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Applied Mathematics and Mechanics

, Volume 32, Issue 12, pp 1577–1586 | Cite as

Boundary-layer non-Newtonian flow over vertical plate in porous medium saturated with nanofluid

  • F. M. Hady
  • F. S. Ibrahim
  • S. M. Abdel-Gaied
  • M. R. EidEmail author
Article

Abstract

The free convective heat transfer to the power-law non-Newtonian flow from a vertical plate in a porous medium saturated with nanofluid under laminar conditions is investigated. It is considered that the non-Newtonian nanofluid obeys the mathematical model of power-law. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The partial differential system governing the problem is transformed into an ordinary system via a usual similarity transformation. The numerical solutions of the resulting ordinary system are obtained. These solutions depend on the power-law index n, Lewis number Le, buoyancy-ratio number N r, Brownian motion number N b, and thermophoresis number N t. For various values of n and Le, the effects of the influence parameters on the fluid behavior as well as the reduced Nusselt number are presented and discussed.

Key words

non-Newtonian free convection nanofluid porous media 

Nomenclature

DB

Brownian diffusion coefficient

DT

thermophoretic diffusion coefficient

f

rescaled nanoparticle volume fraction

g

gravitational acceleration

km

effective thermal conductivity

K

permeability of the porous medium

Le

Lewis number

n

power index of non-Newtonian fluid, n ⩾ 0

Nr

buoyancy ratio

Nb

Brownian parameter

Nt

thermophoresis parameter

Nu

Nusselt number

Nur

reduced Nusselt number, \(Nu/Ra_x^{\tfrac{1} {2}} \)

qw

wall heat flux

Rax

modified Rayleigh number

s

dimensionless stream function

T

temperature

Tw

temperature at the vertical plate

T

ambient temperature attained as y → ∞

u, υ

Darcian velocity components in x- and y-directions

x, y

Cartesian coordinates

Greek symbols

αm

thermal diffusivity

β

volumetric expansion coefficient of the non-Newtonian fluid

ɛ

porosity

η

similarity variable

θ

dimensionless temperature

μ

effective viscosity of non-Newtonian fluid

ρf

density of the non-Newtonian fluid

ρp

nanoparticles mass density

(ρc)f

heat capacity of the fluid

(ρc)m

effective heat capacity of the porous medium

(ρc)p

effective heat capacity of the nanoparticles material

τ

parameter defined by Eq. (6)

ϕ

nanoparticles volume fraction

ϕw

nanoparticles volume fraction at the vertical plate

ϕ

ambient nanoparticles volume fraction attained as y → ∞

ψ

stream function

Chinese Library Classification

O241.81 

2010 Mathematics Subject Classification

676A05 76N20 76R10 

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Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • F. M. Hady
    • 1
  • F. S. Ibrahim
  • S. M. Abdel-Gaied
    • 2
  • M. R. Eid
    • 2
    Email author
  1. 1.Department of Mathematics, Faculty of ScienceAssiut UniversityAssiutEgypt
  2. 2.Department of Science and Mathematics, Faculty of EducationAssiut UniversityAssiutEgypt

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