**NATURAL CONVECTION IN MHD **

**TRANSIENT FLOW PAST AN **

**ACCELERATED VERTICAL PLATE **

**WITH HEAT SINK **

N. AHMED

Department of Mathematics Gauhati University, Guwahati 781014

Assam, India

Email: saheel_nazib@yahoo.com

K. Kr. DAS

Department of Mathematics Gauhati University, Guwahati 781014

Assam, India

Email: kishoredas969@yahoo.in

S. M. DAS

Department of Mathematics Gauhati University, Guwahati 781014

Assam, India

Email: suwagmoni1.618@gmail.com

**Abstract **

The problem of an MHD heat and mass transfer flow past an accelerated infinite vertical plate in a porous medium in presence of chemical reaction, thermal diffusion and first order heat sink is studied. A magnetic field of uniform strength is assumed to be applied normal to the field directed to the fluid region. The resulting system of equations governing the fluid motion is solved by adopting Laplace Transform technique in closed form. The effects of the physical parameters involved in the problem on the flow and the transport characteristics are studied graphs.

**Keywords: Heat absorption, Soret effect, Free convection, permeability. ****AMS 2000 subject classification 76W05 **

**1.** **Introduction **

The study of MHD flow of an electrically conducting fluid is of considerable interest in modern metallurgical and metal – working processes. The study of MHD flow and heat transfer are deemed as of great interest due to the effect of magnetic field on the boundary layer flow control and on the performance of many systems using electrically conducting fluids. Some of the engineering applications are in MHD generators, plasma studies, nuclear reactor, geothermal energy extractions, purifications of metal from non – metal enclosures, polymer technology and metallurgy.

The study of convective fluid flow with mass transfer along a vertical porous plate in the presence of
magnetic field and internal heat generation receiving considerable attention due to its useful applications in
different branches of Science and Technology such a cosmical and geophysical sciences, fire engineering ,
combustion modeling etc. The unsteady fluid flow past a moving plate in the presence of free convection and
radiation were studied by [Monsour,(1990)]. Free convection effect on a flow past a moving vertical plate
embedded in porous medium was analyzed by [Chaudhary and jain, (2007)]. [Israel-Cookey *et al*.,(2010)] have
made an analysis on MHD Natural convention oscillatory Couetee flow of a radiating viscous fluid in a porous
medium with periodic wall temperature. [Sattar and Maleque,(2000)] have studied the unsteady MHD natural
convention flow and mass transfer along an accelerated porous plate in a porous medium. Transient free
convention past a semi-infinite vertical plate with veriable surface temperature has been investigate by [Takhar

*et al*.,(1997)] .

itself. [Muthucumaraswamy and Meenakshisundaram,(2006)] investigated theoretical study of chemical reaction effects on vertical oscillating plate with variable temperature and mass diffusion. [Chamber and Young,(1958)] have presented a first order chemical reaction in the neighbourhood of horizontal plate.

The Soret effect or thermophoresis is a phenomenon observed in mixture of mobile particles where the
different particles where the different particle types exhibit different responses to the force of a temperature
gradient. The term soret effect most often applies to aerosol mixtures, but may also commonly refer to the
phenomenon in all phases of matter. It has been used in commercial precipitators for applications similar to
electro static precipitators, manufacturing of optical fibre in vapour deposition process, facilitating drug
discovery by allowing the detection of aptamer binding by comparison of the bound versus unbound motion of
the target molecule. It is also used to separate different polymers particles in fluid flow fractionation. [Ahmed
and Das,(2013)] investigated Soret effet on an MHD free convective mass transfer flow past an accelerated
vertical plate with chemical reaction. [Anghel and Takhr,(2000)] studied Dufour effect and Soret effect on free
convection boundary layer over a vertical surface embedded in porous medium. [Raju *et al*.,(2008)] studied
Soret effect due to natural convection between heated inclined plates. [Singh and Garg,(2010)] have studied the
radiative heat transfer in MHD oscillatory flow through porous medium bounded by two vertical plates.

