DOUBLE-DIFFUSIVE CONVECTION-RADIATION INTERACTION ON UNSTEADY MHD MICROPOLAR FLUID FLOW OVER A...

Journal of Engineering and Technology 19

A. A. Bakr, Z. A. S. Raizahb DOUBLE-DIFFUSIVE CONVECTION-RADIATION INTERACTION ON UNSTEADY MHD MICROPOLAR FLUID FLOW OVER A VERTICAL MOVING POROUS PLATE WITH HEAT GENERATION AND SORET EFFECTS

DOUBLE-DIFFUSIVE CONVECTION-RADIATION INTERACTION ON UNSTEADY MHD MICROPOLAR FLUID FLOW OVER A VERTICAL MOVING POROUS PLATE WITH HEAT

GENERATION AND SORET EFFECTS A. A. Bakr, Z. A. S. Raizahb

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt Department of Mathematics, Girls College of education. Abha, King Khaled University, Saudi Arabia

Abstract This work is focused on the study of combined heat and mass transfer on double-diffusive convection for a viscous fluid past a semi-infinite vertical moving porous plate embedded in a porous medium in the presence of a first-order chemical reaction and heat generation. The fluid is considered to be a gray, absorbing-emitting but non-scattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The plate moves with constant velocity in the direction of fluid flow while the free stream velocity is assumed to follow the exponentially increasing small perturbation law. A uniform magnetic field acts perpendicular to the porous surface, which absorbs the fluid with a suction velocity varying with time. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. Graphical results for velocity, temperature and concentration profiles of both phases based on the analytical solutions are presented and discussed. INTRODUCTION The theory of micropolar fluid has received great attentions during the recent years, because of traditional Newtonian fluids cannot precisely describe the characteristic of fluid with suspended particles. Physically, micropolar fluids may present the non-Newtonian fluid consisting of dumb-bell molecules or short rigid cylindrical elements, polymer fluids, fluids suspensions and animal blood. The presence of dust or smoke particular in a gas may also be modeled using micropolar fluid dynamics. In addition, due to its practical application to boundary layer control and thermal protection in high energy flow by means of wall velocity and mass transfer, considerable attention has been paid to the thermal boundary layer flows over moving boundaries [1]. The theory of micropolar fluids which takes into account the inertial characteristics of the substructure particles which allowed to undergo rotation has been proposed by Eringen [2]. Micropolar fluids are fluids

with microstructure belonging to a class of fluids non-symmetrical stress tensor. Physically, they represent fluids consisting of randomly oriented particles suspended in a viscous medium. Khonsari and Brewe [3] have examined the thermal aspects of a journal bearing lubricated with micropolar fluid. They conclude that the existence of microstructure, according to the micropolar theory, tends to enhance the load carrying capacity and friction coefficient. Kim and Lee [4] have analyzed the effect of non-zero values of micro-gyration vector on the semi-infinite moving porous plate with a constant velocity. 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 drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat and the 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 chemical reaction within or at the boundary. There are two types of reactions. A homogeneous reaction is one that occurs uniformly throughout a give phase. The species generation in a homogeneous reaction is analogous 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 therefore be treated as a boundary condition similar to the constant heat flux condition in heat transfer. Muthucumaraswamy and Ganesan [5] studied effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate. Deka et al. [6] studied the effect of the first order homogeneous chemical reaction on the process of an unsteady flow past an infinite vertical plate with a constant heat and mass transfer. Chamkha [7] studied the MHD flow of a numerical of uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction. Soundalgekar and Patti [8] studied the problem of the flow past an impulsively started isothermal infinite vertical plate with mass transfer effects. The effect of foreign mass on the free-convection flow past a semi-infinite vertical plate were studied [9]. Chamkha [10] assumed that the plate is embedded in a uniform porous medium and moves with a constant velocity in the flow direction in the presence of a transverse magnetic field. Raptis [11] investigate the steady flow of a viscous fluid through a very porous medium bounded by a porous plate subjected to a constant suction velocity by the presence of thermal radiation. Ibrahim et al [12] studied effect of The Chemical Reaction and Radiation Absorption on the Unsteady MHD Free Convection Flow Past a Semi Infinite Vertical Permeable Moving Plate with Heat Source

