rakesh ijamm 2

Int. J. of Appl. Math and Mech. 7 (6): 19-35, 2011.

UNSTEADY MHD FLOW OF RADIATING AND REACTING FLUID

PAST A VERTICAL POROUS PLATE WITH COSINUSOIDALLY

FLUCTUATING TEMPERATURE

R. Kumar1 and K. D. Singh

2

1Department of Mathematics, Govt. College for Girls (RKMV), Shimla 171 001, India

2Department of Mathematics (ICDEOL), H.P. University, Shimla 171 005, India

Email: [email protected]

Received 11 March 2010; accepted 10 February 2011

ABSTRACT

An analysis of an unsteady MHD free convective flow of a viscous, incompressible,

electrically – conducting and radiating fluid past an infinite hot vertical porous plate with

chemical reaction of first order has been carried out by taking into account the effect of

viscous dissipation. The temperature of the plate is assumed to be spanwise cosinusoidally

fluctuating with time. The governing equations are solved by perturbation technique.

Numerical evaluation of the analytical results is performed. Graphical results for transient

velocity and transient temperature profiles and tabulated results for skin-friction coefficient

and Nusselt number are presented and discussed. It is found that velocity and temperature

decrease for destructive chemical reactions and increase for generative chemical reactions.

Keywords: Spanwise Cosinusoidally Fluctuating Temperature, Chemical Reaction, Thermal

Radiation, Unsteady Flow

1 INTRODUCTION

Process involving coupled heat and mass transfer occur frequently in nature. It occurs not

only due to temperature difference, but also due to concentration difference or the

combination of these two. 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 mass

transfer occur simultaneously. Possible applications of this type of flow can be found in many

industries. Representative applications of interest include: solidification of binary alloy and

crystal growth, dispersion of dissolved materials or particulate water in flows, drying and

dehydration operations in chemical and food processing plants and combustion of atomized

liquid fuels. Muthucumaraswamy and Ganesan (2001) pointed out that chemically reacting

flows are classified as heterogeneous or homogeneous depending on whether they occur at an

interface or as a single phase volume. Furthermore, the presence of a foreign mass in air or

water causes some kind of chemical reaction. During a chemical reaction between two species

heat is also generated. The effect of mass transfer on flow past an impulsively started infinite

vertical plate with constant heat flux and chemical reaction were studied by Das et al. (1994).

R. Kumar and K. D. Singh


20

Anderson et al. (1994) studied the flow and mass diffusion of a chemical species with first

order and higher order reactions over a linearly stretching surface. Fan et al. (1998) studied

the mixed convective heat and mass transfer over a horizontal moving plate with a chemical

reaction effect. El-Kabeir et al. (2007) investigated heat and mass transfer on MHD flow over

a vertical isothermal cone surface with heat sources and chemical reactions.

Muthucumaraswamy and Vijayalakshmi (2008) investigated the effects of heat and mass

transfer on flow past an oscillating vertical plate with variable temperature. Ahmed (2009)

studied free and forced convective three dimensional flows with heat and mass transfer.

Mansour et al. (2008) studied the effects of chemical reaction and viscous dissipation on

MHD natural convection flows saturated in porous media. Moreover chemical reaction effects

on heat and mass transfer laminar boundary layer flow have been studied by many scholars

e.g. Chamkha (2003), Kandasamy et al. (2005), Afify (2004), Takhar et al. (2000), Raptis and

Perdikis (2006) and Anjalidevi and Kandasamy (2000) etc.

Radiative convective flows are encountered in countless industrial and environment processes

e.g. heating and cooling chambers, fossil fuel combustion energy processes, evaporation from

large open water reservoirs, astrophysical flows, solar power technology and space vehicle re-

entry. Radiative heat and mass transfer play an important role in manufacturing industries for

the design of reliable equipment. Nuclear power plants, gas turbines and various propulsion

devices for air craft, missiles, satellites and space vehicles are examples of such engineering

applications. Radiation effect on mixed convection along an isothermal vertical plate was

studied by Hossain and Takhar (1996). Raptis and Perdikis (2003) investigated the effects of

thermal radiation on a moving vertical plate in the presence of mass diffusion. Prasad and

Reddy (2008) studied the radiation effects on unsteady MHD flow with viscous dissipation.

