oscillatory chemically-reacting mhd free convection heat ...€¦ · convection heat and mass...

Int. J. of Appl. Math and Mech. 5 (6): 15-37, 2009.

OSCILLATORY CHEMICALLY-REACTING MHD FREE

CONVECTION HEAT AND MASS TRANSFER IN A POROUS

MEDIUM WITH SORET AND DUFOUR EFFECTS: FINITE ELEMENT

MODELING

R. Bhargava

1, R. Sharma

2 , and O.A. Bég

3

1 Mathematics Department, Indian Institute of Technology, Roorkee-247667, India.

2 Mathematics Department, Indian Institute of Technology, Roorkee-247667, India.

3 Engineering Magnetohydrodynamics and Heat Transfer Research, Thermofluids Group,

Mechanical Engineering Department, Sheaf Building, Sheffield Hallam University, Sheffield,

South Yorkshire, S1 1WB, England, UK.

Email: [email protected]

Received 12 August 2008; accepted 8 December 2009

ABSTRACT

The effects of thermo-diffusion (Soret effect) and diffuso-thermal gradients (Dufour effect) on

the unsteady incompressible magneto-hydrodynamic (MHD) free convection flow with mass

transfer past a semi-infinite vertical plate in a Darcian porous medium in the presence of

significant thermal radiation, first order homogenous chemical reaction and viscous heating

are analyzed. The governing differential equations are non-dimensionalized using a similarity

transformation rendering a system of coupled, nonlinear partial differential equations which

are solved numerically using the robust, extensively-validated finite element method.

Dimensionless velocity (U) is decreased with increasing magnetic parameter (M). An increase

in Eckert number (Ec) causes greater mechanical energy to be dissipated as thermal energy

and enhances fluid temperatures (θθθθ). An increase in chemical reaction parameter (γ) increases

velocity (U), temperature (θ) and also concentration value (φ). Temperatures (θ) are elevated

substantially with decreasing Soret number (Sr) and simultaneous increasing Dufour number

(Du). Concentration values (φ) are conversely enhanced with increasing Soret number (Sr)

and a concurrent decrease in Dufour number (Du). Both temperature and velocity are

suppressed with a rise in heat absorption parameter ( Φ ); On the other hand an increase in

thermal radiation absorption parameter (Q1) generates an increase in both velocity and

temperature fields. Increasing Schmidt number (Sc) is found to cause a decrease in both

temperature and concentration profiles. Finally, the numerical values of local skin friction,

local rate of heat transfer parameter and local mass transfer parameter are also presented in

tabular form. The present problem has significant applications in chemical engineering

materials processing, solar porous wafer absorber systems and metallurgy.

Keywords: Porous medium, oscillatory flow, MHD, free convection, heat transfer, mass

transfer, finite elements, Soret and Dufour effects, chemical engineering systems.

1 INTRODUCTION

16 R. Bhargava et al.


Boundary layer convection heat and mass transfer flows abound in many areas of chemical

engineering. For example in the molecular evaporator (Lutisan et al 2002), combined heat and

mass transfer exerts a significant role in liquid film performance characteristics. Other

applications include polymer processing (Kleinstreuer and Wang 1989), catalytic slab systems

(Mihail and Teodorescu 1978), electrochemical phenomena (Probstein 1989) and adsorption

processes (Yang 1990). Many physical phenomena also involve free convection driven by

heat generation or absorption as encountered in for example chemical reactor design and

dissociating fluids. Heat generation effects may alter the temperature distribution and

therefore, the particle deposition rate in such systems. Heat generation may also be critical in

nuclear reactor cores, electronic chips, semi conductor wafers and also fire dynamics (Bég

2006). In the recent years, the effect of magnetic field on heat and mass transfer flows through

a porous medium has also stimulated considerable interest owing to diverse applications in

film vaporization in combustion chambers, transpiration cooling of re-entry vehicles, solar

wafer absorbers, manufacture of gels, magnetic materials processing, astrophysical flows and

hybrid MHD power generators. Porous media abound in chemical engineering systems and

magnetic fields are frequently used to control transport phenomena in electrically-conducting

fluent media. Iliuta and Larachi (Iliuta and Larachi 2003) used a Kozeny-Carman porous

hydrodynamic model to study the isothermal hydromagnetic two-fluid flow in trickle bed

reactors subjected to a homogeneous transverse magnetic field. Other interesting analyses of

hydromagnetic flow in porous media include the studies by Al-Nimr and Hader (Al-Nimr and

Hader 1999) concerning open-ended vertical porous channels, Iliuta and Larachi (Iliuta and

Larachi 2003) who investigated multi-phase porous media hydromagnetics under spatially

uniform magnetic-field gradients in a process intensification context, Dahikar and Sonolikar

(2006) who considered hydromagnetic flows in a circulating fluidized bed, Takhar and Bég

(1997) who studied numerically the effects of Hartmann number and inertial porous drag on

flat-plate hydromagnetic boundary layer convection and Geindreau and Auriault (Geindreau

and Auriault 2002) who derived a modified permeability tensor for magnetohydrodynamic

flow in a Darcian porous medium. Bég et. al. (Bég et. al. 2001) used a numerical difference

method to analyze the two-dimensional steady free convection magneto-viscoelastic flow in a

Darcy-Brinkman porous medium. Also very recently Bég et al. (Bég et. al. 2008) used the

network thermodynamic simulation approach to study the hydromagnetic convection flow

from an isothermal sphere to a non-Darcian porous medium with heat generation or

absorption effects. Extensive studies have also materialized pertaining to the effects of

chemical reaction on coupled heat and mass transfer flows in porous media owing to potential

applications in drying technologies, distribution of temperature and moisture over agricultural

fields and groves of fruit trees, energy transfer in wet cooling towers, flow in desert coolers

and the drying and/or burnout of processing aids in the colloidal processing of advanced

ceramic materials. Gatica et al. (Gatica et al.1989) studied free convection boundary layer

flow in a porous medium with chemical reaction effects. Stangle and Aksay (Stangle and

Aksay 1990) studied simultaneous reactive momentum, heat and mass transfer phenomena

in disordered porous media with applications in optimization of processing conditions in the

design of an improved binder removal process. Souza et al. (Souza et al. 2003) studied mass

transfer in a packed-bed reactors including dispersion in the main fluid phase, internal

diffusion of the reactant in the pores of the catalyst, and surface reaction inside the catalyst.

