Effect of chemical reaction on an unsteady MHD free convective flow

Proceedings of ICFM 2015 International Conference on Frontiers in Mathematics 2015

March 26-28, 2015, Gauhati University, Guwahati, Assam, India Available online at http://www.gauhati.ac.in/ICFMGU

ISBN: 978-81-928118-9-5 204

Effect of chemical reaction on an unsteady MHD free convective flow past a porous plate with ramped temperature

S. SINHA

Department of Mathematics, Assam Down Town University, Guwahati-781026 [email protected]

Abstract: A parametric study to investigate the effect of first order chemical reaction on an unsteady MHD free convective flow of an incompressible viscous, incompressible, electrically conducting and heat absorbing fluid past an infinite vertical porous plate with ramped wall temperature in presence of uniform transverse magnetic field is presented. The Magnetic Reynolds number is considered small enough to neglect the induced hydro magnetic effects. The equations governing the flow are solved by adopting Laplace Transform technique in closed form. Detailed computations of the influence of chemical reaction on the variations in the fluid velocity, fluid temperature, fluid concentration, and skin friction, Nusselt number and Sherwood number at the plate are demonstrated graphically for various values of the physical parameters involved in the problem and the results are physically interpreted. The investigation reveals the fact that the chemical reaction has significant effect in controlling the flow. Keywords: Chemical reaction, porous medium, ramped temperature, Sherwood number I. INTRODUCTION MHD is concerned with the study of the interaction of magnetic fields and electrically conducting fluids in motion. There are numerous examples of applications of MHD principles including MHD generators, MHD pumps and MHD flow meters etc. MHD principles find its applications in Medicine and Biology also. The present form of MHD is due to the pioneer contributions of several notable authors like Cowling, 1957; Shercliff, 1965; Ferraro and Plumpton, 1966 and Crammer and Pai, 1978. Radiative convective flows are encountered in countless industrial and environment processes like heating and cooling chambers, evaporation from large open water reservoirs, astrophysical flows and solar power technology. Due to importance of the above physical aspects, several authors have carried out model studies on the problems of free convective flows of incompressible viscous fluid under different flow geometries taking into account of the thermal radiation. Some of them are Mansor, 1990; Raptis and Perdikis, 1999; Ganesan and Logonathan, 2002; Mbeledogue et al., 2007; Makinde, 2005 and Sattar and Kalim, 1996. Investigation of problems on natural convective radiating flow of electrically conducting fluid past an infinite plate becomes very interesting and fruitful when a magnetic field is applied normal to the plate. Comprehensive literature on various aspects of free convective radiative MHD flows and its applications can be found in Sattar and Maleque, 2000; Samad and Rahman, 2006; Prasad et al., 2006; Takhar et al., 1996; Ahmed and Sarmah, 2009 and Ahmed, 2012. The analysis of an unsteady MHD free convective flow of a viscous incompressible, electrically conducting fluid past an infinite vertical porous plate subjected to a constant suction and heat sink was made by Sahoo et al., 2003. In many times, it has been observed that the foreign mass reacts with the fluid and in such a situation chemical reaction plays an important role in chemical industry. The study of effect of chemical reaction on heat and mass transfer in a flow is of