Cookey et al. (2003) have investigated unsteady two-dimensional flow of a radiating and

chemically reacting MHD fluid with time dependent suction and Ogulu et al. (2008) extended

it for micro-polar fluid. Alam et al. (2009) investigated the effects of variable reaction,

thermophoresis and radiation on MHD free convection flows. Loganathan et al. (2008)

investigated first order chemical reaction past a semi-infinite vertical plate for optically thin

gray gas neglecting viscous dissipation.

The objective of the present paper is to investigate the effects of radiation and first order

chemical reaction on an unsteady MHD free convective heat and mass transfer flow past a hot

vertical non-conducting porous plate with constant suction normal to the plate when the plate

temperature is spanwise cosinusoidally fluctuating with time. Solution of the flow problem is

obtained by using perturbation technique by assuming the Eckert number as the perturbation

parameter.

2 MATHEMATICAL FORMULATION

We consider the flow past an infinite, hot porous plate lying vertically on ** zx plane. The

*x axis is oriented in the direction of the buoyancy force and *y axis is taken

perpendicular to the plane of the plate. A magnetic field of uniform strength 0H is introduced

normal to the plane of the plate. Let )w,v,u( *** be the components of velocity in the

)z,y,x( *** directions respectively. The plate is being considered infinite in *x direction;

hence all physical quantities will be independent of *x . Further, since the plate is subjected to

Unsteady MHD Flow Of Radiating And Reacting Fluid Past A Vertical Porous Plate


21

a constant suction velocity, i.e. Vv* , thus, following Singh and Chand (2000), *w is

independent of *z and so we assume 0*w throughout. The temperature of the plate is

considered to vary spanwise cosinusoidally fluctuating with time and assumed to be of the

form

tz

cosTTTt,zTw

00, (1)

where ** T,T 0 and *

wT are the mean, ambient temperature and wall temperature of the plate

respectively, * is the frequency, *t is the time, is the wave length and is a small

parameter i.e., 1 .

We further assume that (i) the magnetic Reynolds number is small so that the induced

magnetic field is negligible in comparison to the applied magnetic field, (ii) the fluid is

considered to be gray; absorbing – emitting radiation but a non-scattering medium, (iii) the

effects of Joule heating is negligible as small velocity usually encountered in the free

convection flows, (iv) no external electric field is applied and effect of polarization of ionized

fluid is negligible, therefore electric field is assumed to be zero, (v) there exists a first order

chemical reaction between the fluid and species concentration, (vi) the level of species

concentration is very low so that the heat generated during chemical reaction can be

neglected.

Using the Boussinesq and boundary–layer approximation, the governing equations for this

problem can be written as follows:

00

V,Vv

y

v *

*

*

, (2)

**

*

*

*

*

*

**

*

*

TTgz

u

y

u

y

uv

t

u

2

2

2

2

*e**

c uH

CCg

2

0

2

, (3)

*

*

*

*

*

*

*

*

*

*

*

**

*

*

Py

q

z

u

y

u

z

T

y

Tk

y

Tv

t

TC

22

2

2

2

2

, (4)

**

*

*

*

*

*

**

*

*

CCKz

C

y

CD

y

Cv

t

C

12

2

2

2

, (5)

where *

cPe

** q,,,,,C,k,,D,H,,C,T,g 0 and

1K are acceleration due to gravity,

fluid temperature, species concentration, magnetic permeability, magnetic field, chemical

molecular diffusivity, electrical conductivity, thermal conductivity, specific heat at constant

pressure, kinematic viscosity, density, coefficient of volume expansion for heat transfer,

volumetric coefficient of expansion with species concentration, radiative heat flux and

chemical reaction parameter.



