j. anand rao chemical reaction effect on an s. … no3...paper received: 18 december, 2010 paper...

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Chem. Ind. Chem. Eng. Q. 17 (3) 249−257 (2011) CI&CEQ

249

J. ANAND RAO1

S. SHIVAIAH2 1Department of Mathematics, Os-

mania University, Hyderabad, India 2Department of Mathematics,

Padmasri Dr. B.V. Raju Institute of Technology, Narsapur, Medak,

India

SCIENTIFIC PAPER

UDC 530.1:532

DOI 10.2298/CICEQ101218011A

CHEMICAL REACTION EFFECT ON AN UNSTEADY MHD FREE CONVECTION FLOW PAST AN INFINITE VERTICAL POROUS PLATE WITH CONSTANT SUCTION

This paper focuses on the effects of chemical reaction on an unsteady magne-tohydrodynamic free convection fluid flow past an infinite vertical porous plate with constant suction. The dimensionless governing equations are solved nu-merically by a finite element method. The effects of the various parameters on the velocity, temperature and concentration profiles are presented graphically and values of skin-friction coefficient, Nusselt number and Sherwood number for various values of physical parameters are presented through tables. Key words: chemical reaction, MHD, free convection, constant suction.

Combined heat and mass transfer problems with chemical reaction are of importance in many processes and have, therefore, received a considerable amount of attention in recent years. In processes such as dry-ing, evaporation at the surface of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occur simultaneously. Possible applications of this type of flow can be found in many industries. For example, in the power industry, among the methods of generating electric power is one in which electrical energy is extracted directly from a moving conducting fluid.

Many practical diffusive operations involve the molecular diffusion of a species in the presence of che-mical reaction within or at the boundary. There are two types of reactions, homogeneous reaction and hetero-geneous reaction. A homogeneous reaction is one that occurs uniformly throughout a given phase. The spe-cies generation in a homogeneous reaction is analo-gous to internal source of heat generation. In contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a phase. It can there-fore be treated as a boundary condition similar to the constant heat flux condition in heat transfer. The study of heat and mass transfer with chemical reaction is of great practical importance to engineers and scientists

Correspondening author: S. Shivaiah, Department of Mathema-tics, Padmasri Dr. B.V. Raju Institute of Technology, Narsapur, Medak, India. E-mail: [email protected] Paper received: 18 December, 2010 Paper accepted: 20 March, 2011

because of its almost universal occurrence in many branches of science and engineering. The flow of a fluid past a wedge is of fundamental importance since this type of flow constitutes a general and wide class of flows in which the free stream velocity is proportional to a power of the length coordinate measured from the stagnation point.

In many transport processes in nature and in in-dustrial applications in which heat and mass transfer is a consequence of buoyancy effects caused by diffu-sion of heat and chemical species. The study of such processes is useful for improving a number of chemical technologies, such as polymer production and food processing. In nature, the presence of pure air or water is impossible. Some foreign mass may be present either naturally or mixed with the air or water. The present trend in the field of chemical reaction with vis-cosity 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 chemical reaction, heat and mass transfer is of considerable importance in chemical and hydrometallurgical industries. Chemical reaction can be codified as either heterogeneous or homogeneous processes. This depends on whether they occur at an interface or as a single phase volume reaction.

Chamkha et al. [1] 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. Ibrahim et al. [2] studied the effects of chemical reac-

J.A. RAO, S. SHIVAIAH: CHEMICAL REACTION EFFECT ON AN UNSTEADY MHD… CI&CEQ 17 (3) 249−257 (2011)

250

tion and radiation absorption on transient hydro mag-netic natural convection flow with wall transpiration and heat source. Bejan and Khair [3] investigated the ver-tical free convection boundary layer flow in porous me-dia owing to combined heat and mass transfer. The suction and blowing effects on free convection coupled heat and mass transfer over a vertical plate in a sa-turated porous medium was studied by Raptis et al. [4] and Lai and Kulacki [5] respectively.

Hydromagnetic flows and heat transfer have be-come more important in recent years because of its varied applications in agriculture, engineering and pet-roleum industries. Raptis [6] studied mathematically the case of time varying two dimensional natural con-vective flow of an incompressible electrically conduc-ting fluid along an infinite vertical porous plate em-bedded in a porous medium. Soundalgekar [7] ob-tained an approximate solution for two dimensional flow of an incompressible viscous flow past an infinite porous plate with constant suction velocity, the diffe-rence between temperature of the plate and the free convection is moderately large causing free convection currents. Takhar and Ram [8] studied the MHD free convection heat transfer of water at 4 °C through a porous medium. Soundalgekar et al. [9] analyzed the problem of the free convection effects on stokes pro-blems for a vertical plate under the action of transver-sely applied magnetic field with mass transfer. Elbash-beshy [10] studied heat and mass transfer along a vertical plate under the combined buoyancy effects of thermal and species diffusion in the presence of mag-netic field.

