heat and mass transfer on unsteady mhd flow past an ... · medium over an infinite vertical plate....

Heat and Mass transfer on unsteady MHD flow past an infinite vertical

plate through a porous medium with time varying pressure gradient

M.Chitra1 ,M.Suhasini2

1Associate Professor, 2 Research Scholar

Department of Mathematics, Thiruvalluvar University, Serkkadu, Vellore.

ABSTRACT

In this paper , we investigate the effect of unsteady MHD free convection two dimensional flow of an

incompressible viscous electrically conducting fluid with simultaneous heat and mass transfer over an

infinite vertical plate through porous medium with time varying pressure gradient, oscillatory suction

velocity and permeability of the porous medium. The closed forms of analytical solution are obtained

for the Momentum, Energy and concentration Equations. The effect of various flow parameters like

Hartmann number, prandtl number, Schmidt number, thermal Grashof number and mass Grashof

number on velocity profile, temperature, concentration, wall shear stress, and the rate of heat and mass

transfer are obtained and their behaviour are discussed graphically.

KEYWORDS: MHD flow, porous medium, heat and mass transfer, time varying pressure gradient,

suction velocity.

INTRODUCTION

The study of MHD flow of electrically conducting fluid has a lot of application in the field of

astrophysics and geophysics to study the stellar and solar structures, interstellar matter and radio

propagation through the ionosphere. In engineering field, it has its application in MHD pumps, MHD

bearings, nuclear reactors, geothermal energy extraction and in boundary layer control in the field of

aerodynamics. Raptis et al 1982 [1] studied the hydromagnetic free convection flow through a porous

medium between two parallel plates. Makinde et al 2005 [2] studied the heat transfer to MHD oscillatory

flow in a channel filled with porous medium. D.S Chauhan et al 2009 [3] discussed the effect of MHD

flow in a channel partially filled with a porous medium in a rotating system. Ashaf .A.Moniem et al

2013 [8] studied the solution of MHD flow past a vertical porous plate through a porous medium under

oscillatory suction. G.Prabhakaran et al 2005 [9] studied the effect of heat transfer on steady MHD

rotating flow through porous medium in a parallel plate channel. D.Dastagiri Babu et al 2017 [10]

discussed the heat and mass transfer on MHD free convective flow of second grade fluid through porous

medium over an infinite vertical plate.

The effect of chemical reaction on heat and mass transfer has attracted many researches to this flied

because of its application in many branches of science and engineering. Reddy et al 2012 [5] studied

the chemical reaction and radiation effect on unsteady MHD free convective flow near a moving vertical

plate. Reddy et al 2012 [6] discussed the effect of slip condition, radiation and chemical reaction on

unsteady MHD periodic flow of a viscous fluid through a saturated porous medium in a planer channel.

The concept of fluid flow with heat and mass transfer through porous medium plays an important role

in the field of geophysics, petrochemical engineering, metrology, oceanography and aeronautics.

Several researchers have analyzed the incompressible flow with MHD free convective flow through

porous medium of a rotating or non-rotating fluid with heat and mass transfer.

International Journal of Pure and Applied MathematicsVolume 117 No. 20 2017, 17-31ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)url: http://www.ijpam.euSpecial Issue ijpam.eu

17

Raju and Varma 2011 [4] discussed an unsteady MHD free convection oscillatory coquette flow through

a porous medium with periodic wall temperature. Rao et al 2011 [7] studied the effect os unsteady MHD

mixed convection of a viscous double diffusion fluid over a vertical plate in porous medium with

chemical reaction, thermal radiation and joule heating. P.Gurivi reddy 2017 [11] studied the MHD

boundary layer flow of a rotating fluid past a vertical porous plate.

In view of the above studies, in the paper we have considered the effect of heat and mass transfer on

convective two dimensional flow of a viscous incompressible electrically conducting fluid past an

infinite vertical plate through porous medium under the influence of uniform transverse magnetic field

with time varying permeability, oscillatory suction and time varying pressure gradient [12].

MATHEMATICAL FORMULATION:

We considered the unsteady MHD free convection two dimensional flow of an incompressible viscous

electrically conducting fluid with heat and mass transfer over an infinite vertical plate through porous

medium with time varying pressure gradient, time varying permeability and oscillatory suction under

the influence of uniform transverse magnetic field. The z-axis is taken along the plate and x- axis is

perpendicular to it. u and v are the velocity components along the x- axis and z-axis respectively. The

plate and the fluid is assumed to be at the same temperature and the concentration of species is raised

or lowered. The permeability of the porous medium is assumed to be 𝐾(𝑡) = 𝑘(1 + 𝜖𝑒𝑖𝜔𝑡) .

