analytical solution of thermal radiation and chemical ...and r. a. mohamed1 1 mathematics...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2011, Article ID 205181, 18 pagesdoi:10.1155/2011/205181

Research ArticleAnalytical Solution of Thermal Radiation andChemical Reaction Effects on Unsteady MHDConvection through Porous Media with HeatSource/Sink

Abdel-Nasser A. Osman,1 S. M. Abo-Dahab,1, 2

and R. A. Mohamed1

1 Mathematics Department, Faculty of Science, SVU, Qena 83523, Egypt2 Department of Mathematics, Faculty of Science, Taif University, Taif 21974, Saudi Arabia

Correspondence should be addressed to S. M. Abo-Dahab, [email protected]

Received 29 March 2011; Accepted 27 May 2011

Academic Editor: Moran Wang

Copyright q 2011 Abdel-Nasser A. Osman et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

This paper analytically studies the thermal radiation and chemical reaction effect on unsteadyMHD convection through a porous medium bounded by an infinite vertical plate. The fluidconsidered here is a gray, absorbing-emitting but nonscattering medium, and the Rosselandapproximation is used to describe the radiative heat flux in the energy equation. The dimensionlessgoverning equations are solved using Laplace transform technique. The resulting velocity,temperature and concentration profiles as well as the skin-friction, rate of heat, and mass transferare shown graphically for different values of physical parameters involved.

1. Introduction

Thermal radiation of a gray fluid which is emitting and absorbing radiation in a nonscatteringmedium has been investigated by Ali et al. [1], Ibrahim and Hady [2], Mansour [3], Hossainet al. [4, 5], Raptis and Perdikis [6], Makinde [7], and Abdus-Sattar and Hamid Kalim [8]. Allthese studies have investigated the unsteady flow in a nonporous medium. From the previousliterature survey about unsteady fluid flow, we observe that few papers were done in a porousmedium. The radiative flows of an electrically conducting fluid with high temperature in thepresence of a magnetic field are encountered in electrical power generation, astrophysicalflows, solar power technology, space vehicle, nuclear engineering application, and otherindustrial areas. The analytical solution of unsteady MHD laminar convective flow withthermal radiation of a conducting fluid with variable properties through a porous medium inthe presence of chemical reaction and heat source or sink has not been investigated.

2 Mathematical Problems in Engineering

Combined heat and mass transfer problems with chemical reaction are of importancein many processes and have, therefore, received a considerable amount of attention in therecent years. In processes such as drying, evaporation at the surface of a water body, energytransfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occursimultaneously. Possible applications of this type of flow can be found in many industries,for example, in the electric power industry, among the methods of generating electric poweris one in which electrical energy is extracted directly from a moving conducting fluid. Manypractical diffusion operations involve the molecular diffusion of a species in the presence ofchemical reaction within or at the boundary. There are two types of reactions. A homogeneousreaction is one that occurs uniformly throughout a give phase. The species generation ina homogeneous reaction is analogous to internal source of heat generation. In contrast, aheterogeneous 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 conditionin heat transfer. Combined heat and mass transfer with chemical reaction in geometric withand without porous media has been studied by others [9–19].

This paper deals with the study of thermal radiation and chemical reaction effects onthe unsteady MHD convection through a porous medium bounded by an infinite verticalplate with heat source/sink. The governing equations are solved by Laplace transformtechnique. The results are obtained for velocity, temperature, concentration, skin-friction, rateof heat and mass transfer. The effects of various material parameters are discussed on flowvariables and presented by graphs.

2. Formulation of the Problem

Consider the unsteady free convection flow of an incompressible viscous fluid due to heatand mass transfer through a porous medium bounded by an infinite vertical plate under theaction of an external transfer magnetic field of uniform strength B0. The fluid considered is agray, absorbing-emitting radiation but nonscattering medium. It is assumed that there existsa homogeneous first-order chemical reaction between the fluid and species concentration.Then, by usual Boussinesq’s approximation, the unsteady flow is governed by the followingequations:

∂v′

∂y′ = 0, (2.1)

∂u′

∂t′+ v′ ∂u

′

∂y′ = gβ(T ′ − T ′

∞)+ gβ∗

(C′ − C′

∞)+ ν

∂2u′

∂y′2−(

σB2o

ρ+

ν

K

)

u′, (2.2)

∂T ′

∂t′+ v′ ∂T

′

∂y′ = α∂2T ′

∂y′2+Q(T ′ − T ′

∞) − 1

ρcp

∂q∗r∂y′ , (2.3)

∂C′

∂t′+ v′ ∂C

′

∂y′ = D∂2C′

∂y′2− R∗(C′ − C′

∞). (2.4)

For constant and uniform suction, (2.1) integrates to

v′ = −v0, (2.5)

where the negative sign indicates that suction is towards the plate.

