radiation and chemical reaction effects on mhd convective flow past ...

RADIATION AND CHEMICAL REACTION EFFECTS ONMHD CONVECTIVE FLOW PAST A MOVING VERTICAL

POROUS PLATE

S. MOHAMMED IBRAHIM, T. SANKAR REDDY AND N. BHASKAR REDDY

ABSTRACT: This paper focuses on the effects of radiation on unsteady combined freeconvection boundary layer and mass transfer flow of a viscous incompressible electricallyconducting and radiating fluid past a moving vertical porous plate in the presence ofchemical reaction. The governing non linear partial differential equations and theirboundary conditions are reduced into a system of ordinary differential equations by asimilarity transformation. This system is solved numerically using Runge-Kutta fourthorder method along with shooting technique. The behavior of the velocity, temperature,concentration, skin-friction coefficient, Nusselt number and Sherwood number forvariations in the governing thermo physical parameters are computed, analyzed anddiscussed.

Keywords: Thermal radiation, Free convection, Mass transfer, Porous plate, Boundarylayer flow, Chemical reaction.

1. INTRODUCTION

The effect of free convection flow of a viscous incompressible fluid past an infinitevertical plate has many important technological applications in the astrophysical,geophysical and engineering problems. Pohlhausen (Pohlhausen et al., 1921) first studiedthe steady free convection flow past a semi-infinite vertical plate by integral method.But the similarity solution to steady free convection flow past a semi-infinite verticalplate was presented by Ostrach (Ostrach 1953) who solved the ordinary non-linearequations by a numerical method. Seigel (Seigel 1958) was the first to study the transientfree convective flow past a semi-infinite vertical plate by integral method. The sameproblem was studied by Gebhart (Gebhart 1961) by an approximate method.Soundalgekar (Soundalgekar 1977) presented convection effects on the Stokes problemfor infinite vertical plate.

Many transport processes exist in nature and in industrial applications in which thesimultaneous heat and mass transfer occur as a result of combined buoyancy effects ofthermal diffusion and diffusion of chemical species. A few representative fields ofinterest in which combined heat and mass transfer plays an important role are designingof chemical processing equipment, formation and dispersion of fog, distribution oftemperature and moisture over agricultural fields and groves of fruit trees, crop damagedue to freezing, and environmental pollution. In this context, Soundalgekar

IJAMAA, Vol. 7, No. 1, (January-June 2012,), pp. 1-16 © Serials PublicationsISSN: 0973-3868

2 S. MOHAMMED IBRAHIM, T. SANKAR REDDY AND N. BHASKAR REDDY

(Soundalgekar 1979) extended his own problem of Soundalgekar (Soundalgekar 1977)to mass transfer effects. Callahan and Marner (Callahan and Marner 1976) consideredthe transient free convection flow past a semi-infinite vertical plate with mass transfer.Unsteady free convective flow on taking into account the mass transfer phenomenonpast an infinite vertical plate was studied by Soundalgekar and Wavre (Soundalgekarand Wavre 1977 ).

The study of magnetohydrodynamic (MHD) flows plays an important role inagriculture, engineering and petroleum industries. The problem of free convention underthe influence of a magnetic field has attracted the interest of many researchers in viewof its applications in geophysics and astrophysics. Soundalgekar et al. (Soundalgekaret al., 1977) analyzed the problem of free convection effects on Stokes problem for avertical plate under the action of transversely applied magnetic field. Helmy (Helmy1998) presented an unsteady two-dimensional laminar free convection flow of anincompressible, electrically conducting (Newtonian or polar) fluid through a porousmedium bounded by an infinite vertical plane surface of a constant temperature. Zueco(Zueco 2006) analyzed the hydromagnetic con-vection past a flat plate. Elabashbeshy(Elabashbeshy 1997) studied the heat and mass transfer along a vertical plate under thecombined buoyancy effects of thermal and species diffusion, in the presence of magneticfiled. Bala Anki reddy and Bhaskar reddy (Anki reddy and Bhaskar reddy 2011) studiedthe finite difference analysis of radiation effects on unsteady MHD flow of chemicallyreacting fluid with time-dependent suction.