great practical importance to the Engineers and Scientists because of its almost universal occurrence in many branches of Science and technology. In processes such as drying, distribution of temperature and moisture over agricultural fields, energy transfer in a wet cooling tower and flow in a cooler heat and mass transfer occur simultaneously. Possible applications of this type of flow can be found in many industries. Many investigators have studied the effect of chemical reaction in different convective heat and mass transfer flows of whom Apelblat, 1982 and Anderson et al., 1994 are worth mentioning. Chambre and Young, 1958 have presented a first order chemical reaction in the neighbourhood of a horizontal plate. Muthucumaraswamy, 2002 presented heat and mass transfer effects on a continuously moving isothermal vertical surface with uniform suction by taking in to account the homogeneous chemical reaction of first order. Muthucumaraswamy and Meenakshisundaram, 2006 investigated theoretical study of chemical reaction effects on vertical oscillating plate with variable temperature and mass diffusion. In view of the practical importance of the above fields, an attempt has been made in the present work to study the problem of a transient MHD free convective flow of an incompressible viscous electrically conducting chemically reacting, thermal radiating fluid past a vertical porous plate with ramped wall temperature. Here our main objective is to study the effect of chemical reaction on the flow characteristics. However the effects of the other physical parameters will also be taken into account to some extent in the preview of our investigation. This work is a generalization to the work done by Nandkeolyar et al., 2013 to consider the effect of chemical reaction. In absence of chemical reaction, the solutions and the results obtained in the present work are consistent with those of Nandkeolyar et al., 2013.

II. MATHEMATICAL FORMULATION OF THE PROBLEM We now consider an unsteady radiative MHD free convective flow of an incompressible viscous and electrically conducting heat absorbing fluid past vertical porous plate in its own place with temporary ramped temperature in presence of a magnetic field of uniform strength

applied normal to the plate directed

into the fluid region. Our investigation is restricted to the following assumptions:

i) All the fluid properties are considered constants except the influence of the variation in density in the buoyancy force term.

ii) The viscous and Ohmic dissipations of energy are negligible.

iii) The magnetic Reynolds number is so small for that the induced magnetic field can be neglected in comparison to the applied magnetic field.

iv) The plate is electrically non -conducting. v) The radiation heat flux in the direction of the plate

velocity is considered negligible in comparison to that in the normal direction.

B0



ISBN: 978-81-928118-9-5 205

vi) No external electric field is applied for which the

polarization voltage is negligible leading to Initially the plate and the surrounding fluid were at rest at the same temperature and concentration . At time ,

the plate begins to move with a uniform velocity along its own plane along the x-axis and the temperature of the plate is raised to when and the

constant temperature and concentration

is maintained at .

We now introduce a coordinate system with -

axis along the plate in the upward vertical direction, -axis normal to the plate directed into the fluid region and -axis

along the width of plate. Let denote the fluid

velocity and be the applied magnetic field in the

fluid. Under the foregoing assumptions, the equations that describe the physical situation are given by:

, which yields (1)

(2)

(3)

(4)

The appropriate initial and boundary conditions are:

(5)

(6)

(7)

(8)

(9) Proceeding with the analysis, we introduce the following non-dimensional variables and similarity parameters to normalize the flow model:

u = !uU 0

, η =U0 !yυ, t = !t

t 0,Q1 =

υQ0ρCpU 0

2, M =

σB02υ

ρU02,

,

(10)

All the physical quantities are defined in the Nomenclature. By virtue of transformations cum definitions (10), the equations (2), (3) and (4) in normalized form become,

(11)

(12)

(13)

Subject to relevant initial and boundary conditions: (14)

(15)

(16) (17)

(18) III. METHOD OF SOLUTION Taking Laplace Transform of the equations (11), (12) and (13), the combined initial and boundary value problem reduce to a boundary value problem governed by the equations:

(19)

(20)

(21)

Subject to the boundary conditions

(22)

(23) The equations (19)-(21) are three ordinary coupled second order differential equations and the solutions of these equations subject to the conditions (22) and (23) are as follows:

(24)

(25)

E!"= 0!"

