Behavior of stratifications and convective phenomena in mixed convection flow of 3D Carreau nanofluid with radiative heat flux

  • M. IrfanEmail author
  • M. Khan
  • W. A. Khan


Nanoliquids, the engineered liquids with isolated effectual nanoparticles have disclosed a surprising thermo-physical effects and added functionalities and therefore have supported an extensive sort of essential applications. In particular, nanoliquids have displayed pointedly improved aptitude of heat transfer as equated to traditional functioning liquids. The notable intention of current scrutiny is to explore the features of combined convective and stratification phenomena by utilizing Brownian and thermophoresis nanoparticles on 3D mixed convection flow of magnetite Carreau fluid influenced by a bidirectional stretching surface. The heat transport phenomenon is also betrothed in the manifestation of thermal radiation and the heat sink/source. By means of suitable conversions the nonlinear PDEs transformed into nonlinear ODEs. To identify the behavior of numerous somatic parameters, numerically bvp4c tactic has been worked to elucidate the governing ODEs. The graphical depiction is delineated and tables are organized for diverse physical parameters on Carreau nanofluid. It is scrutinized that the impact of magnetic parameter on both the velocity components is analogous and diminishes both the velocities for shear thinning/thickening liquids. Moreover, the present exploration reports that the mixed convection and thermal stratification parameters decline the liquid temperature and allied thickness of the thermal boundary layer for both shear thickening/thinning liquids.


3D Carreau nanofluid Mixed convection Thermal radiation Heat sink/source Double stratification Combined convective conditions 

List of symbols

\({\mathbf{S}}^{ * }\)

Cauchy stress tensor




Identity tensor

\(\dot{\gamma }\)

Shear rate


Material rate constant

\((\mu_{0} ,\mu_{\infty } )\)

Zero and infinity shear rate viscosities


First Rivlin–Ericksen tensor


Power law index

u, v, w

Velocity components

x, y, z

Space coordinates


Kinematic viscosity


Electrical conductivity


Fluid density


Strength of magnetic field


Gravitational acceleration


Thermal diffusivity


Nanofluid thermal conductivity

\(\left( {\beta_{{T}} ,\beta_{{C}} } \right)\)

Thermal and concentration coefficients expansion

(T, C)

Temperature and concentration of fluid


Effective heat capacity ratio


Brownian diffusion coefficient


Thermophoresis diffusion coefficient

\((T_{\infty } ,C_{\infty } )\)

Nanofluid ambient temperature and concentration

\((T_{0} ,C_{0} )\)

Reference temperature and concentration

\((d,d_{1} ,e,e_{1} )\)

Dimensionless constants


Radiative heat flux


Mean absorption coefficient

\(\sigma^{ * }\)

Stefan–Boltzmann constant


Heat source/sink coefficient

\(U_{w} (x),\;V_{w} (x)\)

Stretching velocities

a, b

Positive constants

\(\left( {h_{{f}} ,h_{{m}} } \right)\)

Heat and mass wall transfer coefficient

\(\left( {T_{{f}} ,C_{{f}} } \right)\)

Heated fluid temperature and concentration


Dimensionless variable

(We1, We2)

Local Weissenberg numbers


Magnetic parameter


Mixed convection parameter


Buoyancy ratio parameter


Thermal radiation

(S1, S2)

Thermal and mass stratification parameters

\((\gamma_{1} ,\gamma_{2} )\)

Thermal and mass Biot numbers


Brownian motion parameter


Thermophoresis parameter


Heat source/sink parameter


Lewis number


Ratio of stretching rates parameter

\((\tau_{{xz}} ,\tau_{{yz}} )\)

Surface shear stresses along x- and y-directions

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

Skin friction coefficients

\(\left( {Nu_{x} ,Sh_{x} } \right)\)

Local Nusselt and Sherwood numbers


Local Reynolds number

(f, g)

Dimensionless velocities


Dimensionless temperature


Dimensionless concentration



Ordinary differential equations


Partial differential equations


Three dimensional


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan
  2. 2.Department of Mathematics and StatisticsHazara UniversityMansehraPakistan

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