In all these investigations, the radiation effects are neglected. For some industrial applications such as glass production and furnace design and in space tech-nology applications, such as cosmic flight aerodyna-mics rockets, propulsion systems, plasma physics and space craft reentry aero thermodynamics which ope-rate at higher temperatures, radiation effects can be significant. Alagoa et al. [11] studied radiative and free convection effects on MHD flow through porous me-dium between infinite parallel with time-dependent suc-tion. Bestman and Adjepong [12] analyzed unsteady hydro magnetic free convection flow with radiative heat transfer in a rotating fluid.

In all these investigations, the viscous dissipation is neglected. The viscous dissipation heat in the natural convection flow is important when the flow field is of extreme size or at low temperature or in high gravita-tional field. Gebhart [13] has shown the importance of viscous dissipative heat in free convection flow in the case of isothermal and constant heat flux at the plate. Soundalgekar [14] analyzed the effect of viscous dis-

sipative heat on the two-dimensional unsteady, free convective flow past an infinite vertical porous plate when the temperature oscillates in time and there is constant suction at the plate. Israel – Cookey et al. [15] investigated the influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time dependent suction.

The object of the present paper is to analyze the chemical reaction effect on an unsteady magnetohyd-rodynamic free convection flow past an infinite vertical plate by taking constant suction into account. The go-verning equations are transformed by using unsteady similarity transformation and the resultant dimension-less equations are solved by using the finite element method. The effects of various governing parameters on the velocity, temperature, concentration, skin-fric-tion coefficient, Nusselt number and Sherwood number are shown in figures and tables and discussed in detail. From computational point of view it is identified and proved beyond all doubts that the finite element me-thod is more economical in arriving at the solution.

FORMULATION OF THE PROBLEM

An unsteady two-dimensional laminar free con-vective boundary layer flow of a viscous, incompres-sible, electrically conducting, chemically reacting of the fluid flow past an infinite vertical porous plate is con-sidered. The x’-axis is taken along the vertical porous plate and the y’-axis normal to the plate. The physical sketch and geometry of the problem is shown in Figure 1. A uniform magnetic field is applied in the direction perpendicular to the plate. It is assumed that there is no applied voltage, which implies the absence of an electric field. The transverse applied magnetic field and magnetic Reynolds number are assumed to be very small so that the induced magnetic field. The concen-tration of the diffusing species in the binary mixture is assumed to be very small in comparison with the other chemical species which are present, and hence the Soret and Dufour effects are negligible. Since the plate is of infinite length, all the physical variables are func-tions of y’ and t’ only. Now, under the usual Bous-sinesq’s approximation, the governing boundary layer equations are:

′∂ =′∂

0vy

(1)

( )

( ) ( )

2

2

2 2e 0

u u U uv v g T T

t y t yH

g C C u UK

β

σμ νβρ

∞

∗∞

′ ′ ′ ′∂ ∂ ∂ ∂′+ = + + −′ ′ ′ ′∂ ∂ ∂ ∂

′ ′+ − − + − ′

(2)


251

22

2p p

1T T k T q uv

t y c y k y c yν

ρ′ ′ ∂ ∂ ∂ ∂ ∂′+ = − + ′ ′ ′ ′ ′∂ ∂ ∂ ∂ ∂

(3)

22 3

s23 16 0

q Tq T

y yα σ α ∞

′∂ ∂′− − =′ ′∂ ∂

(4)

22

r2

C C Cv D k C

t y y∂ ∂ ∂′ ′+ = −

′ ′ ′∂ ∂ ∂ (5)

where u’ and v’ are the velocity components in x’ and y’ direction, respectively, t’- the time, ρ - the fluid density, ν - the kinematic viscosity, cp - the specific heat at constant pressure, g – the acceleration due to gravity, β and β* - the thermal and concentration expansion coef-ficient respectively, σ - fluid electrical conductivity, μe - magnetic permeability, H0

2 - constant transverse mag-netic field, q’ - the radiative heat flux, σs - the Stefan- -Boltzmann constant, ke - mean absorption coefficient, α - the fluid thermal diffusivity, K’ - the permeability of the porous medium, T - the dimensional temperature, C - the dimensional concentration, k - the thermal con-ductivity, μ - coefficient of viscosity, D - the mass diffu-sivity, kr’ - the chemical reaction parameter, u – dimen-sionless velocity, θ - dimensionless temperature, φ - di-mensionless concentration.