22

The boundary conditions of the problem are:

yasCC,TT,u

yatCC,tz

cos)TT(TT,u

*****

*****

*****

0

00 000

. (6)

For the case of an optically thin gray gas the local radiant absorption is expressed by

44

4 ****

*

*

TTay

q

, (7)

where *a is the mean absorption coefficient and * is Stefan – Boltzmann constant.

We assume that the temperature differences with in the flow are sufficiently small such that 4*T may be expressed as a linear function of the temperature. This is accomplished by

expanding 4*T in a Taylor series about *T and neglecting higher-order terms, thus

434

34 *** TTTT

(8)

By using equations (6) and (7), equation (4) reduces to

22

2

2

2

2

*

*

*

*

*

*

*

*

*

**

*

*

Pz

u

y

u

z

T

y

Tk

y

Tv

t

TC ***** TTTa

3

16 (9)

Introducing the following non-dimensional parameters

,,,***

V

uu

zz

yy

,**

0

**

TT

TT (Dimensionless temperature)

,**

0

**

CC

CCC (Dimensionless concentration)

,**tt (Dimensionless time)

**

P TTC

VEc

0

2

(Eckert number),

k

CPr

p (Prandtl number),



23

VRe (Reynolds number),

3

0

V

TTgGr

**

(Grashoff number),

3

0

V

)CC(gGm

**

c

(Modified Grashoff number),

2*

, (Dimensionless frequency of oscillation)

DSc

(Schmidt number),

22

0

2

2 HM e (Hartmann number),

1

2 K (Chemical reaction parameter),

2

2 3

16

Vk

TaR

**

(Radiation parameter)

into the equations (3), (5) and (9), we get

uMGmCGrRez

u

y

u

y

uRe

t

u 22

2

2

2

2

, (10)

Pr

RRe

z

u

y

uEc

zyPryRe

t

222

2

2

2

21

, (11)

Cz

C

y

C

Scy

CRe

t

C

2

2

2

21

. (12)

The boundary conditions (6) reduce to

yasC,,u

yatC,tzcos,u

000

0110

(13)



24

3 SOLUTION OF THE PROBLEM

Assume

tzie)y(f)y(f)t,z,y(f

10 , (14)

where f stands for u, and C . Then substituting equation (14) into equations (10) to (12) and

equating the like powers of, we get

Zeroth-order equations

00

2

0

20

2

0

2

GmCGrReuMyd

udRe

yd

ud , (15a)

0

2

0

0

20

2

0

2

yd

duPrEcRRe

yd

dPrRe

yd

d

, (15b)

00

0

2

0

2

ScCyd

dCScRe

yd

Cd . (15c)

The corresponding boundary conditions reduce to

yasC,,u

yatC,,u

000

0110

000

000

(16)

First-order equations

11

2

1

221

2

1

2

GmCGrReuiMyd

udRe

yd

ud , (17a)

yd

du

yd

duEcPriRRe

yd

dPrRe

yd

d 10

1

221

2

1

2

2

, (17b)

01

21

2

1

2

CSciScyd

dCScRe

yd

Cd . (17c)

The corresponding boundary conditions reduce to

yasC,,u

yatC,,u

000

0010

111

111

(18)



25

The equations (15c) and (17c) are ordinary second order differential equations and solved

under the boundary conditions given in (16) and (18), respectively. Hence the expressions of

)y(C0 and )y(C1 are given by

y

e)y(C 1

0

, (19a)

01 )y(C . (19b)

Since equations (15a), (15b), (17a) and (17b) are coupled differential equations, therefore

approximate solution is obtained by perturbation technique for small values of Ec as the

Eckert number is small for incompressible fluid flows. Hence assuming

2

11101

2

01000

EcOEcFFF

EcOEcFFF, (20)

where F stands for u and .