They employed volume averaging and Darcy’s law for a spatially periodic porous medium.

Prud’homme and Jasmin (Prud’homme and Jasmin 2006) studied biochemical free

convection flow in a porous medium with internal heat generation from microbial oxidation,

as a simulation of a composting reactor, for Rayleigh numbers equal to 0.25 and 25. Silva et

al. (Silva et al. 2007) also used a volume averaging transport model to simulate two reactive

processes in porous media characterizing the porous medium by different length scales. Zueco

Oscillatory Chemically-Reaction MHD Free Convection Heat and Mass Transfer


17

et al. (Zueco et al. 2008) recently examined the influence of chemical reaction on the

hydromagnetic heat and mass transfer boundary layer flow from a horizontal cylinder in a

Darcy-Forchheimer regime using network simulation. Bég et al. (Bég et al. 2007) have also

analyzed the chemical reaction rate effects on steady buoyancy-driven dissipative micropolar

free convective heat and mass transfer in a Darcian porous regime. Effects of chemical

reaction on free convection flow of a polar fluid through a porous medium in the presence of

internal heat generation are examined by Patil and Kulkarni (Patil and Kulkarni 2008). When

heat and mass transfer occur simultaneously in a moving fluid, the relations between the

fluxes and the driving potentials may be of a more intricate nature. An energy flux can be

generated not only by temperature gradients but by composition gradients also. The energy

flux caused by a composition gradient is termed the Dufour or diffusion-thermo effect. On the

other hand, mass fluxes can also be created by temperature gradients and this embodies the

Soret or thermal-diffusion effect. Such effects are significant when density differences exist in

the flow regime. For example, when species are introduced at a surface in a fluid domain,

with a different (lower) density than the surrounding fluid, both Soret (thermo-diffusion) and

Dufour (diffuso-thermal) effects can become influential. Soret and Dufour effects are

important for intermediate molecular weight gases in coupled heat and mass transfer in fluid

binary systems, often encountered in chemical process engineering and also in high-speed

aerodynamics. Dursunkaya and Worek (Dursunkaya and Worek 1992) have studied

Soret/Dufour effects on transient and steady natural convection from vertical surface. Both

free and forced convection boundary layer flows with Soret and Dufour have been addressed

by Abreu et al. (Abreu et al. 2006). Recently Bég et al. (Bég et al. 2008) used the local

nonsimilarity method with a shooting procedure to analyze mixed convective heat and mass

transfer from an inclined plate with Soret/Duofur effects with applications in solar energy

collector systems. Bég et al. (Bég et al. 2008) extended this study to include chemical

reaction effects. Anghel et. al. (Anghel et. al. 2000) have discussed the composite Dufour and

Soret effects on free convection boundary layer over a vertical surface embedded in a Darcian

porous medium. Postelnicu (Postelnicu 2004) has presented numerical solutions for the effect

of magnetic field on buoyancy-driven heat and mass transfer from a vertical surface in a

porous medium including Soret and Dufour effects. Bég et al. (Bég et al. 2008) obtained finite

element solutions for coupled heat and mass transfer in a micropolar fluid-saturated Darcy-

Forchheimer porous medium with Soret and Dufour effects. Rawat et al. (Rawat et al. 2008)

considered the corresponding problem for a Newtonian fluid. The above studies neglected the

presence of thermal radiation which becomes important when high temperatures are reached.

Many chemical engineering and industrial processes invoke simultaneous convective and

radiative heat transfer modes including combustion chambers, glass production and

metallurgy. Makinde (Makinde 2005) studied the free convection flow with thermal radiation

and mass transfer past a moving vertical porous plate. Sattar and Kalim (Sattar and Kalim

1996) investigated the transient natural convection-radiation boundary layer flow along a

vertical porous plate. Chamkha et al. (Chamkha et al. 2001) used the Blottner difference

method to study laminar free convection flow of air past a semi-infinite vertical plate in the

presence of chemical species concentration and thermal radiation effects. Duwairi and

Damseh (2004) analyzed the combined effects of forced and natural convection heat transfer

in the presence of transverse magnetic field form a vertical surfaces with radiation heat

transfer. Very recently Ibrahim et al. (Ibrahim et al. 2008) studied the effects of chemical

reaction and radiation absorption on transient hydromagnetic natural convection flow with

wall transpiration and heat source. Several studies have also described thermal radiation

effects on convection flows in porous media. Takhar et al. (Takhar et al. 1998) used a non-

gray gas differential model to study numerically the radiation effects on convective boundary



layer flow in a Darcy-Brinkman-Forchheimer nonlinear porous medium. Takhar et al. (Takhar

et al. 2003) later employed the Rosseland diffusion radiative algebraic model to investigate

mixed radiation-convection flow in a non-Darcy porous medium. Seddeek et al. (Seddeek et

al. 2007) more recently reported on the effect of chemical reaction and variable viscosity on

hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through a Darcian

porous media in the presence of radiation and magnetic field. Bég et al. (Bég et al. 2008) used

the network thermodynamic simulation method to investigate the transient free convection

heat and mass transfer boundary layer flow in a Darcy-Forchheimer isotropic porous medium.

Another effect which bears great importance on heat transfer processes is viscous dissipation.

The determination of the temperature distribution when the internal friction is not negligible is

of great significance in different industrial fields, such as chemical and food processing, oil

exploration and bio-engineering. In such scenarios the effects of viscous dissipation must be

included in the energy equation. Several authors have recently studied numerically viscous

dissipation effects in purely fluid and also porous media regimes. Zueco (Zueco 2007)

considered the transient magnetohydrodynamic natural convection boundary layer flow with

suction, viscous dissipation and thermal radiation effects. Bég et al. (Bég et al. 2008)

presented the first analysis of dissipative third grade viscoelastic flow in a Darcy-Forchheimer

porous domain using finite elements. Bég et al. (Bég et al. 2009) have also studied

numerically both viscous heating and Joule (Ohmic) dissipation effects on transient

Hartmann-Couette convective flow in a Darcian porous medium channel also including Hall

current and ionslip effects. In the present paper, we extend the problem investigated in

Ibrahim et al. (Ibrahim et al. 2008) by including viscous dissipation, Darcian porous drag and

also Soret and Dufour effects. The extended conservation equations are solved using the

highly efficient finite element method. Such a study constitutes an important addition to

numerical multi-physical fluid dynamics simulations and has not appeared thusfar in the

literature.