In all the above studies the effect of heat source/sink were not studied in the presence of soret effect and chemical reaction. In electronic system, a heat sink is a passive component that cools a device by dissipating heat into the surrounding air. Therefore, in this paper we will extend the work of Ahmed and Das through introducing heat source/sink parameter. The main objective of this paper is to investigate the natural convection in MHD transient flow with the heat sink

**2.** **Nomenclature **

Bo strength of the applied magnetic field

C species concentration

### C

_{}

###

species concentration in free steam w### C

###

species concentration at the plate p### C

specific heat at constant pressure M### D

mass diffusivity r### G

Grashof number for heat transfer Gc Grashof number for mass transfer K chemical reaction parameterT

### K

thermal diffusion ratio k thermal conductivityK permeability of porous medium M magnetic parameter

Pr Prandtl number

### Q

###

rate of absorptionQ heat absorption parameter Sr Soret number

Sc Schmidt number T temperature

w

### T

###

temerature at the plate### T

_{}

###

temperature at the free stream M### T

mean fluid temperature### t

###

timef plate acceleration

kinametic viscosity

electrical conductivity

###

_{ }fluid density

###

temperature###

coefficient of viscosity porosity parameter

###

### x , y

###

###

Cartesian parameter y non dimensional normal coordinate
**3.** **Mathematical Analysis **

We consider a two dimensional flow of an incompressible viscous electrically conducting fluid past an infinite vertical plate embedded in porous medium. The

### X

###

axis is taken along the infinite vertical plate and### Y

###

axis normal to it. Initially the plate and the fluid were at same temperature### T

_{}

###

with concentration level### C

_{}

###

at all points. At time### t

>0, the plate temperature is suddenly raised to### T

w###

and the concentration level at the plate gets raised to### C

_{w}

###

. A magnetic field of uniform strength is applied perpendicular to the plate. We assume that the plate is started to with velocity### u

###

###

### f t

###

in its own plane at time### t

>0. With the foregoing assumptions and under usual boundary layer and Boussinesq of approximation, the governing equations reduce to:Continuity equation:

### u

### 0

### x

###

###

_{}

###

###

(1)Momentum equation :

###

###

###

###

2 022

### B

### u

### u

### g

### T

### T

### g

### C

### C

### u

### u

### t

### y

### K

###

###

###

_{ }

_{}

_{}

_{}

_{ }

_{ }

_{}

_{}

_{ }

###

_{}

_{}

_{}

###

_{}

###

###

###

###

###

###

(2)Energy equation:

###

###

2 2

### T

### T

### Cp

### k

### Q T

### T

### t

### y

###

###

###

###

_{ }

_{}

###

###

###

###

###

###

###

###

(3)Species continuity equation:

###

###

2 2

M T

M 2 2

M

### D K

### C

### C

### T

### D

### K C

### C

### t

### y

### T

### y

###

###

###

###

_{}

###

_{}

_{ }

_{}

_{}

_{}

###

###

###

###

###

###

###

(4)The initial and boundary conditions for the flow problems are

### t

###

###

### 0, u

###

###

### 0 , T

###

###

### T

_{}

###

### , C

###

###

### C

_{}

###

### for all y

w w

### u

### f t , T

### T , C

### C

### at y

### 0

### t

### 0,

### u

### 0 , T

### T , C

_{}

### C , y

_{}

###

###

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###

###

_{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

_{}

_{ }

###

###

###

(5) We introduce the non-dimensional quantities2

0 o

0 w

2

w 0

### t u

### y u

### T

### T

### u

### u

### , t

### , y

### ,

### ,

### u

### T

### T

### C

### C

### Q

### C

### , Q =

### C

### C

### Cp u

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

Gr (Ratio of thermal buoyancy force to viscous force) = w_{3}
o

### g

### ( T

### T )

### u

###

###

Gc (Ratio of solutal buoyancy force to viscous force) = _{3}w
o

### g

### ( C

### C )

### u

###

###

###

###

###

Pr (Ratio of viscous diffusivity force to thermal diffusivity) =

### Cp

### k

###

Sc (Ratio of momentum diffusivity to mass diffusivity) = M

### D

###

Sr (Relative effectiveness of temperature gradient to concentration

gradient) = M T w

M w

### D K (T

### T )

### T (C

### C )

###

###

###

###

###

###

###

M (Ratio of magnetic force to viscous force)= 2 0 2 0

### B

### u

###

###

###

,2 0

### K

### K

### u

###

###

###

,2 o 2

### u K

###

###

(6)Where,

### u (characteristic velocity) (f

_{0}

###

### )

1/3With the help of non dimensional quantities (6), equations (2) to (4) reduce to:

2 2

### u

### u

### 1

### Gr

### Gc C

### ( M

### ) u

### t

### y

###

###

###

###

###

###

###

###

###

###

(7)2 2

### 1

### Q

### t

### Pr

### y

###

_{}

###

_{ }

###

###

(8)2 2

2 2

### C

### 1

### C

### K C Sr

### t

### Sc

### y

### y

###

###

###

###

###

###

###

###

###

(9)Subject to the initial and boundary conditions

### u

### t,

### 1, C 1, at y

### 0

### t

### 0

### u

### 0,

### 0, C

### 0, y

###

###

###

###

###

###

###

_{}

_{}

###

###

###

###

###

###

(10)All physical quantities are defined in the nomenclature.

**4.** **Method of Solution **

On taking Laplace Transforms of the equations (7), (8) and (9), the following ordinary differential equations are obtained :

###

###

2 2

### d

### Pr s Q

### 0

### dy

###

_{}

_{}

_{ }

** (**11)

###

###

###

###

y s Q Pr2 2

### d C

### e

### Sc s K C

### Sr Sc Pr s Q

### dy

### s

###

###

###

###

**(**12

**)**

###

###

2 2

### d u

### s a u

### Gr Gc

### C

### dy

###

###

###

**(13)**

Subject to the boundary conditions:

2

### 1

### 1

### 1

### u

### , C

### ,

### at y

### 0

### s

### s

### s

### u

### 0

### , C

### 0 ,

### 0

### at y

###

###

###

###

###

_{}

###

###

###

###

###

_{ }

(14)
Where

### a

###

### M

### 1

### , u

###

### L u y, t

###

###

###

### , C

###

### L C y, t ,

###

###

###

###

### L

###

###

###

### y, t

###

###

_{ }

The solutions of the equations (11) to (13) under the conditions (14) are as follows:

y s Q Pr

### 1

### e

### s

###

(15)###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

###

y s K Sc

1 2

y s Q Pr

1 2

### A

### A

### 1

### C

### e

### s

### s Q Pr

### s K Sc

### s s Q Pr

### s K Sc

### A

### A

### e

### s Q Pr

### s K Sc

### s s Q Pr

### s K Sc

###

###

###

###

###

###

###

###

###

###

###

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###

###

###

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###

(16)###

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###

6 2 3 6 1 1 2 3 8 1 1 2 3

2

7 7 2 5 2 y s a

2 8 3 4 5

6 2 3 6 1 1 2 3 y s k Sc

7 7 2

8 1 1 2 3 2 8 3 4 y s

5 2 5

### a 1 A a

### a

### A a

### A a

### a

### A a

### A a

### 1

### s

### s s a

### s a

### s a

### s a

### s a

### u

### e

### A a a

### a

### s s a

### a 1 A a

### a

### A a

### A a

### e

### s s a

### s a

### s a

### a

### A a

### A a

### A a a

### a

### e

### s a

### s a

### s s a

###

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###

Q Pr (17)Taking inverse Laplace transforms of the equations (15). (16) and (17). We derive the expressions for the representative temperature, concentration and velocity field as follows:

###

### y, t

2###

###

(18)###

###

a t2###

###

2 3 1 2 3 2 1 1 2 3 3 4

### C y, t

###

### 1 A a

###

### A a

###

### (A a

###

### A a )e

###

(19)###

0### y

### u y, t

### 4 a

###

###

2 8 3 4 6###

2 3###

55 7

### a 1 A a

### A a a

### a

### t

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_{}

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a t_{5}

8 1 1 2 3 2 8 3 4

6

5 5 2

### a

### A a

### A a

### A a a

### a

### e

### a

### a

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a t76 2 3 6 1 1 2 3

7

7 7 2

### a 1 A a

### a

### A a

### A a

### e

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_{2}

6 1 1 2 3 8 1 1 2 3 a t

8

7 2 5 2

### a

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### A a

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### e

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28 1 1 2 3 a t 4

5 2

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a t58 1 1 2 3 2 8 3 4

9

5 5 2

### a

### A a

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2 8 3 4 2 5

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### a

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a t76 2 3 6 1 1 2 3

10

7 7 2

### a 1 A a

### a

### A a

### A a

### e

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_{}

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6

###

2 3###

6###

1 1 2 3###

a t21 3

7 7 2

### a 1 A a

### a

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### A a

### e

### a

### a

### a

###

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(20)Where,

1

1 2 1 1 2 3

2

4 5 6 7 8

### a

### 1

### K Sc Q Pr

### A

### Sr Sc Pr , A

### A Q , a

### , a

### , a

### ,

### Pr Sc

### Pr Sc

### a

### Gr

### a

### Q Pr

### Gc

### a

### K Sc

### Gc

### a

### , a

### , a

### , a

### , a

### Pr 1

### Pr 1

### Sc 1

### Sc 1

### Pr 1

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y a y a

0

### y

### y

### e

### erfc

### at

### e

### erfc

### at

### 2 t

### 2 t

###

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1

### K, y, Sc, t ,

2### Q, y, Pr, t ,

3### K

### a , y, Sc, t ,

2###

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###

4

### Q a , y, Pr, t

2###

###

’

###

5###

### a, y, 1, t ,

###

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6###

### a

### a , y, 1, t ,

5###

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7

### a

### a , y, 1, t ,

7 8### a

### a , y, 1, t ,

2###

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9

### Q

### a , y, Pr, t ,

5 10### K

### a , y, Sc, t

7###

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### 1

y x z### y

### z

y x z### y

### z

### x , y , z , t

### e

### erfc

### x t

### e

### erfc

### x t

### 2

### 2

### t

### 2

### t

###

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###

**5.** **Skin friction coefficient: **

The non dimensional skin friction at the plate in the direction of flow is given by:

###

###

###

###

###

###

###

###

###

5

7

6 2 3

2 8 3 4

5

5 7

y 0

a t

8 1 1 2 3

2 8 3 4

6

5 5 2

a t

6 2 3 6 1 1 2 3

7

7 7 2

6

a 1 A a A a a a

u 1

erf at + t

y 2 a a a

a A a A a A a a a

e

a a a

a 1 A a a A a A a

e

a a a

a A
_{} _{} _{} _{}
_{} _{}
_{} _{}

###

###

###

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###

###

###

###

###

###

2 5 21 1 2 3 8 1 1 2 3 a t

8

7 2 5 2

a t

8 1 1 2 3

2 8 3 4

9

5 5 2

8 1 1 2 3 a t

2 8 3 4

2 4

5 5 2

6 2 3

a A a a A a A a

e

a a a a

a A a A a A a a a

+ e

a a a

a A a A a A a a a

e

a a a

a 1 A a
+
_{} _{}
_{} _{}
_{}
_{} _{} _{} _{}

###

###

###

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###

###

7 2 a t6 1 1 2 3

10

7 7 2

6 2 3 6 1 1 2 3 a t

1 3

7 7 2

a A a A a e

a a a

a 1 A a a A a A a

e

a a a

_{}
_{} _{} _{} _{}
** **
**(**21)

**6.** **Heat transfer coefficient: **

The coefficient of heat transfer at the plate in terms of Nusselt number is given by:

2 y 0

### Nu

### y

_{}

###

###

###

###

_{}

_{}

###

###

###

###

(22)**7.** **Mass transfer coefficient: **

The mass flux in terms of Sherwood number is given by

###

###

a t2###

###

2 3 1 2 3 2 1 1 2 3 3 4

y 0

### C

### Sh

### 1 A a

### A a

### (A a

### A a )e

### y

_{}

###

###

###

###

_{}

_{}

###

###

###

###

###

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###

###

(23)Where,

###

###

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###

###

###

###

###

###

###

###

1 2 3 2

4 2 5 6 5

7 7 8 2

9 5 10 7

### Sc, K, t ,

### Pr, Q, t ,

### Sc, K+a , t ,

### Pr, Q+a , t ,

### 1, a, t ,

### 1, a+a , t ,

### =

### 1, a+a , t ,

### =

### 1, a+a , t ,

### =

### Pr, Q+a , t ,

### =

### Sc, K+a , t

###

###

###

###

###

###

###

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###

###

###

###

###

###

###

_{x , z , t}

###

### x

_{e}

z t _{x z erf}

###

_{z t}

### t

###

###

###

###

**8.** **Results and Discussions **

In order to have a physical view of the problem, we have computed the numerical calculations for non dimensional velocity field, concentration field, temperature field, skin friction, Sherwood number, Nusselt number at the plate for different values of the physical parameters involved and their effects have been demonstrated in graphs. Through out our investigation the value of solutel Grashof number Gc has been fixed at 4. The values of Prandtl number Pr are taken as 0.71 and 7 which correspond to air and water respectively at 200 C and the values of Schmidt number Sc are choosen as 0.22, 0.3, 0.6, 0.92 which corresponds to H2 ,He, H2O

and CO2 respectively and the respective values of the other physical parameter namely magnetic field parameter

M, chemical reaction parameter K, Soret number Sr and heat absorption parameter Q are chosen arbitrarily.

Figures 1-6 demonstrate the behaviour of the fluid velocity distribution against normal coordinate Y under the influence of magnetic field parameter M, chemical reaction parameter K, Schmidt number Sc, soret number Sr, heat absorption parameter Q and solutel Grashof number Gc. We observe that there is a steady fall in fluid velocity for increasing values of M, K, Sc and Q. That is to say that the fluid flow is retarded due to the imposition of the transverse magnetic field. This observation is in excellent agreement to the physical fact that the Lorentz forces that appears due to interaction of the fluid velocity and the transverse magnetic field, acts as a resistive force to the flow for which the fluid motion is decelerated. The rise in the fluid velocity may also be controlled to some extent by increasing the chemical reaction rate or increasing the values of the heat absorption parameter. It is inferred from figure 3 that the fluid velocity decreases comprehensively when Sc increases. We recall that a rise in Sc indicates that mass diffusivity causes the flow to accelerate substantially. Figure 4 and figure 6 show that a rise in Soret number Sr or solute Grashof number Gc leads the fluid velocity to increase considerably. All these figures uniquely establish the fact that the fluid velocity first rises in a very thin layer adjacent to the plate and thereafter it declines asymptotically to its minimum value as y→ ∞. This is due to the effect of the buoyancy force which is more pronounced in the fluid region adjacent to the plate and it effect gets nullified as move away from the plate.

The effects of the Sr ,Sc and K on the concentration field are illustrated in figures 7, 8 and 9. From these figures, we see that concentration level of the fluid marginally auguments under the thermal diffusion effect, whilst an increase in Sr or K results in a significant fall in the fluid concentration. Further the concentration falls asymptotically from its maximum value 1 with minimum value 0 as y varies from 0 to ∞ irrespective of choice of the values of Sr, Sc and K.

Figures 10, 11 and 12 show how the fluid temperature is affected by the variations in the values of Q, Pr and t. It is marked from the figures that there is a steady fall in the temperature field for increasing values of Q and Pr whereas as time progresses the temperature rises significantly. The observation that the temperature drops as Q rises is consistent with the physical fact that when heat is absorbed by sink, there is a natural tendency of fall in temperature field. Like concentration field the temperature field also decreases asymptotically from its maximum values at y = 0 with minimum values at y → ∞.