In the context of space technology and in the processes involving high temperatures,the effects of radiation are of vital importance. Recent developments in hypersonic flights,missile re-entry, rocket combustion chambers, power plants for inter planetary flight andgas cooled nuclear reactors, have focused attention on thermal radiation as a mode ofenergy transfer, and emphasized the need for improved understanding of radiative transferin these processes. The interaction of radiation with laminar free convection heat transferfrom a vertical plate was investigated by Cess (Cess 1966) for an absorbing, emittingfluid in the optically thick region, using the singular perturbation technique. Arpaci (Arpaci1968) considered a similar problem in both the optically thin and optically thick regionsand used the approximate integral technique and first-order profiles to solve the energyequation. Raptis (Raptis 1998) analyzed the thermal radiation and free convection flowthrough a porous medium bounded by a vertical infinite porous plate by using a regularperturbation technique. Hossain and Takhar (Hossain and Takhar 1996) studied theradiation effects on mixed convection along a vertical plate with uniform surfacetemperature using the Keller Box finite difference method. In all these papers, the flow isconsidered to be steady. The unsteady flow past a moving plate in the presence of freeconvection and radiation were presented by Mansour (Mansour 1990). Radiation andmass transfer effects on two-dimensional flow past an impulsively started isothermal

RADIATION AND CHEMICAL REACTION EFFECTS ON MHD CONVECTIVE FLOW PAST... 3

vertical plate were analyzed by Ramachandra Prasad et al. (Ramachandra Prasad et al.,2007). Abdus Sattar and Hamid kalim (Abdus Sattar and Hamid kalim 1996) investigatedthe unsteady free convection interaction with thermal radiation in boundary layer flowpast a vertical porous plate. Makinde (Makinde 2005) discussed radiation and mass transfereffects on free convection flow past a moving vertical porous plate. Suneetha etal.(Suneetha et al., 2008) reported radiation effects on MHD free convection flow pastan impulsively started vertical plate with variable surface temperature and concentration.Sankar reddy (Sankar Reddy et al., 2010) analyzed the radiation effects on MHD mixedconvection flow of a micropolar fluid past a semi-infinite vertical plate in a porous mediumwith heat absorption.

Combined heat and mass transfer problems with chemical reaction are of importancein many processes and have, therefore, received a considerable amount of attention inrecent years. In processes such as drying, evaporation are the surface of a water body,energy transfer in wet cooling tower and the flow in desert cooler, heat and mass transferoccur simultaneously. Possible applications of this type of flow can be found in manyindustries. For example, in the power industry, among the methods of generating electricpower is one in which electrical energy is extracted directly from a moving conductingfluid. Many practical diffusive operations involve the molecular diffusion of a species inthe presence of chemical reaction within or at the boundary. The study of heat of heatand mass transfer with chemical reaction is of great practical importance to engineersand scientists because of its almost universal occurrence in many branches of scienceand engineering. Das et al., (Das et al., 1994) studied the effects of mass transfer on flowpast an impulsively started infinite vertical plate with constant heat flux and chemicalreaction. Muthucumaraswamy (Muthucumaraswamy 2002) has studied the effects ofreaction on a moving isothermal vertical infinitely long surface with suction.

However, the interaction of radiation and mass transfer on a hydromagnetic flowwith chemical reaction past a moving vertical porous plate has received little attention.Hence, an attempt is made to analyze the radiation and mass transfer effects on anunsteady MHD free convection flow of a viscous incompressible fluid past a movingvertical porous plate in the presence of chemical reaction. The governing equations aretransformed by using unsteady similarity transformation and the resultant dimensionlessequations are solved numerically using shooting technique. The effects of variousgoverning parameters on the velocity, temperature, concentration, skin-frictioncoefficient, Nusselt number and Sherwood number are obtained.