2 FORMULATION AND MATHEMATICAL MODELLING

We consider the transient, incompressible, two-dimensional, coupled, convective heat and

mass transfer of a Newtonian, viscous, electrically-conducting and heat absorbing fluid along

an infinite, porous, vertical plate embedded in an isotropic, homogenous, Darcian porous

medium in the presence of thermal radiation, viscous dissipation and homogenous chemical

reaction with Soret and Dufour effects. The x-axis is directed along the infinite plate and the

y-axis is transverse to this. A magnetic field B0 of uniform strength is applied transversely to

the direction of the flow. For 0t ≥ , the plate starts moving impulsively in its own plane with

constant velocity, up, with plate temperature raised toT and the concentration level at the plate

raised to .CC

The physical model and geometrical coordinates are shown in Fig. 1. A heat

source is placed within the flow to simulate possible heat absorption effects. Under the above

assumptions, the physical variables are functions of y and t only. With the usual boundary

layer and Boussinesq approximations, the governing equations may be written as follows:

Continuity Equation:

0v

y

∂=

∂ (1)

Momentum Conservation Equation:



19

( ) ( )22

2

o

p

Bu u uv u g T T g C C

t y y k

σ νν β β

ρ∗

∞ ∞

∂ ∂ ∂+ = − + + − + − ∂ ∂ ∂

(2)

Energy Conservation Equation:

( ) ( )22 2

012 2

m T

p p s p

Q D kT T T u Cv T T Q C C

t y y c y c c c y

να

ρ∞ ∞

∂ ∂ ∂ ∂ ∂′+ = + − − + − + ∂ ∂ ∂ ∂ ∂

(3)

Species Diffusion Equation:

( )2 2

2 2

m Tm l

m

D kC C C Tv D k C C

t y y T y∞

∂ ∂ ∂ ∂+ = − − +

∂ ∂ ∂ ∂ (4)

In equation (2) the first term on the left hand side is the temporal velocity gradient, the second

is the convective inertial (acceleration) term. On the right hand side of equation (2), the first

term simulates the viscous shear effects (momentum diffusion), the first term in the square

brackets represents the Lorentzian magnetohydrodynamic retarding force (acting transverse

to the magnetic field), the second term in the square brackets is the Darcian linear porous

drag force, the penultimate term on the right hand side of equation (2) is the thermal

buoyancy force and the final term designates species buoyancy. In equation (3) the first term

on the left is the temporal thermal gradient, the second being the thermal convective term. On

the right hand side of equation (3), the first term represents thermal diffusion, the second is

the viscous dissipation term (due to internal friction in the fluid), the third term is the heat

source term, the fourth is the thermal radiation source term and the final term represents

Dufour concentration gradient effects on the temperature field. In equation (4) the first term

on the left hand side signifies the temporal concentration gradient, and the second term is the

convective mass transfer term. On the right hand side of equation (4), the first term is the

mass (species) diffusion term, the second is the chemical reaction effect term and the final

term represents the contribution from Soret temperature gradients on the concentration field.

The corresponding boundary conditions on the vertical surface and in the free stream can be

defined now as follows:

0 : , ( ), ( ) , ( )nt nt

p w w w wy u u v v t T T T T e C C C C eε ε∞ ∞= = = = + − = + −

: 0, ,y u T T C C∞ ∞→ ∞ = → → (5)

where u is the x -direction velocity, v is the y -direction velocity, up is the plate translational

velocity, t is the time, ρ is fluid density, ν is kinematic viscosity, σ is the fluid electrical

conductivity, 0B is the magnetic field intensity, pk is the permeability of the porous

medium,α is the thermal diffusivity, 0Q is the heat absorption coefficient, 1Q′ is the radiation

absorption coefficient, g is gravity, lk is the chemical reaction parameter, β and β ∗ are the

thermal and concentration expansion coefficients respectively, mD is coefficient of mass

diffusivity, pc is specific heat at constant pressure, sc is the concentration susceptibility, mT is

mean fluid temperature, Tk is thermal diffusion ratio, ,T C are the dimensional temperature and



concentration respectively, wT , wC are the wall temperature and concentration respectively,

T∞ andC∞ are the free stream temperature and concentration respectively and n denotes the

frequency of oscillations. The last terms on the right-hand side of the energy equation (3) and

concentration equation (4) signify respectively, the Dufour or diffusion-thermo effect and the

Soret or thermo-diffusion effect. We assume that the solution of the equation (1) takes the

following form:

0( ) (1 )ntv v t V Aeε= = − + (6)

where A is a real positive constant, ε is constant ( )1ε < and 0V is scale of suction velocity at

the plate surface.

3 TRANSFORMATION OF MODEL

To obtain the non-dimensional form of the governing equations, we now introduce the

following dimensionless variables:

2

0 0

2

0 0 0 0

, , , , , , ,p

p

w w

uV y V t T T C Cu v nU V Y T U N

V V V V T T C C

νθ φ

ν ν∞ ∞

∞ ∞

− −= = = = = = = =

− − (7)

where U is dimensionless x-direction velocity, V is dimensionless y-direction velocity, Y is

dimensionless y coordinate (normal to the plate), T is dimensionless time, Up is

dimensionless plate translational velocity, N is dimensionless frequency of oscillation, θ is

dimensionless temperature function and φ is the dimensionless concentration function.