Figures 13 to 18 depict the variation of the coefficient of skin fraction under the influence of Hartman number M, chemical reaction parameter K, Schmidt number Sc, Soret number Sr, heat absorption parameter Q and solutel Grashof number Gc. It is observed from these figures that the coefficient of skin fraction increases due to increase of the values of solutal Grashof number Gc and soret number Sr indicating the fact that the viscous drag at the plate is increased under the effect of thermal diffusion as well as buoyancy force for mass transfer. The figures further demonstrate that the coefficient of skin friction is reduced under the effect of M, Q, K and Sc. In other words, the viscous drag at the plate may be controlled by imposing the transverse magnetic field or by increasing the rate of heat absorption. An increase in the rate of first order homogeneous chemical reaction has also some contribution in reducing the drag force on the plate due to viscosity. Figure 18 depicts that the frictional resistance at the plate slowly and steadily increases due to mass diffusivity.

From figure 19 it is observed that the thermal diffusion leads to a fall in mass flux at the plate. However, figure 20 exhibits that an increase in Schmidt number cause a corresponding augmentation in the mass flux at the plate. Hence the rate of mass transfer is enhanced on account of decrease in chemical molecular diffusivity. Figure 21 exhibits a rise in mass flux corresponding to increase in consumption of the chemical species. Figures 22 and 23 demonstrates that the rate of mass of transfer at the plate drops owing to an increase in sink strength and Prandtl number. It is further noted that the mass flux at the plate steadily declines with increasing intervals of time.

From figure 24, it is observed that a rise in Prandtl number augments the rate of heat transfer at the plate. The heat flux decreases as time increases

Figure1: Velocity distribution versus y for Gr=4,

Gc=4, Sr=.5, Sc=.3, Pr=.71, K=1, Q=1, α=1, t=.5

Figure2: Velocity distribution versus y for Gr=4,

Gc=4, Sr=.5, Sc=.3, Pr=.71, M=1, Q=1, α=1, t=.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.40.81.21.6 2 2.42.83.23.6 4 4.44.85.25.6

**u**

**→**

**y→**

M=1 M=2 M=3

0 0.2 0.4 0.6 0.8 1

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6

**u**

** **

**→**

**y→**

K=1 K=2 K=3

Figure3: Velocity distribution versus y for Gr=4,

Gc=4, Sr=.5, Pr=.71, M=1, K=1, Q=1, α=1, t=.5

Figure4: Velocity distribution versus y for Gr=4,

Gc=4, Sc=.3, Pr=.71, M=1, K=1, Q=1, α=1, t=.5

Figure5: Velocity distribution versus y for Gr=4,

Gc=4, Sr=.5, Sc=.3 Pr=.71, M=1, K=1, α=1, t=.5

‐0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 0.40.81.21.6 2 2.42.83.23.6 4 4.44.85.25.6 6

