \( \left( {C_{\text{fx}} {\text{Re}}_{\text{x}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} ,\,\,C_{\text{fy}} {\text{Re}}_{\text{x}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} } \right), \) local Nusselt number \( \left( {{{{\text{Nu}}_{\text{x}} } \mathord{\left/ {\vphantom {{{\text{Nu}}_{\text{x}} } {{\text{Re}}_{\text{x}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}} \right. \kern-0pt} {{\text{Re}}_{\text{x}}^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}} \right) \) and local Sherwood number \( \left( {{{{\text{Sh}}_{\text{x}} } \mathord{\left/ {\vphantom {{{\text{Sh}}_{\text{x}} } {{\text{Re}}_{\text{x}}^{{\,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}} \right. \kern-0pt} {{\text{Re}}_{\text{x}}^{{\,{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} }}} \right) \) are tabulated graphically and discussed in details. This study specifies that nanoparticles in the base fluid offer a potential in increasing the convective heat transfer performance of various liquids.


MHD Three-dimensional flow Nanofluid Velocity slip Convective boundary condition Optimal HAM 

List of symbols

a, b



Biot number

\( B_{\text{o}} \)

Constant magnetic field


Heat capacity


Nanoparticle concentration

\( C_{\text{w}} \)

Concentration of nanoparticle

\( C_{\infty } \)

Ambient concentration

\( C_{\text{f}} \)

Friction coefficient

\( C_{\text{i}} \)

Constant in Eq. (24)


Brownian diffusion coefficient

\( D_{\text{T}} \)

Thermophoretic diffusion coefficient

\( f(\eta ),\;g(\eta ) \)

Velocity similarity functions


Convective heat transfer coefficient


Thermal conductivity



\( {\text{Le}} \)

Lewis number


Magnetic parameter


Nusselt number


Brownian motion parameter


Thermophoresis parameter

n, m



Prandtl number


Heat flux and embedding factor in Eqs. (25)–(32)


Auxiliary function


Reynolds number


Sherwood number



\( T_{\text{f}} \)

Convective surface temperature

\( T_{\infty } \)

Ambient temperature

u, v, w

Velocity components in x, y, z directions

x, y, z

Cartesian coordinates


\( \alpha \)

Thermal diffusivity

\( \gamma \)

Velocity slip parameter

\( \gamma_{0} \)

Slip length

\( \varepsilon \)

Total squared residual error

\( \eta \)

Similarity parameter

\( \theta (\eta ) \)

Temperature distribution

\( \lambda \)

Stretching rate ratio parameter

\( \nu \)

Kinematic viscosity

\( \rho \)

Fluid density

\( \sigma \)

Electrical conductivity

\( \tau \)

Skin friction

\( \phi (\eta ) \)


\( \chi \)

Auxiliary parameter

\( {\mathcal{L}} \)

Auxiliary linear operator

\( {\mathcal{N}} \)

Nonlinear operator in Eqs. (25)–(32)



Financial support of Ferdowsi University of mashhad under Contract No. 2/40473 is acknowledged.


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2018

Authors and Affiliations

  1. 1.Faculty of EngineeringFerdowsi University of MashhadMashhadIran

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