2. MATHEMATICAL ANALYSIS

An unsteady two-dimensional free convection flow of a viscous incompressibleelectrically conducting, radiating and chemically reacting fluid past a moving verticalporous plate is considered. The x-axis is taken along the plate in the upward direction


and y-axis is taken normal to the plate. The fluid is considered to be a gray, absorbingemitting radiation but non-scattering medium and the Rosseland approximation is usedto describe the radiation heat flux in the energy equation. A uniform magnetic field isapplied in the direction perpendicular to the plate. The fluid is assumed to be slightlyconducting, and hence the magnetic Reynolds number is much less than unity and theinduced magnetic field is negligible in comparison with the applied magnetic field. Itis assumed that the external electrical field is zero and the electric field due to thepolarization of charges is negligible. Initially, the plate and the fluid are at the sametemperature T� and the concentration C�. At time t > 0, the plate temperature andconcentration are raised to T

w and C

w respectively and are maintained constantly

thereafter. It is also assumed that all fluid properties are constant except that the influenceof the density variation with temperature and concentration in the body force term(Boussinesq’s approximation). Also, there is chemical reaction between the diffusingspecies and the fluid. The foreign mass present in the flow is assumed to be at lowlevel and hence Soret and Dufour effects are negligible. Under these assumptions, thegoverning boundary layer equations of the flow field are:

Conservation of mass

��

�0

vy

(1)

Conservation of momentum:

� ��

� � � � � � � � � ��

22* 0

2( ) ( )

Bu u uv g T T g C C u

t y y(2)

Conservation of energy (Heat):

� � � � ��

� � ��

2

2rT T T q

vt y k yy

(3)

Conservation of Species (Concentration):

��

� � � ��

2

2( )r

C C Cv D K C C

t y y(4)

where u and v are the velocity components in x and y directions respectively, �-thefluid density, g-the acceleration due to gravity, �, �*-the thermal and concentrationexpansion coefficients respectively, T-the temperature of the fluid in the boundary layer,�-the kinematic viscosity, �-the electrical conductivity of the fluid, T�-the temperatureof the fluid far away from the plate, �-the thermal diffusivity, C-the species


concentration in the boundary layer, C�-the species concentration in fluid far awayfrom the plate, B0-the magnetic induction, k-the thermal conductivity, q

r-the local

radiative heat flux and D-the mass diffusivity and Kr-the chemical reaction parameter.

The second and third terms on the right hand side of the momentum equation (2) denotethe thermal and concentration buoyancy effects respectively.

By using the Rosseland approximation (Brewster 1992), the radiative heat flux isgiven by

� ��

�

443

sr

e

Tq

k y(5)

where �s is the Stefan-Boltzmann constant and k

e-the mean absorption coefficient.

The boundary conditions for the velocity, temperature and concentration fields are:

t � 0 : u = 0, v = 0, T = T�, C = C�

t > 0 : u = U, v = v (t), T = Tw, C = C

wat y = 0 (6)

u � 0, v � 0, T � T�, C � C� as y � �

where U is the plate characteristic velocity. We introduce similarity variables and thedimensionless quantities, i.e.

� ��2

y

t, � �( )u Uf , �

�

��

�w

T T

T T, �

�

��

�w

C C

C C, ��

�316 ( )

3s w

e

T TR

k k

��Sc

D,

��

Pr ,�

��

204 B t

M , �4

rr

KK

t, ��

�4 ( )w

rg t T T

GU

(7)

��

*4 ( )wc

g t C CG

U, �

��

�w

TN

T Twhere f is the dimensionless velocity.

From equation (1), � is either a constant or a function of time. Following (Singhand Soundalgekar 1990), we choose

��

12

v ct

(8)

where c > 0 is the suction parameter.

