Theoretical investigation of an unsteady MHD free convection heat and mass transfer flow of a...

Theoretical investigation of an unsteady MHD free convection heat and mass transfer flow of a non-Newtonian fluid flow past a permeable

moving vertical plate in the presence of thermal diffusion and heat sink

V. Ravikumar1, a, M. C. Raju2, b* and G. S. S. Raju3,c 1, 2 Department of Humanities and Sciences (Mathematics), Annamacharya Institute of Technology

and Sciences, (Autonomous), Rajampet- 516126, A.P, India

3 Department of Mathematics, JNTUA College of Engineering Pulivendula, Pulivendula, A.P, India.

[email protected], b*[email protected], [email protected] * Corresponding author

Keywords: MHD, Rivlin-Ericksen fluid, heat and mass transfer, source or sink and vertical moving porous plate.

Abstract: The problem of unsteady, two-dimensional, laminar, boundary-layer flow of a viscous,

incompressible, electrically conducting and heat-absorbing Rivlin-Ericksen flow fluid along a semi-

infinite vertical permeable moving plate has been investigated. A uniform transverse magnetic field

is applied in the direction of the flow. The presence of thermal and concentration buoyancy effects

is considered. The plate is assumed to move with a constant velocity in the direction of fluid flow

while the free stream velocity is assumed to follow the exponentially increasing small perturbation

law. Time-dependent wall suction is assumed to occur at the permeable surface. The dimensionless

governing equations for this investigation are solved analytically using two-term harmonic and non-

harmonic functions. Numerical evaluation of the analytical results is performed and some graphical

results for the velocity, temperature and concentration distributions within the boundary layer are

presented. Skin-friction coefficient, Nusselt number and Sherwood number are also discussed with

the help of the graphs. Local skin-friction coefficient increases with an increase in the permeability

parameter, and Soret number whereas reverse effects is seen in the case of dimensionless

viscoelasticity parameter of the Rivlin-Ericksen fluid. Nusselt number decreases in the presence of

heat absorption. The presence of Soret number Sherwood number increases.

1. Introduction:

The problem of boundary layer flow over a continuously moving solid surface is an

important type of flow occurring in many industrial processes, such as heat treated materials

traveling between a feed roll and a wind-up roll or materials manufactured by extension, glass fiber

and paper production. Convection of a heated or cooled vertical plate is one of the fundamental

problems in heat and mass transfer studies in recent times. If the existing free convection is

accompanied by an external flow, the combined mode of free and forced convection exists.

Magneto hydrodynamics (MHD) is important in many engineering applications such as, in MHD

power generators, cooling of nuclear reactors, and the boundary layer control in aerodynamics and

crystal growth. Transport processes in porous media play a significant role in various applications,

such as thermal insulation, energy conservation, petroleum industries, solid matrix heat exchangers,

geothermal engineering, chemical catalytic reactors, and underground disposal of nuclear waste

materials. In many transport processes in nature and in industrial applications, the heat and mass

transfer with variable viscosity is a consequence of buoyancy effects caused by the diffusion of heat

and chemical species. The study of such processes is useful for improving a number of chemical

technologies, such as polymer production and food processing. In nature, the presence of pure air or

water is impossible. Some foreign mass may be presented either naturally or mixed with air or

water. The problem of MHD laminar flow through a porous medium has become very important in

recent years because of its possible applications in many branches of science and technology,

particularly in the field of agricultural engineering to study the underground water resources,

seepage of water in river beds; in chemical engineering for filtration and purification process; in

International Journal of Engineering Research in Africa Vol. 16 (2015) pp 90-109 Submitted: 2014-11-10© (2015) Trans Tech Publications, Switzerland Revised: 2015-04-20doi:10.4028/www.scientific.net/JERA.16.90 Accepted: 2015-04-27

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petroleum technology, to study the movement of natural gas, oil and water through the oil

reservoirs. Kim [1] investigated an unsteady MHD convective heat transfer past a semi-infinite

vertical porous moving plate with variable suction. Later Chamkha [2] extended this work; he

discussed unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable

moving plate with heat absorption. Pal et al. [3] studied combined effects of Joule heating and

chemical reaction on unsteady magneto hydrodynamic mixed convection of a viscous dissipating

fluid over a vertical plate in porous media with thermal radiation. Chen [4] discussed heat and mass

transfer in MHD flow by natural convection from a permeable, inclined surface with variable wall

temperature and concentration. Singh et al. [5] discussed fluctuating heat and mass transfer on

unsteady MHD free convection flow of radiating and reacting fluid past a vertical porous plate in

slip- flow regime. Guedda et al. [6] discussed analytical and ChPDM analysis of MHD mixed

convection over a vertical flat plate embedded in a porous medium filled with water at 4℃. Israel-

Cookey et al. [7] discussed MHD oscillatory Couette flow of a radiating viscous fluid in a porous

medium with periodic wall temperature. Ahamed et al. [8] discussed, the combined heat and mass

transfer by mixed convection MHD flow along a porous plate with chemical reaction in presence of

heat source. Chaudhaury et al. [9] addressed combined heat and mass transfer effects on MHD free

convection flow past an oscillating plate embedded in porous medium. Ravikuamr et al. [10]

considered heat and mass transfer effects on MHD flow of viscous fluid through non-homogeneous

porous medium in presence of temperature dependent heat source. Ravikumar et al. [11]

investigated a problem of MHD three dimensional coquette flows past a porous plate with heat

transfer. Mbeledogu et al. [12] considered, an unsteady MHD free convection flow of a

compressible fluid past a moving vertical plate in the presence of radioactive heat transfer. Israel-

Cookey et al. [13] discussed influence of viscous dissipation on unsteady MHD free convection

flow past an infinite heated vertical plate in porous medium with time-dependent suction. The

density of sea water is determined by both its temperature and its salt content or salinity. Whereas

added heat makes water lighter, added salt makes it denser, so both must be considered when

evaluating the gravitational stability of the water column. That is why, a given column of water will

‘convert’ or overturn if dense waters overlie lighter waters. In many parts of the world ocean, the

distributions of temperature and salinity are opposed in their effects on density. This arises because

of the tendency of warm water to evaporate easily in low latitudes, the predominance of rainfall in

cold, high latitude regions, and the deep circulation patterns that bring the cold waters to lower

latitudes. The opposing effects of temperature and salinity on density, and the fact that the

molecular conductivity of heat is about 100 times as large as the diffusivity of salt in water, makes

possible a variety of novel convective motions that have come to be known as double diffusive

convection. In the following, the oceanic double-diffusive mixing phenomena: ‘salt Rngers’,