and Suction. Mohamed [13] investigate the effect of a first-order homgeneous chemical reaction, thermal radiation, heat source and thermal diffusion on the unsteady MHD double-diffusive free convection fluid flow past a vertical porous plate in the presence of mass blowing or suction. Recently, Bakr [14] investigate free convection heat and mass transfer adjacent to moving vertical porous infinite plate for incompressible, micropolar fluid in a rotating frame of reference in the presence of heat generation or absorption effects, a first-order chemical reactions. In spite of all these studies, the unsteady MHD double-diffusive free convection of a micropolar fluid for a heat generating fluid with thermal radiation and chemical reaction has received little attention. Hence, the main object of the present investigation is to study the effect of a first-order homogeneous chemical reaction, thermal radiation, heat source and thermal diffusion on the unsteady MHD double-diffusive free convection of a micropolar fluid flow past a vertical porous plate in the presence of mass blowing or suction. It is assumed that the plate is embedded in a uniform porous medium and moves with a constant velocity in the flow direction in the presence of a transverse magnetic field. It is also assumed that the free stream to consist of a mean velocity and temperature over which are superimposed an exponentially varying with time. MATHEMATICAL ANALYSIS Consider unsteady two-dimensional flow of a laminar, viscous, electrically conducting and heat-generation absorption fluid past a semi-infinite vertical moving plate embedded in a uniform porous medium and subjected to a uniform transverse magnetic field in the presence of a pressure gradient with double-diffusive free convection flow of a micropolar fluid, thermal diffusion, chemical reaction, and thermal radiation effects. The flow is assumed to be in the *x direction, which is taken along the plate in the upward direction and *y -axis is normal to it. The fluid is assumed to be



a gray, absorbing-emitting but non-scattering medium. The radiative heat flux in the *x direction is considered negligible in comparison with that in the

*y - direction [15]. It is assumed that there is no applied voltage of which implies the absence of an electric filed. The transversely applied magnetic field and magnetic Reynolds number are very small and hence the induced magnetic field is negligible [16]. Viscous and Darcy resistance terms are taken into account the constant permeability porous medium. The MHD term is derived form an order-of magnitude analysis of the full Navier-stokes equation. It is assumed here that the hole size of the porous plate is significantly larger than a characteristic microscopic length scale of the porous medium. We regard the porous medium as an assemblage of small identical spherical particles fixed in space, following Yamamoto and Iwamura [17]. It is also assumed that there exists a homogeneous first-order chemical reaction between the fluid and species concentration. The chemical reactions are taking place in the flow and all thermophysical properties are assumed to be constant of the linear momentum equation which is approximated according to the Boussinesq approximation. Due to the semi-infinite plane surface assumption, the flow variables are functions of *y and the time *t only. Under these assumptions, the equations that describe the physical situation are given by

,0*

*

yv

(1)

*

**

*202*

*2

*

*

*

**

*

*2)(

yu

KBg

yu

xP

yuv

tu

rr

(2)

2*

*2*

*

**

*

*

ylyv

t

(3) )(1 **0

*

*

2*

*2

*

**

*

*TT

cQ

y

qcy

Tck

yTv

tT

p

r

pp

(4)

CCky

TDy

CDyCv

tC

lTm*

2*

*2

2*

*2

*

**

*

*

(5) where ***, tandyx are the dimensional distances along and perpendicular to the plate and dimensional time, respectively. *u and *v are the components of

dimensional velocities along *x and *y directions, respectively, is the fluid density, is the viscosity, r is the rotational viscosity, pc is the specific heat at constant pressure, the electrical conductivity of the fluid, B0 is the magnetic induction,

*K is the permeability of the porous medium, *T is the dimensional temperature, mD is coefficient of chemical molecular diffusivity, TD is the coefficient of thermal diffusivity, g is the gravitational acceleration and k is the thermal conductivity of the fluid, *C is the dimensional concentration, wT and

wC are the temperature and concentration at the wall, T and C are the free stream dimensional temperature and concentration , **, Rqr are the local radiative heat flux, the reaction rate constant respectively. The term ) _T(TQ **

