# Numerical simulation for three-dimensional flow of Carreau nanofluid over a nonlinear stretching surface with convective heat and mass conditions

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## Abstract

This article addresses (3D) flow of Carreau liquid in the presence of nanomaterials induced by a nonlinearly extendable surface. A nonlinear extendable surface generates the flow. Heat and mass transport via convective process is considered. The novel characteristics in regard to Brownian dispersion and thermophoresis are retained. The variation in partial differential framework to nonlinear ordinary differential framework is done through reasonable transformations. The graphical representation of transformed nonlinear ordinary differential framework is developed for both situations (*n* < 1 and *n* > 1). An efficient numerical solver namely NDSolve is used to tackle the governing nonlinear framework. The contributions of various interesting variables are studied graphically. Physical amounts like surface drag coefficients, transfer of heat and mass rates are portrayed by numeric esteems.

## Keywords

3D flow Carreau liquid Nanomaterials Convective heat and mass conditions Nonlinear stretched surface## List of symbols

- \( u, v,w \)
Velocity components

- \( \mu \)
Dynamic viscosity

- \( \nu \)
Kinematic viscosity

*T*Temperature

- \( T_{\infty } \)
Ambient fluid temperature

*n*Flow behavior index

- \( \left( {\rho c} \right)_{\text{p}} \)
Effective heat capacity of nanoparticles

- \( D_{\text{B}} \)
Brownian diffusion coefficient

- \( U_{w} , V_{w} \)
Surface velocities

- \( \varGamma \)
Material time constant

- \( h_{2} \)
Mass transfer coefficient

- \( \theta \)
Dimensionless temperature

- \( f^{\prime}, g' \)
Dimensionless velocities

- \( We \)
Local Weissenberg number

*m*Positive constant

- \( \alpha \)
Ratio of stretching rates

- \( N_{\text{b}} \)
Brownian motion parameter

*Sc*Schmidt number

- \( Nu_{x} \)
Local Nusselt number

- \( x,y,z \)
Coordinate axes

- \( \rho_{\text{f}} \)
Density of base fluid

- \( \alpha^{*} \)
Thermal diffusivity

*C*Concentration

- \( C_{\infty } \)
Ambient fluid concentration

*k*Thermal conductivity

- \( \left( {\rho c} \right)_{\text{f}} \)
Heat capacity of fluid

- \( D_{\text{T}} \)
Thermophoretic diffusion coefficient

- \( a, b \)
Positive constants

- \( h_{1} \)
Heat transfer coefficient

- \( \zeta \)
Dimensionless variable

- \( \phi \)
Dimensionless concentration

- \( \gamma_{1} \)
Thermal Biot number

- \( \gamma_{2} \)
Concentration Biot number

- \( Sh_{x} \)
Local Sherwood number

- \( Pr \)
Prandtl number

- \( N_{\text{t}} \)
Thermophoresis parameter

- \( C_{\text{fx}} , C_{\text{fy}} \)
Skin friction coefficients

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

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