‘diffusive convection’, and ‘intrusions’, are discussed in turn. Observational evidence suggests their

importance in all the oceans, and models indicate a substantial impact on water mass structure and

the thermohaline circulation. Awad et al. [14] examined the linear stability analysis of a Maxwell

fluid with double-diffusive convection. Patil et al. [15] considered a double diffusive mixed

convection flow over a moving vertical plate in the presence of internal heat generation and a

chemical reaction. Chamkha et al. [16] discussed a double-diffusive convection in an inclined orous

enclosure with opposing temperature and concentration gradients. Ravikumar et al. [17] studied

analytically MHD double diffusive and chemically reactive flow through porous medium bounded

by two vertical plates. Israel-Alam et al. [18] discussed effects of variable suction and

thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow over

a semi-infinite permeable inclined plat in the presence of thermal radiation. Rahman et al. [19]

discussed numerical study of the combined free forced convection and mass transfer flow past a

vertical porous plate in a porous medium with heat generation and thermal diffusion.

A Newtonian fluid is considered in the above studies. Motivated by the previous

investigations, we have considered a well known non Newtonian fluid, namely Rivlin-Ericksen

fluid in this paper. The Rivlin-Ericksen elastic –viscous fluid has relevance and importance in

geophysical fluid dynamics, chemical technology and industry (e.g. manufacture of various items

International Journal of Engineering Research in Africa Vol. 16 91

mentioned above). Sunil et al. [20] considered, Hall effects on thermal instability of Rivlin-Ericksen

fluid. Gupta et al. [21] discussed on Rivlin-Erickson Elastico-Viscous fluid heated and solution

from below in the presence of compressibility, rotation and hall currents. Uwanta et al. [22]

addressed the effects of mass transfer on hydro magnetic free convective Rivlin-Ericksen flow

through a porous medium with time dependent suction. Rana et al. [23] studied, thermal instability

of compressible Rivlin-Ericksen rotating fluid permeated with suspended dust particles in porous

medium. Noushima Humera et al. [24] discussed hydromagnetics free convective Rivlin-Ericksen

flow through a porous medium with variable permeability. Takhar et al. [25] examined, dissipation

effects on MHD free convection flow past a semi-infinite vertical plate. Ravikumar et al. [26]

studied, combined effects of heat absorption and MHD on convective Rivlin-Ericksen flow past a

semi-infinite vertical porous plate with variable temperature and suction. Raju et al. [27] discussed

MHD convective flow through porous medium in a horizontal channel with insulated and

impermeable bottom wall in the presence of viscous dissipation and joule heating. In this research,

we have considered an unsteady magneto hydrodynamic, double diffusive, mixed convective

Rivlin-Ericksen fluid flow past a moving porous plate. The dimensionless governing equations are

solved by using a regular perturbation technique. The obtained results are compared with the

existing results and found good agreement with the results of Chamkha [2] in the absence of

Revlin–Ericksen fluid.

2. Problem formulation:

We have considered a viscous incompressible electrically conducting non Newtonian fluid,

namely Rivilin-Ericksen fluid past a semi infinite vertical permeable moving porous plate. A

transversely applied magnetic field of uniform strength Bo is applied perpendicular to the plate in

the direction of the flow. Because of the absence of the electrical field, it is assumed that there is no

applied voltage. Since, it is assumed that the transversely applied magnetic field and magnetic

Reynolds number are very small, so that the induced magnetic field and as well as the Hall Effects

are negligible.

Concentration Boundary Layer

g Thermal Boundary Layer

Momentum Boundary Layer

*v

O

Fig.1 Physical model and coordinate system of the problem

Considering the above assumptions, the governing equations for this investigation are based on the

balances of conservation of mass, conservation of momentum, energy and species concentration is

as follows.

y∗

B0

( ) ( ),n t n t

T T T T e C C C C ew ww wε ε

∗ ∗ ∗ ∗∗ ∗ ∗ ∗= + − = + −∞ ∞

,pU x∗ ∗

oo

oo

oo

oo

oo

oo

oo

oo

oo

oo

oo

, ,U T C∗ ∗ ∗∞ ∞ ∞

92 International Journal of Engineering Research in Africa Vol. 16

Equation of Continuity

0v

y

∗

∗

∂=

∂ (1)

Equation of Momentum

( ) ( )2

2 3

2

2 3 3

01 *

1T c

u u p uv v g T T g c c

t y x y

Bu u uv u v

K t y y

β βρ

σβ

ρ

∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

∞ ∞∗ ∗ ∗ ∗

∗ ∗ ∗∗ ∗

∗ ∗ ∗

∂ ∂ ∂ ∂+ = − + + − + −

∂ ∂ ∂ ∂

∂ ∂− − − +

∂ ∂ ∂

(2)

Equation of Energy

( )2

2

0

p

QT T Tv T T

t y cyα

ρ

∗ ∗ ∗∗ ∗ ∗

∞∗ ∗ ∗

∂ ∂ ∂+ = − −

∂ ∂ ∂ (3)

Equation of Mass diffusion

2 2

2 2

1

c c c Tv D D

t y y y

∗ ∗ ∗ ∗∗

∗ ∗ ∗ ∗

∂ ∂ ∂ ∂+ = +

∂ ∂ ∂ ∂ (4)

where ,x y∗ ∗ and t∗ are the dimensional distance along and perpendicular to the plate and

dimensional time, respectively. u∗and v∗ are the components of dimensional velocities along x∗

and y∗ directions, respectively, ρ is the fluid density, ν is the kinematic viscosity, pc is the specific

heat at constant pressure , σ is the fluid electrical conductivity, 0B is the magnetic induction , K ∗ is

the permeability of the porous medium, T ∗ is the dimensional temperature, Q0 is the dimensional

heat absorption coefficient, c∗ is the dimensional concentration, α is the thermal diffusivity , D is

the mass diffusivity, g is the gravitational acceleration and βT and βc are the thermal and

concentration expansion coefficients, respectively. The magnetic and viscous dissipations are

neglected in this study. The third and fourth terms on the RHS of the momentum Eq. (2) denote the

thermal and concentration buoyancy effects, respectively. Also, the last term of the energy Eq. (3)

represents the heat absorption effect. It is assumed that the permeable plate moves with a constant

velocity in the direction of fluid flow, and the free stream velocity follows the exponentially

increasing small perturbation law. In addition, it is assumed that the temperature and the

concentration at the wall as well as the suction velocity are exponentially varying with time.

