International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 9 (2019) pp. 2107-2120

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Heat and Mass transfer on MHD Rotating flow of a Visco-elastic fluid

through porous medium

Gorakhnath Waghmode 1* and S.V.Suneetha2

1Research Scholar, Department of Mathematics, Rayalaseema University, Kurnool, Andhra Pradesh – 518007, India.

2Department of Mathematics, Rayalaseema University, Kurnool, Andhra Pradesh – 518007, India.

Abstract

We discussed the heat and mass transfer on MHD free

convective rotating flow of a visco-elastic incompressible

electrically conducting fluid past a vertical porous plate

through a porous medium with suction and heat source. We

analyze the effect of time dependant fluctuative suction on a

visco-elastic fluid flow The coupled nonlinear partial

differential equation are turned to ordinary by super imposing

a solution with steady and time dependant transient part.

Finally, the set of ordinary differential equations is solved

with a perturbation scheme to meet the inadequacy of

boundary condition. Elasticity of the fluid and the Lorentz

force reduce the velocity and it is more pronounced in case of

heavier species. Most interesting observation is the fluctuation

of velocity appears near the plate due to the presence of sink

and presence of elastic element as well as heat source reduces

the skin friction.

Keywords: Convective flows, heat and mass transfer, porous

medium, visco-elastic fluids, oscillatory permeability.

1. INTRODUCTION

An important class of two dimensional time dependent flow

problem dealing with the response of boundary layer to

external unsteady fluctuations of the free stream velocity

about a mean value attracted the attention of many

researchers. MHD flow with heat and mass transfer has been a

subject of interest of many researchers because of its varied

application in science and technology. Such phenomena are

observed in buoyancy induced motions in the atmosphere, in

water bodies, quasi-solid bodies such as earth, etc. In natural

processes and industrial applications many transportation

processes exist where transfer of heat and masstakes place

simultaneously as a result of thermal diffusion and diffusion

of chemical species. Several researchers have analyzed the

free convective and mass transfer flow of a viscous fluid

through porous medium. The permeability of the porous

medium is assumed to be constant while the porosity of the

medium may not be necessarily being constant. Kim [1]

studied the unsteady MHD convective heat past a semiinfinite

vertical porous moving plate with variable suction. The

problem of three dimensional free convective flow and heat

transfer through porous medium with periodic permeability

has been discussed by Singh and Sharma [2]. Singh and Singh

[3] have analyzed the heat and mass transfer in MHD flow of

a viscous fluid past a vertical plate under oscillatory suction

velocity. Bathul [4] discussed the heat transfer in a three

dimensional viscous flow over a porous plate moving with

harmonic disturbance. Postelnicu [5] studied numerically the

influence of a magnetic field on heatand mass transfer by

natural convection from vertical surfaces in porous media

considering Soret and Dufour effects. Singh and Gupta [6]

have investigated the MHD free convective flow of a viscous

fluid through a porous medium bounded by oscillatory porous

plate in slip flow regime with mass transfer. Ogulu and

Prakash [7] considered heat transfer to unsteady magneto

hydrodynamic flow past an infinite vertical moving plate with

variable suction. Das et al. [8] analyzed the mass transfer

effect on unsteady flow past an accelerated vertical porous

plate with suction employing numerical methods. The basic

equations of incompressible MHD flow are non-linear. But

there are many interesting cases where the equations became

linear in terms of the unknown quantities and may be solved

easily. Linear MHD problems are accessible to exact solutions

and adopt the approximations that the density and transport

properties be constant. No fluid is incompressible but all may

be treated as such whenever the pressure changes are small in

comparison with the bulk modulus. Ferdows et al. [9]

analyzed free convection flow with variable suction in

presence of thermal radiation. Alam et al. [10] studied Dufour

and Soret effect with variable suction on unsteady MHD free

convection flow along a porous plate. Mishra et al. [16] have

studied fluctuating flow of a non-Newtonian fluid past a

porous flat plate with time varying suction. Majumdar and

Deka [11] gave an exact solution for MHD flow past an

impulsively started infinite vertical plate in the presence of

thermal radiation. Muthucumaraswamy et al. [12] studied

unsteady flow past an accelerated infinite vertical plate with

variable temperature and uniform mass diffusion. Asghar et

al. [13] have reported the flow of a non-Newtonian fluid

induced due to oscillations of a porous plate. Moreover, Dash

et al. [14] have studied free convective MHD flow of a visco-

elastic fluid past an infinite vertical porous plate in a rotating

frame of reference in the presence of chemical reaction. The

specific area of application of heat and mass transfer

phenomena is common in chemical process industries such as

food processing and polymer production. Recently, Gireesh

Kumar and Satyanarayana [17], Sivraj and Rushi Kumar [18]

have considered MHD flow of visco-elastic fluid with short

memory (Walters model). Many authors [20-24] discussed

MHD flows through porous medium in planar channels.