0 is assumed to be

amount of heat generated or absorbed per unit volume 0Q is a constant, which may take on either positive or

negative values. When the wall temperature *T exceeds the free stream temperature *T , the source term 0Q0 and heat sink when 0Q0 . The magnetic and viscous dissipations are neglected in this study. It is assumed that the porous plate moves with a constant velocity *

pu in the direction of fluid flow, and the free stream velocity *U follows the exponentially increasing small perturbation law. In addition, it is assumed that the temperature and concentration at the wall as well as the suction velocity are exponentially varying with time. The quantity *

rq in the right-hand side of equation (4) represents the radiative heat flux in the - direction



*y In order to simplify the physical problem, the optically thick radiation limit is considered in the present analysis. Thus the radiative heat flux term is simplified by using the Rosseland diffusion approximation [18] for an optically thick fluid according to,

*

4***

34

yT

kq

er

(6) where * and ke are the Stefan-Boltzmann constant and the mean absorption coefficient, respectively. It should be noted that by using the Rosseland approximation we limit our analysis to optically thick fluids. If the temperature differences within the flow are sufficiently small, then Eq.(6) can be linearized by expanding 4*T into the Taylor series about *T and

neglecting higher order terms [19], we have, 4**3*4* 34 TTTT

(7) In view of equations (6) and (7), equation (4) reduces to

)(316 **0

2*

*23**

*

**

*

*TT

cQ

y

Tkc

Tck

yTv

tT

pepp

(8) The boundary conditions for the velocity, temperature, and concentration fields are given as follows:

******0

**

*************

****

,,0),1(

)(,)(,,

yatCCTTeUUu

eCCCCeTTTTyumuu

tn

tnww

tnwwp

(9) in which *n is a scalar constant, and 0U is a scale of free stream velocity. The last equation in (9) is the boundary condition for microrotation variable * that describes its relationship with the surface stress. In this equation, the parameter m is a number between 0 and 1 that relates the micro-gyration vector to the shear stress. The value 0m corresponds to the case where the particle density is sufficiently large so that

microelements close to the wall are unable to rotate. The value 5.0m is indicative of weak concentrations, and when 1m flows are believed to represent turbulent boundary layers [20]. From the continuity equation (1), it is clear that the suction velocity normal to the plate is a function of time only and we shall take it in the form:

)1(0* ntAevv ,

(10) where A is real positive constant, is constant ( < 1) and 0v is a scale of suction velocity which has non-zero positive constant. Outside the boundary layer, Eq. (2) gives

*20**

*

*

*UB

Kg

dtdU

xP

(11) by making use of the equation of state [21] (12) Where T and c are the thermal and concentration volumetric coefficients, respectively. Upon substituting the expressions in (11) and (12) into equation (2), we have

*

***

*

20

**2*

*2

*

*

*

**

*

*

2)(

)()()(

yuU

K

B

CCgTTgy

utd

dUyuv

tu

r

cTr

(13) The third term on the RHS of the momentum equation (13) denote body force due to nonuniform temperature, the fourth term denote body force due to nonuniform concentration. On the introducing the dimensionless quantities:

CC

CCCTT

TTU

Vnn

Vtt

UUU

Uuu

Vyy

ww,,,,,,,

20*

20

*20

**0

0

*0

*

(14) In view of the above non-dimensional variables, the basic field equations (3), (5), (8) and (13) can be expressed in non dimensional form as



yuUM

KyuGmCGr

dtdU

yuAe

tu nt 2))(1()1()1( 2

2

(15)

2

2)1(

yyAe

tnt

(16)

2

2)

341(

Pr1)1(

yR

yAe

tnt

(17)