Under these assumptions, the appropriate boundary conditions for the velocity, temperature and

concentration fields are

pu U∗ ∗= , ( ) n t

w wT T T T eε∗ ∗∗ ∗

∞= + − , ( ) n t

w wc c c c eε∗ ∗∗ ∗

∞= + − at y*=0 (5)

( )0 1 n tu U U eε∗ ∗∗ ∗

∞→ = + , T T∞→ , c c∞→ as y*→∞

where pU ∗ , cw and Tw are the wall dimensional velocity, temperature and concentration,

respectively. U ∗∞ , c∞ and T∞ are the free stream dimensional velocity, concentration and

temperature respectively. U0 and n* are constants. It is clear from Eq. (1) that the suction velocity at

the plate surface is a function of time only. Assuming that it takes the following exponential form:

0 (1 )n tv V Aeε∗ ∗∗ = − + (6)

where A is a real positive constants, ε and εA are small less than unity, and V0 is a scale of suction

velocity which has non-zero positive constant. Outside the boundary layer Eq. (2) gives

2

0

1 dU vU B U

x dt K

ρ σρ ρ

∗∗∗ ∗∞∞ ∞∗ ∗ ∗

∂− = + +

∂ (7)


It is convenient to employ the following dimensionless variables: 2

0 0

0 0 0

, , , , , ,p

p

U uV y t Vu vu v U U t

U V v U U vη ∞

∗ ∗∗ ∗∗ ∗

∞∗

= = = = = =

0

2 2

0

, , , ,p

w w

v cK VT T c c n v vC n K Pr

T T c c V v k

ρθ

α

∗ ∗∗ ∗ ∗ ∗ ∗∞ ∞

∗ ∗ ∗ ∗∞ ∞

− −= = = = = =

− − (8)

( ) ( )2

0 0

2 2 2 2

0 0 0 0 0 0

, , , ,T w c w

p

v g T T v g c cB v vQM Gr Gc Q

V U V U V pc V

β βσρ

∗ ∗ ∗ ∗∞ ∞− −

= = = =

( )( )

2*1 0 1

2 2

0

4 1, , , ,Sc ,

w

w

T TV Dvw vRm w N M So

v V K v Dc c

β∗ ∗

∞

∗ ∗∞

−= = = + = =

−

In view of Eq. (6) - (8), Eq. (2) - (4) reduce to the following dimensionless form:

( ) ( )

( )

2

2

3 3

2 3

1

1

nt

nt

dUu u uAe Gr GcC N U u

t dt

u uRm Ae

t

ε θη η

εη η

∞∞

∂ ∂ ∂− + = + + + + −

∂ ∂ ∂

∂ ∂− − + ∂ ∂ ∂

(9)

( )2

2

11 ntAe Q

t Pr

θ θ θε θ

η η∂ ∂ ∂

− + = −∂ ∂ ∂

(10)

( )2 2

2 2

11 ntC C C

Ae Sot Sc

θε

η η η∂ ∂ ∂ ∂

− + = +∂ ∂ ∂ ∂

(11)

where 1

N MK

= + and Gr, Gc, Pr, and Q are the solutal Grashof number, thermal Grashof

number , Prandtl number, dimensionless heat absorption coefficient, and the Schmidt number ,

respectively. By setting Rm and So equal to zero, Eqs. (9) - (11) reduce to those reported by

Chamkha [2].

The dimensionless form of the boundary conditions (5) become

, 1 , 1 , 1 0

, 0, 0, 0

nt nt nt

pu U e C e U e at

u U C U as

θ ε ε ε η

θ η∞

= = + = + = + =

→ → → → →∞ (12)

3. Solution of the problem

Eq. (9), (10) and (11) represent a set of partial differential equations that cannot be solved in

closed form. However, it can be reduced to a set of ordinary differential equations in dimensionless

from that can be solved analytically. This can be done, by representing the velocity and temperature

and concentration in terms of harmonic and non harmonic functions as follows:

( ) ( ) ( )2

0 1 .............ntU f e fη ε η ε= + + +○ (13)

( ) ( ) ( )2

0 1 .............ntg e gθ η ε η ε= + + +○ (14)

( ) ( ) ( )2

0 1 .............ntC h e hη ε η ε= + + +○ (15)

Substituting Eq. (13) - (15) into eq. (9) - (11), equating the harmonic and non-harmonic terms, and

neglecting and higher- order terms of ( )2ε○ , one obtains the following pairs of equations for

( )0 0 0, ,f g h and ( )1 1 1, ,f g h .

/ / / / / /

0 0 0 0 0 0Rmf f f Nf Grg Gch N+ + − = − − − (16)

( )/ / / / / / / / / /

1 1 1 1 1 1 1 0 01Rmf nRm f f Nf nf n Grg Gch N RmAf Af+ − + − − = − − − − − − (17)

/ / /

0 0 0Pr Pr 0g g Q g+ − = (18)


/ / / /

1 1 1 1 0Pr Pr Pr Prg g n g Q g A g+ − − = − (19)

/ / / / /

0 0 oh Sch ScSog+ = − (20) / / / / / /

1 1 1 1oh Sch nSch ASch ScSog+ − = − − (21)

where a prime denotes ordinary differentiation with respect to η. The corresponding boundary

conditions can be written as

0 1 0 1 0 1

0 1 0 1 0 1

, 0, 1, 1, 1, 1 0

1 , 1, 0, 0 , 0, 0

pf U f g g h h at

f f g g h h at

η

η

= = = = = = =

= = → → → → →∞ (22)

Without going into detail, the solution of Eqs. (16)- (21) subject to Eq. (22) can be shown to be 2

0

mg e η−= (23)

4 2

1 2 1

m mg k e k eη η− −= + (24)

6 2

0 4 3

m mh k e k eη η− −= + (25)

8 6 2 4

1 10 5 11 7

m m m mh k e k e k e k eη η η η− − − −= + + + (26)

Equations (16) and (17) are third order D.Es when Rm≠ 0 and we have two boundary conditions, so

we follow bears and Walters as

( )2

0 01 02mf f R f Rm= + +○ (27)

( )2

1 11 12mf f R f Rm= + +○ (28)

Substituting Esq. (27) and (28) into (16) and (18), equating different powers of Rm and

neglecting ( )2Rm○ .