!T∞ !C

∞!t = 0

U0

!T∞+ !Tw − !T

∞( ) !t / t 0 !t < t 0

!Tw !Tw > !T∞( )

!Cw !Cw > !C∞( ) !t > t 0

!x , !y , !z( ) X

YZ

q!= !u ,o,o( )

B!"= 0,B0 ,0( )

∂ "u∂ "x

= 0 !u = !u !y , !t( )∂ "u∂ "t

= υ∂2 "u∂ "y 2

+ gβ "T − T∞"( )

+ gβ* "C − C∞"( )− σB0

2

ρ"u − υ "u

"K1∂ "T∂ "t

=KT

ρC p

∂2 "T∂ "y 2

−Q0ρC p

"T − "T∞( )

∂ "C∂ "t

= DM∂2 "C∂ "y 2

− K "r "C − "C∞( )

!u = 0, !T = !T∞, !C = !C

∞∀ !y ≥ 0, !t ≤ 0

!u = U0 , !C = !Cw at !y = 0 for !t > 0

!T = !T∞+ !Tw − !T

∞( ) !t / t 0 at !y = 0 for 0 < !t ≤ t0

!T = !Tw at !y = 0 for !t > t0!u → 0, !T → !T

∞!C → !C

∞at !y →∞, !t > 0

Gr =υg β #Tw − #T

∞( )U 0

3,Gm =

υg β* #Cw − #C∞( )

U03

,

θ =#T − #T

∞

#Tw − #T∞

, φ =#C − #C

∞

#Cw − #C∞

, Pr =µCp

KT

,

Kr = K !r υU02,K1 =

!K1U 02

υ2, Sc = υ

DM

, t 0=υ

U02

∂u∂ t

=∂2u∂η2

+Gr θ +Gm φ − Mu − uK1

∂θ∂ t=1Pr

∂2θ

∂η2−Q1θ

∂φ∂ t

=1Sc

∂2φ

∂η2− Kr φ

u = 0, θ = 0, φ = 0 ∀η≥ 0 and t ≤ 0u = 1, φ = 1 at η = 0 for t > 0θ = t at η = 0 for 0 ≤ t ≤ 1θ = 1 at η= 0 for t > 1u = 0, θ = 0, φ = 0 as η→∞, t > 0

d2udη2

− s+N( )u = − Grθ −Gmφ

d2θdη2

− s + Q1( )Pr θ = 0

d2φdη2

− Sc s + Kr( ) φ = 0

u = 1s, θ = 1

s21− e−s( ) , φ = 1s at η = 0

u = 0, θ = 0, φ = 0 at η→∞

θ =1s21− e−s( )e− Pr s+Q1( )η

φ =1se− Sc s+Kr( ) η



ISBN: 978-81-928118-9-5 206

(26)

Taking inverse Laplace Transforms of the equations (24), (25) and (26) we have:

(27)

(28)

(29)

IV. COEFFICIENT OF SKIN FRICTION The viscous drag at the plate per unit area in the direction of the plate velocity is given by the Newton’s law of viscosity in the form:

The coefficient of skin friction at the plate is given by

(30)

V. COEFFICIENT OF RATE OF HEAT TRANSFER

The heat flux from the plate to the fluid is given by the Fourier law of conduction in the form

(31)

The co-efficient of the rate of heat transfer from the plate to the fluid in terms of Nusselt number is given by

(32)

VI. COEFFICIENT OF RATE OF MASS TRANSFER (33)

VII. RESULTS AND DISCUSSION In order to get clear insight of the physical problem, numerical computations from the analytical solutions for the representative velocity field, temperature field, concentration field, the coefficient of skin friction, the coefficient of rate of heat transfer in terms of Nusselt number and the coefficient of rate of mass transfer in terms of Sherwood number have been carried out by

Figure 1: Velocity versus η for Kr=1, Gr=5, Gm=5, Pr=.71, K=1, Sc=.60, t=.8, Q1 =1

Figure 2: Velocity versus η for M=1, Gr=5, Gm=5, Pr=.71, K=1, Sc=.60, t=.8, Q1 =1

Figure 3: Velocity versus η for Kr=1, Gr=5, Gm=5, Pr=.71, K=1, Sc=.60, M=1, Q1 =1

u =

1s−GE2

1s+E

−1s+Es2

"

#$

%

&' 1− e−s( )