The suction velocity is assumed to be 𝑤(𝑡) = −𝑤0(1 + 𝜖𝑒𝑖𝜔𝑡). The time varying pressure gradient is

assumed to be 𝜕𝑝

𝜕𝜉= 𝑃(1 + 𝜖𝑒𝑖𝜔𝑡). All the physical variables are functions of z and t since the plate is

extended to infinite length.

The governing equations for the unsteady MHD free convective flow of an incompressible viscous

electrically conducting fluid with heat and mass transfer over an infinite vertical plate through porous

medium under the influence of uniform transverse magnetic flied using Boussinesq’s approximation

solution are given by

0u v

x y

(1)

2 22 3

2 2

1( ) ( )

( )

e oHu u p u u vw v u u g T T g C C

t z x z z t K t

(2)

2 22 3

2 2

1

( )

e oHv v p v v vw v v v

t z y z z t K t

(3)

2

12( )

C C Cw D K C C

t z z

(4)

2

12( )

T T Tw S T T

t z z

(5)

Where 𝑢, 𝑣 is the velocity components in the 𝑥 , 𝑧 directions respectively, 𝛽 is the coeffeicient of

volume expansion due to temperature, g is the acceleration due to gravity, 𝜌 is the density of the fluid,

𝜎 is the electrical conductivity of the fluid, T is the temperature of the fluid 𝑇∞ is the infinite plate

International Journal of Pure and Applied Mathematics Special Issue

18

temperature of the fluid, 𝐶 is the concentration of the fluid, 𝐶∞ is the infinite plate concentration of

the fluid, D is the mass diffusivity,

Let F u iv , x iy ,Combining the equations (2) and (3) , we get

2 22 3

2 2

1( ) ( )

( )

e oHF F p F F vw v u F g T T g C C

t z z z t K t

(6)

The boundary conditions are

𝐹(𝑧, 𝑡) = 𝑇(𝑧, 𝑡) = 𝐶(𝑧, 𝑡) = 𝑓(𝑡) 𝑎𝑡 𝑧 = 0 (7)

𝐹(𝑧, 𝑡) = 𝑇(𝑧, 𝑡) = 𝐶(𝑧, 𝑡) = 0 𝑎𝑡 𝑧 → ∞ (8)

Where, 𝑓(𝑡) = (1 + 𝜖𝑒𝑖𝜔𝑡) , Introducing the non-dimensional variables

*

0

FF

w ,

* 0w zz

v ,

2* 0w t

tv

, *

2

0

v

w

,

w

T T

T T

,

w

C C

C C

(9)

Using the above non – dimensional quantities, the governing equation can be reduced to the following

non- dimensional form by Dropping the asterisks

2 3 2

0 2 2 2

1(1 ) (1 )

1 (1 )

i t i t

i t

F F F F Mw e p e F Gr Gc

t z z z t m K e

(10)

2

0 2Pr (1 )Pr Pri tw e S

t z z

(11)

2

0 2(1 )i tSc w e Sc KcSc

t z z

(12)

The corresponding boundary conditions are

( , ) ( , ) ( , ) 1 i tF z t z t z t e at 0z (13)

( , ) ( , ) ( , ) 0F z t z t z t at z (14)

Where

22 0

2

0

B vM

w

is the Hartmann number ,

2

2

0

vK

Kw is the permeability parameter(porosity or

Darcy parameter),

2

1 0

2

w

v

is the fluid parameter, Pr

v

is the prandtl number,

vSc

D is the

Schmidt number, 1

2

0

K vKc

w is the chemical reaction parameter,


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1

2

0

S vS

w is the Heat source parameter,

3

0

( )wg v T TGr

w

is the thermal Grashof number,

*

3

0

( )wg v C CGm

w

is the mass Grashof number, e em is the Hall parameter.

In order to solve the equations (10) - (12) using boundary condition (13) and (14), we assume the

solution of the following form, because the amplitude ( 1) of permeability is very small.

0 1( , ) ( ) ( ) i tF z t F z F z e (15)

0 1( , ) ( ) ( ) i tz t z z e (16)

0 1( , ) ( ) ( ) i tz t z z e (17)

Substituting the equations (15)-(17) in to equations (10) – (12) respectively and equate the harmonic

and non-hormonic terms to obtain the Zeroth order and first order for momentum , temperature and

concentration distributions.