Mathematical Problems in Engineering 3

The initial and boundary conditions are

u′ = 0, T ′ = T ′∞, C′ = C′

∞, for t ≤ 0,

u′ = 0, T ′ = T ′w, C′ = C′

w, for y′ = 0, t > 0,

u′ −→ 0, T ′ −→ T ′∞, C′ −→ C′

∞, for y′ −→ ∞, t > 0.

(2.6)

The radiative heat flux under term by using the Rosseland approximation is given by

q∗r = −4σ∗

3k∗∂T

′4

∂y′ . (2.7)

We assume that the temperature differences within the flow are sufficiently small suchthat T

′4 may be expressed as a linear function of the temperature. This is accomplished byexpanding T

′4 in a Taylor series about T ′∞ and neglecting higher-order terms, thus T

′4 can beexpressed as

T′4 ∼= 4T

′3∞T

′ − 3T′4∞. (2.8)

By using (2.7) and (2.8), (2.3) reduces to

∂T ′

∂t′= α

∂2T ′

∂y′2+Q(T ′ − T ′

∞)+

16σ∗T′3∞

3k∗ρcp

∂2T ′

∂y′2. (2.9)

To present solutions which are independent of geometry of the flow regime, we introduce thedimensionless variable as follows:

u =u′

vo, y =

y′vo

ν, t =

t′v2o

ν,

θ =T ′ − T ′

∞T ′w − T ′∞

, φ =C′ − C′

∞C′

w − C′∞.

(2.10)

Substituting from (2.10) into (2.2), (2.9), and (2.4), we obtain

∂u

∂t− ∂u

∂y=

∂2u

∂y2+Grθ +Gmφ −

(1k+M

)u,

(1 +

4R3

)∂2θ

∂y2− Pr(∂θ

∂t− ∂θ

∂y

)+ Prηθ = 0,

∂2φ

∂y2− Sc(∂φ

∂t− ∂φ

∂y

)− Scδφ = 0,

(2.11)


where

Gr =gβν(T ′

w − T ′∞)

v3o

, Gm =gβ∗ν(C′

w − C′∞)

v3o

, δ =R∗ν

v2o

,

k =Kν2

o

ν2, M =

σB2oν

ρv2o

, Sc =ν

D, η =

νQ

v2o

,

δ =R∗ν

v2o

, Pr =ρcpν

k1=

ν

α, R =

4σ∗T′3∞

k∗k1.

(2.12)

The initial and boundary conditions in nondimensional form are

u = 0, θ = 0, φ = 0, ∀y, t ≤ 0,

u = 0, θ = 1, φ = 1, at y = 0, t > 0,

u −→ 0, θ −→ 0, φ −→ 0 as y −→ ∞, t > 0.

(2.13)

All the physical variables are defined in the nomenclature. The solutions are obtained forhydrodynamic flow field in the presence of thermal radiation, chemical reaction and heatsource/sink.

3. Analytic Solution

In order to obtain the solution of the present problem, we will use the Laplace transformtechnique.

Applying the Laplace transform to the system of (2.11), and the boundary conditions(2.13), we get

∂2θ

∂y2+ F2

∂θ

∂y− F2(s − η)θ = 0, (3.1)

∂2φ

∂y2+ Sc

∂φ

∂y− Sc(s + δ)φ = 0, (3.2)

∂2u

∂y2+∂u

∂y− (s +M′)u +Grθ +Gmφ = 0, (3.3)

where,

M′ =1k+M, F2 =

Pr(1 + (4/3)R)

, (3.4)


s is Laplace transformation parameter, u, θ, and φ are Laplace transformation of u, θ, and φ,respectively,

u = 0, θ = φ =1s

at y = 0, t > 0,

u = θ = φ = 0, as y −→ ∞, t > 0.

(3.5)

Solving the system of (3.1)–(3.3), with the help of the result in (3.5), we get

θ(y, s)=

1s

Exp(λθy), (3.6)

φ(y, s)=

1s

Exp(λφy), (3.7)

u(y, s)= u1(y, s)+ u2(y, s)+ u3(y, s)+ u4(y, s), (3.8)

where

u1(y, s)=

−Gr

s

[eλθy

λ2θ+ λθ − B4

]

=−Gr

seλθy

⎡

⎢⎣

(αs + γ

) − β√s + B2

1

α2[(

(s +W)2 +Q2)]

⎤

⎥⎦,

u2(y, s)=

−Gm

s

⎡

⎣ eλφy

λ2φ + λφ − B4

⎤

⎦ =−Gm

seλφy

⎡

⎢⎣

(α1s + γ1

) − β1

√s + B2

2

α21

[((s +W1)2 +Q2

1

)]