Implementing the transformations (7) into the equations (2) to (4), we obtain the following

dimensionless coupled partial differential equations:

2

2

1(1 )NTU U U

Ae M U Gr GmT Y Y K

ε θ φ∂ ∂ ∂

− + = − + + + ∂ ∂ ∂

(8)

22 2

12 2

1(1 )

Pr

NT UAe Ec Q Du

T Y Y Y Y

θ θ θ φε θ φ

∂ ∂ ∂ ∂ ∂ − + = + − Φ + +

∂ ∂ ∂ ∂ ∂ (9)

2 2

2 2

1(1 )NT

Ae SrT Y Sc Y y

φ φ φ θε γφ

∂ ∂ ∂ ∂− + = − +

∂ ∂ ∂ ∂ (10)

where 2

0

2

0

BM

V

σ ν

ρ= is the magnetic field parameter simulating the relative effects of the

magnetic drag and the viscous hydrodynamic force,

2

0

2

pk VK

ν= is the permeability parameter

which represents the hydraulic conductivity of the fluid percolating in the porous medium,

( )3

0

wg T TGr

V

ν β ∞−= is the Grashof number which simulates the relative influence of thermal



21

buoyancy to viscous hydrodynamic forces,( )*

3

0

wg C CGm

V

ν β ∞−= is the solutal (species)

Grashof number which embodies the ratio of species buoyancy to viscous hydrodynamic

forces , Prν

α= is the Prandtl number which represents the ratio of momentum to thermal

diffusivity ,( )

2

0

p w

VEc

c T T∞

=−

is the Eckert number which simulates the relationship between

the kinetic energy of the flow and enthalpy, and is used to characterize

dissipation, 0

2

0p

Q

c V

ν

ρΦ = is the heat source parameter,

( )( )1

1 2

0

w

w

Q C CQ

T T V

ν∞

∞

′ −=

− is the radiation

absorption parameter,m

ScD

ν= is the Schmidt number which defines the ratio of the shear

component for diffusivity viscosity/density to the diffusivity for mass transfer i.e. the ratio of

momentum diffusivity (viscosity) and mass diffusivity,2

0

lk

V

νγ = is the chemical reaction

parameter,( )( )

m T w

s p w

D k C CDu

c c T T

∞

∞

−=

− is the Dufour number and

( )( )

m T w

m w

D k T TSr

T C Cν∞

∞

−=

− is the Soret

number. The corresponding transformed boundary conditions are:

0 : , 1 , 1NT NT

pY U U e eθ ε φ ε= = = + = +

: 0, 0, 0Y U θ φ→ ∞ → → → (11)

In for example the design of chemical engineering systems, the following parameters are

useful to compute. The skin friction coefficient which signifies the surface shear stress is

defined as:

2

wf

w

Cu

τ

ρ= , 2

0

0

(0)w

y

uV U

yτ µ ρ

=

∂′= =

∂ i.e. 4 (0)fC U ′= (12)

The local Nusselt number which embodies the ratio of convective to conductive heat transfer

across (normal to) the boundary and is a quantification of the surface temperature gradient

(heat transfer rate at the wall) is defined as:

0

( ) ,( )

u

w y

x TN x

T T y∞ =

− ∂=

− ∂ i.e.

( )(0)

Re

u

x

N xθ ′= − (13)

Finally the local Sherwood number which encapsulates the ratio of convective to diffusive

mass transport and simulates the surface mass transfer rate, is defined as:

0

( ) ,( )

h

w y

x CS x

C C y∞ =

− ∂=

− ∂then

( )(0)

Re

h

x

S xφ ′= − (14)



where 0Rex

V x

ν

−= is the local Reynolds number.

4 NUMERICAL SOLUTION BY THE FINITE ELEMENT METHOD

The equations are solved using the finite element method (FEM) as described by Reddy

(Reddy 2006). The authors have implemented this excellent method in a wide spectrum of

magnetohydrodynamic and also numerous non-magnetic transport phenomena problems of

interest in energy, chemical and biomechanical engineering systems. For example Naroua et

al. (Naroua et al. 2007) studied the influence of heat source and Hall and ionslip currents on

rotating unsteady plasma flow. Takhar et al. (Takhar et al. 2007) analysed third grade

hydrodynamic flow in a Darcy-Forchheimer porous medium. Bhargava et al. (Bhargava et al.

2007) investigated coupled heat and mass transfer in micropolar boundary layer flow from a

nonlinear stretching sheet. Further studies utilizing the present finite element approach

include Bég et al. (Bég et al. 2008) who studied heat transfer in biomagnetic micropolar flow

in Darcy-Forchheimer porous media, and Bhargava et al. (Bhargava et al. 2008) who

considered micropolar heat and mass transfer from a cylinder. The present finite element code

has therefore been extensively validated with other numerical schemes including finite

difference solvers, asymptotic methods and network numerical simulation (Bhargava et al.

2007, Bhargava et al. 2008). It is based on a conservative approach and detailed convergence

tests have been conducted to guarantee monotonicity. Further details are provided later in the

paper. FEM is extremely effective in solving nonlinear multiple degree partial and ordinary

differential equation systems. The fundamental steps comprising the method are as follows:

Step 1: Discretization of the domain into elements:

The whole domain is divided into finite number of sub-domains, a process known as

discretization of the domain. Each sub-domain is termed a finite element. The collection of

elements is designated the finite element mesh.

Step 2: Derivation of the element equations:

The derivation of finite element equations i.e. algebraic equations among the unknown

parameters of the finite element approximation, involves the following three steps:

a. Construct the variational formulation of the differential equation.

b. Assume the form of the approximate solution over a typical finite element.

c. Derive the finite element equations by substituting the approximate solution into

variational formulation.

These steps results in a matrix equation of the form { } { }e e eK c F = , which defines the finite

element model of the original equation.

Step 3: Assembly of element equations:



23

The algebraic equations so obtained are assembled by imposing the inter-element continuity

conditions. This yields a large number of algebraic equations, constituting the global finite

element model, which governs the whole flow domain.

Step 4: Impositions of boundary conditions:

The physical boundary conditions defined in equation (11) are imposed on the assembled

equations.

Step 5: Solution of the assembled equations:

The final matrix equation can be solved by a direct or indirect (iterative) method. For

computational purposes, the coordinate y is varied from 0 to ymax = 8, where ymax represents

infinity i.e. external to the momentum, energy and concentration boundary layers. The whole

domain is divided into a set of 80 line elements of equal width 0.1, each element being two-

noded. Monotonic convergence is achieved.