Zeroth order :

2

0 00 02

0Scw KcScz z

(18)

2

0 00 02

Pr Pr 0w Sz z

(19)

220 0

0 0 0 02

1F Fw M F p Gr Gm

z z K

(20)

The corresponding boundary conditions are

0 1F , 0 1 , 0 1 at 0z (21)

0 0F , 0 0 , 0 0 at z (22)

First order:

2

01 10 1 02

( )Scw Sc Kc i Scwz z z

(23)

2

01 10 1 02

Pr Pr( ) Prw S i wz z z

(24)

22 01 1

0 1 0 1 12

1(1 )

FF Fi w M i F p w Gr Gm

z z K z

(25)


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The corresponding boundary condition are

1 1F , 1 1, 1 1 at 0z (26)

1 0F , 1 0, 1 0 at z (27)

Solving the equations (18) – (20) with the boundary condition (21) – (22) ,we obtain the zeroth order

concentration, temperature and velocity

1

0

m ze (28)

3

0

m ze

(29)

5 3 5

0 122 3

1

m z m z m zGr Gm pF Ae e e

A AM

K

(30)

Solving the equations (22) – (24) with the boundary condition (25) – (26) ,we obtain the first order

concentration, temperature and velocity

1

20 1 0 11 2 2

1 0 1 1 0 1

1( ) ( )

m zm zScw m Scw m e

em Scw m Sc Kc i m Scw m Sc Kc i

(31)

4

40 3 0 31 2 2

3 0 3 3 0 3

Pr Pr1

Pr Pr( ) Pr Pr( )

m zm zw m w m e

em w m S i m w m S i

(32)

344

5 3 1

2 12

13 1311 1 2 3

2 2 3 4 4 5

1

15 15

6 6 7

1

m zm zm zm z m z m z

m z m zm z

A e A eBp Gr Gm eAC e C e C e Gr

B B A A C C CF

B A e A eeGm

C C C

(33)

The wall shear stress at the plate is given by

0 1

00 zz

F FF

z z z

(34)

The rate of heat transfer (Nusselt number) at the plate is given by

0 1

00 zz

Nuz z z

(35)


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The rate of mass transfer (Sherwood number) at the plate is given by

0 1

00 zz

Shz z z

(36)

RESULTS AND DISCUSSION:

In this paper , we studied the effect of unsteady MHD free convection two dimensional flow of an

incompressible viscous electrically conducting fluid with simultaneous heat and mass transfer over an

infinite vertical plate through porous medium with time varying pressure gradient, oscillatory suction

velocity and permeability of the porous medium. The effect of various flow parameters like Hartmann

number(M), prandtl number(Pr), Schmidt number(Sc), thermal Grashof number (Gr)and mass Grashof

number(Gc) on velocity profile, temperature, concentration, wall shear stress, and the rate of heat and

mass transfer are obtained and their behaviour are discussed graphically.

The figures (1) &(2) shows that an increase in thermal Grashof number(Gr) leads to increase in both

the velocities u and v. The increase in thermal Grashof number(Gr) indicates the supplementary

heating and a reduced amount of density. The figures (3) & (4) shows that the velocities u and v

increases as the increase of the mass Grashof number(Gm) for fixed M=2, α=0.3, K=1,Pr=0.75,

w0=0.4,Gr=3,ω=π / 8 .The figures (5) & (6) shows that the velocities u and v increases as the increase

of the Hartmann number. This is because of the reason that effect of a transverse magnetic field on an

electrically conducting fluid doesn’t gives rise to a resistive type force called Lorentz force similar to

drag force and upon increasing the values of M decreases the drag force which has the tendency to slow

down the motion of the fluid. The figures (7) & (8) shows that an increase in permeability of the porous

medium leads to decreases in both the velocities u and v . The Figure (9),(10)&(11) shows that an

increase in Schmidt number(Sc), chemical reaction (Kc) and the suction velocity (w0) leads to increases

the concentration profile. The Figure (12)&(13) shows that an increase in Prandtl number(Pr) and the

suction velocity (w0) leads to increases the temperature profile. Figure (14), shows the wall shear stress

for different values of thermal Grashof number(Gr) . It is clear from the figure that an increase in the

thermal Grashof number(Gr) increases the wall shear stress. . Figure (15), shows the wall shear stress

for different values of porous permeability parameter(K). It is clear from the figure that an increase in

the porous permeability parameter(K) decreases the wall shear stress.