Figure 1. Physical sketch and geometry of the problem.

The boundary conditions for the velocity, tempe-rature and concentration fields are:

( )( )( )

w

w

0

0, e ,

at 0

1 , , as

n t

n t

n t

u T T T TC C C C e y

u U U e T T C C y

εε

ε

′ ′∞ ∞

′ ′∞ ∞

′ ′∞ ∞

′ = = + −′= + − =

′ ′ ′= = + → → → ∞(6)

where Tw and Cw are the wall dimensional temperature and concentration respectively, T∞and C∞ are the free stream dimensional temperature and concentration respectively, U’ - velocity of the plate, U0 - mean velo-

city, w - condition at the wall, ∞ - free stream con-ditions, n’ - the constant.

From Eq. (1), it is clear that suction velocity nor-mal to the plate is either a constant or function of time, hence:

( )0 1 en tv V Aε ′ ′′ = − + (7)

where A is a real positive constant, ε and εA are small values less than unity and V0 is scale of suction ve-locity which is non zero positive constant. The negative sign indicates that the suction is towards the plate.

Since the medium is optically thin with relatively low density and α << 1 the radiative heat flux given equation (4), in the spirit of Cogley et al. [16] becomes:

( )24q

T Ty

α ∞′∂ = −′∂

(8)

where:

2

0

BT

α δλ∞

∂=′∂ (9)

with B being Planck’s function. In order to write the governing equations and the

boundary conditions in dimensionless form, the follow-ing non-dimensional quantities are introduced.

0

0 0 02

0

4, , , ,

, , ,4 w w

u v V y Uu v y U

U V v Ut V T T C C

t Cv T T C C

θ ∞ ∞

∞ ∞

′ ′ ′ ′= = = =

′ − −= = =− −

22

2 20 0

2 202

2 30

4, , Pr ,

, , ,4

p

e e

s

v Cn vn

V K V kv H kk

Sc M ND V T

ρν νχα

μ σρ σ ∞

′= = = =

′

= = =

( )

*w w

2 20 0 0 0

2 20 r2

r 2p w 0

( ) ( ), ,

,

v g T T v g C CGr Gm

U V U VU k

Ec Kc T T V

β β

ν

∞ ∞

∞

− −= =

′= =

−

(10)

In view of Eqs. (4), (7)–(10), Eqs. (2), (3) and (5) reduce to the following dimensionless form.

( )

2

2

2 2

1 1(1 e )

4 4( )

ntu u U uA Gr

t y dt yGm M u U

ε θ

ϕ χ

∂ ∂ ∂ ∂− + = + +∂ ∂ ∂

+ − + − (11)

22

2

1 1 4(1 e ) 1

4 Pr 3nt u

A Ect y N y yθ θ θε

∂ ∂ ∂ ∂ − + = + + ∂ ∂ ∂ ∂ (12)

22r2

1 1(1 e )

4ntA K

t y Sc yϕ ϕ ϕε ϕ∂ ∂ ∂− + = −

∂ ∂ ∂ (13)


252

where Gr, Gm, M, χ, Pr, N, Ec, Sc and Kr are the thermal Grashof number, solutal Grashof number, magnetic parameter, permeability parameter, Prandtl number, radiation parameter, Eckert number, Schmidt number and chemical reaction parameter, respectively.

The corresponding boundary conditions are:

0, 1 e , 1 at 01 e , 0, 0 as

nt nt

ntu e yu U y

θ ε ϕ εε θ ϕ

= = + = + =→ = + → → → ∞

(14)

SOLUTION OF THE PROBLEM

The set of differential Eqs. (11)–(13) subject to the boundary conditions, Eq. (14), are highly nonlinear, coupled and therefore it cannot be solved analytically. Hence, following Reddy [17] and Bathe [18], the finite element method is used to obtain an accurate and ef-ficient solution to the boundary value problem under consideration. The fundamental steps comprising the method are as follows.

Step 1. Discretization of the domain into ele-ments. 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 ele-ment. The collection of elements is designated the fi-nite 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 dif-ferential equation.

