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

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## Abstract

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.

## Keywords

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

- \({\mathbf{S}}^{ * }\)
Cauchy stress tensor

*p*Pressure

**I**Identity tensor

- \(\dot{\gamma }\)
Shear rate

- \(\varGamma\)
Material rate constant

- \((\mu_{0} ,\mu_{\infty } )\)
Zero and infinity shear rate viscosities

- \({\mathbf{A}}_{1}\)
First Rivlin–Ericksen tensor

*n*Power law index

*u*,*v*,*w*Velocity components

*x*,*y*,*z*Space coordinates

- \(\nu\)
Kinematic viscosity

- \(\sigma\)
Electrical conductivity

- \(\rho_{{f}}\)
Fluid density

*B*_{0}Strength of magnetic field

*g*Gravitational acceleration

- \(\alpha_{1}\)
Thermal diffusivity

*k*Nanofluid thermal conductivity

- \(\left( {\beta_{{T}} ,\beta_{{C}} } \right)\)
Thermal and concentration coefficients expansion

- (
*T*,*C*) Temperature and concentration of fluid

- \(\tau\)
Effective heat capacity ratio

*D*_{B}Brownian diffusion coefficient

*D*_{T}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

*q*_{r}Radiative heat flux

*k******Mean absorption coefficient

- \(\sigma^{ * }\)
Stefan–Boltzmann constant

*Q*_{0}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

- \(\eta\)
Dimensionless variable

- (
*We*_{1},*We*_{2}) Local Weissenberg numbers

*M*Magnetic parameter

- \(\lambda^{*}\)
Mixed convection parameter

*N**Buoyancy ratio parameter

*R*Thermal radiation

- (
*S*_{1},*S*_{2}) Thermal and mass stratification parameters

- \((\gamma_{1} ,\gamma_{2} )\)
Thermal and mass Biot numbers

*N*_{b}Brownian motion parameter

*N*_{t}Thermophoresis parameter

- \(\delta\)
Heat source/sink parameter

*Le*Lewis number

*α*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

- \(Re_{x}\)
Local Reynolds number

- (
*f*,*g*) Dimensionless velocities

- \(\theta\)
Dimensionless temperature

- \(\varphi\)
Dimensionless concentration

## Abbreviations

- ODEs
Ordinary differential equations

- PDEs
Partial differential equations

- 3D
Three dimensional

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