COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
Chemical Reaction, Heat and Mass Transfer on Nonlinear MHD
Boundary Layer Flow through a Vertical Porous Surface with
Thermal Stratification in the Presence of Suction
R. Kandasamy 1*, K. Periasamy 2, K. K. Sivagnana Prabhu 3
1
Department of Mathematics, Institute of Road and Transport Technology, Erode -638 316, India
Department of Chemistry, Institute of Road and Transport Technology, Erode -638 316, India
3
Department of Chemical Engineering, R. M. K. Engineering College, Chennai, India
2
Email: [email protected]
Abstract In this paper a similarity analysis is made for the forced and free convection boundary layer flow
in a semi infinite expanse of an electrically conducting viscous incompressible fluid past a semi-infinite non
conducting porous plate with chemical reaction, heat and mass transfer in the presence of suction. A
magnetic field is applied transversely to the direction of the flow. Adopting the similarity transformation,
governing nonlinear partial differential equations of the problem are transformed to nonlinear ordinary
differential equations. Then the numerical solution of the problem is derived using Gill method, for different
values of the dimensionless parameters. The results obtained show that the flow field is influenced
appreciably by the presence of chemical reaction, buoyancy effect and magnetic field.
Key words: chemical reaction, magnetic field, thermal stratification, porous plate, buoyancy effects
INTRODUCTION
Magneto-hydrodynamic (MHD) in the branch of continuum mechanics which deals with the flow of
electrically conducting fluids in electric and magnetic fields. Many natural phenomena and engineering
problems are worth being subjected to an MHD analysis. Magneto-hydrodynamic equations are ordinary
electromagnetic and hydrodynamic equations modified to take into account the interaction between the
motion of the fluid and the electromagnetic field. The formulation of the electromagnetic theory in a
mathematical form is known as Maxwell’s equation. In many mixed flows of practical importance in nature
as well as in many engineering devices, the environment is thermally stratified. The discharge of hot fluid
into enclosed regions often results in a stable thermal stratification with lighter fluid overlying denser fluid.
The thermal stratification effects of heat transfer over a stretching surface is of interest in polymer extrusion
processes where the object, after passing through a die, enters the fluid for cooling below a certain
temperature. The rate at which such objects are cooled has an important bearing on the properties of the final
product. In the process of cooling the fluids, the momentum boundary layer for linear stretching of sheet was
first studied by Crane [1]. The present trend in the field of chemical reaction analysis is to give a
mathematical model for the system to predict the reactor performance. A large amount of research work has
been reported in this field. In particular, the study of heat and mass transfer with chemical reaction is of
considerable importance in chemical and hydrometallurgical industries. In order to study the thermal
stratification effects over the above mentioned problem, an attempt has been made have to analyze the
nonlinear hydromagnetic flow with heat and mass transfer over a vertical stretching surface with chemical
reaction and thermal stratification effects.
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In the past decades, the penetration theory of Highie 1935 had been widely applied to unsteady state
diffusional problems with and without chemical reaction. As far as we can ascertain, all the solutions with
chemical reaction were obtained for the case of a semi-infinite body of liquid, although physical absorption
into a finite film was considered. Among some of the interesting problems which were studied are the
analysis of laminar forced convection mass transfer with homogeneous chemical reaction, Goddard and
Acrivos [2]. The effect of different values of Prandtl number of the fluid along the surface was analysed by
Gebhart[3]. A study on heat and mass transfer over a stretching surface with suction or blowing was carried
out by Gupta and Gupta[4]. The same type of problem with inclusion of constant surface velocity and
power-law temperature variations were studied by Soundalgekar and Ramamurthy[5]. Gubka and Bobba[6]
studied the power-law temperature variations in the case of a stretching continuous surface. Chen and
Char[7] investigated the effect of power-law temperature and power-law surface heat flux in the heat transfer
characteristics of a continuous linear stretching surface. Bestman [8,9] analyzed the effect of natural
convection boundary layer with suction and mass transfer in a porous medium. Sattar[10] investigated the
effect of free and forced convection boundary layer flow through a porous medium with large suction.