Recently, Krishna and Gangadhar Reddy [25] discussed the

unsteady MHD free convection in a boundary layer flow of an

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electrically conducting fluid through porous medium subject

to uniform transverse magnetic field over a moving infinite

vertical plate in the presence of heat source and chemical

reaction. Krishna and Subba Reddy [26] have investigated the

simulation on the MHD forced convective flow through

stumpy permeable porous medium (oil sands, sand) using

Lattice Boltzmann method. Krishna and Jyothi [27] discussed

the Hall effects on MHD Rotating flow of a visco-elastic fluid

through a porous medium over an infinite oscillating porous

plate with heat source and chemical reaction. Reddy et al.[28]

investigated MHD flow of viscous incompressible nano-fluid

through a saturating porous medium. Recently, Krishna et al.

[29-32] discussed the MHD flows of an incompressible and

electrically conducting fluid in planar channel. Veera Krishna

et al. [33] discussed heat and mass transfer on unsteady MHD

oscillatory flow of blood through porous arteriole. The effects

of radiation and Hall current on an unsteady MHD free

convective flow in a vertical channel filled with a porous

medium have been studied by Veera Krishna et al. [34]. The

heat generation/absorption and thermo-diffusion on an

unsteady free convective MHD flow of radiating and

chemically reactive second grade fluid near an infinite vertical

plate through a porous medium and taking the Hall current

into account have been studied by Veera Krishna and

Chamkha [35]. Veera Krishna et al. [36] discussed the heat

and mass transfer on unsteady, MHD oscillatory flow of

second-grade fluid through a porous medium between two

vertical plates under the influence of fluctuating heat

source/sink, and chemical reaction. Veera Krishna et al. [37]

investigated the heat and mass transfer on MHD free

convective flow over an infinite non-conducting vertical flat

porous plate. Veera Krishna and Jyothi [38] discussed the

effect of heat and mass transfer on free convective rotating

flow of a visco-elastic incompressible electrically conducting

fluid past a vertical porous plate with time dependent

oscillatory permeability and suction in presence of a uniform

transverse magnetic field and heat source. Veera Krishna and

Subba Reddy [39] investigated the transient MHD flow of a

reactive second grade fluid through a porous medium between

two infinitely long horizontal parallel plates. Veera Krishna et

al. [40] discussed heat and mass-transfer effects on an

unsteady flow of a chemically reacting micropolar fluid over

an infinite vertical porous plate in the presence of an inclined

magnetic field, Hall current effect, and thermal radiation taken

into account. Veera Krishna et al.[41] discussed Hall effects

on steady hydromagnetic flow of a couple stress fluid through

a composite medium in a rotating parallel plate channel with

porous bed on the lower half. Veera Krishna et al. [42]

discussed Hall effects on unsteady hydromagnetic natural

convective rotating flow of second grade fluid past an

impulsively moving vertical plate entrenched in a fluid

inundated porous medium, while temperature of the plate has

a temporarily ramped profile. Veera Krishna and Chamkha

[43] discussed the MHD squeezing flow of a water-based

nanofluid through a saturated porous medium between two

parallel disks, taking the Hall current into account. Veera

Krishna et al. [44] discussed Hall effects on MHD peristaltic

flow of Jeffrey fluid through porous medium in a vertical

stratum.

In the above studies mentioned, therefore, the objective of the

present study is to analyze the effect of permeability variation

and oscillatory suction velocity in free convective mass

transfer rotating flow of a visco-elastic fluid (Walters

B’fluid model) past an infinite vertical porous plate through

a porous medium in presence of a uniform transverse

magnetic field and heat source.

2. FORMULATION AND SOLUTION OF THE

PROBLEM

The unsteady free convective rotating flow of a visco-elastic

(Walters B’) fluid past an infinite vertical porous plate in a

porous medium with time dependent oscillatory suction in

presence of a transverse magnetic field is considered. Let x-

axis be along the plate in the direction of the flow and y-axis

normal to it.