−HF1s−1s+ F

"

#$

%

&'

(

)

*****

+

,

-----

e− s+N( )η

+GE2

1s+E

−1s+Es2

"

#$

%

&' 1− e−s( )e− Pr s+Q1( )η

+HF1s−1s+ F

"

#$

%

&'e

− Sc s+Kr( )η

θ = f1 − f2φ = φ1

u = ψ1 − Je−E ψ2 −ψ3( )− ψ1 −ψ4( )+E #φ1 − #φ2( )

%

&

''

(

)

**

−K ψ1 − e−Fψ5( )

+ Je−E ψ6 −ψ7( )− ψ8 −ψ9( )+E f1 − f2( )

%

&

''

(

)

**

+K φ1 − e−Fψ10( )

!τ = µ∂ !u∂ !η

%

&'!η=0

=µt0

∂u∂η

%

&''η=0

τ ="τ

µt0

=∂u∂η

%

&'η=0

= τ1 − J

e−E τ2 − τ3( )− τ1 − τ4( )+E τ5 − τ6( )

)

*

++++

%

&

''''

−K τ1 − e−Fτ7( )

+ J

e−E τ8 − τ9( )− τ10 − τ11( )+E τ12 − τ13( )

)

*

++++

%

&

''''

+K τ14 − e−Fτ15( )

q∗

q∗ = −KT∂ $T∂ $η

&

'($η=0

= −KT

U0t0"Tw − "T

∞( ) ∂θ∂η

'

()η=0

Nu =q∗U0t0

KT "Tw − "T∞( )= −

∂θ∂η

(

)*η=0

= τ13 − τ12

Sh = − τ14



ISBN: 978-81-928118-9-5 207

assigning some arbitrarily chosen specific values to the physical parameters involved in the problem. Throughout our investigation, the value of the Pr have been kept fixed at 0.71, both the values of Gr and Gm are fixed at 5 as the numerical computations are concerned. We recall that Pr=0.71 corresponds physically to air. The numerical results computed from the analytical solutions of the problem have been displayed in figure 1-9. The velocity profiles under the influence of Hartmann number M, Chemical reaction parameter Kr and time t versus are exhibited in figures 1-3. It is inferred from figure 1 that an increase in Magnetic parameter M has an inhibiting effect on the fluid velocity. The fluid velocity is continuously reduced with increasing M. In other words the imposition of the transverse magnetic field tends to retard the fluid flow. This phenomenon

Figure 4: Temperature versus η for Pr=.71, t=0.8

Figure 5: Temperature versus η for Pr=.71, t=1.2

Figure 6: Concentration versus η for Sc=0.60, t=.8 has an excellent agreement with the physical fact that the Lorentz force generated in the present flow model due to interaction of the transverse magnetic field and the fluid velocity acts as a resistive force to the fluid flow which serves to decelerate the flow. It is further revealed from the figure that the fluid velocity

first increases in a thin layer adjacent to the plate and there after it decreases asymptotically as we move away from the plate indicating the fact that the buoyancy force has a significant role on the flow near the plate and its effect is nullified in the free stream. Figure 2 presents how the fluid velocity is affected by chemical reaction. The figure indicates that the fluid motion is retarded on account of chemical reaction. It is observed that the fluid velocity quickly increases up to some thin layer of the fluid adjacent to the plate and then decreases asymptotically towards

as . i.e in the free stream effect. It is also inferred that the buoyancy effects (due to concentration and temperature difference) are significant near the hot plate. Figure 3 simulates

Figure 7: Skin friction versus t for M=1, Gr=5, Gm=5, Pr=.71, K=1, Sc=.60, 1Q =1

Figure 8: Nusselt number versus t for Q1 =1

Figure 9: Sherwood number versus t for Sc=0.60

the effect of time t on velocity profile versus . It is observed from this figure that as time progresses the fluid velocity increases near the plate and as we move further away from the plate it decreases asymptotically.