In view of equations (7) and (8), the equations (2)-(4) reduce to

�� 2( ) r cf c f G G Mf (9)


�� 2 2 32( ) Pr [3( ) ( ) ]c R N N (10)

�� 2( ) rc Sc K Sc (11)

The corresponding dimensionless boundary conditions are

f = 1, � = 1, � = 1 at � = 0

f � 0, � � 0, � � 0 as � � � (12)

where Pr-the Prandtl number, Sc-Schmidt number, Gr-thermal Grashof number,Gc- solutal Grashof number, N-the temperature difference parameter, R-the radiationparameter, M-the magnetic field parameter and K

r-chemical reaction pa1rameter. The

prime indicates the differentiation with respect to.

For the type of flow under consideration, the physical quantities such as the wallshear stress, surface heat flux and the surface mass flux are very important and theseare given by

�

� �� 0

wy

uy

(13)

�

� �� 0

wy

Tq k

y(14)

�

� �� 0

wy

CM D

y(15)

where �- is the viscosity and k-the thermal conductivity.

Hence, the skin-friction coefficient, Nusselt number and Sherwood number nearthe plate are given by

Cf = �

�

� � �� 1

0

2 22(Re) (0)w

y

uf

yU vt U vt(18)

Nu =� � �

� �� 0

2 2(0)

( ) ( )w

w w y

q vt k vt Tk T T k T T y

(19)

Sh =� � �

� �� 0

2 2(0)

( ) ( )w

w w y

M vt D vt CD C C D C C y

(20)

where ��Re Ut is the Reynolds number.


3. SOLUTION OF THE PROBLEM

The set of coupled non-linear governing boundary layer equations (9)-(11) togetherwith the boundary conditions (12) are solved numerically by using Runge-Kutta fourthorder technique along with shooting method. First of all, higher order non-lineardifferential Equations (9)-(11) are converted into simultaneous linear differentialequations of first order and they are further transformed into initial value problem byapplying the shooting technique (Jain et al., 1985). The resultant initial value problemis solved by employing Runge-Kutta fourth order technique. The step size �� = 0.05 isused to obtain the numerical solution with decimal place accuracy as the criterion ofconvergence. From the process of numerical computation, the skin-friction coefficient,the Nusselt number and the Sherwood number, which are respectively proportional tof �(0), – ��(0) and – ��(0), are also sorted out and their numerical values are presented ina tabular form.

4. RESULTS AND DISCUSSION

In the preceding sections, the problem of radiation and mass transfer effects on anunsteady MHD free convection flow of a viscous incompressible electrically conductingand chemically reacting fluid past a vertical porous plate was formulated and thedimensionless governing equations were solved by means of shooting method. As aresult of the numerical calculations, the dimensionless velocity, temperature andconcentration distributions for the flow under consideration are obtained and theirbehavior have been discussed for variations in the governing parameters viz., the thermalGrashof number Gr, solutal Grashof number Gc, magnetic filed parameter M, Prandtlnumber Pr, thermal radiation parameter R, Schmidt number Sc and suction parameterc. In the present study we adopted the following default parametric values: Gr = 20,Gc = 10, M = 1.0, Pr = 0.71, R = 0.5, N = 0.1, Sc = 0.6, K

r = 0.5, c = 0.5. All the graphs

therefore correspond to these values unless specifically indicated on the appropriategraph.

The influence of thermal Grashof number Gr on velocity is shown in Fig. 1. Theflow is accelerated due to the enhancement in buoyancy force corresponding to anincrease in the thermal Grashof number i.e., free convection effects. The positive valuesof Gr correspond to cooling of the plate by natural convection. Heat is thereforeconducted away from the vertical plate into the fluid which increases the temperatureand thereby enhances the buoyancy force. In addition, it is seen that the peak values ofthe velocity increases rapidly near the plate as thermal Grashof number increases andthen decays smoothly to the free stream velocity.

Figure 2 presents typical velocity profiles in the boundary layer for various valuesof the solutal Grashof number Gc. It is noticed that the velocity increases with increasingvalues of the soltal Grashof number.


The effect of magnetic filed parameter M on the velocity is shown in Fig. 3. Thevelocity decreases with an increase in the magnetic field parameter. It is because that theapplication 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 magnetic parameter.