2

2

2

21)1(y

SyScy

Aet o

nt

(18) The boundary condition are

yaseu

yateeyumUu

ti

ntntp

0,00),1(

01,1,, (1

9) where Gr is the Grashof number, Gm is the solutal Grashof number, Pr is the Prandtl number, M is the magnetic field parameter, K is the permeability parameter, is the Chemical reaction parameter, Sc is the Schmidt number, fD is the Dufour parameter

epw

wTo

l

wcwTpr

kkTR

Vc

QkVK

CC

TTDS

V

K

V

BM

IVU

CCgBGm

VU

TTgBGr

kc

DSc

3*

20

02

*20

**

**

20

20

20

*

200

200

4,,,

)(

)(,,

,)(

,)(

,Pr,,

The mathematical statement of the problem is now complete and embodies the solution of Equations (15-(18) subject to boundary conditions (19). Solution In order to reduce the above system of partial differential equations to a system of ordinary differential equations in dimensionless form, we may represent the liner velocity, microrotation, temperature and concentration as

),()()(),(

),()()(),(

),()()(),(

)(),()(),(

210

210

210

210

OyCeyCtyC

Oyeyty

Oyeyty

Oyueyutyu

iwt

iwt

iwt

iwt

(20) By substituting the above Eqs.(20) into Eqs.(15-19), equating the harmonic and non-harmonic terms, and neglecting the higher order of )( 2O , and simplifying we obtains the following pairs of equations for

0000 ,,, Cu and 1111 ,,, Cu

000000 2)1( GmGrNNuuu (21)

000 (22)

0Pr3Pr3)43( 000R (23)

0000 ScSScSc o (24) subject to the boundary conditions

yasu

yatumuu p

0,0,0,1

01,1,,

0000

00000

(25) for O(1) equations, and

1110111 2)()()1( GmGruAnNuNnuu (26)

0111 An (27)

0111 Pr3Pr)(3Pr3)43( AnR (28)

01111 )( AScScSnScSc o (29) The prime denotes differentiation with respect to y. with the boundary conditions

yasuyatumu

0,0,0,101,1,,0

1111

11111



(30) Without going into detail, the solutions of Equations (21)-(24) and (26)-(29) subject to Equations (25),(30) can be shown to be

)

1(1),(4152141311211131029

1817154133216yRymymyHyRyR

yRyLntymyRyRyL

eLeLLeLeLeLeL

eLeLeeLLeLeLeLtyu

(31)

)(),( 1521415ymymntym e

nAL

eLeeLty

(32) )(),( 13113 yHyRyHntyR eeZeeety

(33)

)

)(),(1534

13263121

yHyR

yRyRntyRyRyR

eZeZ

eZeZeeeZety (3

(34) The exponential indices and the coefficients appearing in the Eqs.(31)-(34) are given in the Appendix. The physical quantities of interest are which the local wall shear stress w , the local surface heat wq and mass flux. Given the velocity field in the boundary layer, we can now calculate the local wall shear stress ( i.e. skin- friction) is given by

00

0**

*

yyw y

uUVyu

(35) therefore, the local friction factor fC is given by

15414132121111103

9281715411332160

(

LRLLmLmLHLR

LRLRLLeLLmRLRLLLUV

C ntwf

(36) Knowing the temperature field, it is interesting to study the effect of the free convection and radiation on the rate of heat transfer *

wq . This is given by

0**

4**

0**

**

34

yeyw y

Tky

Tkq

(37) by using equation (7), we can write equation (37) as follow

0**

*3**

316

yew y

Tk

Tkq

(38) which is written in dimensionless form as;

0

0**

*3

41)(

y

ww y

RVTTkq

(39) The definition of the local Nusselt number is given by

kx

TTq

Nuw

wx

*

(40) one can write

)((3

413

41Re 131130

1 HRZHeRRy

RNu nt

yx

(41) Where xV

x0Re is the local Reynolds number. The

local Sherwood number is given by; 154331622321

1 )1()0(Re HZZRZRZReZRZRSh ntx

(42) RESULTS AND DISCUSSION The formulation of the problem that accounts for the effects of thermal radiation, chemical reaction, thermal diffusion and heat source on the flow and heat transfer of an incompressible micropolar fluid along a semi-infinite, moving vertical porous plate was accomplished out in the preceding sections. This enables us to carry out the numerical computations for the velocity, microrotation, temperature and concentration fields for various values of the flow conditions and fluid properties. This is done in order to illustrate the special features of the solutions. In the present study we have chosen ,5.0A ,2.0 ,1t