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

c. Derive the finite element equations by substi-tuting the approximate solution into variational formu-lation.

Step 3. Assembly of element equations. The al-gebraic equations so obtained are assembled by im-posing the inter-element continuity conditions. This yields a large number of algebraic equations, consti-tuting the global finite element model, which governs the whole flow domain.

Step 4. Impositions of boundary conditions. The physical boundary conditions defined in Eq. (14) are imposed on the assembled equations.

Step 5. Solution of the assembled equations. The final matrix equation can be solved by a direct or indi-rect (iterative) method. For computational purposes, the coordinate y is varied from 0 to ymax = 2, where ymax represents infinity, i.e., external to the momentum, energy and concentration boundary layers. Numerical solutions for these equations are obtained by C-pro-

gram. In order to prove the convergence and stability of finite element method, the same C-program was run with slightly changed values of h and k and no signi-ficant change was observed in the values of u, θ and φ. This process is repeated until the desired accuracy of 0.0005 is obtained. Hence, the finite element method is stable and convergent.

The skin-friction, Nusselt number and Sherwood number are important physical parameters for this type of boundary layer flow.

The skin-friction at the plate in the non-dimensional form is given by:

wf

0 0 0y

uC

U V yτ

ρ =

′ ∂= = ∂ (15)

The rate of heat transfer coefficient in the non- -dimensional form, in terms of the Nusselt number, is given by:

0 1

w 0

Reyx

y

Ty

Nu x NuT T y

θ′= −

∞ =

∂ ′∂ ∂= − = − − ∂

(16)

The rate of mass transfer coefficient in the non-dimensional form, in terms of the Sherwood number, is given by:

0 1

w 0

Reyx

y

Cy

Sh x ShC C y

ϕ′= −

∞ =

∂ ′∂ ∂= − = − − ∂

(17)

where 0RexV x

ν= is the local Reynolds number.

RESULTS AND DISCUSSION

In the preceding sections, the problem of che-mical reaction effects on an unsteady MHD free con-vection flow past an infinite vertical porous plate with constant suction was formulated and the dimension-less governing equations were solved by means of a finite element method. In the present study we adopted the following default parameter values of finite element computations: Gr = 2.0, Gm = 2.0, M = 1.0, χ = 0.5, Pr = 0.71, Ec = 0.001, N = 0.5, Sc = 0.6 and Kr = 0.5, A = 0.01, ε = 0.01, n = 0.1, t = 1.0. All graphs therefore correspond to these values unless specifically indi-cated on the appropriate graph.

The influence of thermal Grashof number, Gr, on the velocity is shown in Figure 2. The thermal Grashof number signifies the relative effect of the thermal buo-yancy force to the viscous hydrodynamic force. The flow is accelerated due to the enhancement in buo-


253

yancy force corresponding to an increase in the ther-mal Grashof number. The positive values of Gr corres-pond to cooling of the plate by natural convection. Heat is therefore conducted away from the vertical plate into the fluid which increases the temperature and thereby enhances the buoyancy force. In addition, it is seen that the peak values of the velocity increases rapidly near the plate as thermal Grashof number increases and then decays smoothly to the free stream velocity.

Figure 2. Effect of Gr on velocity profiles.

Figure 3 presents typical velocity profiles in the boundary layer for various values of the solutal Grashof number Gm. The solutal Grashof number Gm defines the ratio of the species buoyancy force to the viscous hydrodynamic force. It is noticed that the velocity in-creases with increasing values of the solutal Grashof number.

The effect of magnetic field parameter M on the velocity is shown in Figure 4. The velocity decreases with an increase in the magnetic parameter. It is be-cause that the application of transverse magnetic field will result a resistive type force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity. Also, the boundary layer thickness decreases with an increase in the magnetic parameter.

Figure 5 shows the velocity profiles for different values of the permeability parameter χ. Clearly, as χ increases, the velocity tends to decrease.

For different values of the radiation parameter N, the velocity and the temperature profiles are shown in

Figures 6a and 6b. It is noticed that an increase in the radiation parameter results a decrease in the velocity and temperature within the boundary layer, as well as decreased the thickness of the velocity and tempe-rature boundary layers.

Figure 3. Effect of Gm on velocity profiles.

Figure 4. Effect of M on velocity profiles.