4.1. Variational Formulation

The variational formulation associated with equations (8) to (10) over a typical two-noded

linear element 1( , )e eY Y + is given by

1 2

1 2

1(1 ) 0

e

e

Y

NT

Y

U U Uw Ae M U Gr Gm dY

T Y Y Kε θ φ

+ ∂ ∂ ∂ − + − + + − − =

∂ ∂ ∂ ∫ (15)

1 22 2

2 12 2

1(1 ) 0

Pr

e

e

Y

NT

Y

Uw Ae Ec Q Du dY

T Y Y Y Y

θ θ θ φε θ φ

+ ∂ ∂ ∂ ∂ ∂ − + − − + Φ − − = ∂ ∂ ∂ ∂ ∂

∫ (16)

1 2 2

3 2 2

1(1 ) 0

e

e

Y

NT

Y

w Ae Sr dYT Y Sc Y Y

φ φ φ θε γφ

+ ∂ ∂ ∂ ∂− + − + − =

∂ ∂ ∂ ∂ ∫ (17)

where1 2,w w and

3w are arbitrary test functions and may be viewed as the variation in ,U θ and

φ respectively. After reducing the order of integration and non-linearity, we arrive at the

following system of equations:

1

1

11 1

1

1 1 1

(1 )

01

ee

ee

NT

YY

YY

wU U Uw Ae w

T Y Y Y UdY w

YM wU Grw Gmw

K

ε

θ φ

+

+

∂∂ ∂ ∂ − + + ∂ ∂ ∂ ∂ ∂ − =

∂ + + − −

∫ (18)



1

1

22 2

22 2 1 2 2

2

1(1 )

Pr

0Pr

ee

ee

NT

YY

YY

ww Ae w

T Y u Y

wU UEcw w Q w dY Duw

Y Y Y Y

wDu

u Y

θ θ θε

θ φθ φ

φ

+

+

∂∂ ∂ ∂ − + + ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂

− + Φ − − + = ∂ ∂ ∂ ∂ ∂ ∂

+ ∂ ∂

∫ (19)

1

1

33 3

33

33

1(1 )

0ee

ee

NTYY

YY

ww Ae w

wT Y Sc Y YdY Srw

w Sc Y Yw Sr

Y Y

φ φ φε

φ θ

θγ φ

+

+

∂∂ ∂ ∂ − + + ∂ ∂ ∂ ∂ ∂ ∂

− + = ∂ ∂ ∂ ∂ + + ∂ ∂

∫ (20)

4.2 Finite-Element Formulation

The finite-element model may be obtained from equations (18) to (20) by substituting finite

element approximations of the form:

2 2 2

1 1 1

, ,e e e e e e

j j j j j j

j j j

U U ψ θ θ ψ φ φ ψ= = =

= = =∑ ∑ ∑ (21)

with ( )1 2 3 1,2 ,e

iw w w iψ= = = = where e

jU , e

jθ and e

jφ are the velocity, temperature and

concentration respectively at jth

node of typical eth

element 1( , )e eY Y + and e

iΨ are the shape

functions for this typical element 1( , )e eY Y + and are taken as:

1 21

11 1

,Y Y Y Ye ee e Y Y Ye eY Y Y Ye ee e

ψ ψ− −+= = ≤ ≤ +− −+ +

(22)

The finite element model of the equations for eth

element thus formed is given by

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

{ }

111 12 13 11 12 13

21 22 23 21 22 23 2

31 32 33 31 32 33 3

e e e

e e e

e e e

U U bK K K M M M

K K K M M M b

K K K M M M b

θ θ

φ φ

′ ′ + = ′

(23)

where { },mn mn

K M and { } { } { } { }{ , , , ,

e e e eU Uθ φ ′ { } { } { }},

e e meand bθ φ′ ′ ( , 1,m n =

)2,3 are the set of matrices of order 2 2× and 2 1× respectively and ′(dash) indicates d/dY.

These matrices are defined as follows:

( )1 1 1

11 1(1 )

e e e

e e e

Y Y Ye eej jNT e e ei

ij i i j

Y Y Y

K Ae dY dY M dYY Y Y K

ε+ + + ∂Ψ ∂Ψ∂Ψ

= − + Ψ + + + Ψ Ψ ∂ ∂ ∂ ∫ ∫ ∫



25

1 1

12 13,

e e

e e

Y Y

e e e e

ij i j ij i j

Y Y

K Gr dY K Gm dY+ +

= − Ψ Ψ = − Ψ Ψ∫ ∫1

11 12 13, 0

e

e

Y

e e

ij i j ij ij

Y

M dY M M+

= Ψ Ψ = =∫

1

21e

e

Y e

je

ij i

Y

UK Ec dY

Y Y

+ ∂Ψ∂= − Ψ ∂ ∂ ∫

( )1 1 1

22 1(1 )

Pr

e e e

e e e


ij i i j

Y Y Y

K Ae dY dY dYY Y Y

ε+ + + ∂Ψ ∂Ψ∂Ψ

= − + Ψ + + Φ Ψ Ψ ∂ ∂ ∂ ∫ ∫ ∫

1

23

1 ,e

e

Y eeje e i

ij i j

Y

K Q Du dYY Y

+ ∂Ψ∂Ψ= − Ψ Ψ + ∂ ∂ ∫

1

21 23 220,e

e

Y

e e

ij ij ij i j

Y

M M M dY+

= = = Ψ Ψ∫

1

31 320,

e

e

Y eeji

ij ij

Y

K K Sr dYY Y

+ ∂Ψ∂Ψ= =

∂ ∂∫

( )1 1 1

33 1(1 )

e e e

e e e


ij i i j

Y Y Y

K Ae dY dY dYY Sc Y Y

ε γ+ + + ∂Ψ ∂Ψ∂Ψ

= − + Ψ + + Ψ Ψ ∂ ∂ ∂ ∫ ∫ ∫

1

31 32 330,e

e

Y

e e

ij ij ij i j

Y

M M M dY+

= = = Ψ Ψ∫

1 1

1

1 2

2

1,

Pr

1

e e

e e

e

e

Y Y

e e e e e

i i i i i

Y Y

Y

e e e

i i i

Y

UAnd b b Du

Y Y Y

b SrSc Y Y

θ φ

φ θ

+ +

+

∂ ∂ ∂ = Ψ = Ψ + Ψ

∂ ∂ ∂

∂ ∂ = Ψ + Ψ

∂ ∂

(24)

where 2

1

e e

i iU U

i= Ψ∑

=. The whole domain is divided in to a set of 80 intervals of equal

length, 0.1. At each node 3 functions are to be evaluated; hence after assembly of the elements,

we obtain a set of 243 equations. The system of equations after assembly of the elements, are

nonlinear and consequently an iterative scheme is employed to solve the matrix system. The

system is linearized by incorporating known functionU , which are solved using the Gauss

elimination method maintaining an accuracy of 0.0005.