The corresponding boundary conditions are

01 02 11 12

01 02 11 12

, 0, 0, 0 0

1, 0, 1, 0

pf U f f f at

f f f f as

η

η

= = = = =

= = = = →∞ (29)

We get zeroth order and first order equations of Rm. 10 62

01 17 15 13 1m mmf k e k e k e

η ηη− −−= + + + (30)

10 612 2

02 21 18 19 20

m mm mf k e k e k e k eη ηη η− −− −= − + + + (31)

8 6 1014 4 2

11 36 33 25 34 27 35 30

m m mm m mf k e k k e k e k e k e k eη η ηη η η − − −− − −= − + + + + + + (32)

16 10 6 812 2 14 4

12 51 44 45 46 47 48 49 50

m m m mm m m mf k e k e k e k e k e k e k e k eη η η ηη η η η− − − −− − − −= − + + + + + + + (33)

In view of the above solutions, the velocity ,temperature and concentration distributions in the

boundary layer become

( )10 6 10 62 12 2

8 6 10 1614 4 2

1012

0 1

17 15 13 21 18 19 20

36 33 25 34 27 35 30 51

44 45

, ( ) ( )

[( 1) ( )]

[( ) (

nt

m m m mm m m nt

m m m mm m m

mm

U t f e f

k e k e k e Rm k e k e k e k e e

k e k k e k e k e k e k e Rm k e

k e k e

η η η ηη η η

η η η ηη η η

ηη

η η ε η

ε− − − −− − −

− − − −− − −

−−

= +

= + + + + − + + + +

− + + + + + + + − +

+ + 6 82 14 4

46 47 48 49 50 )]m mm m mk e k e k e k e k eη ηη η η− −− − −+ + + +

(34)

( )2 4 2

0 1

2 1

, ( ) ( )

[ ]

nt

m m mnt

t g e g

e e k e k eη η η

θ η η ε η

ε− − −

= +

= + + (35)

( )

( )6 8 62 2 4

0 1

4 3 10 5 11 7

, ( ) ( )

[ ]

nt

m m mm m mnt

C t h e h

k e k e e k e k e k e k eη η ηη η η

η η ε η

ε− − −− − −

= +

= + + + + + (36)

The skin-friction coefficient and Nusselt number are important physical parameters for this type of

boundary-layer flow. These parameters can be defined and determined as follows:

( )52 53 54 55

0

( )ntuk Rmk e k Rmk

η

τ εη

=

∂= = + + + ∂

(37)


56 57

0

( )ntNu k e kη

θε

η=

∂= = + ∂

(38)

58 59

0

( )ntCSh k e k

η

εη

=

∂= = + ∂

(39)

Where

2

1

4

2

pr pr Qprm

− + += ,

2

2

4

2

pr pr Qprm

+ += ,

( )2

3

4

2

pr pr Qpr nprm

− + + += ,

( )2

4

4

2

pr pr Qpr nprm

+ + += ,

( )2

1 2

2 2

P r

P r P r P r

A mk

m m n Q=

− − +, 2 11k k= − , 5 0m = ,

6m Sc= − , 2

23 2

2 2

ScSomk

m Scm

−=

−, 4 31k k= − ,

2

7

4

2

Sc Sc nscm

− + += ,

2

8

4

2

Sc Sc nscm

+ += ,

6 45 2

6 6

AScm kk

m m Sc nSc=

− −, 2 3

6 2

2 2

AScm kk

m m Sc nSc=

− −,

2

4 27 2

4 4

SoScm kk

m m Sc nSc

−=

− −,

2

2 18 2

2 2

SoScm kk

m m Sc nSc

−=

− −

9 5 6 7 8k k k k k= + + + , 10 91k k= − , 11 6 8k k k= + , 9

1 1 4

2

Nm

− + += , 10

1 1 4

2

Nm

+ += ,

12 2

2 2

G rk

m m N

−=

− −, 4

13 2

6 6

Grkk

m m N

−=

− −, 3

14 2

2 2

Gckk

m m N

−=

− −, 15 12 14k k k= + , 16 13 15 1k k k= + + ,

17 16k Up k= − ,11

1 1 4

2

Nm

− + += ,

12

1 1 4

2

Nm

+ += ,

3

1 0 1 71 8 2

1 0 1 0

m kk

m m N=

− −,

3

2 1519 2

2 2

m kk

m m N=

− −,

3

6 1320 2

6 6

m kk

m m N=

− −, 21 18 19 20k k k k= + + , 22 2 7k Grk Gck= + , 23 1 11k Grk Gck= + ,

( )13

1 1 4

2

N nm

− + + += ,

( )14

1 1 4

2

N nm

+ + += , 24

nk

N n=

+,

( )22

25 2

4 4

kk

m m N n

−=

− − +,

( )23

26 2

2 2

kk

m m N n

−=

− − +,

( )10

27 2

8 8

Gckk

m m N n

−=

− − +,

( )5

2 8 2

6 6

G ckk

m m N n

−=

− − +,

2 9

Nk

N n=

+,

( )10 17

30 2

10 10

Am kk

m m N n=

− − +,

( )2 15

31 2

2 2

Am kk

m m N n=

− − +,

( )6 13

32 2

6 6

Am kk

m m N n=

− − +, 33 24 29k k k= + , 34 26 31k k k= + , 35 28 32k k k= + ,

36 33 25 34 27 35 30k k k k k k k= + + + + + , 37 21 12k Ak m= − , 3 3 2

38 18 10 17 10 30 10 30 10k Ak m Ak m k m nk m= + + + , 3 3 2

39 19 2 15 2 34 2 34 2k Ak m Ak m k m nk m= + + + , 3 3 2

40 20 6 13 6 35 6 35 6k Ak m Ak m k m nk m= + + + ,

3 2

41 36 14 36 14k k m nk m= − − , 3 2

42 25 4 25 4k k m nk m= + , 3 2

43 27 8 27 8k k m nk m= + , ( )

3744 2

12 12

kk

m m N n=

− − +,

( )38

45 2

10 10

kk

m m N n=

− − +,

( )39

46 2

2 2

kk

m m N n=

− − +,

( )40

47 2

6 6

kk

m m N n=

− − +,

( )41

48 2

14 14

kk

m m N n=

− − +,

( )42

49 2

4 4

kk

m m N n=

− − +,

( )43

50 2

8 8

kk

m m N n=

− − +, 51 44 45 46 47 48 49 50k k k k k k k k= + + + + + +

52 17 10 15 2 13 6k k m k m k m= − − − , 53 21 12 18 10 19 2 20 6k k m k m k m k m= − − − ,

54 36 14 25 4 34 2 27 8 35 6 30 10k k m k m k m k m k m k m= − − − − − ,

55 51 16 44 12 45 10 46 2 47 6 48 14 49 4 50 8k k m k m k m k m k m k m k m k m= − − − − − − − , 56 2k m= − , 57 2 4 1 2k k m k m= − − ,

58 4 6 3 2k k m k m= − − , 59 10 8 5 6 11 2 7 4k k m k m k m k m= − − − − .


