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MHD mixed convection–radiation interaction along a permeable surface immersed in a porous medium in the presence of Soret and Dufour’s Effects

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Abstract

This work is focused on the numerical modeling of steady, laminar, heat and mass transfer by MHD mixed convection from a semi-infinite, isothermal, vertical and permeable surface immersed in a uniform porous medium in the presence of thermal radiation and Dufour and Soret effects. A mixed convection parameter for the entire range of free-forced-mixed convection is employed and the governing equations are transformed into non-similar equations. These equations are solved numerically by an efficient, implicit, iterative, finite-difference scheme. The obtained results are checked against previously published work on special cases of the problem and are found to be in excellent agreement. A parametric study illustrating the influence of the thermal radiation coefficient, magnetic field, porous medium inertia parameter, concentration to thermal buoyancy ratio, and the Dufour and Soret numbers on the fluid velocity, temperature and concentration as well as the local Nusselt and the Sherwood numbers is conducted. The obtained results are shown graphically and the physical aspects of the problem are discussed.

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Abbreviations

B 0 :

magnetic field strength

C :

dimensionless concentration, C = (c−c ∞)/(c w−c ∞)

c :

concentration at any point in the flow field

c s :

concentration susceptibility

c w :

concentration at the wall

c ∞ :

concentration at the free stream

D :

mass diffusivity

D f :

Dufour number, D f =  Dk T (c w − c ∞)/[α e c s c p (T w − T ∞)]

F :

inertia coefficient of the porous medium

f :

dimensionless stream function, f =  ψ /[ α e (Pe 1/2 x +  Ra 1/2 x ) ]

g :

gravitational acceleration

h :

local convective heat transfer coefficient

h m :

local mass transfer coefficient

K :

permeability of the porous medium

k e :

porous medium effective thermal conductivity

k T :

thermal–diffusion ratio

k*:

mean absorption coefficient

L :

characteristic length

Le :

Lewis number, Le  = αe/D

M :

square of the Hartmann number, M = (σ B 0 K)/(ɛ μ)

N :

buoyancy ratio, N = β c (c w−c ∞)/[β T (T w−T ∞)]

Nu x :

local Nusselt number, Nu x = hx/k e

p :

Fluid pressure

Pe x :

local Peclet number, Pe x  = V ∞ x/αe

Pe L :

Peclet number at x = L

q r :

radiative heat flux

Ra x :

local Rayleigh number, Ra x  = x/αe[ρ g β T |T w−T ∞|K/μ]

Ra L :

Rayleigh number at x = L

S r :

Soret number, S r =  Dk T (T w − T ∞)/[α e T m (c w − c ∞)]

Sh x :

local Sherwood number, Sh x  = h m x/D

T :

temperature at any point

T m :

mean fluid temperature

T w :

wall temperature

T ∞ :

free stream temperature

u :

tangential or x-component of velocity

v :

normal or y-component of velocity

v 0 :

wall mass transfer coefficient

V ∞ :

free stream velocity

x :

distance along the plate

y :

distance normal to the plate

ε:

porosity of porous medium

Γ:

dimensionless porous medium inertia coefficient, Γ =  2FK(Pe 1/2 L + Ra 1/2 L )/(μ L)

αe :

effective thermal diffusivity of the porous medium

β c :

concentration expansion coefficient

β T :

thermal expansion coefficient

η:

coordinate transformation in terms of x and y, η = y(Pe 1/2 x +  Ra 1/2 x )/x

χ:

mixed convection parameter, \(\chi = [1 + ({Ra_{x} } \mathord{\left/ {\vphantom {{Ra_{x} } {Pe_{x} }}} \right. \kern-\nulldelimiterspace} {Pe_{x} })^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ]^{{ - 1}} \)χ = [1 + (Ra x /Pe x )1/2]−1

ψ:

stream function

θ:

dimensionless temperature, θ = (T−T ∞)/(T w−T ∞)

ρ:

fluid density

σ:

fluid electrical conductivity

σ* :

Stefan–Boltzmann constant

ξ:

transformed suction or injection parameter, ξ =  v 0 x (Pe 1/2 x +  Ra 1/2 x )− 1 /αe

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Authors and Affiliations

  1. Manufacturing Engineering Department, The Public Authority for Applied Education and Training, P. O. Box 42325, Shuweikh, 70654, Kuwait

    Ali J. Chamkha

  2. Mechanical Power and Refrigeration Department, The Public Authority for Applied Education and Training, P. O. Box 42325, Shuweikh, 70654, Kuwait

    Abdullatif Ben-Nakhi

Authors
  1. Ali J. Chamkha
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  2. Abdullatif Ben-Nakhi
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Correspondence to Ali J. Chamkha.

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Chamkha, A.J., Ben-Nakhi, A. MHD mixed convection–radiation interaction along a permeable surface immersed in a porous medium in the presence of Soret and Dufour’s Effects. Heat Mass Transfer 44, 845–856 (2008). https://doi.org/10.1007/s00231-007-0296-x

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