5 VALIDATION AND MONOTONIC CONVERGENCE

Benchmarking of the source FEM code has been performed against finite difference methods.

Excellent agreement was found. Details have been omitted however for brevity. The present

program has been adapted for over 50 different nonlinear boundary value problems by

the authors and is therefore extremely reliable. Previous validations have been performed



rigorously and compared with published results in the literature. The reader is referred to

Agarwal et al. (Agarwal et al. 1990), Takhar et al. (Takhar et al. 1998), Takhar et al. (Takhar

et al. 2000), Takhar et al. (Takhar et al. 2001), Bhargava et al. (Bhargava et al. 2003) and

Bhargava et al. (Bhargava et al. 2004). The line elements employed achieved rapid

convergence. A monotonic convergence criterion was also established for which the two-

nodes line (“rod-type”) elements employed were selected to ensure that the mesh was

compatible. When monotonic convergence is achieved the accuracy of the solution results in a

continuous increase with further refinement of the finite element mesh. As such mesh

refinement is executed by delineating a priori utilized elements into two or more elements,

resulting in “embedding” in the new mesh. Effectively, as documented by Bathe (Bathe 1996)

the new space of finite element interpolation functions encapsulates the previously utilized

space and with mesh refinement the dimension of the finite element solution space is

enhanced continuously to embody the exact solution. Excellent convergence was achieved in

the present study.

6 RESULT AND DISCUSSION

We are primarily interested in examining the influence of thermal-diffusion and diffusion-

thermo effects i.e. Soret number (Sr) and Dufour number (Du) on the flow variables.

Additionally we have computed the influence of the magnetic parameter (M), chemical

reaction parameter (γ), Eckert number (Ec), Schmidt number (Sc), heat absorption parameter

( Φ ) and radiation absorption parameter (Q1). The values of other parameters are taken to be

fixed as follows: A = 0.5, ε = 0.2, N (dimensionless frequency of oscillations) = 0.1, plate

translational velocity (Up) = 0.5, Prandtl number (Pr) = 0.71 (air), Schmidt number (Sc) =

0.22 (hydrogen at 25 degrees Celsius and 1 atmosphere pressure, following Gebhart and Pera

(1971)), thermal Grashof number (Gr) = 5, solutal Grashof number (Gm) = 5, permeability

parameter ( K ) = 0.5 and T (dimensionless time) = 1. The permeability in all the figures

plotted is set at 0.5 which corresponds to a highly porous regime i.e. weak Darcian bulk drag

associated with the medium fibers. Such a situation may accurately simulate the properties of

foams or loosely arranged arrays of particles in a filtration material regime. Sc = 0.22

physically corresponds to hydrogen gas diffusing in air. Such data therefore corresponds to

hydrogen gas diffusing in air percolating in a highly permeable isotropic, homogenous

porous medium under the action of weak thermal and species buoyancy forces. Generally

weak magnetic field (e.g. M = 0.3 in most of our graphs) is also studied, which negates the

need to consider Hall current or ionslip current effects, as indicated by Sutton (1965). The

values of Sr and Du have been selected to ensure that the product Sr Du is constant, assuming

that the mean temperature is constant. In the present analysis for conservation of space we

have excluded plots for the effects of Gr, Gm, Pr and K . The variation of skin friction, local

heat transfer parameter and local mass transfer parameter with respect to the Soret number (Sr)

and Dufour number (Du), Eckert number (Ec), chemical reaction parameter (γ), heat

absorption parameter ( Φ ) and radiation absorption parameter (Q1) are presented in tables 1 to

3.

Table 1 indicates that the skin friction coefficient, (0)U ′ decreases with a decrease in Soret

number (Sr) and an increase in Dufour number (Du). The rate of heat transfer, (0)θ ′− ,

increases as Sr decreases from 2.0 to 1.0 and 0.5; thereafter however it decreases with a



27

subsequent lowering in Sr from 0.2 through to the least value of 0.12. The rate of mass

transfer, (0)φ ′− , increases continuously with a decrease in Sr i.e. the maximum mass transfer

rate corresponds to the minimum Sr value of 0.12 (and the maximum Du value of 0.75).

Table 1: Distribution of skin friction { (0)U ′ }, the rate of heat transfer { (0)θ ′− } and the rate

of mass transfer { (0)φ ′− } with different values of Soret number (Sr) (or Dufour number Du)

for M=0.3, Φ = 1, Q1 = 1, Sc= 0.22, γ =1, Ec = 0.2.

Sr Du (0)U ′ (0)θ ′− (0)φ ′−

2.0 0.03 5.11446 0.42148 0.682547

1.0 0.06 4.87215 0.51076 0.698430

0.5 0.12 4.75604 0.54243 0.720574

0.2 0.30 4.72882 0.51557 0.738615

.12 0.50 4.76921 0.45930 0.744047

.08 0.75 4.83689 0.38232 0.746810

Table 2 indicates that skin friction increases with an increase in Eckert number (Ec) but

decreases with a rise in chemical reaction parameter (γ). Heat transfer rate however decreases

with a rise in Ec but increases with increasing γ. Increasing Eckert number implies more

thermal energy is added to the fluid so that heat is conducted from the plate into the fluid i.e.

causing a decrease in heat transfer at the wall. The rate of mass transfer { (0)φ ′− } is increased

both with a rise in Ec and γ.


of mass transfer { (0)φ ′− } with different values of Eckert number (Ec) and Chemical reaction

parameter (γ).

M = 0.3, Φ = 1, Q1= 1, Sc = 0.22, γ =1

Du=0.06, Sr =1.0

M=0.3, Ec = 0.2, Q1= 1, Sc = 0.22

Du = 0.06, Sr = 1.0, Φ = 1

Ec (0)U ′ (0)θ ′− (0)φ ′− γ (0)U ′ (0)θ ′− (0)φ ′−

0 4.75526 0.93469 0.61048 1 4.87215 0.51076 0.69843

0.5 5.06401 -

0.21104

0.84808 5 3.92488 0.81795 1.30339

1.0 5.43659 -

1.70303

1.15681 10 3.39101 0.96268 1.76454

1.5 5.89385 -

3.69256

1.56763 15 3.08461 1.03525 2.10955

Table 3 indicates that skin friction, (0)U ′ , decreases with an increase in heat absorption

parameter (Φ ) but is enhanced with an increase in the radiation absorption parameter (Q1).