Using (20) in (15a), (15b), (17a) and (17b) and equating like powers of Ec, we obtain:

Zeroth-order equations

000

2

00

2

0000 GmCGrReuMuReu , (21a)

10

2

10

22

1010 GrReuiMuReu , (21b)

000

2

0000 RRePrRe , (21c)

010

22

1010 PriRRePrRe . (21d)

The corresponding boundary conditions are

yas,u,,u

yat,u,,u

0000

01010

10100000

10100000

(22)

First - order equations

01

2

01

2

0101 GrReuMuReu , (23a)

11

2

11

22

1111 GrReuiMuReu , (23b)

2

0001

2

0101 uPrRRePrRe , (23c)

100011

22

1111 2 uu)PriR(RePrRe , (23d)



26

where prime denotes differentiation with respect to y.

The corresponding boundary conditions are

yas,u,,u

yat,u,,u

0000

00000

11110101

11110101

(24)

Solving (21a) to (21d) and (23a) to (23d) under the boundary conditions (22) and (24)

respectively, we get

y

e 2

00

, (25)

y

e 3

10

, (26)

yyyyeeAeeAu 1424

2100

, (27)

yyeeAu 35

310

, (28)

2 1 2 1 4 2 1 42

01 1 4 7 5 6 8 9

y y y y y y y yB A e A e e A e A e A e A e e , (29)

3 52 1 4 2 1 4

11 2 10 11 12 13 14 15

y yy y y y y yB A e A e A e e A e A e A e e , (30)

yyyy

yyyy

eAeeAeAA

eeAeAeABu

1212

4124

2

18201716

222119301

, (31)

yyyy

yyyy

eeAeAeAA

eeAeAeABu

3412

5412

26252423

292827411

. (32)

The solutions obtained in equations (25) to (32) are in complex variable notations and only

the real part of it will have the physical significance. From equation (14) we get the

expressions for velocity and temperature profiles as

)tz(sinf)tz(cosf)y(f)t,z,y(f ir 0 , (33)

where

ir fiff 1 .

Hence we can obtain the expressions for the transient velocity and transient temperature

profiles from (33) for 0z and 2

t as



27

if)y(f,,yf

0

20

. (34)

Skin-friction: Knowing the velocity field, the expression for the skin-friction coefficient at

the plate in the *x direction is given by:

0

yy

u . (35)

Nusselt number: From the temperature field, the rate of heat transfer coefficient in terms of

the Nusselt number Nu at the plate is given by:

0

yy

Nu

. (36)

4 PARTICULAR CASES

(1) Our results are similar to the results of Singh and Chand (2000) in the absence of mass

transfer, chemical reaction, viscous dissipation and radiation effect.

(2) In the absence of viscous dissipation, chemical reaction and radiation effect for mixed

convective MHD flow with constant wall temperature past an accelerated infinite

vertical porous plate our results reduce to the results of Reddy et al. (2009).

(3) In the absence of radiation effects and in the presence of heat generation/absorption

for an ordinary medium our results are found in good agreement with Singh and

Kumar (2010).

5 RESULTS AND DISCUSSION

The numerical values of the transient velocity, transient temperature, coefficient of skin-

friction and Nusselt number are computed for different parameters like modified Grashoff

number, Grashoff number, Hartmann number, Reynolds number, Prandtl number, Eckert

number, Schmidt number, frequency parameter, chemical reaction parameter and radiation

parameter. To be realistic, the values of Prandtl number Pr , are chosen 710.Pr and

007.Pr to represent air and water respectively. The values of Schmidt number are taken for

Oxygen ( 600.Sc ) and Ammonia ( 780.Sc ).



28

Figure 1: Transient velocity for 2

20

t,. , 20. and 0z .

The transient velocity profiles and transient temperature profiles are shown in Figure 1 and

Figure 2 respectively. It is observed from these figures that the velocity and temperature

profiles decrease with the increase of Hartmann number, Reynolds number, Prandtl number,

Schmidt number and radiation parameter, however, modified Grashoff number, Grashoff

number, Eckert number and frequency parameter have opposite effect on velocity and

temperature profiles.