Fig. 1: The Physical model

Let us consider the magnetic Reynolds number is much less

than unity so that the induced magnetic field is neglected in

comparison with the applied transverse magnetic field. The

basic flow in the medium is, therefore, entirely due to the

buoyancy force caused by the temperature difference between

the wall and the medium. It is assumed that initially, at t > 0,

the plate as well as fluids is at the same temperature and also

concentration of the species is very low so that the Soret and

Dofour effects are neglected. When t > 0, the temperature of

the plate is instantaneously raised to Tw and the concentration

of the species is set to Cw. The physical configuration of the

problem as shown in Fig. 1.

Let the suction velocity be of the form

0( ) (1 )i tw t w e (1)

where 0 0w and 1 are positive constants.

Under the above assumption with usual Boussinesq’s

approximation, the governing equations with respect to the

rotating frame are given by

0u vx y

(2)

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2 3 3

0

2 2 3

12

ku u p u u uw vt z x z z t z

2

*0 ( ) ( )p

B u u g T T g C CK

(3)

2

2

23 3

0 0

2 3

2

1

p

v vw ut z

p vy z

k Bv vv v vz t z K

(4)

2

12( )

T T Tw k S T Tt z z

(5)

2

2

C C Cw Dt z z

(6)

The boundary conditions are

0, ,

at 0

i tw

i tw

u T T T T e

C C C C e y

(7)

0, , asu T T C C y (8)

Combining (1) & (2) we obtain, let

andq u iv x iy

3 3 3

0

2 2 3

2

1

q qw i qt z

kp q q qz z t z

2

0 ( ) *( )p

B vq q g T T g C CK

(9)

Introducing the non-dimensional quantities,

2

0 0

2

0 0

4* , * , * , * ,

4

, ,

w w

w z w t qz t qw w

T T C pPT T C

Making use of non-dimensional variables, the governing

equations reduces to

22

2

3 3

2 3

1 1(1 ) 2

4

(1 )

i t

i t

q q qe iEq P M qt z z K

q qRc et z z

Gr Gc (11)

2

2

1 1(1 )

4 Pr

i te St z z

(12)

2

2

1 1(1 )

4 Sc

i tet z z

(13)

The boundary conditions are,

0, 1 , 1 at 0

0, 0, 0 as

i t i tq e e zq z

(14)

Where,

1

2

0

SSv

is the Heat source parameter,

2

0

2

pw KK

is the

Permeability parameter, 2

2 0

2

0

BMw

is the Hartmann

number, 2

0

Ew

is the rotation parameter, Prk

is the

Prandtl number, 3

0

*( )Gr wg T T

w

is the Thermal

Grashof number, 3

0

( )Gc wg C C

w

is the mass

Grashof number, ScD

is the Schmidt number,

0 0

24

k wRc

is the visco-elastic parameter.

In view of periodic suction, temperature and concentration at

the plate let us assume the velocity, temperature,

concentration in the neighbourhood of the plate be

0 1, i tq y t u y u y e (15)

0 1, i ty t T y T y e (16)

0 1, i ty t C y C y e (17)

The above method of solution has been adopted by Mishra et

al. [16], Das et al. [19] and Gireesh Kumar and Satyanarayana

[17] to solve the periodically fluctuating flow problems.

Substituting Eqs. (15) – (17) into (11) – (13) and equating the

non-harmonic and harmonic terms, we get

'" '" ' 2

0 0 0 0

0 0

1Rc 2

Gr Gc

u u u M iE uK

T C (18)

'" '" ' 2

1 1 1 1

3

0 0 01 13

1Rc 1 Rc 2

Rc Gr Gc

u i u u M iE uK

d u du uT Cdy dy K

(19)

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2110

" '

0 0 0Pr Pr 0T T ST (20)

" ' '

1 1 1 0Pr Pr4

iT T P S T T

(21)

" '

0 0Sc 0C C (22)

" ' '

1 1 1 0Sc Sc Sc 4

iC C T C (23)

The boundary conditions reduce to

0 1 0 1 0 1

0 1 0 0 1

0, 1, 0 at 0,

0, 0, 0, as

u u T T C C yu u T T C C y

(24)

Eqs. (18) and (19) are of third order but two boundary

conditions are available. Therefore the perturbation method

has been applied using 1Rc Rc , the elastic parameter

as the perturbation parameter.