⎤

⎥⎦,

u3(y, s)=

Gr

s

[eλuy

λ2θ+ λθ − B4

]

=Gr

seλuy

⎡

⎢⎣

(αs + γ

) − β√s + B2

1

α2[(

(s +W)2 +Q2)]

⎤

⎥⎦,

u4(y, s)=

Gm

s

⎡

⎣ eλuy

λ2φ+ λφ − B4

⎤

⎦ =Gm

seλuy

⎡

⎢⎣

(α1s + γ1

) − β1

√s + B2

2

α21

[((s +W1)2 +Q2

1

)]

⎤

⎥⎦,

(3.9)

where

B21 =

F2

4− η, λθ = −F2

2−√F2

√s + B2

1 , B22 =

Sc4

+ δ,

λφ = −Sc2

−√

Sc√s + B2

2 , B23 =

14+M′, λu = −1

2−√s + B2

3 ,

α = F2 − 1, β = α√F2, γ =

F2

2α − F2η −M′, W =

2αγ − β2

2α2,


Q =

√γ2 − β2B2

1

α2−W2, Q1 =

√√√√γ2

1 − β21B

22

α21

−W21 , α1 = Sc − 1,

β1 = α1√

Sc, γ1 =Sc2

2+ Sc(δ − 1

2

)−M′, W1 =

2α1γ1 − β21

2α21

, B4 = S +M′.

(3.10)

The inverse Laplace transformation of (3.6) is

θ(y, t)=

e−(y/2)F2

2

⎧⎨

⎩ey√

F2√

F2/4−η erf c

⎛

⎝y

2

√F2

t+ B1

√t

⎞

⎠

+e−y√

F2√

F2/4−η erf c

⎛

⎝y

2

√F2

t− B1

√t

⎞

⎠

⎫⎬

⎭.

(3.11)

The analytic solution of (3.2) can be obtained by taking the inverse transforms of (3.7). So,the solution of the problem for the concentration φ(y, t) for t > 0

φ(y, t)=

e−(y/2)Sc

2

⎧⎨

⎩ey

√ScB2 erf c

⎛

⎝y

2

√Sct+ B2

√t

⎞

⎠ + e−y√

ScB2 erf c

⎛

⎝y

2

√Sct− B2

√t

⎞

⎠

⎫⎬

⎭.

(3.12)

Similarly, the general solution of (3.3) can be obtained by taking the inverse Laplacetransform of (3.8). The expressions for velocity, u, velocity gradient, ∂u/∂y|y=0, temperaturegradient, ∂θ/∂y|y=0 and concentration gradient, ∂φ/∂y|y=0 are shown in Appendix A.

4. Numerical Results and Discussion

In order to get physical insight into the problem, the numerical calculations are carriedout to study the variations in velocity u, temperature θ and concentration φ. The variationin skin-friction (shear stress at the wall), rate of heat and mass transfer are computed.These variations involve the effects of time t, heat generation parameter η, chemical reactionparameter δ, Schmidt number Sc, Prandtl number Pr, thermal radiation parameter R,magnetic field parameter M and permeability parameter k. The values of Prandtl numberPr are chosen to be 3, 7, and 10. The values of Schmidt number Sc are chosen to be 0.22, 0.62,and 0.78, which represent hydrogen, water vapour and ammonia, respectively.

Representative velocity, temperature and concentration profiles across the boundarylayer for different values of the dimensionless time t are presented in Figure 1. As thedimensionless time increases, the velocity, temperature and concentration profiles increase.


0 1 2 30

0.2

0.4

0.6

t 0.1, 0.2, 0.3, 0.4V

eloc

ityu

y

:=

(a)

0 10.2 0.4 0.6 0.80

0.5

1

Tem

pera

ture

θ

t 0.1, 0.2, 0.3, 0.4

y

:=

(b)

0 2 4 6 80

0.5

1

y

t :=0.1, 0.2, 0.3, 0.4

Con

cent

rati

on

(c)

Figure 1: Velocity, temperature and concentration profiles against y for various values of t with Pr = 10,M = 0.2, η = −2, k = 1, Sc = 0.22, Gr = Gm = 5, δ = 0.1, and R = 0.1.

0 1 2 30

0.2

0.4

0.6

y

η − 5, − 4, 0.5, 1

Vel

ocit

yu

:=

(a)

0 10.2 0.4 0.6 0.80

0.5

1

y

η − 5, − 4, 0.5, 1:=

Tem

pera

ture

θ

(b)

Figure 2: Velocity and temperature profiles against y for various values of η with t = 0.1, Pr = 10, M = 0.2,k = 1, Sc = 0.22, Gr = Gm = 5, δ = 0.1, and R = 0.1.