**u**

**→**

**y→**

Sc=.22 Sc=.3 sc=.6 Sc=.92

0 0.2 0.4 0.6 0.8 1 1.2

0 0.40.81.21.6 2 2.42.83.23.6 4 4.44.85.25.6 6 6.4

**u **

**→**

**y →**

Sr=.5 Sr=1.5 Sr=2.5

0 0.2 0.4 0.6 0.8 1 1.2

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6

**u**

**→**

**y→**

Figure6: Velocity distribution versus y for Gr=4,

Sr=.5, Sc=.3, Pr=.71, M=1, K=1, Q=1, α=1, t=.5

Figure7: Concentration distribution versus y for

Sc=.3, Pr=.71, K=1, Q=1, t=.5

Figure8: Velocity distribution versus y for

Sr=.5, Pr=.71, K=1, Q=1, t=.5

0 0.2 0.4 0.6 0.8 1 1.2

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6

**u**

**→**

**y→**

Gc=3 Gc=4 Gc=5

0 0.2 0.4 0.6 0.8 1 1.2

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6 6.4

**C**

**→**

**y →**

Sr=.5 Sr=1 Sr=1.5

‐0.2 0 0.2 0.4 0.6 0.8 1 1.2

0 0.40.81.21.6 2 2.42.83.23.6 4 4.44.85.25.6 6 6.4

**C **

**→**

**y →**

Sc=.22 Sc=.3 Sc=.6 Sc=.92

Figure9: Concentration distribution versus y for

Sr=.5, Sc=.3, Pr=.71, Q=1, t=.5

Figure10: Temperature distribution versus y for

Pr=.71, t=.5

Figure11: Temperature distribution versus y for

Q=1, t=.5

0 0.2 0.4 0.6 0.8 1 1.2

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6

**C**

**→**

**y→**

K=1 K=2 K=3

0 0.2 0.4 0.6 0.8 1 1.2

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8

**θ→**

**y→**

Q=.5 Q=1.5 Q=2.5

0 0.2 0.4 0.6 0.8 1 1.2

‐0.2 0.2 0.6 1 1.4 1.8 2.2 2.6 3 3.4 3.8 4.2 4.6 5

**θ→**

**y→**

Figure12: Temperature distribution versus y

Pr=.71, Q=1

Figure13: skin friction versus t for Gr=4, Sr=.5,

Sc=.3, Pr=.71, M=1, K=1, Q=1, α=1

Figure14: skin friction versus t for Gr=4, Gc=4,

Sr=.5, Sc=.3, Pr=.71, K=1, Q=1, α=1

0 0.2 0.4 0.6 0.8 1 1.2

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4

**θ**

**→**

**y→**

t = .2 t =.5 t =.8

0 0.5 1 1.5 2 2.5 3

0.1 0.2 0.3 0.4 0.5 0.6

**ζ**

**→**

**t →**

Gc=3 Gc=4 Gc=5

0 0.5 1 1.5 2 2.5

0.1 0.2 0.3 0.4 0.5 0.6

**ζ→**

**t →**

M=1 M=2 M=3

Figure15: skin friction versus t for Gr=4, Gc=4,

Sr=.5, Sc=.3, Pr=.71, M=1, K=1, α=1

Figure16: skin friction versus t for Gr=4, Gc=4,

Sr=.5, Sc=.3, Pr=.71, M=1, Q=1, α=1

Figure17: skin friction versus t for Gr=4, Gc=4,

Sc=.3, Pr=.71, M=1, K=1,Q=1, α=1

0 0.5 1 1.5 2 2.5

0.1 0.2 0.3 0.4 0.5 0.6

**ζ**

**→**

**t→**

Q=.5 Q=1.5 Q=2.5

0 0.5 1 1.5 2 2.5

0.1 0.2 0.3 0.4 0.5 0.6

**ζ→**

**t→**

K=1 k=2 k=3

0 0.5 1 1.5 2 2.5

0.1 0.2 0.3 0.4 0.5 0.6

**ζ**

**→**

**t →**

Figure18: skin friction versus t for Gr=4, Gc=4,

Sr=.5, Pr=.71, M=1, K=1, Q=1, α=1

Figure19: Sherwood number versus t for

Sc=.3, Pr=.71, K=1, Q=1

Figure20: Sherwood number versus t for

Sr=.5, Pr=.71, K=1, Q=1

0 0.5 1 1.5 2 2.5

0.1 0.2 0.3 0.4 0.5 0.6

**ζ**

**→**

**t→**

Sc=.22 Sc=.3 Sc=.6

0 0.2 0.4 0.6 0.8 1

0.1 0.2 0.3 0.4 0.5 0.6

**Sh**

**→**

**t→**

Sr=.5 Sr=1 Sr=1.5

0 0.5 1 1.5 2

0.1 0.2 0.3 0.4 0.5 0.6

**Sh**

**→**

**t→**

Sc=.22 Sc=.3 Sc=.6 Sc=.92

Figure21: Sherwood number versus t for

Sr=.5, Sc=.3, Pr=.71, Q=1

Figure22: Sherwood number versus t for

Sr=.5, Sc=.3, Pr=.71, K=1

Figure23: Sherwood number versus t for Sr=.5, Sc=.3, K=1, Q=1

0 0.2 0.4 0.6 0.8 1 1.2

0.1 0.2 0.3 0.4 0.5 0.6

**Sh**

**→**

**t→**

K=1 K=2 K=3

0 0.2 0.4 0.6 0.8 1

0.1 0.2 0.3 0.4 0.5 0.6

**Sh**

**→**

**t→**

Q=.5 Q=1.5 Q=2.5

0 0.2 0.4 0.6 0.8 1

0.1 0.2 0.3 0.4 0.5 0.6

**Sh**

**→**

**t→**

Figure24: Nusselt number versus t for Q=1