Heat transfer rate, (0)θ ′− , however is strongly boosted with an increase in heat absorption

parameter (Φ ) but is considerably decreased with an increase in the radiation absorption

parameter (Q1). The rate of mass transfer { (0)φ ′− } is markedly reduced with an increase in

heat absorption parameter (Φ ) but is substantially boosted with an increase in the radiation

absorption parameter (Q1).




of mass transfer { (0)φ ′− } with different values of heat absorption parameter ( Φ ) and

radiation absorption parameter (Q1).

M = 0.3, Ec = 0.2, Q1= 1, Sc = 0.22, γ =1

Du=0.06, Sr =1.0

M=0.3, Ec = 0.2, γ = 1, Sc = 0.22

Du = 0.06, Sr = 1.0, Φ = 1

Φ (0)U ′ (0)θ ′− (0)φ ′− Q1 (0)U ′ (0)θ ′− (0)φ ′−

0 5.27065 -

0.16144

0.82799 1 4.87215 0.51076 0.69843

1 4.87215 0.51075 0.69843 2 5.35111 -

0.17517

0.83199

2 4.59375 0.99619 0.60228 3 5.81195 -

0.83947

0.96155

Figures 2 to 4 depict the spatial distribution through the boundary layer of velocity,

temperature and concentration functions at a fixed time, T = 1. In figures 2 to 4 the effect of

chemical reaction on the flow variables is shown. In figure 2 velocity, U, is clearly boosted

with stronger chemical reaction i.e. as the chemical reaction parameter, γ, increases from 1

through 5, 10 to 15 (very high rate), profiles are lifted continuously throughout the boundary

layer, transverse to the plate. A distinct velocity escalation occurs near the wall after which

profiles decay smoothly to the stationary value in the free stream. Chemical reaction therefore

boosts momentum transfer i.e. accelerates the flow. A similar response for the non-magnetic

case has been documented by Chamkha et al. (Chamkha et al. 2001) in the presence of

thermal radiation and later for the magnetohydrodynamic case (without porous media, viscous

heating and Soret/Dufour effects) by Ibrahim et al. (Ibrahim et al. 2008). Temperature (θ) and

concentration (φ) are likewise increased in figures 3 and 4, respectively, with an increase in

the chemical reaction parameter (γ) although profiles in both these cases descend from a

maximum at the wall (plate surface) to zero in the freestream i.e. the profiles are monotonic

decays. The presence of chemical reaction is therefore assistive to momentum, heat and mass

transfer processes in the regime.

Figure 1: Flow Configuration and coordinate system

,x u

Semi- infinite Vertical Porous Plate

B0

,y v

Fluid saturated Darcian porous medium pu

g

( )v t

,T C∞ ∞

,T C



29

Figure 2

Figure 3

Figure 4

Figures 2 to 4: Effect of chemical reaction parameter on the dimensionless velocity,

temperature and concentration for air (Pr = 0.71) with M = 0.3, K= 0.5, Gr = 5, Gm = 5, Φ =1,

Q1=1, Ec = 0.2, Du = 0.06, Sr = 1.0 and Sc = 0.22 for T = 1.

Figures 5 to 6 illustrate the variation of velocity, U, and temperature function, θ, for various

values of the radiation absorption parameter (Q1). It is immediately apparent that velocity (U)

as well as temperature (θ) clearly increase as Q1 rises from 1 to 3. Velocity reaches a

maximum in close proximity to the wall and then falls gradually to zero at the edge of the

boundary layer. Inspection of Fig. 6 shows that for a small value of Q1 (Q1<2) temperature

profile continuously decreases from the wall, while for higher values of Q1 it increases

attaining a maximum near the plate boundary and then decreases. As such there is a

noticeable temperature overshoot with Q1> 1 since considerable thermal energy is imparted

via the presence of a thermal radiation source to the fluid causing an elevation in temperatures

near the wall.

Figure 5

Figure 6

Figures 5 to 6: Effect of radiation absorption parameter on the dimensionless velocity and

temperature for air (Pr = 0.71) with M = 0.3, K = 0.5, Gr = 5, Gm = 5, Φ =1, γ = 1, Ec = 0.2,

Du = 0.06, Sr = 1.0 and Sc = 0.22 for T = 1.



Figures 7 to 8 illustrate the variation of velocity and temperature functions with the effects of

heat absorption parameter, Φ. U values are clearly reduced with increasing Φ; again an

overshoot is computed close to the plate both in the presence and absence of heat absorption.

Heat absorption however suppresses the overshoot. For a non-zero value of Φ, temperature

profile, θ, continuously decreases while for Φ = 0 a temperature overshoot occurs very close

to the wall with temperature decaying continuously thereafter to zero in the free stream.

Figure7

Figure 8

Figures 7 to 8: Effect of heat absorption parameter on the dimensionless velocity and

temperature for air (Pr=0.71) with M = 0.3, K = 0.5, Gr = 5, Gm = 5, Q =1, γ = 1, Ec = 0.2,

Du = 0.06, Sr = 1.0 and Sc = 0.22 for T = 1

Figures 9 to 10 show the temperature and concentration distributions with collective variation

in Soret number, Sr, and Dufour number, Du. Sr represents the effect of temperature gradients

on mass (species) diffusion. Du simulates the effect of concentration gradients on thermal

energy flux in the flow domain. We observe from figure 9 that a rise in Du from 0.06 to 0.75

boosts the influence of species gradients on the temperature field, so that θ values are clearly

enhanced i.e. the fluid in the porous medium is heated. The Sr values fall from 1.0 to 0.08

over this range (the product of Sr and Du must stay constant i.e. 0.06). Temperature

continuously decreases as we move into the boundary layer. In figure 10, φ (concentration

function) in the Darcian flow is increased as Sr increases from 0.08 to 1.0, i.e. mass transfer is

boosted as a result of the contribution of temperature gradients. These results concur with the

trends in Anghel et al. (Anghel et al 2000) who considered the non-magnetic Darcian case

and also Postelnicu (Postelnicu 2004) who considered the magnetohydrodynamic Darcian

case.