Figure 2 describe the behavior of velocity and temperature profiles across theboundary layer for different values of the heat generation parameter η. As the heat sourceparameter η increases, the velocity and temperature profiles increase. The volumetric heatsource term may exert a strong influence on the heat transfer and as a consequence, also onthe fluid flow.


0 1 2 30

0.2

0.4

0.6

y

δ 5, 10, 15V

eloc

ityu

:=

(a)

1 2 3 40

0

0.5

1

y

δ 5, 10, 15:=

Con

cent

rati

on

(b)

Figure 3: Velocity and concentration profiles against y for various values of δ with t = 0.1, Pr = 10, M = 0.2,k = 1, η = −2, Sc = 0.22, Gr = Gm = 5, and R = 0.1.

Sc

0 1 2 30

0.2

0.4

0.6

y

:= 0.22, 0.62, 0.78

Vel

ocit

yu

(a)

1 2 3 40

0

0.5

1

y

Con

cent

rati

on Sc := 0.22, 0.62, 0.78

(b)

Figure 4: Velocity and concentration profiles against y for various values of Sc with t = 0.1, Pr = 10,M = 0.2, k = 1, η = −2, Gr = Gm = 5, and R = 0.1.

Figure 3 shows the velocity and concentration profiles for different values of chemicalreaction parameter. As the chemical reaction parameter increases, the velocity increases butthe concentration profile decreases.

Figure 4 displays the effects of Schmidt number on the velocity and concentrationprofiles, respectively. As the Schmidt number increases, the velocity and concentrationdecrease. Reductions in the velocity and concentration profiles are accompanied bysimultaneous reductions in the velocity and concentration boundary layers. These behaviorsare evident from Figure 4.

Figure 5 illustrates the velocity and temperature profiles for different values of Prandtlnumber. The numerical results show that the effect of increasing values of Prandtl numberresults in a decreasing velocity. Also, it is shown that an increase in the Prandtl number resultstend to a decreasing of the thermal boundary layer and in general it lowers the averagetemperature through the boundary layer. The reason is that, the smaller values of Pr areequivalent to increase in the thermal conductivity of the fluid and therefore heat is able todiffuse away from the heated surface more rapidly for higher values of Pr. Hence in the caseof smaller Prandtl numbers, the thermal boundary layer is thicker and the rate of heat transferis reduced.


0

0.2

0.4

0.6

y

Pr := 3, 7, 10

0 1 2 3

Vel

ocit

yu

(a)

0 10.2 0.4 0.6 0.80

0.5

1

y

Pr := 3, 7, 10

Tem

pera

ture

θ

(b)

Figure 5: Velocity and temperature profiles against y for various values of Pr with t = 0.1, Sc = 0.22,M = 0.2, k = 1, η = −2, Gr = Gm = 5, and R = 0.1.

0 1 2 30

0.2

0.4

0.6

y

Vel

ocit

yu

R := 0, 0.1, 0.2, 0.3

(a)

R := 0, 0.1, 0.2, 0.3

0 10.2 0.4 0.6 0.80

0.5

1

y

Tem

pera

ture

θ

(b)

Figure 6: Velocity and temperature profiles against y for various values of R with t = 0.1, Pr = 10, M = 0.2,k = 1, η = −2, Gr = Gm = 5.0, and Sc = 0.22.

For different values of the radiation parameter R, the velocity and temperature profilesare shown in Figure 6. It is noticed that an increase in the radiation parameter results anincrease in the velocity and temperature within the boundary layer, also it increases thethickness of the velocity and temperature boundary layers.

The velocity profiles for different values of the magnetic field parameter anddimensionless permeability are shown in Figure 7. It is clear that the velocity decreases withincreasing of the magnetic field parameter. It is because that the application of transversemagnetic field will result a restrictive type force (Lorentz’s force), similar to drag force whichtends to restrictive the fluid flow and thus reducing its velocity. The presence of porous mediaincreases the resistance flow resulting in a decrease in the flow velocity. This behavior isdepicted by the decrease in the velocity as permeability decreases and when k → ∞ (i.e., theporous medium effect is vanishes) the velocity is greater in the flow field. These behaviorsare shown in Figure 7.

Figure 8 displays the effect of Sc, t, M, k, δ, and Pr on shear stress u′(0, t) with respectto radiation parameter R, it is obvious that there is a slight changes in shear stress, also, it is


M := 0.2, 0.3, 0.5

0 1 2 30

0.2

0.4

0.6

y

Vel

ocit

yu

(a) (b)

Figure 7: Velocity profiles against y for various valuesof M and k with t = 0.1, Pr = 10, M = 0.2, Sc = 0.22,R = 0.1, and η = −2.

seen that shear stress increases with an increasing of k and δ but decreases with an increasingvalues of Sc, t, M, and Pr.