,1.0,1 nt 22.0,2,5.0 ScGrUp and Gm = 1, while ,,,,, oSmR and Pr are varied over a range, which are listed in the figure legends. For different values of radiation parameter R, the translational



velocity, microrotation, temperature and concentration profiles are plotted in Fig.1. It is obvious that velocity and temperature distribution across the boundary layer increases with the increasing values of R-parameter. Fig.1. Velocity, microrotation, temperature and concentration profiles for various values of R with

0 10 20 300.0

0.2

0.4

0.6

0.8

1.0

R=1.2, 1.6, 2, 2.4, 2.8, 3.2

y0 10 20 30

0.0

0.5

1.0

R=1.2, 1.6, 2, 2.4, 2.8, 3.2

y

0 5 10 15-1.8

-1.2

-0.6

0.0

R=1.2, 1.6, 2, 2.4, 2.8, 3.2

y0 10 20 30

0

2

4R =1.2, 1 .6 , 2 , 2 .4 , 2 .8 , 3 .2

u

y

The results also show that the magnitude of microrotation and concentration on the porous plate increases as R-parameter increases.- The effects of viscosity ratio b on the translational velocity and the microrotation profiles across the boundary layer are presented in Fig. 2, it is clear that the velocity distribution is greater for a Newtonian fluid ( 0 ) with the fixed flow and material parameters, as compared with micropolar fluids until its peak value reaches. Fig.2. Velocity and microrotation profiles for various values of β with

0 2 4 6 8 10

-2

-1

0

=0, 0.2, 0.5, 1, 2, 5

y0 2 4 6 8 100

1

2

=0, 0.2, 0.5, 1, 2, 5

u

y The translational velocity distribution shows a decelerating nature near the porous plate as parameter increases, and then decays to the relevant free stream velocity. Furthermore, the magnitude of microrotation at the wall decreased as parameter increases. Fig. 3 shows the effect of m parameter, which is related to microrotation vector and shear stress, on the translational velocity and the microrotation profiles. Fig.3. Velocity and microrotation profiles for various values of m with

0 2 4 6 8 10

-20

-10

0

y0 2 4 6 8 100.0

0.5

1.0

1.5

u

y

It is observed that the magnitude of velocity increases with an increasing of m parameter but the magnitude of microrotation increases with an increasing of m parameter but the magnitude of microrotation increases with an increasing of m parameter. Fig.4. displays results for the velocity, temperature and concentration distributions, respectively. It is seen, that the velocity, temperature and concentration increases with decreasing the chemical reaction parameter c but the magnitude of microrotation increases as increases.



Fig.4. Velocity, microrotation and concentration profiles for various values of So with

0 5-8

-6

-4

-2

0

So=0 0.5, 1, 2, 3, 5

y0 2 4 6 8 100

1

2

3So=0, 0.5, 1, 2, 3, 5

u

y

0 100

2

4

6

So=0, 0.5, 1, 2, 3, 5

y

Also, we observe that the magnitude of the stream wise velocity increases and the inflection point for the velocity distribution moves further away from the surface. The time required to reach the steady state increases with increasing Schmidt number and chemical reaction parameter. This shows that the contribution of mass diffusion to the buoyancy force increases the maximum velocity significantly. For the case of 0 , i.e., for a destructive reaction, increasing values of leads to a fall in velocity profiles. For a generative reaction, 0 , a fall in velocity is also observed for increasing . This is due to the fact that as 0 , the last term in the concentration equation becomes positive and plays a crucial role. The effects of Soret number So on the velocity, microrotation and concentration profiles is shown in Fig.5. From this figure we see that Fig.5. Velocity, microrotation and concentration profiles for various values of with

0 2 4

-2

0

y0 2 4 6 8 100.5

1.0

1.5

u

y

0 2 4 6 8 100.0

0.4

0.8

1.2

y Velocity profiles increase with an increasing of So from which we conclude that the fluid velocity rises due to greater thermal diffusion. Also, we observe that the concentration profiles increase significantly