4. Result and discussion

The non-linear coupled Equations (13) - (21) subject to the boundary condition (22), which

describe, heat and mass transfer flow past an infinite vertical plate immersed in a porous medium in

the presence of heat-absorbing Rivlin-Ericksen flow past a semi-infinite vertical porous moving

plate, under the influence of magnetic field and thermal diffusion are solved analytically by

perturbation technique. In order to get physical insight into the problem, the effects of various

parameters encountered in the equations of the problem are analyzed on velocity, temperature and

concentration fields with the help of figures. These results show the influence of the various

physical parameters such as Grashof number Gr, dimensionless viscoelasticity parameter of the

Rivlin-Ericksen fluid Rm, solutal Grashof number Gc, Magnetic parameter M, Schmidt number Sc,

permeability parameter K, heat absorption parameter Q, Scalar constant ε, Soret number So and

Prandtl number Pr on the velocity, temperature and the concentration distributions. We have also

analyzed the effects of these physical parameters on skin friction coefficient, Nusselt number and

Sherwood number. We can extract interesting insight regarding the influence of all parameters that

govern the problem.

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

η

U

K=0.1,0.2,0.3,0.4,0.5

Fig. 2. Effects of K on velocity profiles.

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

η

U

So=0.5,1,1.5,2,2.5,3

Fig. 3. Effects of So on velocity profiles

Pr=0.71; Q=0.001; A=0.5;

n=0.1; Sc=0.22; So=0.1;

M=1; Gr=5; Up=0.5;

Gc=5; Rm=0.05; ε=0.2; t=1;

Pr=0.71; Q=0.001; A=0.5;

n=0.1; Sc=0.22; M=1;

K=0.5; Gr=5; Up=0.5;

Gc=5; Rm=0.05; ε=0.2; t=1;


The velocity profiles are plotted in Fig. 2 for various values of permeability parameter K.

From this figure, it is noticed that the velocity increases with the increase in the values of the

permeability parameter K. Physically, an increase in the permeability of porous medium leads to

rise, in the flow of fluid through it. When the holes of the porous medium become large, the

resistance of the medium may be neglected. A similar approach is noticed with Raju et al. [27]. Fig.

3 shows the velocity profiles against span wise direction for different values of the Soret number

So. It was found that an increase in the value of So leads to an increase in the velocity distribution

across the boundary layer. This is true, as the Soret number increases, small light molecules and

large heavy molecules get separated under a temperature gradient, which intern increases the

velocity of a fluid. Fig. 4 illustrates the variation of velocity function with span wise coordinate’s η

for several values of dimensionless visco-elastic parameter of a Revlin-Ericksen fluid Rm. It is

found that an increase in Rm leads to a decrease in the velocity distribution across the boundary

layer because; it is due to variability in viscosity. Fig. 5 shows the velocity profiles for different

values of magnetic parameter M. It is observed that the increase in the value of M results a decrease

in the velocity profiles. This is due to the application of a magnetic field to an electrically

conducting fluid which produces a dragline force that causes reduction in the fluid velocity. Similar

approach is noticed in the results of Kim [1], Chamkha [2] and Raju et al. [27]. Fig. 6 shows the

velocity profiles for different values of heat absorption parameter Q. From this graph we observe

that the velocity decreases with an increase in the heat absorption parameter Q. Because, in the

presence of heat absorption, the buoyancy force decreases the velocity profiles. Physically, the

presence of heat absorption coefficient has the tendency to reduce the fluid velocity. This cause the

thermal buoyancy effects to decrease resulting in a net reduction in the fluid velocity (see

Ravikumar et al. [26]). Fig. 7 illustrates the temperature profiles for different values of Prandtl

number Pr. It is observed that the temperature decrease as an increase in the values of Prandtl

number. The reason is that smaller values of Pr are equivalent to increase in the thermal

conductivity of the fluid and therefore heat is able to diffuse away from the heated surface more

rapidly for higher values of Pr. Hence, in the case of smaller Prandtl number the thermal boundary

layer is thicker and the rate of heat transfer is reduced.

0 1 2 3 4 5 6 7 8 9 100.5

1

1.5

2

2.5

3

η

U

Rm=1,2,3,4,5

Fig. 4. Effects of Rm on velocity profiles

Pr=0.71; Q=0.001; A=0.5;

n=0.1; Sc=0.22; So=0.1;

M=1; K=0.5; Gr=5;

Up=0.5; Gc=5; ε=0.2;

t=1;


0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

η

U

M=1,2,3,4,5

Fig. 5. Effects of M on velocity profiles

Fig. 8 shows the temperature profile for different values of heat absorption parameter Q. From this

figure it is noticed that increase in the value of Q results a decrease in the temperature profiles as

expected. It is observed that increase in the heat absorption parameter is to decrease the temperature

in the boundary layer. This effect is more prominent for fluid closer to the porous plate. This is due

to the fact that heat from the fluid will be absorbed by the porous plate and hence, higher the value

of heat absorption parameter, lower the value of temperature profile in the boundary layer ( see Pal

et al. [3], S. Harinath Reddy et al. [29]). The effects of increasing the Soret number So on the

species concentration profiles have been shown in Fig. 9. From this figure it is noticed that increase

in So result in increase in the concentration profiles.