Figure 9 shows the influence of time, heat source parameter and the radiationparameter on the negative values of gradient temperature (i.e., −θ′(0, t)) with respect to thePrandtl number, it is seen that the increasing values of the time, heat source parameter andradiation parameter tend to decreasing in the negative values of the temperature gradient,also, it increases with the increasing of Pr.

Figure 10 displays the influence of time and chemical reaction parameter on thenegative values of concentration gradient (i.e., −φ′(0, t)) respect the Schmidt number, it isconcluded that it increases with the increasing of Sc and chemical reaction δ but decreaseswith an increase of t.

5. Conclusion

A mathematical model has been presented for analytically studies the thermal radiation andchemical reaction effect on unsteady MHD convection through a porous medium boundedby an infinite vertical plate. The fluid considered here is a gray, absorbing-emitting butnonscattering medium and the Rosseland approximation is used to describe the radiativeheat flux in the energy equation. The dimensionless governing equations are solved usingLaplace transform technique. The resulting velocity, temperature and concentration profilesas well as the skin-friction, rate of heat and mass transfer are shown graphically for differentvalues of physical parameters involved. It has been shown that the fluid is accelerated,that is, velocity (u) is increased with an increasing values of time (t), chemical reactionparameter (δ), Prandtl number (Pr), radiation parameter (R), while they show opposite tendswith an increasing values of heat source parameter (η), Schmidt number (Sc) and magneticfield parameter (M). Also, it is shown that velocity (u) does not affected with the variousvalues of the dimensionless permeability k. We conclude that, the negative temperaturegradient (−θ′(0, t)) increases as Pr increases but decreases as t, η, and R increase. Finally, weobvious that, the negative concentration gradient (−φ′(0, t)) increases as Sc and δ increasesbut decreases as t increase.

It is hoped that the results obtained here will not only provide useful information forapplications, but also serve as a complement to the previous studies.


0 0.1 0.2 0.3 0.4 0.5− 6

− 4

− 2

0

R

Sc := 0.22, 0.62, 0.78Sh

ear

stre

ss

(a)

Shea

rst

ress

0 0.1 0.2 0.3 0.4 0.5− 3

− 2.5

− 0.5

− 1.5

− 2

− 1

R

t := 0.1, 0.2, 0.3, 0.4

(b)

Shea

rst

ress

0 0.1 0.2 0.3 0.4 0.5−2.5

−1.5

−2

−1

R

M := 0.2, 0.3, 0.5

(c)

0 0.1 0.2 0.3 0.4 0.5

R

Shea

rst

ress

−0.5

0.5

−1

0

k := 0.3, 0.5, 0.7

(d)

0 0.1 0.2 0.3 0.4 0.5

R

0

200

400

600

800

1000

Shea

rst

ress

δ := 5, 10, 15

(e)

0 0.1 0.2 0.3 0.4 0.5

R

Shea

rst

ress

−1.65

−1.6

−1.55

−1.5

−1.45

−1.4

Pr := 7, 8, 9

(f)

Figure 8: Variation of shear stress u′(0, t) against R for various values of Sc, t, M, k, δ, and Pr with constantPr = 10, M = 0.2, k = 1, Sc = 0.22, Gm = Gr = 5, and η = −2.


0

2

4

6

8

Pr

0 2 4 6 8 10

-Tem

pera

ture

grad

ien

tt := 0.1, 0.2, 0.3, 0.4

(a)

0

2

4

6

8

Pr

0 2 4 6 8 10

η := − 5, − 4, − 3, 0.03

-Tem

pera

ture

grad

ient

(b)

0

5

10

15

Pr

0 2 4 6 8 10

R := 0, 0.1, 0.2, 0.3

-Tem

pera

ture

grad

ient

(c)

Figure 9: Variation of −θ′(0, t) for various values of t, η, and R with Pr = 10, M = 0.2, k = 1, Sc = 0.22,Gr = Gm = 5.0, and η = −2.

0

5

10

15

Sc

0 2 4 6 8 10

t := 0.1, 0.2, 0.3, 0.4

-Con

cent

rati

ongr

adie

nt

(a)

0

5

10

15

20

Sc

0 2 4 6 8 10

-C

once

ntr

atio

ngr

adie

nt

δ := 5, 10, 15

(b)

Figure 10: Variation of −φ′(0, t) for various values of t and δ with Pr = 10, M = 0.2, k = 1, Sc = 0.22,Gr = Gm = 5.0, and η = −2.


Appendix

A.