NGmZGr

323

22

R)R(1L

with an increasing of Soret number but the magnitude of microrotation increases as So decreases. The effects of heat generation on the translational velocity and microrotation profiles across the boundary layer are presented in Fig. 6. Fig.6. Velocity, microrotation, temperature and concentration profiles for various values of with



0 5 100.0

0.4

0.8

1.2

=-0.3, -0.2, -0.15, -0.1, 0, 0.1, 0.15

y0 2 4 6 8 100.0

0.4

0.8

1.2

= -0.3, -0.2, -0.15, -0.1, 0, 0.1, 0.15

y It is shown that the translational velocity across the boundary layer increases with an increasing of . Also, it is appear that there is a slight change on microrotation as increases. At the same time, the temperature and concentration profiles increases with increasing values of . Fig.7(a) shows the velocity profiles across the boundary layer for diffrent values of Prandtl number Pr. Fig.7. Velocity, microrotation, temperature and concentration profiles for various values of Pr with

0 2 4 6 8 10

-2

-1

0

=-0.3, -0.2, -0.15, -0.1, 0, 0.1, 0.15

y0 2 4 6 8 10

1

2=-0.3, -0.2, -0.15, -0.1, 0, 0.1, 0.15

u

y

0 5 100.0

0.4

0.8

1.2

Pr =0.71, 1, 2, 3

y0 5 10

0.0

0.4

0.8

1.2

Pr =0.71, 1, 2, 3

y

0 5 100

1

2

3

Pr =0.71, 1, 2, 3

u

y 0 2 4 6 8 10-8

-6

-4

-2

0

Pr =0.71, 1, 2, 3

y

The results show that the effect of increasing values of Pr results in a decreasing the velocity. Fig. 7(b)

shows the microrotation profiles for different values of the Prandtl number Pr. The numerical results show that the effect of increasing values of Prandtl number results in a decreasing microrotation. Typical variation of the temperature profiles along the spanwise coordinate y are shown in Fig. 7(c) for different values of Prandtl number Pr. The results show that an increase of Prandtl number results in a decreasing the thermal boundary layer thickness and more uniform temperature distribution across the boundary layer. The reason is that smaller values of Pr are equivalent to increasing the thermal conductivities, and therefore, heat is able to differ away from the heated surface more rapidly than for higher values of Pr. Hence, the boundary layer is thicker and the rate of heat transfer is reduced, for gradient have been reduced. Fig.7(d) illustrates the concentration profiles for various values of Pr. We see that the concentration profiles increases (decreases) near (far) from vertical porous plate as parameter Pr increases.

APPENDIX

)Sc 4Sc(21R 2

1 Sc Sc)( 4Sc(21R 2

2 nSc

)14N11(12

1L1

)Pr3

43411(432

Pr3R3R

R

)1n)4(N11(121R4

)Pr3

43)-4(n11(432

Pr3H1R

R

2n 4iSm

2

6S

)Pr(33PrR-)R43(

Pr3Z

321

31

nR

RA



Sc

ScRSo

323

32

ScRRZ

Scn

ScRRZSZA o

)(ScRR

)(Z

323

33124

Scn

ScHRZSo

)(ScHH

))1(Z

321

2131

5

N

GmZ

121

23

R)R(1

)1(L

Nmm

121

14

)m(1

2L

)m-(L-1)R-(L)R-(LL1)-(Upm(L

L114

11331215 mL

L

54326 LL1-UpL LL

)(L)L(1L

121

617

nN

LAL

)(R)R(1L

121

3318

nN

GmZLAR

)(R)R(1L

222

69

nN

GmZ

5436 Z1Z ZZ

)(R)R(1L

323

423110

nN

GmZLARGrZ

REFERENCES R. Bhargava, H. S Takhar. Numerical study of heat transfer characteristics of the micropolar boundary layer near a stagnation point on a moving wall, Int J Eng Sci, 38 (2000) 383-94.