Figure 4(a) and 4(b) illustrate the velocity and temperature profiles for differentvalues of Prandtl number Pr. The numerical results show that the effect of increasingvalues of Prandtl number results in a decreasing velocity. From Fig.4 (b), it is observedthat an increase in the Prandtl number results in a decrease of the thermal boundarylayer thickness and in general lower average temperature with in the boundary layer.The reason is that smaller values of Pr are equivalent to increasing the thermalconductivities, and therefore heat is able to diffuse away from the heated surface morerapidly than for higher values of Pr. Hence in the case of smaller Prandtl numbers asthe boundary layer is thicker and the rate of heat transfer is reduced.

The influence of the thermal radiation parameter R on the velocity and temperatureare shown in Figs. 5(a) and 5(b) respectively. It is obvious that an increase in the radiationparameter R results in an increase in both the velocity and temperature within theboundary layer.

For different values of the Schmidt number Sc, the velocity and concentrationprofiles are plotted in Figs.6 (a) and 6(b) respectively. It physically relates the relativethickness of the hydrodynamic boundary layer and mass transfer (concentration)boundary layer. As the Schmidt number Sc increases the concentration decreases. Thiscauses the concentration buoyancy effects to decrease yielding a reduction in the fluidvelocity. The reductions in the velocity and concentration profiles are accompanied bysimultaneous reductions in the velocity and concentration boundary layers, which isevident from Figs. 6 (a) and 6(b).

Figures 7 (a), 7(b) and 7(c) show the velocity, temperature and concentration profilesfor different values of suction parameter c. it is observed that an increase in the suctionparameter c results in a decrease in the velocity, temperature and concentration.

Figures 8 (a) and 8(b) show the velocity and concentration profiles for differentvalues of chemical reaction parameter K

r. It is observed that an increase in the chemical

reaction parameter Kr results in a decrease in both the velocity and concentration.

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

number Nu and the Sherwood number Sh are shown in Tables 1, 2 and 3. From Table 1,it is noticed that as Gr or Gc increases, the skin-friction coefficient increases. It is obviousthat an increase in magnetic field parameter M reduces the skin-friction. From Table 2, itis observed that as the Prandtl number Pr increases the skin-friction decreases while theNusselt number increases. Also, it is found that an increase in the radiation parameter Rresults in an increase in the skin-friction and a decrease in the Nusselt number. From


Table 3, it is seen that, as the Schmidt number Sc or chemical reaction parameter Kr

increases, the skin-friction decreases while the Sherwood number increases.

0 . 0 0 .5 1 . 0 1 .5 2 . 00 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

f

�

G r= 5 ,1 0 ,1 5 ,2 0

Figure 1: Velocity Profiles for Different Values of Gr

Figure 2: Velocity Profiles for Different Values of Gc

0 . 0 0 .5 1 . 0 1 .5 2 .00 .0

0 .4

0 .8

1 .2

1 .6

f

�

G c= 5 ,1 0 , 1 5 ,2 0

Figure 3: Velocity Profiles for Different Values of M

0 . 0 0 . 5 1 .0 1 .5 2 .00 .0

0 .4

0 .8

1 .2

1 .6

f

�

M =1 .0 ,2 .0 ,3 .0 ,4 .0


0 . 0 0 . 5 1 . 0 1 .5 2 . 00 .0

0 .5

1 .0

1 .5

2 .0

2 .5

f

�

P r = 0 .3 ,0 .7 1 ,1 .0 ,1 .5

0 . 0 0 .5 1 .0 1 .5 2 .00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

�

�

P r = 0 .3 ,0 .7 1 ,1 .0 ,1 .5

0 .0 0 .5 1 .0 1 .5 2 .00 .0

0 .4

0 .8

1 .2

1 .6

�

f

R = 0 .0 ,0 .5 ,1 .0 ,1 .5

Figure 4(a): Velocity Profiles for Different Values of Pr

Figure 4(b): Temperature Profiles for Different Values of Pr

Figure 5(a): Velocity Profiles for Different Values of R


Figure 5(b): Temperature Profiles for Different Values of R

Figure 6(a): Velocity Profiles for Different Values of Sc

Figure 6(b): Temperature Profiles for Different Values of Sc

0 . 0 0 . 5 1 . 0 1 . 5 2 . 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