Figure (16) and (17), shows the Nusselt number (Nu) for different values of prandtl numder(Pr) and the

suction velocity (w0). It is clear from the figures that an increase in the prandtl number(Pr) and the

suction velocity (w0) increases the rate of heat transfer (Nusselt number). Figure (18) and (19), shows

the Sherwood number (Sh) for different values of Schmidt numder(Sc) and the chemical reaction

parameter (Kc). It is clear from the figures that an increase in the Schmidt number(Sc) and the chemical

reaction parameter (Kc) increases the rate of mass transfer (Sherwood number(Sh)).


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Fig 1: Variation of fluid velocity (𝑢) for different values of thermal Grashof number(Gr) for fixed M=2, α=0.3, K=1,Pr=0.75,

w0=0.4,Gm=5,ω=π / 8.

:

Fig 2: Variation of fluid velocity (𝑣) for different values of thermal Grashof number(Gr) for fixed M=2, α=0.3, K=1,Pr=0.75,

w0=0.4,Gm=5,ω=π / 8.

Fig 3: Variation of fluid velocity (𝑢) for different values of mass Grashof number(Gm) for fixed M=2, α=0.3, K=1,Pr=0.75, w0=0.4,Gr=3,

ω=π / 8.


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Fig 4: Variation of fluid velocity (𝑣) for different values of mass Grashof number(Gm) for fixed M=2, α=0.3, K=1,Pr=0.75,

w0=0.4,Gm=5,ω=π / 8.

Fig 5: Variation of fluid velocity (𝑢) for different values of Hartman number(M)for fixed α=0.3, K=1,Pr=0.75, w0=0.4,Gm=5,ω=π / 8.

Fig 6: Variation of fluid velocity (𝑣) for different values of Hartman number(M)for fixed α=0.3, K=1,Pr=0.75, w0=0.4,Gm=5,ω=π / 8.


24

Fig 7: Variation of fluid velocity (𝑢) for different values of porosity parameter(K)for fixed α=0.3, Gr= 4,Pr=0.75, w0=0.4,Gm=5,ω=π / 8.

.

Fig 8: Variation of fluid velocity (𝑣) for different values of porosity parameter(K)for fixed α=0.3, Gr= 4,Pr=0.75, w0=0.4,Gm=5,ω=π / 8.

Fig 9: Variation of fluid concentration (𝜙) for different values of Schmidt number(Sc) for fixed ω=π / 8, Kc=1, t=0.2, 𝑤𝑜 = 0.6


25

Fig 10: Variation of fluid concentration (𝜙) for different values chemical reaction parameter(Kc)for fixed ω=π / 8, Sc=1, t=0.2, 𝑤𝑜 = 0.6.

Fig 11: Variation of fluid concentration (𝜙) for different values of suction velocity(𝑤𝑜)for fixed ω=π / 8, Sc=1, t=0.2

Fig 12: Variation of fluid temperature (𝜃) for different values of Prandtl number (Pr )for fixed ω=π / 8, Sc=1, t=0.2


26

Fig 13: Variation of fluid temperature (𝜃) for different values of suction velocity (𝑤0) for fixed ω=π / 8, S=1, t=0.2.

Fig 14: Variation of skin friction for different values of thermal Grashof number (Gr)for fixed M=2, α=0.3, K=1,Pr=0.75,

w0=0.4,Gm=7,ω=π / 8.

Fig 15: Variation of skin friction for different values of Prandtl number(Pr) fixed M=2, α=0.3, K=1, w0=0.4,Gm=7,ω=π / 8.


27

Fig 16: Variation of Nusselt number (Nu) for different values of Prandtl number(Pr) fixed ω=π / 8, S=1, t=0.2

Fig 17: Variation of Nusselt number (Nu) for different values suction velocity of suction velocity(𝑤𝑜) fixed ω=π / 8, Kc=1, t=0.2

Fig 18: Variation of Sherwood number (Sh ) for different values of Schmidt number (Sc) fixed ω=π / 8, Kc=1, t=0.2


28

Fig 19: Variation of Sherwood number (Sh ) for different values of chemical reaction parameter (Kc) for fixed ω=π / 8, Sc=2, t=0.2, 𝑤𝑜 =

0.6.