2

0 00 01

2

1 10 11

( ) Rc ( ) Rc ,

( ) Rc ( ) Rc

u u y u y O

u u y u y O

(25)

Inserting Eq. (25) into (18) and (19) and equating the

coefficients of Rc0 and Rc to zero we have following sets of

ordinary differential equations zeroth order and first or der;

" ' 2

00 00 00 0 0

12 Gr Gc u u M iE u T C

K

(26)

" ' 2 '"

01 01 01 00

12u u M iE u u

K

(27)

" ' 2

10 10 10

' 0000 1 1

12

Gr Gc

u u M iE uK

uu T CK

(28)

" ' 2

11 11 11

'" " '" 0110 10 00

12

u u M iE uK

uu i u uK

(29)

The corresponding boundary conditions are:

00 01

10 11

0, as 0,

0, as

u u yu u y

(30)

Solving these differential equations with the help of boundary

conditions we get,

4

1

1 22

Sc

1 22

( , )1

2

12

m yo

m y y o

Pq y t a a eM iE

KPa e a e

M iEK

4 4 1 Sc

3 4 5 3 4 5( ) )m y m y m y ycR a a a e a e a e a e

5 34

1 2

5 34

2 1

118 12 13

6

Sc 114 15 16 17

6

25 31 26 27

Sc

25 29 30

( )

m y m ym y

m y m yy

m y m ym y

m y m y y i t

Pa e a e a ea

Pa e a e a e aa

Rc a a e a e a e

a e a e a e e

(31)

3 31 114( , )

m y m ym y m y i timy t e e e e e

(32)

24 4

( , ) 1c cS y S ym y i tc ciS iSy t e e e e

(33)

The skin friction at the plate in terms of amplitude and phase

angle is given by

0

0 010

0

cosy

i ty

y

u uue N ty y y

(34)

Where, , tan ir i

r

NN N iNN

The rate of heat transfer, i.e. heat flux at the (Nu) in terms

of amplitude and phase is given by,

0

0 10

0

0 cos

y

i tu y

y

T TN ey y

T R ty

(35)

, tan ir i

r

RR R iRR

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The mass transfer coefficient, i.e, the Sherwood number (Sh)

at the plate in terms of amplitude and phase is given by

0

0 10

0

0 cos

y

i th y

y

C CS ey y

C Q ty

(36)

Where, , tan ir i

r

QQ Q iQQ

4. RESULTS AND DISCUSSION

The present study considers the effect of mass transfer on free

convective rotating flow and heat transfer of a visco-elastic

incompressible electrically conducting fluid past a vertical

plate through a porous medium with time dependent suction in

presence of a constant transverse magnetic field and heat

source. The significance of the present study is to bring out

the effect of oscillatory characteristics of suction velocity,

plate temperature and concentration. The common feature of

the velocity profile is parabolic with picks near the plate. It is

interesting to note that due to dominating effect of heat source

no oscillatory behaviour of the profile is exhibited but in the

presence of sink an oscillation in the profile is well marked.

The result of the present study is verified by comparing with

that of Das et al. [19] for viscous case Rc = 0. The effect of Rc

is to thinning of velocity boundary layer which may be

attributed due to elastic property of the fluid under

consideration. Our result is in conformity with the result of

Das et al. [19].

The flow governed by the non-dimensional parameters

Hartmann number M, permeability parameter K, rotation

parameter E, thermal Grashof number Gr, mass Grashof

number Gc, Elastic parameter Rc, the frequency of

oscillation, Schmidt number Sc, Prandtl number Pr, Heat

source parameter S, time t. The velocity, temperature and

concentration profiles are shown on Figures (2-12), Figures

(13-16) and Figures (17-18) respectively. Tables (1-3)

represent the skin friction, Nusselt number and Sherwood

number for different variations in the governing parameters.

We noticed that from the Fig. 2, the velocity components u

and v reduce with increasing the Hartmann number M.

Further, due to interaction of applied magnetic field with

conducting fluid the current is generated. Consequently, it

gives rise to an induced magnetic field and ponders motive

force. That force acts opposite to the direction of the flow and

unless an additional electric field is applied to counter act. In

the present study the Lorentz force decelerates the fluid flow

consequently resulted in thinning of boundary layer. The

resultant velocity is also reduces with increasing the intensity

of the magnetic field. Both the velocity components u and v

enhance with increasing permeability parameter K. i.e., both

the real and imaginary parts of the velocity enhance with

increasing the permeability parameter K throughout the fluid

region. We also observe that lower the permeability lesser the

fluid speed in the entire fluid medium (Fig. 3). The similar

behaviour is observed with increasing the thermal Grashof

number Gr, mass Grashof number Gc or rotation parameter E

(Figs. 4-6). It is also observed that, from the Figures (7-8)