ηu = 0 η→∞

η



ISBN: 978-81-928118-9-5 208

It is inferred from Fig. 4 & 5 that, for ramped temperature and isothermal plates, an increase in heat source parameter has

an inhibiting effect on fluid temperature. Increasing shows

an opposing influence on indicating the fact that the fluid temperature falls steadily for high radiation. Figure 6 presents the variation of the species concentration under the influence of chemical reaction. The figure predicts that the species concentration decreases to zero as . This phenomenon physically states that the consumption of chemical species leads to fall in the species concentration. This clearly agrees with the physical laws. Figure 7 shows the effect of the Chemical reaction parameter Kr and time t on the skin friction at the plate in the direction of the plate velocity. Both the figures indicate that the magnitude of shear stress at the plate is considerably increased with the increase in Kr and time t. The effect of Prandtl number Pr on the co-efficient of rate of heat transfer in terms of Nusselt number versus time t is shown in Fig. 8. It is seen from this figure that heat flux raises for low thermal conductivity. The effect of chemical reaction parameter Kr on the co-efficient of rate of mass transfer in terms of Sherwood number has been displayed in Fig. 9. It is observed from this figure that the mass flux is constantly increased for increasing the chemical reaction parameter Kr. This simulates that the high consumption (chemical reaction) leads the substantial rise in the mass transfer rate. VIII. CONCLUSIONS Our investigation leads to the following conclusions:

i) The fluid motion is retarded under the application of transverse magnetic field as well as chemical reaction.

ii) The concentration level of the fluid falls due to increasing chemical reaction. i.e the consumption of chemical species leads to fall in the species concentration field.

iii) Magnitude of shear stress at the plate is considerably increased due to the application of chemical reaction.

iv) Mass flux increases with the increasing values of chemical reaction. i.e high consumption (chemical reaction) leads the substantial rise in the mass transfer rate.

NOMENCLATURE

Symbol Description

Magnetic induction vector

Strength of the applied magnetic field

Concentration

Specific heat at constant pressure

Concentration at the plate

Concentration far away from the plate

Mass diffusivity

Acceleration due to gravity Grashof number for heat transfer Grashof number for mass transfer

Porosity parameter

Thermal conductivity

Kr Chemical reaction parameter

M Hartmann number Nu Nusselt number Pr Prandtl number

Fluid velocity vector

Q1

Heat absorption parameter

Sc Schmidt number

Sh Sherwood number

Non-dimensional time

Time

Characteristic time

Fluid temperature

Reference temperature

Temperature far away from the plate

u Non-dimensional fluid velocity X component of

Free stream velocity

Non dimensional normal coordinate Fluid density

Coefficient of Skin friction

Thermal diffusivity Electrical conductivity

Volumetric co-efficient of expansion with concentration

Non-dimensional concentration Volumetric co-efficient of thermal

expansion Kinematic viscosity Non-dimensional temperature

APPENDIX

,

,

,

Q1Q1

θ

φ

η→∞

!BB0!CCp

!Cw!C∞

DMgGrGmK1KT

!q

t!tt0!T!Tw!T∞

!u!q

U0

ηρτα

σ

β*

φ

β

υθ

f1 = f Pr, Q1, η, t( ) f2 = f Pr, Q1, η, t −1( )H t −1( ),φ1 = φ Sc,Kr ,η ,t( )ψ1 = ψ N ,η, t( ) , ψ4 = ψ N ,η , t −1( )H t −1( )ψ2 = ψ N −E,η, t( )!φ1 = φ N ,η, t( ) , !φ2 = !φ N ,η, t −1( )H t −1( )ψ5 = ψ N − F,η, t( )ψ6 = φ Pr,Q1 −E,η , t( ),ψ7 = φ Pr,Q1 −E,η , t −1( )H t −1( )

ψ8 = φ Pr,Q1,η , t( ), ψ9 = φPr,Q1,

η , t −1

%

&''

