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Natural convection of nanofluid over vertical plate embedded in porous medium: prescribed surface heat flux

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Abstract

The aim of the present paper is to analyze the natural convection heat and mass transfer of nanofluids over a vertical plate embedded in a saturated Darcy porous medium subjected to surface heat and nanoparticle fluxes. To carry out the numerical solution, two steps are performed. The governing partial differential equations are firstly simplified into a set of highly coupled nonlinear ordinary differential equations by appropriate similarity variables, and then numerically solved by the finite difference method. The obtained similarity solution depends on four non-dimensional parameters, i.e., the Brownian motion parameter (N b), the Buoyancy ratio (N r), the thermophoresis parameter (N t), and the Lewis number (Le). The variations of the reduced Nusselt number and the reduced Sherwood number with N b and N t for various values of Le and N r are discussed in detail. Simulation results depict that the increase in N b, N t, or N r decreases the reduced Nusselt number. An increase in the Lewis number increases both of the reduced Nusselt number and the Sherwood number. The results also reveal that the nanoparticle concentration boundary layer thickness is much thinner than those of the thermal and hydrodynamic boundary layers.

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Abbreviations

C :

constant from the integration of Eq. (23)

D B :

Brownian diffusion coefficient

D T :

thermophoretic diffusion coefficient

f :

rescaled nanoparticle volume fraction, nanoparticle concentration

g :

gravitational acceleration vector

K :

permeability of the porous medium

k :

thermal conductivity

k m :

effective thermal conductivity

Le :

Lewis number

N b :

Brownian motion parameter

N r :

buoyancy ratio

N t :

thermophoresis parameter

p :

pressure

q np :

nanoparticle flux

q w :

wall heat flux of the vertical plate

Ra x :

local Rayleigh number

S :

dimensionless stream function

T :

temperature

T ∞ :

ambient temperature

T w :

wall temperature of the vertical plate

U :

reference velocity

u,v :

Darcy velocity components

x,y :

Cartesian coordinates

(ρc) f :

heat capacity of the fluid

(ρc) p :

effective heat capacity of the nanoparticle material

µ:

viscosity of the fluid

α m :

effective thermal diffusivity defined by Eq. (5)

β :

volumetric expansion coefficient of the fluid

ɛ :

porosity

η :

dimensionless distance defined by Eq. (5)

θ :

dimensionless temperature

λ :

index in the power-law variation of the wall heat and mass flux

ρ f :

fluid density

ρ p :

nanoparticle mass density

τ :

parameter defined by Eq. (5)

ϕ :

nanoparticle volume fraction

ϕ ∞ :

ambient nanoparticle volume fraction

ϕ w :

nanoparticle volume fraction at the wall of the vertical plate

ψ :

stream function

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Authors and Affiliations

  1. Department of Mechanical Engineering, Shahid Chamran University of Ahvaz, Ahvaz, 6135743337, Iran

    A. Noghrehabadi, A. Behseresht & M. Ghalambaz

Authors
  1. A. Noghrehabadi
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  2. A. Behseresht
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  3. M. Ghalambaz
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Correspondence to A. Noghrehabadi.

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Noghrehabadi, A., Behseresht, A. & Ghalambaz, M. Natural convection of nanofluid over vertical plate embedded in porous medium: prescribed surface heat flux. Appl. Math. Mech.-Engl. Ed. 34, 669–686 (2013). https://doi.org/10.1007/s10483-013-1699-6

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  • DOI: https://doi.org/10.1007/s10483-013-1699-6

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2010 Mathematics Subject Classification

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