both the velocity components continuously retardation in its

magnitude with increasing elastic parameter Rc or the

frequency of oscillation . The figure (9) reveals that, the

primary velocity component u reduces and secondary velocity

component v increases for 2z and the gradually reduces

with increasing Schmidt number Sc. Likewise, from the

figures (10 & 11) both the velocity components u and v

reduces with increasing Heat source parameter S or Prandtl

number Pr. The resultant velocity is also reduces with

increasing S or Pr. The effect of source parameter is well

marked from Fig. (10). Presence of source reduces the

velocity, significantly in all the layers. Most interesting case is

due to the presence of sink which gives rise to a span wise

oscillatory flow after a few layers from the plate. This is due

to depletion of thermal energy causing a sudden reduction of

velocity. Again it is compensated due to buoyancy effect.

Finally with increasing time the velocity components u, v and

resultant velocity enhance throughout the fluid region Figs.

12). Hence we observed that the free convection due to

thermal buoyancy, mass buoyancy and porosity of the

medium enhances the fluid velocity whereas Schmidt number

and Prandtl number reduce it. Kumar et al. [17] observed that

the velocity profile increases with increase of thermal Grashof

number and Solutal Grashof number and they have also

observed that the velocity decreases with increase in magnetic

parameter. Hence our observation in respect of these

parameters agree with [17] in the absence of porous medium.

Thus, the above result indicates that heavier species with

lower thermal conductivity reduces the fluid flow.

Figs. (13-16) exhibits the effects of Prandtl number (Pr),

source/sink parameter S, the frequency of oscillation and time

on temperature field. The variation is asymptotic in nature.

Sharp fall of temperature is noticed in case of water (Pr = 7.0)

for both source/sink. Absence of source contributes to

increase the thickness of thermal boundary layer. Higher

Prandtl number fluid causes lower thermal diffusivity and

hence reduces the temperature at all points. It is interesting to

note that the effect of source is to lower down the temperature

than the effect of sink. The temperature increases with

increasing frequency of oscillation and time throughout the

fluid region. From Figs. (17-19), it is observed that the

concentration distribution asymptotically decreases with the

heavier species. This result is in good agreement with Das et

al. [19]. Likewise the concentration enhances with increasing

frequency of oscillation and time throughout the fluid region.

We noticed from the Table (1) we present the values of skin

friction. The increase of visco-elastic parameter (Rc) leads to

decrease the skin friction. Further, it is to note that an increase

in Gr, Gc, Pr, and K leads to exert greater skin friction on

the boundary whereas Sc, M, S, t and Rc reduce it.

From the Table 2 it is seen that an increase in Pr or S leads to

an increase in Nusselt number. Nusselt number reduces with

increasing the frequency of oscillation. There is no effect of

time on Nusselt number.

It is noticed from the table (3) that the Schmidt number i.e.

mass transfer coefficient affects Sherwood number in a

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similar manner as that of Prandtl number, the frequency of oscillation and time in heat transfer.

Fig. 2 The velocity profiles for the components u and v against M

0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1K E S t

Fig. 3 The velocity profiles for the components u and v against K

1, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1M E S t

Fig. 4 The velocity profiles for the components u and v against E

1, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1M K S t

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Fig. 5 The velocity profiles for the components u and v against Gr

1, 0.5, 0.5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1M K E S t

Fig. 6 The velocity profiles for the components u and v against Gc

1, 0.5, 0.5,Gr 5,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1,M K E S t

Fig. 7 The velocity profiles for the components u and v against Rc

1, 0.5, 0.5,Gr 5,Gc = 3, / 6, 0.5,Sc 0.22,Pr 0.71, 0.1M K E S t

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Fig. 8 The velocity profiles for the components u and v against

1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, 0.5,Sc 0.22,Pr 0.71, 0.1M K E S t

Fig. 9 The velocity profiles for the components u and v against Sc

1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, 0.5,Sc 0.22,Pr 0.71, 0.1M K E S t

Fig. 10 The velocity profiles for the components u and v against S

1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6,Sc 0.22,Pr 0.71, 0.1,M K E t

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Fig. 11 The velocity profiles for the components u and v against Pr