The inverse laplace transformation of (3.8) is

u(y, t)= u1(y, t)+ u2(y, t)+ u3(y, t)+ u4(y, t), (A.1)

where

u1(y, t)= −Gr

α2

{

α

∫ t

0θ(y, u)e−W(t−u) cosQ(t − u)du +

γ − αW

Q

×∫ t

0θ(y, u)e−W(t−u) sinQ(t − u)du − β

∫ t

0θ(y, u)G1(t − u)du

− β(B2

1 − 2W)∫ t

0θ(y, u)∫ t−u

0G1(τ)e−W(t−u−τ) cosQ(t − u − τ)dτdu

+β

Q

[W(B2

1 − 2W)+(W2 +Q2

)]

×[∫ t

0θ(y, u)∫ t−u

0G1(τ)e−W(t−u−τ) sinQ(t − u − τ)dτdu

]}

,

u2(y, t)= −Gm

α21

{

α1

∫ t

0φ(y, u)e−W1(t−u) cosQ1(t − u)du +

γ1 − α1W1

Q1

×∫ t

0φ(y, u)e−W1(t−u) sinQ1(t − u)du − β1

∫ t

0φ(y, u)G2(t − u)du

− β1

(B2

2 − 2W1

)∫ t

0φ(y, u)∫ t−u

0G2(τ)e−W1(t−u−τ) cosQ1(t − u − τ)dτdu

+β1

Q1

[W1

(B2

2 − 2W1

)+(W2

1 +Q21

)] ∫ t

0φ(y, u)

×∫ t−u

0G2(τ)e−W1(t−u−τ) sinQ1(t − u − τ)dτdu

}

,

u3(y, t)=

Gr

α2

{

α

∫ t

0θ1(y, u)e−W(t−u) cosQ(t − u)du +

γ − αW

Q1

×∫ t

0θ1(y, u)e−W(t−u) sinQ(t − u)du − β

∫ t

0θ1(y, u)G1(t − u)du

− β(B2

1 − 2W)∫ t

0θ1(y, u)∫ t−u

0G1(τ)e−W(t−u−τ) cosQ(t − u − τ)dτdu


+β

Q

[W(B2

1 − 2W)+(W2 +Q2

)] ∫ t

0θ1(y, u)

×∫ t−u

0G1(τ)e−W(t−u−τ) sinQ(t − u − τ)dτdu

}

,

u4(y, t)=

Gm

α21

{

α1

∫ t

0θ1(y, u)e−W1(t−u) cosQ1(t − u)du +

γ1 − α1W1

Q1

×∫ t

0θ1(y, u)e−W1(t−u) sinQ1(t − u)du − β1

∫ t

0θ1(y, u)G2(t − u)du

− β1

(B2

2 − 2W1

)∫ t

0θ1(y, u)∫ t−u

0G2(τ)e−W1(t−u−τ) cosQ1(t − u − τ)dτdu

+β1

Q1

[W1

(B2

2 − 2W1

)+(W2

1 +Q21

)] ∫ t

0θ1(y, u)

×∫ t−u

0G2(τ)e−W1(t−u−τ) sinQ1(t − u − τ)dτdu

}

,

(A.2)

where

G1(t) =1B1

erf(B1

√t),

G2(t) =1B2

erf(B2

√t),

θ1(y, t)=

12e−y/2

{eyB3 erf c

(y

2√t+ B3

√t

)+ e−yB3 erf c

(y

2√t− B3

√t

)}.

(A.3)

θ′(0, t) = −F2

2− B1

√F2 erf

(B1

√t)−√

F2

πte−B

21t, (A.4)

φ′(0, t) = −Sc2

− B2√

Sc erf(B2

√t)−√

Scπt

e−B22t, (A.5)

∂u(y, t)∂y

∣∣∣∣y=0

=4∑

i=1

∂

∂yui(y, t)

∣∣y=0, i = 1, 2, 3, 4,

∂u1

∂y

∣∣∣∣y=0

= −Gr

α2

{

α

∫√t

0θ′(0, x)e−W(t−x2) cosQ

(t − x2

)dx +

γ − αW

Q


×∫√t

0θ′(0, x)e−W(t−x2) sinQ

(t − x2

)dx − β

∫√t

0θ′(0, x)G1

(t − x2

)dx

− β(B2

1 − 2W)∫

√t

0Θ′(0, x)

∫ t−x2

0G1(τ)e−W(t−x2−τ)

× cosQ(t − x2 − τ

)dτdx

+β

Q

[W(B2

1 − 2W)+(W2 +Q2

)] ∫√t

0θ′(0, x)