Atul Kumar Singh [11] analyzed the MHD free convection and mass transfer flow with heat source and
thermal diffusion. The paper deals with the study of free convection and mass transfer flow of an
incompressible, viscous and electrically conducting fluid past a continuously moving infinite vertical plate
in the presence of large suction and under the influence of uniform magnetic field considering heat source
and thermal diffusion. The problem of a stretching surface with constant surface temperature was analyzed
by Noor Afzal [12]. Processes involving the mass transfer effect have long been recognized as important
principally in chemical processing equipment. Recently, Mohammad et.al.[13] have Studied the effect of
similarity solution for MHD flow through vertical porous plate with suction. Acharya et al. [14] have studied
heat and mass transfer over an accelerating surface with heat source in the presence of suction and
blowing.In Sattar[10] made analytical studies on the combined forced and free convection flow in a porous
2
medium. In these studies it has been generally recognized that γ = Grx R e x is the governing parameter for
a vertical plate. In the present work, therefore the effect of suction on the MHD forced and free convection
flow past a vertical porous plate is studied. Solutions to the problem posed are found numerically for the
whole range of the permeability parameter γ that is considered to be the driving force of the whole range of
the combined forced and free convection. No attempt has been made so far to analyze the effect of chemical
reaction, heat and mass transfer on nonlinear MHD boundary layer flow through vertical porous surface with
thermal stratification in the presence suction and hence we have considered the problem of this kind.
MATHEMATICAL ANALYSIS
Two-dimensional steady nonlinear MHD boundary layer flow of an incompressible, viscous, electrically
conducting and Boussinesq fluid flowing over a vertical porous surface in the presence of an uniform
magnetic field has been considered. According to the coordinate system, the x-axis is chosen parallel to the
vertical surface and the y-axis is taken normal to it. A transverse magnetic field of strength B0 is applied
parallel to the y-axis. The fluid properties are assumed to be constant in a limited temperature range. The
value of C∞ is set as zero in the problem as the concentration of species far from the wall, C∞ , is
infinitesimally small (Byron Bird [15]) and hence the Soret and Dufour effects are negligible. The chemical
reactions are taking place in the flow and the physical properties ρ, μ, D and the rate of chemical reaction,
k1 are constant throughout the fluid. It is assumed that the induced magnetic field, the external electric field
and the electric field due to the polarization of charges are negligible. Under these conditions, the governing
boundary layer equations of momentum, energy and diffusion neglecting viscous and Joules dissipation
under Boussinesq's approximation are
∂u ∂ x + ∂v ∂ y = 0
(1)
u∂u ∂x + v∂v ∂y = v∂ 2u ∂y + g β (T − T∞ ) + g β ∗ (C − C∞ ) + (σ B02 ρ )(U − u )
(2)
u ∂T ∂x + v ∂T ∂y = α ∂ 2T ∂y 2
(3)
2
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2
u ∂C ∂x + v ∂C ∂y = D ∂ 2C ∂y − k1C
(4)
The boundary conditions are
u = 0, v = v0 , T = Tw ( x ), C = Cw ( x ) at y = 0
(5)
u = U , T = T∞ ( x ) = (1 − n)T0 + nTw ( x ), C = C∞ as y → ∞
(6)
where v0 is the velocity of suction or injection at the wall of the surface and a is a dimensional constant and
n is a constant which is the thermal stratification parameter and is such that 0 ≤ n < 1 . The n defined as
thermal stratification parameter is equal to m1 (1 + m1 ) of Nakayama and Koyama [16] where m1 is a
constant. T0 is constant reference temperature say, T∞ (0) . The suffixes w and ∞ denote surface and
ambient conditions.