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

3

3.5

η

U

Q=0.0,0.1,0.3,0.5,0.7

Fig. 6. Effects of Q on velocity profiles

Pr=0.71; Q=0.001; A=0.5;

n=0.1; Sc=0.22; So=0.1;

K=0.5; Gr=5; Up=0.5;

Gc=5; Rm=0.05; ε=0.2;

t=1;

Pr=0.71; A=0.5; n=0.1;

Sc=0.22; So=0.1; M=1;

K=0.5; Gr=5; Up=0.5;

Gc=5; Rm=0.05; ε=0.2;

t=1;


0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

η

T

Pr=0.71,1.0,5.0,7.0

Fig. 7. Effects of Pr on temperature profiles

Figs. 10-15 present the variation of the skin friction coefficient τ against the suction velocity

parameter A for various values of, K, So, Rm, M and Q. Fig.10 shows the skin friction on the

porous plate for different values of permeability parameter K. The results display that an increase in

the value of K results in increase in the skin friction profiles. Fig. 11 shows the skin friction profiles

for different values of Soret number So. From this figure it is noticed that an increase in So results

an increase in the skin friction profile. Fig. 12 depicts the variations in dimensionless viscoelasticity

parameter of the Rivlin-Ericksen fluid Rm against the suction velocity parameter A. It has been

observed that the effects of increase in the values of Rm decrease in the skin friction profile. As

shown in Fig. 13, the surface skin friction for different values of Magnetic parameter M, it is

noticed that an increase in M results a decrease in the skin friction profile. Fig. 14 depicts the

surface skin friction for different values of dimensionless heat absorption coefficient Q. From this

figures it is noticed that an increase in Q results a decrease in the skin friction. Fig. 15 shows the

skin friction on the porous plate for the different values constant plate moving velocity Up against

the suction velocity parameter A. It has been observed that the effects of increase in the value of

Up decrease in the skin friction coefficient.

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

η

T

Q=0.0.0.1,0.3,0.5,0.7

Fig. 8. Effects of Q on temperature profiles.

Q=0.001; n=0.1; Sc=0.22;

So=0.1; M=1; K =0.5;

Gr=5; Up=0.5; Gc=5;

Rm=0.05; ε=0.2 ; t=1;

Pr=0.71; A=0.5; n=0.1;

Sc=0.22; So=0.1; M=1;

K =0.5; Gr=5; Up=0.5;

Gc=5; Rm=0.05; ε=0.2 ;

t=1;


0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5

η

C

So=0.5,1,1.5,2,2.5,3

Fig. 9. Effects of So on concentration profiles

0 1 2 3 4 5 6 7 8 9 108

10

12

14

16

18

20

22

A

τ

K=0.1

K=0.2

K=0.3

K=0.4

Fig. 10. Variation of the surface skin friction with the suction velocity parameter A for various

values of K

Fig. 16 presents the variation of the Nusselt number Nu against the suction velocity

parameter A for various values of Q. From this figure we notice that Nusselt number decreases

when the values of dimensionless heat absorption Q increase. Fig. 17 shows the Sherwood number

on the porous plate for different values of Soret number So. The result display that an increase in

the value of So results an increase in the Sherwood number. Fig. 18 depicts the variation of

Sherwood number in the presence of Schmidt number against the suction velocity parameter A. It

has been observed that the effect of increase in the values of Sc decreases the value of Sherwood

number coefficient for all values of suction velocity parameter A.

Pr=0.71; Q=0.001; A=0.5;

n=0.1; Sc=0.22; M=1;

K =0.5; Gr=5; Up=0.5;

Gc=5; Rm=0.05; ε=0.2;

t=1;

Pr=0.71; Q=0.001; n=0.5;

Sc=0.22; So=0.1; M=1;

Gr=5; Up=0.5; Gc=5;

Rm=0.05; ε=0.2; t=1.


0 1 2 3 4 5 6 7 8 9 106

7

8

9

10

11

12

13

14

15

A

τ

So=0.5,1,1.5,2,2.5,3


values of So

0 1 2 3 4 5 6 7 8 9 105

6

7

8

9

10

11

A

τ

Rm=0.01,0.03,0.05,0.07


values of Rm

Pr=0.71;Q=0.001; n=0.1; Sc=0.22; M=1; K=0.5;

Gr=5; Up=0.5; Gc=5; Rm=0.05; ε=0.2; t=1;

Pr=0.71;Q=0.001; n=0.1; Sc=0.22; So=0.1;

M=1;K =0.5; Gr=5; Up=0.5; Gc=5; ε=0.2; t=1;


0 1 2 3 4 5 6 7 8 9 108

10

12

14

16

18

20

22

A

τ

M=1,2,3,4,5


values of M

5. CONCLUSIONS

We have examined theoretically the problem of an unsteady, incompressible MHD

double diffusive convection Rilvin-Ericksen boundary layer flow past a semi-infinite vertical

moving porous plate. The computed values obtained from analytical solutions of velocity,

temperature, concentration fields as well as for the skin friction coefficient, Nusselt number and

Sherwood number are presented graphically. After a suitable transformation, the governing partial

differentiation equations were transformed to ordinary differential equations. These equations were

solved analytically by using perturbation technique. Thus we conclude the following after analyzing

the graphs.

0 1 2 3 4 5 6 7 8 9 103

4

5

6

7

8

9

10

11

A

τ

Q=0.0,0.1,0.3,0.5,0.7


values of Q

Pr=0.71;Q=0.001; n=0.1;

Sc=0.22; So=0.1; K =0.5;

Gr=5; Up=0.5; Gc=5;

Rm=0.05; ε=0.2; t=1;

Pr=0.71; n=0.1; Sc=0.22;So=0.1;

M=1; K =0.5;

Gr=5; Up=0.5; Gc=5; Rm=0.05;

ε=0.2; t=1;


0 1 2 3 4 5 6 7 8 9 10-8

-6

-4

-2

0

2

4

6

8

10

12

A

τUp=0.0

Up=0.5

Up=2.0

Up=5.0


values of Up

The velocity increases with an increase in permeability parameter, Soret number, whereas reverse

trend is seen with dimensionless viscoelasticity parameter of the Rivlin-Ericksen fluid, Magnetic

parameter, heat absorption parameter. The temperature increases with decrease in the value of

Prandtl umber and heat absorption parameter. The concentration is increasing with increase in the

values of Soret number. The value of the local skin-friction coefficient increases with increase in

the permeability parameter, and Soret number whereas reverse effects is seen dimensionless

viscoelasticity parameter of the Rivlin-Ericksen fluid, Magnetic parameter, heat absorption

parameter and constant plate moving velocity Up. The values of the local Nusselt number decreases

in the presence of heat absorption. The values of the local Sherwood number increases with increase

in Soret number whereas reverse effects is seen in the case of Schmidt number.

