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

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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.

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Abbreviations

D B :

Brownian diffusion coefficient

D T :

thermophoretic diffusion coefficient

f :

rescaled nanoparticle volume fraction

g :

gravitational acceleration

k m :

effective thermal conductivity

K :

permeability of the porous medium

Le :

Lewis number

n :

power index of non-Newtonian fluid, n ⩾ 0

N r :

buoyancy ratio

N b :

Brownian parameter

N t :

thermophoresis parameter

Nu :

Nusselt number

Nu r :

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

q w :

wall heat flux

Ra x :

modified Rayleigh number

s :

dimensionless stream function

T :

temperature

T w :

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

α m :

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

ρ f :

density of the non-Newtonian fluid

ρ p :

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

ϕ w :

nanoparticles volume fraction at the vertical plate

ϕ ∞ :

ambient nanoparticles volume fraction attained as y → ∞

ψ :

stream function

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Assiut University, Assiut, 71516, Egypt

    F. M. Hady

  2. Department of Science and Mathematics, Faculty of Education, Assiut University, The New Valley, Assiut, 72111, Egypt

    S. M. Abdel-Gaied & M. R. Eid

Authors
  1. F. M. Hady
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  2. F. S. Ibrahim
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  3. S. M. Abdel-Gaied
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  4. M. R. Eid
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Correspondence to M. R. Eid.

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Hady, F.M., Ibrahim, F.S., Abdel-Gaied, S.M. et al. Boundary-layer non-Newtonian flow over vertical plate in porous medium saturated with nanofluid. Appl. Math. Mech.-Engl. Ed. 32, 1577–1586 (2011). https://doi.org/10.1007/s10483-011-1524-7

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  • DOI: https://doi.org/10.1007/s10483-011-1524-7

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