Figures 7a and 7b illustrate the velocity and tem-perature profiles for different values of Prandtl number, Pr. The numerical results show that the effect of in-creasing values of Prandtl number results in a de-creasing velocity. From Figure 6b, it is observable that an increase in the Prandtl number results a decrease of the thermal boundary layer thickness and in general


254

lower average temperature within the boundary layer. The reason is that smaller values of Pr are equivalent to increasing the thermal conductivities, and therefore heat is able to diffuse away from the heated surface more rapidly than for higher values of Pr. Hence in the case of smaller Prandtl numbers as the boundary layer is thicker and the rate of heat transfer is reduced.

Figure 5. Effect of χ on velocity profiles.

The influence of the viscous dissipation para-meter, i.e., the Eckert number, Ec, on the velocity and temperature are shown in Figures 8a and 8b, respecti-

vely. The Eckert number, Ec, expresses the relation-ship between the kinetic energy in the flow and the enthalpy. It embodies the conversion of kinetic energy into internal energy by work done against the viscous fluid stresses. Greater viscous dissipative heat causes a rise in the temperature as well as the velocity. This behavior is evident from Figures 7a and 7b.

For different values of the Schmidt number, Sc, the velocity and concentration profiles are plotted in Figures 9a and 9b, respectively. The Schmidt number, Sc, embodies the ratio of the momentum diffusivity to the mass (species) diffusivity. It physically relates the relative thickness of the hydrodynamic boundary layer and mass-transfer (concentration) boundary layer. As the Schmidt number increases the concentration de-creases. This causes the concentration buoyancy ef-fects to decrease yielding a reduction in the fluid velo-city. The reductions in the velocity and concentration profiles are accompanied by simultaneous reductions in the velocity and concentration boundary layers, which is evident from Figures 8a and 8b.

Figures 10a and 10b illustrate the behavior velo-city and concentration for different values of chemical reaction parameter, Kr. It is observed that an increase in leads to a decrease in both the values of velocity and concentration. A distinct velocity escalation oc-curs near the wall after which profiles decay smoothly to the stationary value in free stream. Chemical re-action therefore boosts momentum transfer, i.e., ac-celerates the flow.

The effects of various governing parameters on the skin-friction coefficient, Cf, Nusselt number, Nu,

(a) (b)

Figure 6. Effects of N on a) velocity and b) temperature profiles.


255

(a) (b)

Figure 7. Effects of Pr on a) velocity and b) temperature profiles.

(a) (b)

Figure 8. Effects of Ec on a) velocity and b) temperature profiles.

and the Sherwood number, Sh, are shown in Tables 1– –3. From Table 1, it is noticed that as Gr or Gm in-creases, the skin-friction coefficient increases. It is ob-vious that as M or χ increases, the skin-friction co-efficient decreases. From Table 2, it is observed that an increase in the radiation parameter or the Prandtl

number reduces the skin-friction and increases the Nusselt number. Also, it is found that as the Eckert number increases the skin-friction increases and the Nusselt number decreases. From Table 3, it is found that as Sc or Kr increases, the skin-friction coefficient decreases and the Sherwood number increases.


256

(a) (b)

Figure 9. Effects of Sc on a) velocity and b) concentration profiles.

(a) (b)

Figure 10. Effects of Kr on a) velocity and b) concentration profiles.

Table 1. Effect of Gr, Gm, M and χ on Cf (N = 0.5, Pr = 0.71, Ec = 0.001, Sc = 0.60, Kr = 0.5)

Gr Gm M χ Cf

2.0 4.0 2.0 2.0 2.0

2.0 2.0 4.0 2.0 2.0

1.0 1.0 1.0 2.0 1.0

0.5 0.5 0.5 0.5 1.0

4.2441 5.7759 5.3685 3.8646 4.0152

Table 2. Effect of N, Pr and Ec on Cf and Nu (Gr = 2.0, Gm = 2.0, M = 1.0, χ = 0.5, Sc = 0.60, Kr = 0.5)

N Pr Ec Cf Nu

0.5 1.0 0.5 0.5

0.71 0.71 7.0

0.71

0.001 0.001 0.001

0.1

4.2441 4.1610 3.3434 4.2670

0.3556 0.4254 1.7441 0.2988


257

Table 3. Effect of Sc and Kr on Cf and Sh (Gr = 2.0, Gm = 2.0, M = 1.0, χ = 0.5, Pr = 0.71, Ec = 0.001)