1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22, 0.1M K E S t

Fig. 12 The velocity profiles for the components u and v against t

1, 0.5, 0.5,Gr 5,Gc = 3,Rc 0.01, / 6, 0.5,Sc 0.22,Pr 0.71M K E S

Fig. 13 The temperature profiles for against Pr with 0.01, / 6, 0.5, 0.1S t

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Fig. 14 The temperature profiles for against S with 0.01, / 6,Pr 0.71, 0.1t

Fig. 15 The temperature profiles for against with 0.01, 0.5,Pr 0.71, 0.1S t

Fig. 16 The temperature profiles for against t with 0.01, / 6, 0.5,Pr 0.71S

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Fig. 17 The Concentration profiles for against t with 0.01, / 6,Sc 0.22, 0.1t

Fig. 18 The Concentration profiles for against t with 0.01,Sc 0.22, 0.1t

Fig. 19 The Concentration profiles for against t with 0.01, / 6,Sc 0.22

Table 1: Shear stresses

M K S Gr Gc Sc ω Rc Pr t

1 0.5 0.5 5 3 0.22 / 6 0.001 0.71 0.1 0.647996

1.5 0.509526

2 0.400457

1 0.714341

1.5 0.749688

1 0.601219

1.5 0.579154

6 0.651212

7 0.653632

4 0.666612

5 0.682836

0.3 0.621506

0.6 0.544473

/ 4 0.913730

/ 3 1.135170

0.01 0.647996

0.05 0.647996

3 2.666330

7 -0.11972

0.5 0.438557

0.8 0.281477

Table 2: Nusselt number

Pr ω t S Nu

0.71 / 6 0.1 0.5 1.049300

3 3.443800

7 7.491330

/ 4 1.049020

/ 3 1.048870

0.5 1.049300

0.8 1.049300

0.8 1.189220

1 1.270730

Table 3: Sherwood number

Sc ω t Sh

0.22 / 6 0.1 0.220035

0.3 0.300029

0.6 0.600013

/ 4 0.220062

/ 3 0.220087

0.5 0.220035

0.8 0.220035

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4. CONCLUSIONS

We have considered the effect of heat and mass transfer on

free convective rotating flow of a visco-elastic incompressible

electrically conducting fluid past a vertical porous plate

through a porous medium with time dependent suction in

presence of a uniform transverse magnetic field and heat

source. The conclusions are made as the following.

1. Presence of heat source prevents the oscillatory flow

whereas the sink favours it. Elasticity of the fluid is

responsible for thinning of the boundary layer.

2. Application of magnetic field decelerates the fluid

flow.

3. The heavier species with low conductivity reduces

the flow within the boundary layer.

4. An increase in elasticity of the fluid leads to

decrease the velocity which is an established result.

5. The concentration distribution asymptotically

decreases with the heavier species.

6. Elasticity’s of the fluid and heat source reduce the

skin friction providing a favourable condition for

stretching.

Appendix:

12 2

1 1

Gr

12

am m M iE

K

,

22 2

Gc

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aM iE

K

3

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2 2

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( )

12

a a mam m M iE

K

, 3

1 14

2 2

1 1

Gr

12

a mam m M iE

K

3

25

2 2

Gc Sc

1Sc Sc 2

aaM iE

K

,

2

6

12a M iE

K

, 7 4 1 2( )a m a a

18

4ima

,

9

4 sc1

ia

,

10

4 scia

,

1 211

a aak

,

7 1112 2

4 4 6

a aam m a

813 2

3 3 6

Gr(1 )aam m a

,

814 2

1 1 6

.Gr aam m a

’

1015 2

6

Gc

Sc Sc

aaa

,

916 2

2 2 6

Gcaam m a

1117

6

aaa

, 18 12 13 14 15 16 17a a a a a a a

,

3 2

19 18 5 18 5( )a a m i a m

3 2 3 3 4 520 12 4 12 4 1 2 4( )

a a aa a m i a m a a mk

,

3 2

21 13 3 13 3a a m i a m

3 2

22 16 2 16 2a a m i a m ,

3 2 3

23 14 1 14 1 1 1Gra a m i a m a m ,

3 2 3

24 15 15 2Sc Sc Gc Sca a i a a

1925 2

5 5 6

aam m a

,

2026 2

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aam m a

,

2127 2

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,

2228 2

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2329 2

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2430 2

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aaa

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31 25 26 27 28 29 30( )a a a a a a a

2

1

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2

Sm

,

2

2

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2

im

,

2

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2

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,

2

4

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2

M iEKm

,

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5

1 1 4

2

am

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