Table 1: Values of concentration for Sr=.5, Sc=.3, Pr=.71, K=1, t=.5

y Q=.5 Q=1.5 Q=2.5

0 1 1 1

0.4 0.791194 0.798957 0.805526

0.8 0.615314 0.622601 0.62846

1.2 0.467589 0.471736 0.474913

1.6 0.345088 0.346284 0.347117

2 0.246019 0.245431 0.244916

2.4 0.168746 0.167491 0.166498

2.8 0.111074 0.109841 0.108885

3.2 0.0700692 0.069138 0.0684197

3.6 0.0423419 0.0417398 0.0412742

4 0.0245085 0.0241599 0.0238883

4.4 0.0135891 0.0134038 0.0132577

4.8 0.00721753 0.00712559 0.00705214

5.2 0.00367157 0.00362852 0.00359361

5.6 0.00178841 0.00176924 0.00175348

6 0.000833385 0.000825703 0.000818905

**9.** **Conclusion **

1. The fluid flow is retarded due to the imposition of the transverse magnetic field and for the increasing values of heat absorption parameter.

2. The concentration level of the fluid marginally gets enhanced due to Soret effect. 3. There is a steady fall in the temperature effect under the effect of heat sink

4. Viscous drag at the plate is reduced due to imposition of the transverse magnetic field or by increasing the rate of heat absorption

5. The rate of mass transfer is enhanced on account of decreasing chemical molecular diffusivity.

**References **

[1] Ahmed, N.; Das, K. K. (2013): “Soret effect on an MHD free convective mass transfer flow past an accelerated vertical plate with chemical reaction”, Journal of Energy, Heat and Mass Transfer, 35, pp. 133-147.

[2] Anghel, M.; Takhar, H. S.; Pop, I. (2000): “Dufour and Soret effects on free convection boundary layer over a vertical surface embedded in a porous medium”, Studia Universities Babes-Boyai, Mathematica, 45, pp. 11-12.

[3] Chambre, P. L.; Young, J. D. (1958): “On the diffusion of a chemically reactive species in a laminar boundary layer flow, Phys. Fluids Flow, 1, pp. 48-54.

[4] Chaudhary, R. C.; Jain Arpita (2007): “Combined heat and mass transfer effects on MHD free convection flow past an oscillating plate embedded in porous medium, Rom. Journal. of Physics, 52, pp. 505-524.

[5] Israel-Cookey, C.; Amos, E.; Nwaigwe, C. (2010): “MHD oscillatory couette flow of a radiating viscous fluid in a porous medium with periodic wall temperature”, American Journal of Scientific and Industrial Research, 1(2), pp. 326-331.

0 1 2 3 4 5 6 7

0.1 0.2 0.3 0.4 0.5 0.6

**Nu**

** **

**→**

**t →**

Pr =.71 Pr =7

[6] Monsour, M. A.(1990): “Radiation and free convection effects on the oscillating flow past a vertical plate,” Astrophysics and Space Science, 166(2), pp. 269-275.

[7] Muthucumaraswamy, R.; Meenakshisundaram, S. (2006): “Theoretical study of chemical reaction effects on vertical oscillating plate with variable temperature”, Theoret. Appl. Mech., 33(3), pp. 245-257.

[8] Raju, M. C.; Varma, S. V. K.; Reddy, P. V.; Saha, S. (2008): “Soret effect due to normal convection between heated inclined plates”, Journal of Mechanical Engineering, 39, pp. 65-70.

[9] Sattar, M. A.; Maleque, M. A. (2000): “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, pp. 67-72.

[10] Sattar, M. A.; Rahman, M. M.; Alam, M. M. (2000): “Free convection flow and heat transfer through a porous vertical flat plate immersed in a porous medium”, Journal of energy Research, 22(1), pp. 17-21.

[11] Singh, K. D.; Garg, B. P. (2010): “Radiative heat transfer in MHD oscillatory flow through porous medium bounded by two vertical porous plates”, Bull. Cal. Math. Soc., 102, pp. 129-138.