As in Mohammad et al [13] the following change of variables are introduced
12
ψ = (vxU ( x ))
f (η ) ,
12
η = (U ( x ) vx) y
(7)
The velocity components are given by
u = ∂ψ ∂y ,
v = −∂ψ ∂x
(8)
It can be easily verified that the continuity Eq. (1) is identically satisfied. Similarity solutions exist if we
assume that U = ax and introduce the non-dimensional form of temperature and concentration as
θ = (T − T∞ ) (Tw − T∞ )
(9)
φ = (C − C∞ ) (Cw − C∞ )
(10)
Re = Ux v
(Reynolds number)
(11)
Gr = g βU (Tw − T∞ ) x 2 (4vv02 )
(Grashof number)
(12)
Gc = g β ∗U (Cw − C∞ ) x 2 (4vv02 )
(Modified Grashof number)
(13)
Pr = μ C p K
(Prandtl number)
(14)
Sc = ν D
(Schmidt number)
(15)
M 2 = σ B02 x (U ρ S 2 )
(magnetic parameter )
(16)
γ = 2k1 C
(chemical reaction parameter)
(17)
12
S = −2v0 ( x vU )
(suction parameter)
(17a)
where S is suction if S > 0 and injection if S < 0. In this work, temperature variation of the surface is taken
into account and is also given by the power-law temperature, Tw − T∞ = Nx n where N and n are constants.
Also concentration variation is given by Cw − C∞ = N1 x n1 where N1 and n1 are constants. The nonlinear
equations and boundary conditions are obtained as
f ′′′ + S 2 ( P1θ + P2φ ) + S 2 M 2 (1 − f ′) + 12 f f ′′ = 0
(18)
θ ′′ − Prf ′ (θ + n (1 − n)) + Prf θ ′ = 0
(19)
φ ′′ − Sc (φγ Rex + f ′φ) + Sc f φ ′ = 0
(20)
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where P1 = Gr Re2 and P2 = Gc Re2 and P = ( P1 + P2 ) 2 (Buoyancy parameter)
The boundary conditions are given by
f (0) = −S , f ′ (0) = 1, θ (0) = 1, φ (0) = 1
f ′ (∞) = 0, θ (∞) = 0, φ (∞) = 0
(21)
It is interesting to note that when suction is absent, S = 0, Eq. (18) reduces to ordinary Blasius equation. The
solutions of the Blasius equation are referred to as the Blasius solutions. These have also been studied by
Schilchting [17]. On the other hand, if P1 = P2 = 0 , Re is large and the forced convection is dominating, the
Eq. (18) corresponds to the ordinary Falkner and Skan equation. In this case the boundary conditions differ
largely from those of the original Falkner and Skan equation.
Eqs. (18) to (20) with boundary conditions (21) are integrated using Runge - Kutta Gill method. Flow field
temperature and concentration are analysed in detail for different values of buoyancy parameter, chemical
reaction and magnetic parameter in the forthcoming section.
RESULTS AND DISCUSSION
The present study is aimed to extend the work of Mohammad Ferdows et al [13] to the steady state only with
magnetic effect highlighting the chemical reaction and buoyancy effects. However, the analysis of the
present study is different from that of Mohammad Ferdows work. In order to get a clear insight of the
physical problem, numerical results are displayed with the help of graphical illustrations.
Figure 1: Velocity profiles for different magnetic parameter
The dimensionless velocity profiles for different values of magnetic field with constant chemical reaction
parameter and thermal stratification parameter are presented in Fig. 1. It is observed that the velocity of the
fluid decreases with the increase of magnetic parameter.
The dimensionless temperature profiles for different values of buoyancy parameter with constant chemical
reaction parameter and the uniform magnetic field are shown in Fig. 2. It is clear that the temperature of the
fluid decreases with the increase of buoyancy parameter.
Fig. 3 depicts the dimensionless concentration profiles for different values of buoyancy parameter with
constant chemical reaction parameter and thermal stratification parameter. It is clear that the concentration
of the fluid increases with the increase of buoyancy parameter.