C. Eringen. Theory of micropolar fluids, J Math Mech, 16 (1964) 1-18.

M.M. Khonsari, D. E. Brewe. Effects of viscous dissipation on the lubrication characteristics of micropolar fluids, Acta Mech, 105 (1994) 57-68.

Y. J. Kim, J. C. Lee. Analytical studies on MHD oscillatory flow of a micropolar fluid over a vertical porous plate, Surf Coat Technol, 171 (2003) 87-93.

R. Muthucumaraswamy, P. Ganesan. Effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate, J Appl Mech Tech Phys, 42 (2001) 665-71.

R. Deka, U. N. Das, V. M. Soundalgekar. Effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and

Forschung im Ingenieurwesen, 60 chemical reaction, (1994) 28-47

)(H)H(1

)1(L

121

5111

nN

GmZZGr .

)(R)R(1

2

L3

23

215

15412

nNn

mALmLAL

)R-(L)R-(LL1)-(Upm(L 1133121 L

141312111098715 LLLLLLLL1L

)(m)m(1

2L2

22

213

nNm

)R-(m1

)n

mAL)m-(RL)H-(m(L

L4213

15414121411

14 mL

RR

)R-(RL)R-(RL)R-(RL)R-(LLLL 341024914841714



B. Gebhart, L. Pera. The nature of vertical natural convection flow resulting from the combined buoyancy effects of thermal and mass diffusion, J Heat Mass Transfer , 14 (1971) 2025-50.

J. Chamkha. Unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption, Int J Eng Sci, 42 (2004) 217-30.

Raptis. Radiation and free convection flow through a porous medium, Int Comm Heat Mass Transfer, 25 (1998) 289-95.

F. S. Ibrahim , A. M. Elaiw and A. A. Bakr, Effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate with heat source and suction, Communications in Nonliner Sci. and Numerical Simulation, 13 (2008) 1056- 66.

R. A. Mohamed, Double-Diffusive Convection-Radiation Interaction on Unsteady MHD Flow over a Vertical Moving Porous Plate with Heat Generation and Soret E_ects , Applied Mathematical Sciences,, 3 (2009) 629 - 651. Bakr, Effects of chemical reaction on MHD free convection and mass transfer flow of a micropolar fluid with oscillatory plate velocity and constant heat source in a rotating frame of reference , Commun Nonlinear Sci Numer Simulat, 16 (2011) 698-710.

E. M. Sparrow, R. D. Cess, Hemisphere Publ. Stuart. J. T. , Proc. Soc. London, A231 (1955) 116-121.

T. G. Cowling, Magnetohydrodynamics , Interscience Publishers, New York, (1957).

K. Yamamoto, N. Iwamura, Flow with convective acceleration through a porous medium , J. Eng.Math, 10 (1976) 41- 54.

E. M. Sparrow, R. D. Cess, Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington, DC (1978).

Raptis, Flow of a micropolar fluid past a continuously moving plate by the presence of radiation, Int. J. Heat Mass Transfer, 41 (1998) 2865-2866.

D. A. S. Rees and A.P. Bassom , The Blasius boundary-layer flow of a micropolar fluid, Int. J. Engng. Sci., 34 (1996)113-124.

Hassanien, M. H. Obied Allah, Oscillatory hydromagnetic flow through a porous medium with variable permeability in the presence of free convection and mass transfer flow, Int. Comm. Heat mass transfer, 29 (2002) 567-575.

J. Chamkha. MHD flow of a numerical of uniformly stretched vertical permeable surface in the presence of heat generation/ absorption and a chemical reaction, Int Comm Heat Mass Transfer, 30 (2003) 413-22.

V. M. Soundalgekar, M. R. Patti. Stokes problem for a vertical plate with constant heat flux, Astrophys Space Sci, 70 (1980) 179-82.

DOUBLE-DIFFUSIVE CONVECTION-RADIATION INTERACTION ON UNSTEADY MHD MICROPOLAR FLUID FLOW OVER A...

Documents

Transcript of DOUBLE-DIFFUSIVE CONVECTION-RADIATION INTERACTION ON UNSTEADY MHD MICROPOLAR FLUID FLOW OVER A...