(

)**H t −1( ),

ψ10 = φ Sc,Kr − F,η , t( )τ1 = τ N,t( ), τ2 = τ N −E,t( ),τ1 = τ N −E,t −1( )H t −1( ),τ4 = τ N,t −1( )H t −1( ),



ISBN: 978-81-928118-9-5 209

,

,

,

REFERENCES [1] Ahmed, N., “Soret and Radiation effects on transient

MHD free convection from an impulsively started infinite vertical plate”, Journal of Heat Transfer, 134/062701-1-9, DOI: 10. 1115/1.4005749 (2012).

[2] Ahmed, N. and H.K. Sarmah, “Thermal radiation effect on a transient MHD flow with mass transfer past an impulsively fixed infinite vertical plate”, Int. J. of Appl. Math and Mech., 5(5), 87-98 (2009).

[3] Anderson, H.I., O.R. Hansen and B. Holmedal, “Diffusion of a chemically reactive species from a stretching sheet”, International Journal of Heat and Mass Transfer, 37, 659-664 (1994).

[4] Apelblat, A., “Mass transfer with a chemical reaction of the first order, effect of axial diffusion”, The chemical Engineering Journal, 23, 193-203 (1982).

[5] Chambre, P.L. and J.D. Young, “On the diffusion of a chemically reacting species in a laminar boundary layer flow”, Phys. Fluids Flow, 1, 48-54 (1958).

[6] Cowling, T.G., Magnetohydrodynamics, Wiley Inter Science, New York (1957).

[7] Crammer, K.P. and S.L. Pai, Magneto Fluid Dynamics for Engineers and Applied Physics, Mc. Craw Hill book Co., New York (1978).

[8] Ferraro, V.C.A. and C. Plumpton, An Introduction to Magnetic Fluid Mechanics, Clarandon Press, Oxford (1966).

[9] Ganesan, P. and P. Loganathan, “Radiation and mass transfer effects on flow of an incompressible viscous fluid past a moving vertical cylinder”, Int. J. Heat and Mass Transfer, 45, 4281-4288 (2002).

[10] Mansour, M.A., Astrophysics and Science, 166, 269-275, DOI: 10.1007/ BF01094898 (1990).

[11] Makinde, O.D., “Free convection flow with thermal radiation and mass transfer past a moving vertical porous plate”, Int. Communications in Heat and Mass Transfer, 32, 1411-1419 (2005).

[12] Mbeledogu, I.U., A.R.C. Amakiri, and A. Ogulu, “Unsteady MHD free convective flow of a compressible fluid past a moving vertical plate in the presence of radiative heat transfer”, Int. J. Heat and Mass Transfer, 50, 1668- 1674 (2007).

[13] Muthukumaraswami, R., “Effect of a chemical reaction on a moving isothermal surface with suction”, Acta Mechanica, 155, 65-72 (2002).

[14] Muthecumaraswamy, R. and S. Meenakshisundaram, “Theoretical study of chemical reaction effects on vertical oscillating plate with variable temperature”, Theoret. Appl. Mech., 339, 245-257 (2006).

[15] Prasad, V.R., N.B. Reddy and R. Muthucumaraswamy, “Transient radiative hydromagnetic free convection flow past an impulsively started vertical plate with uniform heat and mass flux”, Theoret. Appl. Mech., 33, 31-63 (2006).

[16] Raptis, A. and C. Perdikis, “Radiation and free convection flow past a moving plate”, Applied Mechanics and Engineering, 4, 817-821 (1999).