Figure 9

Figure 10

Figures 9 to 10: Effect of Soret and Dufour number (product stays constant i.e. at 0.06) on the

dimensionless temperature and concentration for air (Pr=0.71) with M = 0.3, K = 0.5, Gr = 5,

Gm = 5, Q = 1, Φ = 1, γ = 1, Ec = 0.2 and Sc = 0.22 for T=1



31

Figure 11 shows that an increase in Eckert number, Ec, from 0 (no viscous heating) through

0.5, 1 to 1.5 (very high viscous heating) clearly boosts temperatures in the porous regime.

Eckert number signifies the quantity of mechanical energy converted via internal friction to

thermal energy i.e. heat dissipation. Increasing Ec values will therefore cause an increase in

thermal energy contributing to the flow and will heat the regime. For all non-zero values of Ec

the temperature overshoot near the wall is distinct; this overshoot migrates marginally further

into the boundary layer with an increase in Ec.

Figure 11: Effect of Eckert number on the dimensionless temperature for air (Pr=0.71) with

M=0.3, K=0.5, Gr=5, Gm=5, Φ =1, Q1=1, Du=0.06, Sc=0.22, Sr=1.0 and =1 for T=1

Figures 12 to 13 illustrate the temperature and concentration field distributions with

transverse coordinate for different Schmidt number, Sc. An increase in Sc causes a

considerable reduction in temperature, θ, in Fig. 12. A much greater reduction is observed in

concentration values, φ, in Fig. 13. An increase in Sc will suppress concentration in the

boundary layer regime. Higher Sc will imply a decrease of molecular diffusivity (D) causing a

reduction in concentration boundary layer thickness. Lower Sc will result in higher

concentrations i.e. greater molecular (species) diffusivity causing an increase in concentration

boundary layer thickness. For the highest value of Sc = 1.0, the momentum and concentration

boundary layer thicknesses are of the same value approximately i.e. both species and

momentum will diffuse at the same rate in the boundary layer.

Figure 12

Figure 13

Figures 12 to 13: Effect of Schmidt number on the dimensionless temperature and

concentration for air (Pr= 0.71) with M= 0.3, K= 0.5, Gr =5, Gm = 5, Φ =1, Q1= 1, Ec = 0.2,

Du = 0.06, Sr =1.0 and γ =1 for T=1

Finally figure 14 shows the variation of velocity function with the magnetic field parameter,

M. The presence of magnetic field in an electrically-conducting flow creates a drag-like force

called the Lorentz force. This type of resistive force tends to slow down the motion of the

fluid in the boundary layer i.e. decelerates the flow, as shown in figures 13 where velocity (U)

clearly decrease as M rises from 0 (electrically non-conducting case) through 1 to the

maximum magnetic field corresponding to M = 2. The relative influence of magnetic field on



the fluid temperature as well as on concentration has also been investigated; the results are

omitted herein for brevity, although it has been found in consistency with other published

studies e.g. Seddeek et al. (Seddeek et al. 2007) and Zueco (Zueco 2007), that the temperature

as well as concentration is slightly increased as magnetic field parameter, M, increases. Hence

magnetic field heats the fluid and aids in species diffusion in the porous regime. We further

note that for the case M = 1, magnetic and viscous forces will have the same order of

magnitude. As such the Hartmann boundary layer will be formed when M = 1 and this

boundary layer will decrease with increase in M i.e. will be less for M = 2, as confirmed by

Sutton (1965).

Figure 14: Effect of Magnetic parameter M on the dimensionless velocity for air (Pr = 0.71)

with Ec= 0.2, K= 0.5, Gr = 5, Gm= 5, Φ =1, Q1= 1, Sc = 0.22, Du = 0.06, Sr = 1.0 and γ = 1

for T=1

7 CONCLUSIONS

A finite element solution has been developed for the oscillatory chemically-reacting,

dissipative, hydromagnetic convection heat and mass transfer in a Darcian porous medium

with heat absorption and thermal radiation effects. The dimensionless solutions have shown

that:

1. An increase in the chemical reaction parameter, γ� decrease the velocity, temperature and

concentration values in the porous regime.

2. Increasing the radiation absorption parameter, Q1, increase the velocity and temperature i.e.

accelerate and heats the flow throughout the entire porous regime.

3. Increasing the heat absorption parameter,Φ, reduces both velocity and temperature i.e.

retards and cools the flow in the porous regime. Therefore a desired temperature can be

maintained by controlling the heat absorption effect in practical chemical engineering

applications.

4. Increasing Dufour number, Du (and simultaneously reducing Soret number, Sr) increases

the temperature in the porous medium

5. Increasing Soret number, Sr, (and simultaneously reducing the Dufour number, Du)

increases the concentration values in the porous regime i.e. enhances species diffusion.

6. Increasing the magnetic parameter (M) decreases the velocity in the regime.



33

7. Increasing Eckert number (Ec) heats the porous regime i.e. increases temperature

8. Increasing Schmidt number (Sc) reduces both temperature and concentration values in the

porous regime.

9. The skin friction coefficient, (0)U ′ decreases with a decrease in Soret number (Sr) i.e.

increase in Dufour number (Du), decreases with a rise in chemical reaction parameter (γ),

decreases with an increase in heat absorption parameter (Φ ) but increases with an increase in

Eckert number (Ec) and an increase in the radiation absorption parameter (Q1).

10. The rate of heat transfer, (0)θ ′− , increases initially as Sr decreases and then decreases

with a subsequent lowering in Sr, decreases with a rise in Ec, strongly decreases with an

increase in the radiation absorption parameter (Q1), but increases with larger chemical

reaction parameter values (γ) and also with an increase in heat absorption parameter (Φ ).

11. The rate of mass transfer, (0)φ ′− , increases continuously with a decrease in Sr i.e. (and

an increase in Du), and is also increased with a rise in Ec, γ and also with an increase in the

radiation absorption parameter (Q1); however it is decreased with an increase in heat

absorption parameter (Φ )

8 ACKNOWLEDGMENTS

One of the authors (R. Sharma) would like to thank Ministry of Human Resource

Development (MHRD), Government of India, for its financial support through the award of a

research grant. The authors are also grateful to the reviewers for their comments which have

helped to improve the article.

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