Sc Kr Cf Sh

0.22 0.60 0.22

0.5 0.5 1.0

4.5638 4.2441 4.4089

0.4366 0.7800 0.5960

CONCLUSIONS

In this paper, the chemical reaction effects on an unsteady MHD free convection flow past an infinite vertical porous plate with constant suction was consi-dered. The non-dimensional governing equations are solved with the help of finite element method. The con-clusions of the study are as follows:

1. The velocity increases with the increase in thermal Grashof number and solutal Grashof number.

2. The velocity decreases with an increase in the magnetic parameter.

3. The velocity increases with an increase in the permeability parameter.

4. An increase in the Eckert number increases the velocity and temperature.

5. An increase in the Prandtl number decreases the velocity and temperature.

6. An increase in the radiation parameter leads to decreases the velocity and temperature.

7. The velocity as well as concentration de-creases with an increase in the Schmidt number.

8. The velocity as well as concentration de-creases with an increase in the chemical reaction para-meter.

REFERENCES

[1] A.J. Chamkha, H.S. Takhar, V.M. Soundalgekar, J. Chem. Eng. 84 (2001) 335-342

[2] F.S. Ibrhim, A.M. Elaiw, A.A Bakr, Commun. Nonlinear Sci. Numerical Simulation 13 (2008) 1056-1066

[3] A. Bejan, K.R. Khair, Int. J. Heat Mass Transfer 28 (1985) 909-918

[4] A. Raptis, G. Tzivanidis, N. Kafousias, Lett. Heat Mass Transfer 8 (1981) 417-424

[5] F.C. Lai, F.A. Kulacki, Int. J. Heat Mass Transfer 34 (1991) 1189-1194

[6] A. Raptis, Int. J. Energy Res. 10 (1986) 97-100

[7] V.M. Soundagekar, J. Roy. Soc. London 333 (1982) 25-36

[8] H.S. Takhar, P.C. Ram, Int. Comm. Heat Mass Transfer 21 (1994) 371-376

[9] V.M. Soundalgekar, Gupta, N.S. Birajdar, Nucl. Eng. Design 53 (1979) 309-346

[10] E.M.A. Eilbashbeshy, Int. J. Eng. Sci. 35 (1997) 515-522

[11] K.D. Alagoa, G.Tay, T.M. Abbey, Astrophysics Space Sci. 260 (1991) 455-468

[12] A.R. Bestman, S.K. Adjepong, Astrophysics Space Sci. 143 (1998) 73-80

[13] B. Gebhar, J. Fluid Mech. 14 (1962) 225- 232

[14] V.M. Sundalgekar, Int. J. Heat Mass Transfer 15 (1972) 1253-1261

[15] C. Israel-Cookey, A. Ogulu, V.B. Omubo-Pepple, Int. J. Heat Mass Transfer 46 (2003) 2305-2311

[16] A.C. Cogley, W.G. Vincenti, S.E. Gill, AIAA J. 6 (1968) 551-553

[17] J.N. Reddy, An Introduction to the Finite Element Method. McGraw-Hill, New York, 1985

[18] K.J. Bathe, Finite Element Procedures, Prentice-Hall, New Jersey, 1996.

J. ANAND RAO1

S. SHIVAIAH2

1Department of Mathematics, Os-mania University, Hyderabad, India

2Department of Mathematics, Padmasri Dr. B.V. Raju Institute of

Technology, Narsapur, Medak, India

NAUČNI RAD

UTICAJ HEMIJSKE REAKCIJE NA NESTACIO-NARNU MAGNETNOHIDRODINAMIČKU PRIRODNU KONVEKCIJU PREKO BESKONAČNE VERTIKALNE POROZNE PLOČE SA KONSTANTNIM PROSISAVANJEM

U radu su proučavani uticaji hemijske reakcije na nestacionarno magnetnohidrodina-mičko (MHD) prirodno konvektivno strujanje fluida preko beskonačne vertikalne porozne ploče sa konstantnim prosisavanjem. Važeće bezdimenzione jednačine su rešene nu-merički pomoću metode konačnih elemenata. Uticaji različitih parametara na profile br-zine strujanja, temperature i koncentracije su prikazani grafički, dok su vrednosti koefici-jenta trenja, Nuseltovog broja i Šervudovog broja za različite vrednosti fizičkih parame-tara predstavljene tabelarno.

Ključne reči: Hemijska reakcija, MHD, prirodna konvencija, konstantno prosisa-vanje.

j. anand rao chemical reaction effect on an s. … no3...paper received: 18 december, 2010 paper...

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