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Non-Newtonian nanofluid free convection flow subject to mixed thermal boundary conditions about a vertical cone

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Abstract

A similarity analysis for natural convection past a vertical cone in Darcy porous medium saturated with a non-Newtonian nanofluid is considered assuming that the cone is subjected to a mixed thermal boundary condition. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The generalized governing equations derived in this work can be applied to the cases of prescribed surface temperature and prescribed heat flux. The effects of the power-law viscosity index and the similarity exponent on the heat transfer characteristics under mixed thermal boundary conditions have been studied. Moreover, numerical solutions of the boundary layer equations are obtained for several values of the nanofluid parameters (Brownian motion number N b, thermophoresis parameter N t, buoyancy ratio number N r). A comparison with previously published work on a special case of the general problem was performed and the results were found to be in excellent agreement.

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Abbreviations

C :

Nanoparticle volume fraction

D :

Brownian diffusion coefficient

\( \hat{D} \) :

Thermophoretic diffusion coefficient

f :

Dimensionless stream function

g :

Acceleration due to gravity

k :

Thermal conductivity

K :

Modified permeability of the porous medium

Le :

Lewis number

m :

Similarity exponent

n :

Power-law index

N b :

The Brownian motion parameter

N r :

The buoyancy ratio

N t :

The thermophoresis parameter

Nu :

Local Nusselt number

q w :

Heat transfer rate

q m :

Mass transfer rate

r :

Local radius of the cone

Ra :

Rayleigh number

S :

Dimensionless nanoparticle volume fraction

Sh :

Local Sherwood number

T :

Temperature of the fluid

(u, v):

Components of velocity of the fluid

(x, y):

Coordinate axes

α :

Thermal diffusivity of porous medium

β :

Volumetric coefficient of thermal expansion of fluid

\( \mu^{{ \star }} \) :

Viscosity of the fluid

τ :

The ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid

ρ f :

Fluid density

ρ p :

Nanoparticle mass density

(ρc)f :

Heat capacity of the fluid

(ρc)p :

Effective heat capacity of nanoparticle material

Ω:

Half angle of the cone

λ :

Mixed thermal boundary condition parameter

ψ :

Stream function

θ :

Dimensionless temperature

η :

Similarity parameter

w :

Conditions at the wall

∞:

Conditions in the free stream

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Acknowledgments

The authors wish to express their very sincere thanks to the reviewers for their valuable comments and suggestions.

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Authors and Affiliations

  1. Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt

    A. Mahdy

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  1. A. Mahdy
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Correspondence to A. Mahdy.

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Technical Editor: Monica Feijo Naccache.

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Mahdy, A. Non-Newtonian nanofluid free convection flow subject to mixed thermal boundary conditions about a vertical cone. J Braz. Soc. Mech. Sci. Eng. 36, 951–960 (2014). https://doi.org/10.1007/s40430-014-0132-4

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  • DOI: https://doi.org/10.1007/s40430-014-0132-4

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