Figure 2: Transient temperature for 2

20

t,. 20. and 0z .

Gm Gr M Re Pr Ec Sc R

5 5 1 2 0.71 0.01 0.60 5 2 I

10 5 1 2 0.71 0.01 0.60 5 2 II

5 10 1 2 0.71 0.01 0.60 5 2 III

5 5 2 2 0.71 0.01 0.60 5 2 IV

5 5 1 4 0.71 0.01 0.60 5 2 V

5 5 1 2 7.00 0.01 0.60 5 2 VI

5 5 1 2 0.71 0.02 0.60 5 2 VII

5 5 1 2 0.71 0.01 0.78 5 2 VIII

5 5 1 2 0.71 0.01 0.60 10 2 IX

5 5 1 2 0.71 0.01 0.60 5 5 X

y

t,z,yu

Gm Gr M Re Pr Ec Sc R

5 5 1 2 0.71 0.01 0.60 5 2 I

10 5 1 2 0.71 0.01 0.60 5 2 II

5 10 1 2 0.71 0.01 0.60 5 2 III

5 5 2 2 0.71 0.01 0.60 5 2 IV

5 5 1 4 0.71 0.01 0.60 5 2 V

5 5 1 2 7.00 0.01 0.60 5 2 VI

5 5 1 2 0.71 0.02 0.60 5 2 VII

5 5 1 2 0.71 0.01 0.78 5 2 VIII

5 5 1 2 0.71 0.01 0.60 10 2 IX

5 5 1 2 0.71 0.01 0.60 5 5 X

t,z,y

y



29

Figure 3: Effect of chemical reactions on transient velocity for ,.202

t and 0z .

Figure 3 and Figure 4 show the effect of chemical reaction on velocity and temperature

profiles respectively. It is clear from these Figures that velocity and temperature decrease for

destructive chemical reactions ( 0 ) and increase for generative chemical reactions ( 0 ).

Figure 4: Effect of chemical reactions on transient temperature for ,.202

t and 0z .

The values of skin-friction coefficient and Nusselt number at the plate are shown in Table 1. It

is clear from Table 1 that both coefficient of skin-friction increases and Nusselt number

Curve I 20.

Curve II 0

Curve III 20.

y

t,z,yu

25

600010710

2155

R,

,.Sc,.Ec,.Pr

,Re,M,Gr,Gm

Curve I 20.

Curve II 0

Curve III 20.

y

t,z,y

25

600010710

2155

R,

,.Sc,.Ec,.Pr

,Re,M,Gr,Gm



30

decreases with the increase of modified Grashoff number, Grashoff number, Eckert number

and frequency parameter.

Table 1: Values of Skin-friction coefficient and Nusselt number Nu at the plate when

220

t,. and 0z .

Gm Gr M Re Pr Ec Sc R Nu

5 5 1 2 0.71 0.01 0.60 5 0.2 2 16.853 3.3736

10 5 1 2 0.71 0.01 0.60 5 0.2 2 28.812 3.0267

5 10 1 2 0.71 0.01 0.60 5 0.2 2 22.096 3.2740

5 5 2 2 0.71 0.01 0.60 5 0.2 2 12.116 3.4678

5 5 1 4 0.71 0.01 0.60 5 0.2 2 41.087 6.6543

5 5 1 2 7.00 0.01 0.60 5 0.2 2 13.382 12.901

5 5 1 2 0.71 0.02 0.60 5 0.2 2 16.964 3.2035

5 5 1 2 0.71 0.01 0.78 5 0.2 2 14.773 3.4207

5 5 1 2 0.71 0.01 0.60 10 0.2 2 16.909 3.2919

5 5 1 2 0.71 0.01 0.60 5 0 2 17.540 3.3562

5 5 1 2 0.71 0.01 0.60 5 -0.2 2 18.464 3.3312

5 5 1 2 0.71 0.01 0.60 5 0.2 5 15.358 5.0514

However, Hartmann number, Prandtl number, Schmidt number and radiation parameter have

opposite effect on coefficient of skin-friction and Nusselt number. It is interesting to note that

both coefficient of skin-friction increases and Nusselt number increase with the increase of

Reynolds number. It is also observed from Table 1 that Nusselt number increases and skin-

friction coefficient decreases for destructive chemical reactions ( 0 ) and Nusselt number

decreases and skin-friction coefficient increases for generative chemical reactions ( 0 ).