0 1 2 3 4 5 6 7 8 9 10-2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

A

Nu

Q=0.0

Q=0.1

Q=0.3

Q=0.5

Q=0.7

Fig. 16. Variation of the surface heat flux with the suction velocity parameter A for various values

of Q

Pr=0.71; Q=0.001; n=0.1; Sc=0.22; So=0.1;

M=1; K=0.5; Gr=5; Gc=5; Rm=0.05; ε=0.2;

t=1;

Pr=0.71; n=0.1; Sc=0.22; So=0.1; M=1; K=0.5;

Gr=5; Up=0.5; Gc=5; Rm=0.05; ε=0.2; t=1;


0 1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

A

Sh

So=0.5,1,1.5,2,2.5,3

Fig. 17. Variation of the surface mass flux with the suction velocity parameter A for various values

of So

0 1 2 3 4 5 6 7 8 9 10-40

-35

-30

-25

-20

-15

-10

-5

0

A

Sh

Sc=0.60

Sc=0.78

Sc=0.96

Sc=0.22

Fig. 18. Variation of the surface mass flux with the suction velocity parameter A for various values

of Sc

Table

Comparison of our results with the existing results of Chamkha [2] in the absence of Rivlin-

Ericksen fluid: Effects of Gc on τ, Nu and Sh for A=0.5,Pr=0.71, Q=0.6, n=0.1, Sc=0.22, So=0,

M=0, K=0.1, Gr=1, Up=0.5, Rm=0, ε=0.2 and t=2.

Results of Chamkha [2] Results of present study

Gc Τ Nu Sh Gc τ Nu Sh

0

1

2

3

4

2.7200

3.3772

3.8343

4.3915

4.9487

-1.7167

-1.7167

-1.7167

-1.7167

-1.7167

-0.8098

-0.8098

-0.8098

-0.8098

-0.8098

0

1

2

3

4

2.9740

3.3247

3.8507

4.0261

4.7274

-1.7190

-1.7190

-1.7190

-1.7190

-1.7190

-0.8151

-0.8151

-0.8151

-0.8151

-0.8151

Pr=0.71; Q=0.001; n=0.1; Sc=0.22; M=1;K=0.5;

Gr=5; Up=0.5;Gc=5; Rm=0.05; ε=0.2; t=1;

Pr=0.71; Q=0.001; n=0.1; So=0.1; M=1;K=0.5;

Gr=5; Up=0.5; Gc=5; Rm=0.05; ε=0.2; t=1;


6. Nomenclature

A suction velocity parameter

B0 magnetic induction

c concentration

Cp specific heat at constant pressure

C dimensionless concentration

D mass diffusion coefficient

Gc solutal Grashof number

Gr Grashoff number

g acceleration due to gravity

K Permeability of the porous medium

M magnetic field parameter

N dimensionless material parameter

n dimensionless exponential index

Nu Nusselt number

Sc Schmidt number

So Soret number

Sh Sherwood number

Pr Prandtl number

Q0 heat absorption coefficient

T temperature

t dimensionless time

U0 scale of free stream velocity

U*, v

* components of velocities along and Perpendicular to

the plate, respectively

V0 scale of suction velocity

x*, y

* distances along and perpendicular to the plate, respectively

Q dimensionless heat absorption coefficient

Rm dimensionless viscoelasticity parameter of the

Rivlin-Ericksen fluid

Up constant plate moving velocity

Greek symbols

α fluid thermal diffusivity

βc coefficient of volumetric concentration expansion

βT coefficient of volumetric thermal expansion

β1 coefficient of volumetric expansion of the working fluid

ε scalar constant (<<1)

η dimensionless normal distance

σ electrical conductivity

ρ fluid density

µ fluid dynamic viscosity

ν fluid kinematic viscosity

τ friction coefficient

θ dimensionless temperature κ Thermal conductivity

τ skin-friction coefficient

Superscripts

/ Differentiation with respect to η


* Dimensional properties

Subscripts

p plate

w wall condition

∞ Free stream condition

7. References

[1] Kim. J Y, (2000), “Unsteady MHD convective heat transfer past a semi-infinite vertical porous

moving plate with variable suction”, International Journal of Engineering Sciences 38, pp.833-

845.

[2] Chamkha A.J, (2004), “ Usteady MHD convective heat and mass transfer past a semi infinite

vertical permeable moving plate with heat absorption”, Int. J. Engg. Sci., 42, pp. 217-230. DOI:

10.1016/s0020-7225(03)00285-4.

[3] Pal. D and Babulal. T, (2011), “Combined effects of joule heating and chemical reaction on

unsteady magneto hydrodynamic mixed convection of a viscous dissipating fluid over a vertical

plate in porous media with thermal radiation” Mathematical and computer Modelling 54,

pp.3016-3036. DOI: 10.1016/j.mcm.2011.07.030.

[4] Chen C. H, (2004), “Heat and mass transfer in MHD flow by natural convection from a

permeable, inclined surface with variable wall temperature and concentration”, Acta. Mech.,

Vol. 172, pp. 219-235. DOI:10.1007/s00707-004-0155-5

[5] Singh. K.D and Kumar. R, (2011), “Fluctuating heat and mass transfer on unsteady MHD free

convection flow of radiating and reacting fluid past a vertical porous plate in slip- flow

regime”, Journal of Applied Fluid Mechanics, Vol. 4, No. 4, pp. 101-106.

[6] Guedda. M, Aly. E and Quahsine. A, (2011), “Analytical and ChPDM analysis of MHD mixed

convection over a vertical flat plate embedded in a porous medium filled with water at 4℃,”

Applied Mathematical Modeling, Vol. 35, No. 10, pp. 5182-5197.

Doi:10.1016/j.apm.2011.04.014.

[7] Israel-Cookey. C, Amos. E and Nwaigwe. C, (2010), “MHD oscillatory coquette flow of a

radiating viscous fluid in a porous medium with periodic wall temperature,” American Journal

of Scientific and Industrial Research, Vol. 1, No. 2, pp. 326-331.

Doi:10.5251/ajsir.2010.1.2.326.331.