Fig. 4 represents the dimensionless velocity profiles for different values of suction with constant chemical
reaction parameter and the uniform magnetic field. It is observed that the velocity of the fluid decreases with
the increase of suction at the wall of the surface.
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Figure 2: Buoyancy effect over the temperature profiles
Figure 3: Buoyancy effect over the concentration profiles
Figure 4: Suction effects over the velocity profiles
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The dimensionless concentration profiles for different values of suction with constant chemical reaction
parameter and thermal stratification parameter are demonstrated in Fig.5. It is seen that the concentration of
the fluid increases with the increase of suction.
Figure 5: Suction effect over the concentration profiles
Fig. 6 demonstrates the dimensionless temperature profiles for different values of thermal stratification
parameter with constant chemical reaction parameter and the uniform magnetic field. It is seen that the
temperature increases with the increase of thermal stratification parameter.
The concentration of the fluid decreases with the increase of chemical reaction parameter and this is noted
through Fig. 7.
Figure 6: Thermal stratification over the temperature profiles
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Figure 7: Chemical reaction over the concentration profiles
CONCLUSION
We have examined the governing equations for the steady incompressible fluid past a semi-infinite vertical
porous plate embedded in a porous medium and subjected to the presence of a transverse magnetic field.
Numerical results are presented to illustrate the details of the viscous parameters. The value of driving
parameter P1 and P2 as shown above, however, correspond to three regimes, namely, the predominantly
free convection regime, the mixed convection regime and predominantly forced convection regime. For
P1 = P2 = 0 , the gravity induced free convection is absent and the flow is completely forced over the surface.
For low values of P forced convection dominates and the local similarity solutions are the same as those in
the case of force convection only, which was studied by Narain and Uberoi [18]. The large value of P are
interesting from a physical point of view. For this purpose, the value of P = 5.0 can be essentially treated as
the free convection representation.
1) Due to the uniform magnetic field and suction at the wall of the surface with constant thermal
stratification parameter, the concentration of the fluid decreases with the increase of chemical reaction
parameter. It is observed that the effect of destructive reaction on concentration profiles are much more
pronounced than that of the generative reaction.
2) The velocity boundary layer becomes thin as M increases, but concentration boundary layer becomes
thick. Hence velocity profiles become more steep with increasing M, but concentration profiles less steep. In
the presence of the magnetic field, the velocity boundary layer is thinner than the concentration boundary
layer but for M = 0, an opposite trend is observed.
3) Due to uniform magnetic field with suction at the wall of the surface and constant chemical reaction
parameter, the temperature of the fluid decreases and the concentration of the fluid increases with the
increase of buoyancy parameter. All these physical phenomena are due to the combined effect of the strength
of the magnetic field and chemical reaction.
4) In the presence of uniform magnetic field with suction at the wall of the surface and constant chemical
reaction parameter, the temperature of the fluid increases with the increase of thermal stratification
parameter.
5) In the case of uniform magnetic field with suction at the wall of the surface and constant chemical reaction
parameter, the velocity of the fluid decreases and the concentration of the fluid increases with increase of
suction at the wall of the surface. All these physical phenomena reactions are due to the combined effects of
the buoyancy force and porosity of the plate.
6) A comparison of velocity profiles shows that the velocity increases near the plate and thereafter decreases.
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7) It is to note that an increase in magnetic field leads to a rise in temperature at slow rate in comparison to
the velocity profiles.
It is hoped that the present investigation of the study of physics of flow over a vertical surface can be utilized
as the basis for many scientific and engineering applications and for studying more complex vertical
problems. The findings may be useful for the study of movement of oil or gas and water through the
reservoir of an oil or gas field, in the migration of underground water and in the filtration and water
purification processes. The results of the problem are also of great interest in geophysics in the study of
interaction of the geomagnetic field with the fluid in the geothermal region.
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