×∫ t−x2

0G1(τ)e−W(t−x2−τ) sinQ

(t − x2 − τ

)dτdx

}

,

∂u2

∂y

∣∣∣∣y=0

= −Gm

α21

{

α1

∫√t

0φ′(0, x)e−W1(t−x2) cosQ1

(t − x2

)dx +

γ1 − α1W1

Q1

×∫√t

0φ′(0, x)e−W1(t−x2) sinQ1

(t − x2

)dx − β1

∫√t

0φ′(0, x)G2

(t − x2

)dx

− β1

(B2

2 − 2W1

)∫√t

0φ′(0, x)

∫ t−x2

0G2(τ)e−W1(t−x2−τ)

× cosQ1

(t − x2 − τ

)dτdx

+β1

Q1

[W1

(B2

2 − 2W1

)+(W2

1 +Q21

)] ∫√t

0φ′(0, x)

×∫ t−x2

0G2(τ)e−W1(t−x2−τ) sinQ1

(t − x2 − τ

)dτdx

}

,

∂u3

∂y

∣∣∣∣y=0

=Gr

α2

{

α

∫√t

0θ′

1(0, x)e−W(t−x2) cosQ

(t − x2

)dx +

γ − αW

Q

×∫√t

0θ′(0, x)e−W(t−x2) sinQ

(t − x2

)dx − β

∫√t

0θ′(0, x)G1

(t − x2

)dx

− β(B2

1 − 2W)∫

√t

0θ′(0, x)

∫ t−x2

0G1(τ)e−W(t−x2−τ)

× cosQ(t − x2 − τ

)dτdx

+β

Q

[W(B2

1 − 2W)+(W2 +Q2

)] ∫√t

0θ′(0, x)

×∫ t−x2

0G1(τ)e−W(t−x2−τ) sinQ

(t − x2 − τ

)dτdx

}

,


∂u4

∂y

∣∣∣∣y=0

=Gm

α21

{

α1

∫√t

0θ′

1(0, x)e−W1(t−x2) cosQ1

(t − x2

)dx +

γ1 − α1W1

Q1

×∫√t

0θ′

1(0, x)e−W1(t−x2) sinQ1

(t − x2

)dx − β1

∫√t

0θ′

1(0, x)G2

(t − x2

)dx

− β1

(B2

2 − 2W1

)∫√t

0θ′

1(0, x)∫ t−x2

0G2(τ)e−W1(t−x2−τ)

× cosQ1

(t − x2 − τ

)dτdx

+β1

Q1

[W1

(B2

2 − 2W1

)+(W2

1 +Q21

)] ∫√t

0θ′

1(0, x)

×∫ t−x2

0G2(τ)e−W1(t−x2−τ) sinQ1

(t − x2 − τ

)dτdx

}

,

θ′(0, x) = xθ′(0, t)∣∣t=x2 ,

φ′(0, x) = xφ′(0, t)∣∣t=x2 ,

Θ′1(0, t) = −1

2− B3 erf

(B3

√t)−√

1πt

e−B23t,

Θ′1(0, x) = xΘ′

1(0, t)∣∣t=x2 .

(A.6)

Nomenclatures

B0: Magnetic inductionC′

∞: Concentration of the fluid for away from the plate C in the free streamC′: Concentration of the fluidD: Chemical molecular diffusivityg: Gravitational accelerationGm: Solutal Grashof numberGr : Grashof numberK: Permeability of the porous mediumk: Dimensionless permeabilityk1: Thermal conductivity of the fluidk∗: Mean absorption coefficientM: Magnetic field parameterPr: Prandtl numberq∗r : Radiative heat fluxQ: Hea tsource/sink coefficientR: Radiation parameterR∗: First-order chemical reaction rateSc: Schmidt numbert: Dimensionless time


t′: Dimensional timeT ′: Temperature of the fluidT ′w, C′

w : Surface temperature and concentrationT ′∞: Temperature of the fluid for away from the plate (in the free stream)

u: Dimensionless velocityu′, v′: Components of dimensional velocities along x′ and y′ directionsx′, y′: Dimensional distances along and perpendicular to the platey: Nondimensional distanceα: Thermal diffusivityβ: Coefficient of thermal expansionβ∗: Coefficient of expansion with concentrationη: Heat source parameterθ: Dimensionless temperatureν: Kinematic coefficient of viscosityδ: Chemical reaction parameterρ: Fluid densityσ: Electrical conductivity of the fluidσ∗: Stefan-Boltzmann fluxφ: Dimensionless concentration.

References

[1] M. M. Ali, T. S. Chen, and B. F. Armaly, “Natural convection-radiation interaction in boundary-layerflow over horizontal surfaces,” AIAA Journal, vol. 22, no. 12, pp. 1797–1803, 1984.