APPENDIX A:

𝑚1 =−𝑃𝑟𝑊0 + √(𝑃𝑟𝑊0) 2 − 4Pr

2 ; 𝑚2 =

−𝑃𝑟𝑊0 − √(𝑃𝑟𝑊0) 2 − 4Pr

2

𝑚3 =−𝑆𝑐𝑊0 + √(𝑆𝑐𝑊0) 2 − 4(𝐾𝑐𝑆𝑐)

2 ; 𝑚4 =

−𝑆𝑐𝑊0 − √(𝑆𝑐𝑊0) 2 − 4(𝐾𝑐𝑆𝑐)

2

𝑚5 =−𝑊0 + √(𝑊0) 2 − 4(𝑀2 + 1 𝑘⁄ )

2 ; 𝑚6 =

−𝑊0 − √(𝑊0) 2 − 4(𝑀2 + 1 𝑘⁄ )

2

𝑚7 =−𝑆𝑐𝑊0 + √(𝑆𝑐𝑊0) 2 + 4Sc(Kc + iω)

2 ; 𝑚8 =

−𝑆𝑐𝑊0 − √(𝑆𝑐𝑊0) 2 + 4Sc(Kc + iω)

2

𝑚9 =−𝑃𝑟𝑊0 + √(𝑃𝑟𝑊0) 2 + 4Pr(s + iω)

2

𝑚10 =−𝑃𝑟𝑊0 − √(𝑃𝑟𝑊0) 2 + 4Pr(s + iω)

2

𝑚11 =−𝐵1 + √(𝐵1) 2 + 4B2

2

𝑚12 =−𝐵1 − √(𝐵1) 2 + 4B2

2

𝑐1 = 𝑚5

𝑚52 + 𝐵1𝑚5 − 𝐵2

; 𝑐2 = 𝑚3

𝑚32 + 𝐵1𝑚3 − 𝐵2

; 𝑐3 = 𝑚1

𝑚12 + 𝐵1𝑚1 − 𝐵2

𝑐4 = 1

𝑚42 + 𝐵1𝑚4 − 𝐵2

; 𝑐5 = 1

𝑚32 + 𝐵1𝑚3 − 𝐵2

; 𝑐6 = 1

𝑚22 + 𝐵2𝑚2 − 𝐵2

𝑐7 = 1

𝑚12 + 𝐵2𝑚1 − 𝐵2


29

REFERENCES:

[1]. Raptis. A , Massias,.C and Tzivanidis .G , Hydromagnetic free convection flow through a porous

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[2]. Makinde, O.D and Mhone, P.Y, Heat transfer to MHD oscillatory flow in a channel filled with porous

medium, Romanian J.Physics 50(9- 10) ,2005 ,Pp.931-938.

[3]. D.S.Chauhan and P.Rastogi, Hall current and heat transfer effects on MHD flow in a channel

partially filled with a porous medium in a rotating system, Turkish Journal Eng.Env.Sci. 33, 2009,

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[4].M. C. Raju, S. V. K. Varma, Unsteady MHD free convection oscillatory Couette flow through

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[5].T. S. Reddy, S. V. K. Varma & M. C. Raju; Chemical reaction and radiation effects on unsteady MHD

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[6] .T. S. Reddy, M. C. Raju & S.V. K. Varma, “ The effect of slip condition, Radiation and

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[7] .B. M. Rao, G.V. Reddy, M. C. Raju, S.V. K. Varma, Unsteady MHD free convective heat and

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radiation, chemical reaction and Soret effects, International Journal of Engineering Sciences &

Emerging Technologies, October 2013. Volume 6, Issue 2, pp: 241-257.

[8]. Ashaf A. Moniem, W.S.Hassanin, Solution of MHD flow past a vertical porous plate through a porous medium under oscillatory suction, Applied Mathematics, (2013), Vol.4 Pp. 694-702.

[9]. G.Prabhakara Rao, M.Naga sasikala and P.Gayathri ,Heat transfer on steady MHD rotating flow

through porous medium in a parallel plate channel, Int. journal of Engineering Research and

Applications, April 2015,Vol.5, issue 4, Pp.29-38.

[10]. D. Dastagiri Babu1, S.Venkateswarlu2 and E.Keshava Reddy, Heat and Mass Transfer on MHD Free convective flow of Second grade fluid through Porous medium over an infinite vertical plate, IOP Conf. Series: Materials Science and Engineering, 2017, doi:10.1088/1757-899X/225/1/012267.

[11]. P. Gurivi Reddy, MHD boundary layer flow of a rotating fluid past a vertical porous plate, 2017 Volume 12, pp. 579-593. [12]. 1K.Rajkumar, 2 S.A.Mohammed Uveise, “A NETWORK LIFETIME ENHANCEMENT METHOD FOR SINK RELOCATION AND ITS ANALYSIS IN WIRELESS SENSOR NETWORKS”, International Journal of Innovations in Scientific and Engineering Research (IJISER), ISSN: 2347-971X (online), ISSN: 2347-9728(print), Vol.2, No.4, pp.77-82, 2015, http://www.ijiser.com/.


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