�

�

R = 0 .0 ,0 .5 ,1 .0 ,1 .5

0 . 0 0 .5 1 . 0 1 .5 2 . 00 .0

0 .3

0 .6

0 .9

1 .2

1 .5

1 .8

f

�

S c = 0 .2 ,0 .6 ,0 .7 8 ,1 .0

0 .0 0 .5 1 .0 1 . 5 2 . 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

�

�

S c = 0 .3 ,0 .6 ,0 .7 8 ,1 .0


Figure 7(a): Velocity Profiles for Different Values of Kr

Figure 7(b): Concentration Profiles for Different Values of Kr

Figure 8(a): Velocity Profiles for Different Values of c

0 . 0 0 . 5 1 . 0 1 .5 2 . 00 .0

0 .3

0 .6

0 .9

1 .2

1 .5

f

�

K r= 0 .0 .0 .5 ,1 .0 ,1 .5

0 . 0 0 . 5 1 . 0 1 . 5 2 . 00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

�

�

K r= 0 .0 ,0 .5 ,1 .0 ,1 .5

0 . 0 0 . 5 1 . 0 1 .5 2 . 00 .0

0 .3

0 .6

0 .9

1 .2

1 .5

1 .8

f

�

c = 0 .2 5 ,0 .5 ,0 .7 5 ,1 .0


Figure 8(b): Temperature Profiles for Different Values of c

Figure 8(c): Concentration Profiles for Different Values of c

0 .0 0 . 5 1 .0 1 .5 2 .00 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

�

�

c = 0 .2 5 ,0 .5,0 .7 5 ,1 .0

0 .0 0 .5 1 .0 1 .5 2 .00 .0

0 .2

0 .4

0 .6

0 .8

1 .0

�

�

c=0.25,0.5 ,0 .75,1 .0

Table 1Numerical Values of the Skin-Friction Coefficient C

f for

Pr = 0.71, R = 0.5, Sc = 0.6, c = 0.5, Kr = 0.5, N = 0.1

Gr Gc M Cf

5.0 5.0 1.0 1.71219

7.0 5.0 1.0 2.51375

5.0 7.0 1.0 2.45035

5.0 5.0 2.0 1.14244


REFERENCES

[1] Pohlhausen E., Der Warmeaustrausch Zwischen Festen Korpen and Flussigkeiten mit, KleinerReibung under Kleiner Warmeleitung, ZAMM, 1, (1921), 115-121.

[2] Ostrach S., An Analysis of Laminar Free Convection Flow and Heat Transfer Along a FlatPlate Parallel to the Direction of the Generating Body Force, NACA Report, 1111, (1953).

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[5] Soundalgekar V. M., Convection Effects on the Stokes Problem for Infinite Vertical Plate,ASME. J. Heat Transfer, 99, (1977), 499-501.

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[9] Soundalgekar V. M., Effects on Mass Transfer and Free Effects on MHD Stokes Problemfor a Vertical Plate, Nuclear Eng. Des., 53, (1977), 309-346.


f and Nusselt Number Nu for

Gr = 5, Gc = 5, M = 1.0, Sc = 0.6, c = 0.5, N = 0.1, Kr = 0.5

Pr R Cf

Nu

0.71 0.5 1.71219 0.930642

1.0 0.5 1.42462 1.17372

0.71 0.7 1.78951 0.822859


f and Sherwood Number Sh for

Gr = 5, Gc = 5, M = 1, Pr = 0.71, N = 0.1, R = 0.5, c = 0.5

Kr Sc Cf

Sh

0.5 0.6 1.71219 1.4158

0.5 0.78 1.51012 1.68296

1.0 0.6 1.64496 1.53272


[10] Helmy K. A., MHD Unsteady Free Convection Flow Past a Vertical Porous Plate, ZAMM,78, (1998), 255-270.