[8] Sahin Ahamed and Zueco. J, (2010), “ Combined heat and mass transfer by mixed convection

MHD flow along a porous plate with Chemical reaction in presence of heat source”, Applied

Mathematics and Mechanics, Vol. 31(10),pp.1217-1230.

[9] Chaudhaury R.C & Arpita .J, (2007), “Combined heat and mass transfer effects on MHD free

convection flow past an oscillating plate embedded in porous medium”, Rom. Journ. Phys.,

Vol. 52, pp.505-524.

[10] Ravilumar.V, Raju M.C and Raju G.S.S, (2012), “ Heat and Mass Transfer effects on MHD

flow of viscous fluid through Non-Homogeneous porous medium in presence of temperature

dependent heat source”, International journal of contemporary Mathematical sciences, Vol. 7,

pp. 29-32.

[11] Ravilumar.V, Raju M.C and Raju G.S.S, (2012), “MHD Three dimensional coquette flow past

a porous plate with heat transfer” IOSR Journal of Mathematics Vol.1(3), pp.03-09.

[12] Mbeledogu. I. U, Amakiri. A. R. C and Ogulu. A, (2007), “Unsteady MHD free convection

flow of a compressible fluid past a moving vertical plate in the presence of radiative heat

transfer,” International Journal of Heat and Mass Transfer, Vol. 50, No. 9-10, pp. 326-331.

Doi:10.1016/j.ijheatmasstransfer.2006.10.032.


[13] Israel-Cookey, Ogulu A and Omubo-Pepple V.B, (2003), “Influence of viscous dissipation on

unsteady MHD free convection flow past an infinite heated vertical plate in porous medium

with time-dependent suction. Int. J. Heat mass transfer, Vol. 46, pp.2305-2311.

[14] Awad. F.G, Sibanda .P and Sandile S. Motsa, (2010), “On the linear stability analysis of a

Maxwell fluid with double-diffusive convection”, Applied Mathematical Modelling 34, pp.

3509–3517.

[15] Patil .A, Roy.s and Chamkha.A.J, (2009), “Double diffusive mixed convection flow over a

moving vertical plate in the presence of internal heat generation and a chemical reaction”,

Turkish J. Eng. Env. Sci.,Vol. 33, pp.193-205.

[16] Chamkha Ali J and Hameed Al-Naser, (2001), “Double-diffusive convection in an inclined

orous enclosure with opposing temperature and concentration gradients”, Int. J. Therm. Sci.

(2001) 40, pp. 227–244.

[17] Ravikumar. V, Raju .M.C, Raju .G.S.S and Chamkha A. J, (2013), “MHD double diffusive and

chemically reactive flow through porous medium bounded by two vertical plates”, International

Journal of Energy & Technology, 5 (4), pp. 1–8.

[18] Alam M.S, Rahman M.M and Sattar M.A ,(2008), “ Effects of variable suction and

thermophoresis on steady MHD combined free-forced convective heat and mass transfer flow

over a semi-infinite permeable inclined plat in the presence of thermal radiation”, International

Journal of Thermal Sciences, Vol.47(6),pp.758-765.

[19] Alam M.S, Rahman M.M and Samad M.A, (2006), “Numerical study of the combined free

forced convection and mass transfer flow past a vertical porous plate in a porous medium with

heat generation and thermal diffusion”, Nonlinear Analysis: Modelling and control, Vol.11(4),

pp.331-343.

[20] Sharma R.C, Sunil, Suresh chand, (2000), “Hall effects on thermal instability of Rivlin-

Ericksen fluid”, Indian.J.Pure.Appl.Math.Vol. 3(1), pp.49-59.

[21] Urvashi Gupta and Gauray Sharma, (2007), “On Rivlin-Erickson Elastico-Viscous fluid heated

and solution from below in the presence of compressibility, rotation and hall currents”, Journal

Application Mathematical and Computing, Vol. 25(1-2), pp. 51-66.

[22] Uwanta. J and Hussaini. A, (2012), “Effects of mass transfer on hydro magnetic free convective

Rivlin-Ericksen flow through a porous medium with time dependent suction”, International

Journal of Engineering and Sciences Vol. 1(4), pp.21-30.

[23] Rana. G. C, (2012), “Thermal instability of compressible Rivlin-Efficksen rotating fluid

permeated with suspended dust particles in porous medium”, International Journal of Applied

mathematics and mechanics, Vol.8 (4), pp.97-110.

[24] Noushima Humera, Ramana Murthy M.V, Chenna Krishna Reddy, M. Rafiuddin, Ramu. A and

Rajender. S, (2010), “Hydromagnetics free convective Rivlin-Ericksen flow through a porous

medium with variable permeability”, Int. J. of computational and Applied Mathematics, Vol.5,

No.3, pp.267-275.

[25] Takhar H. S. and Soundalgekar V. M, (1980), “Dissipation effects on MHD free convection

flow past a semi-infinite vertical plate”, Applied Scientific Research, Volume 36, Issue 3, pp.

163-171. DOI: 10.1007/BF00386469.

[26] Ravikumar .V, Raju. M.C and Raju G.S.S, (2014), “Combined effects of heat absorption and

MHD on convective Rivlin-Ericksen flow past a semi-infinite vertical porous plate with

variable temperature and suction”, Ain Shams Eng J, 5(3), 867-875,

doi.org/10.1016/j.asej.2013.12.014.


[27] Raju K.V.S, Reddy. T.S, Raju M. C, Satya Narayana .P.V and Venkataramana .S, (2013), “

MHD convective flow through porous medium in a horizontal channel with insulated and

impermeable bottom wall in the presence of viscous dissipation and joule heating”, Ain Shams

Eng J, 5(2), 543-551, doi.org/10.1016/j.asej.2013.10.007.

[28] Mamtha, B., Raju, M.C., Varma, S.V.K, Thermal diffusion effect on MHD mixed convection

unsteady flow of a micro polar fluid past a semi-infinite vertical porous plate with radiation and

mass transfer, International Journal of Engineering Research in Africa, Vol. 13 (2015) pp 21-

37.

[29] S. Harinath Reddy, M. C. Raju, E. Keshava Reddy, Unsteady MHD free convection flow of a

Kuvshinski fluid past a vertical porous plate in the presence of chemical reaction and heat

source/sink, International Journal of Engineering Research in Africa Vol. 14 (2015) pp. 13-

27,doi:10.4028/www.scientific.net/JERA.14.13.


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