[2] F. S. Ibrahim and F. M. Hady, “Mixed convection-radiation interaction in boundary-layer flow overhorizontal surfaces,” Astrophysics and Space Science, vol. 168, no. 2, pp. 263–276, 1990.

[3] M. A. Mansour, “Radiative and free-convection effects on the oscillatory flow past a vertical plate,”Astrophysics and Space Science, vol. 166, no. 2, pp. 269–275, 1990.

[4] M. A. Hossain, M. A. Alim, and D. A. S. Rees, “The effect of radiation on free convection from aporous vertical plate,” International Journal of Heat and Mass Transfer, vol. 42, no. 1, pp. 181–191, 1998.

[5] M. A. Hossain, K. Khanafer, and K. Vafai, “The effect of radiation on free convection flow of fluidwith variable viscosity from a porous vertical plate,” International Journal of Thermal Sciences, vol. 40,no. 2, pp. 115–124, 2001.

[6] A. Raptis and C. Perdikis, “Radiation and free convection flow past a moving plate,” InternationalJournal of Applied Mechanics and Engineering, vol. 4, no. 4, pp. 817–821, 1999.

[7] O. D. Makinde, “Free convection flow with thermal radiation and mass transfer past a moving verticalporous plate,” International Communications in Heat and Mass Transfer, vol. 32, no. 10, pp. 1411–1419,2005.

[8] M. D. Abdus-Sattar and M. D. Hamid Kalim, “Unsteady free-convection interaction with thermalradiation in a boundary-layer flow past a vertical porous plate,” Journal of Mathematical and PhysicalSciences, vol. 30, pp. 25–37, 1996.

[9] U. N. Das, R. Deka, and V. M. Soundalgekar, “Effects of mass transfer on flow past an impulsivelystarted infinite vertical plate with constant heat flux and chemical reaction,” Forschung imIngenieurwesen, vol. 60, no. 10, pp. 284–287, 1994.

[10] A. J. Chamkha, “MHD flow of a uniformly streched vertical permeable surface in the presence ofheat generation/absorption and a chemical reaction,” International Communications in Heat and MassTransfer, vol. 30, no. 3, pp. 413–422, 2003.

[11] R. Muthucumaraswamy and P. Ganesan, “On impulsive motion of a vertical plate with heat flux anddiffusion of chemically reactive species,” Forschung im Ingenieurwesen, vol. 66, no. 1, pp. 17–23, 2000.

[12] R. Muthucumaraswamy and P. Ganesan, “First-order chemical reaction on flow past an impulsivelystarted vertical plate with uniform heat and mass flux,” Acta Mechanica, vol. 147, no. 1-4, pp. 45–57,2001.


[13] R. Muthucumaraswamy, “Effects of suction on heat and mass transfer along a moving vertical surfacein the presence of chemical reaction,” Forschung im Ingenieurwesen, vol. 67, no. 3, pp. 129–132, 2002.

[14] A. Raptis and C. Perdikis, “Viscous flow over a non-linearly stretching sheet in the presence of achemical reaction and magnetic field,” International Journal of Non-Linear Mechanics, vol. 41, no. 4, pp.527–529, 2006.

[15] M. A. Seddeek, A. A. Darwish, and M. S. Abdelmeguid, “Effects of chemical reaction and variableviscosity on hydromagnetic mixed convection heat and mass transfer for Hiemenz flow throughporous media with radiation,” Communications in Nonlinear Science & Numerical Simulation, vol. 12,no. 2, pp. 195–213, 2007.

[16] F. S. Ibrahim, A. M. Elaiw, and A. A. Bakr, “Effect of the chemical reaction and radiation absorptionon the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate withheat source and suction,” Communications in Nonlinear Science & Numerical Simulation, vol. 13, no. 6,pp. 1056–1066, 2008.

[17] A. Postelnicu, “Influence of chemical reaction on heat and mass transfer by natural convection fromvertical surfaces in porous media considering Soret and Dufour effects,” Heat and Mass Transfer, vol.43, no. 6, pp. 595–602, 2007.

[18] R. A. Mohamed, I. A. Abbas, and S. M. Abo-Dahab, “Finite element analysis of hydromagnetic flowand heat transfer of a heat generation fluid over a surface embedded in a non-Darcian porous mediumin the presence of chemical reaction,” Communications in Nonlinear Science & Numerical Simulation, vol.14, no. 4, pp. 1385–1395, 2009.

[19] R. A. Mohamed and S. M. Abo-Dahab, “Influence of chemical reaction and thermal radiation on theheat and mass transfer in MHD micropolar flow over a vertical moving porous plate in a porousmedium with heat generation,” International Journal of Thermal Sciences, vol. 48, no. 9, pp. 1800–1813,2009.

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