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[12] Elbashbeshy M. A., Heat and Mass Transfer Along a Vertical Plate Surface Tension andConcentration in the Presence of Magnetic Field, Int. J. Eng. Sci., 34(5), (1997), 515-522.

[13] Bala Anki Reddy P., and Bhaskar Reddy N., Finite Difference Analysis of Radiation Effectson Unsteady MHD Flow of Chemically Reacting Fluid with Time-Dependent Suction, Int.J. of Appl. Math. and Mech., 7(9), (2011), 96-105.

[14] Cess R. D., The Interaction of Thermal Radiation with Free Convection Heat Transfer,Int. J. Heat Mass Transfer, 9, (1966), 1269-1277.

[15] Arpaci V. S., Effect of Thermal Radiation on the Laminar Free Convection from a HeatedVertical Plate, Int. J. Heat Mass Transfer, 11, (1968), 871-881.

[16] Raptis A., Radiation and Free Convection Flow Through a Porous Medium, Int. Comm.Heat Mass Trasfer, 25(2), (1998), 289-295.

[17] Hosaain M. A., and Takhar H. S., Radiation Effects on Mixed Convection Along a VerticalPlate with Uniform Surface Temperature, Heat Mass Transfer, 31, (1996), 243-248.

[18] Mansour M. H., Radiative and Free Convection Effects on the Oscillatory Flow Past a VerticalPlate, Astrophysics and Space Science, 166, (1990), 26-75.

[19] Ramachandra Prasad V., Bhaskar Reddy N. and Muthucumaraswamy R., Radiation and MassTransfer Effects on Two-Dimensional Flow Past an Impulsively Started Isothermal Verticalplate, Int. J. Thermal Sciences., 46(12), (2007), 1251-1258.

[20] Abdus Sattar M. D., and Hamid Kalim M. D., J. Math. Phys. Sci., 30(1), (1996), 25.

[21] Makinde O. D., Free Convection Flow with Thermal Radiation and Mass Transfer Past aMoving Vertical Porous Plate, Int. Commu. Heat and Mass Transfer, 32, (2005), 1411-1419.

[22] Suneetha S., Bhaskar Reddy N., and Ramachandra Prasad V., Thermal Radiation Effects onMHD Free Convection Flow Past an Impulsively Started Vertical Plate with Variable SurfaceTemperature and Concentration, Journal of Naval Architecture and Marine Engineering, 2,(2008), 57-70.

[23] Sankar Reddy T., Ramachandra Prasad V., Roja P., and Bhaskar Reddy N., Radiation Effectson MHD Mixed Convection Flow of a Micropolar Fluid Past a Semi-Infinite Vertical Platein a Porous Medium with Heat Absorption, Int. J. of Appl. Math and Mech., 6(18), (2010),80-101.

[24] Das U. N., Deka, R. K., and Soundalgekar, V. M., Forschung in Inge, 80, (1994), 284.

[25] Muthcumaraswamy R., Acta Mechanica, 155, (2002), 65.

[26] Brewster M. A., Thermal Radiative Transfer and Properties, John Wiley & Sons, New York,(1992).


[27] Singh A. K., and Soundalgekar V. M., Int. J. Energy Res., 14, (1990), 413.

[28] Jain M. K., Iyengar S. R. K., and Jain R. K., Numerical Methods for Scientific and EngineeringComputation, Wiley Eastern Ltd., New Delhi, India, (1985).

S. Mohammed IbrahimAssistant Professor in Mathematics,Priyadarshini College of Engineering and Technology,Nellore-524 004, (A.P.), India.E-mail: [email protected]

T. Sankar ReddyDept .of Science and Humanities,R.V.P. Engineering College for Women,Kadapa-516 003, (A.P.), India.

N. Bhaskar ReddyDepartment of Mathematics,Sri Venkateswara University,Tirupati-517502, (A.P.), India.E-mail: [email protected]

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