6 SUMMARY

The governing equations for unsteady MHD free convective heat and mass transfer flow of

thermally radiating and chemically reacting fluid past a vertical porous plate with viscous

dissipation was formulated. The plate temperature is taken spanwise cosinusoidally

fluctuating with time. The resulting partial differential equations were transformed into a set

of ordinary differential equations using two-term series and solved in closed-form. The

conclusions of the study are as follows:

1) The velocity, temperature, skin-friction coefficient decrease and Nusselt number increases

with increasing radiation parameter.

2) The temperature decreases tremendously with increasing Prandtl number as compared to

Hartmann number.

3) The velocity increases tremendously with increasing modified Grashoff number as

compared to Grashoff number.



31

4) Velocity, temperature, skin-friction coefficient increase and Nusselt number decreases

with increasing values of Grashoff number or modified Grashoff number.

5) Velocity, temperature, skin-friction coefficient increase and Nusselt number decreases for

generative chemical reactions ( 0 ). However, destructive chemical reactions ( 0 )

have opposite effect on velocity, temperature, skin-friction coefficient and Nusselt

number.

ACKNOWLEDGEMENT

The authors are thankful to the learned referees for their valuable suggestions towards the

definite improvement of the paper.

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33

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APPENDIX

PrRe1 ,

RRe 2

2 ,

iM 22

3 ,

PriRRe 22

4 ,

2

422

1

ScScReScRe

,

2

4 2

2

11

2

,

2

4 4

2

11

3

,

2

4 22

4

MReRe ,

2

4 3

2

5

ReRe,

2

2

2

2

2

1MRe

GrReA

,

2

1

2

1

2

2MRe

GmReA

,

33

2

3

2

3

Re

GrReA ,

221

2

2

2

2

2

14

24

APrA ,



34

211

2

1

2

1

2

25

24

APrA ,

241

2

4

2

4

2

216

24

AAPrA ,

2211

2

21

21217

2

AAPrA ,

2421

2

42

422118

2

AAAPrA ,

2411

2

41

412129

2

AAAPrA ,

4321

2

32

3213

10

2

AAA ,

4311

2

31

3123

11

2

AAA ,

4431

2

43

43213

12

2

AAAA ,

4521

2

52

5213

13

2

AAA ,

4511

2

51

5123

14

2

AAA ,

4541

2

54

54213

15

2

AAAA ,

2

2

2

2

1

2

16MRe

BGrReA

,

2

2

2

2

4

2

1724 MRe

AGrReA

,

2

1

2

1

5

2

1824 MRe

AGrReA

,



35

2

4

2

4

6

2

1924 MRe

AGrReA

,

2

21

2

21

7

2

20MRe

AGrReA

,

2

42

2

42

8

2

21MRe

AGrReA

,

2

41

2

41

9

2

22MRe

AGrReA

,

33

2

3

2

2

23

Re

BGrReA ,

33232

10

2

24

Re

AGrReA ,

33131

11

2

25

Re

AGrReA ,

34343

12

2

26

Re

AGrReA ,

35252

13

2

27

Re

AGrReA ,

35151

14

2

28

Re

AGrReA ,

35454

15

2

29

Re

AGrReA ,

9876541 AAAAAAB ,

1514131211102 AAAAAAB ,

222120191817163 AAAAAAAB ,

292827262524234 AAAAAAAB .

rakesh ijamm 2

Documents

du drey du

spanwise cosinusoidally fluctuating

reacting fluid past

destructive chemical reactions

destructive chemical reactions

generative chemical reactions

nusselt number nu

unsteady mhd flow