τ5 =β N,t( ), τ6 =β N,t −1( )H t −1( ),τ7 = τ N − F, t( ),τ8 = ϕ Pr,Q1 −E,t( )

τ9 = ϕ Pr,Q1 −E,t( )H t −1( )τ10 = ϕ Pr,Q1, t( ),τ11 = ϕ Pr,Q1, t −1( )H t −1( ),

τ12 =β1 Pr,Q1, t( ),τ13 =β1 Pr,Q1, t −1( )H t −1( ),

φ4 = ϑ M1 −K1, t( )

τ14 = ϕ Sc,Kr, t( ), τ15 = ϕ Sc,Kr − F, t( ),

φ ξ,ς,η,t( ) = 12

e ξςηerfc η ξ

2 t+ ςt

%

&''

(

)**

+e− ξςηerfc η ξ

2 t− ςt

%

&''

(

)**

,

-

.

.

.

.

.

.

/

0

111111

f x ,ξ,η,t( ) = t2+xη

4 ξ

#

$%%

&

'((e

ξzerfc xη

2 t+ ξt

#

$%

&

'(

+t2−xη

4 ξ

#

$%%

&

'((e

− ξzerfc xη

2 t− ξt

#

$%

&

'(

ψ ξ,η,t( ) = 12

e ξηerfc η

2 t+ ξt

$

%&

'

()

+e− ξηerfc η

2 t− ξt

$

%&

'

()

+

,

-----

.

/

00000

!φ ξ,η,t( ) = t2+

η

4 ξ

%

&''

(

)**e

ξzerfc η

2 t+ ξt

%

&'

(

)*

+t2−

η

4 ξ

%

&''

(

)**e

− ξzerfc η

2 t− ξt

%

&'

(

)*

ϕ ξ,η,t( ) = −ξ

πte−ηt − ξη erf ηt( )

&

'((

)

*++

β η,t( ) = −tπe−ηt − tη+ 1

2 η

%

&''

(

)**erf ηt( )

+

,--

.

/00

τ ξ,t( ) = −1πte−ξt − ξ erf ξt( )

%

&'

(

)*

β1 ω,η,t( ) =−tωπe−ηt − tηω+ ω

2 η

&

'((

)

*++

erf ηt( )

,

-

.

.

.

.

/

0

1111

H t −1( ) = 0, t <11, t >1

"#$

%$

N =M+1K1,A = Pr−1, B= PrQ1 −N,

C =Sc−1,D =ScKr −N,E = BA, F = D

C

G = −GrA, H = −Gm

C, J = G

E2, K =

HF



ISBN: 978-81-928118-9-5 210

[17] Sahoo ,P.K., N. Datta and S. Biswal, “Magnetohydrodynamic unsteady free convection past an infinite vertical plate with constant suction and heat sink”, Indian J. Pure Appl. Math., 34(1), 145-155 (2003).

[18] Samad, M.A. and M.M. Rahman, “Thermal radiation interaction with unsteady MHD flow past a vertical porous plate immersed in a Porous medium”, Journal of Naval Architecture and Marine Engineering, 3, 7-14 (2006).

[19] Sattar, M.A. and H. Kalim, “Unsteady free convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate”, Journal of Mathematical Physics, 30, 25-37 (1996).

[20] Sattar, M.A. and M.A. Meleque, “Unsteady MHD natural convection flow along an accelerated porous

plate with hall current and mass transfer in a rotating porous medium”, Journal of Energy, Heat and Mass Transfer, 22, 67-72 (2000).

[21] Shercliff, J.A., A text book of Magnetohydrodynamics, Pergomon Press, London (1965).

[22] Takhar, H.S., R.S.R. Gorla and V.M. Sundalgekar, “Short communication radiation effects on MHD free convection flow of a gas past a semi—infinite vertical plate”, Int. J. Numerical Methods Heat Fluid Flow, 6, 77-83 (1996).

[23] Nandkeolyar, R., M. Das and P. Sibanda, “ Exact solutions of unsteady MHD free convection in a heat absorbing fluid flow past a flat plate with ramped wall temperature”, Boundary Value Problems, 2013:247 (2013).

Documents

Effect of chemical reaction on an unsteady MHD free convective flow