International Journal of Scientific & Engineering Research Volume 2, Issue 6, June-2011 - 1 - ISSN 2229-5518

Hydrodynamic Free Convection Flow of A Rotating Visco-elastic Fluid Past An Isothermal Vertical Porous Plate With Mass Transfer

S. Biswal, G.S. Ray, A. Mishra

Abstract - The effect of free convection and mass transfer in the unsteady flow of an incompressible electrically conducting visco-elastic past an isothermal vertical porous plate with constant suction normal to the plate has been studied. The effects of perm eability parameter (Kp) of the porous medium, rotation paramet er (R), Grashof number for heat transfer (Gr), Grashof number for mass transfer (Gm), frequency parameter ( ro) and the heat source param eter (a0) on the transient primary and secondary velocity field, temperature field and the rat e of heat transfer have been investigat ed with the help of graphs and tables.

Key words - Hydrodynamic flow, mass transfer, rotating fluid, porous medium, isotherm al plate.

—————————— • ——————————

1. INTRODUCTION

THE problem of hydrodynamic free convection flow of a rotating viscoelastic fluid has

received a considerable attention of many researchers because of its applications in cosmical and geophysical science. These problems are of general interest in the field of atmospheric and oceanic circulations, nuclear reactors, power transformers and in the field of scientific and industrial research. Permeable porous plates are used in the filtration process and also for a heated body to keep its temperature constant and to make the heat insulation of the surface more effective.
Several authors have discussed the flow of a viscous fluid in a rotating system in the presence and absence of magnetic field. Reptis and Singh [1] have reported the effect of rotation on the free convection MHD flow past an accelerated vertical plate. Singh[2] has studied the unsteady free convection flow of a viscous liquid through a rotating porous medium. Dash
and Biswal [3] have investigated the effect of
at infinite vertical porous plate with time dependent temperature and concentration.
Rath and Bastia[7] have analysed the steady flow and heat transfer in a visco-elastic fluid between two coaxial rotating disks. Mukherjee and Mukherijee[8] have studied the unsteady axisymmetric rotational flow of elastico-viscous liquid. Datta and Jana[9] have investigated the problem of flow and heat transfer in an elastico-viscous liquid over an oscillating plate in a rotating frame.
The present study considers the
simultaneous effect of heat and mass transfer on the hydrodynamic free convection flow of a rotating viscoelastic fluid past an infinite vertical isothermal porous plate.
2. Formulation of the problem
Let us consider the flow of a rotating viscoelastic incompressible fluid cr in a medium past an infinite vertical isothermal porous plate. Let x and y-axis be in the plane of the plate and

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z-axis normal to the plate and let u, v, w be the velocity components of the fluid in x, y and z
direction respectively. Both the liquid and the
z* = w0

K * = K ' w0


z , t* = w0

v



t , u* = u ,

v U 0

plate are considered in a state of rigid body

p p v 2

rotation about z-axis with uniform angular
Where

K 'p

is the dimensional
velocity Q. Further, Let us assume that the
permeability parameter and

* is the non-

constant heat source Q (absorption type Q = -Q0

dimensional permeability parameter.

(T-Too) is at z = 0 and the suction velocity at the


q = u

i v ,

T= =T - Too , P =

vPC p ,

plate w = -w0 where w0 is a positive real number.
Here we have neglected buoyancy effect. Since

Qv

U 0 U 0

2

0 0

Tw - Too K

R= , Rc= 2
the plate is infinite in extent, all physical
variables depend on z and t only. Considering u

2

0

Pg * (C - C )

vg

+ iv = q and using non-dimensional quantities the equations governing the flow (dropping the

asterisks) are

Gm =

V= v'

0

w oo

2

0 0

, Gr =

(Tw - Too ),

0 0

Where Pr : Prandtl number
2 3 Gr : Grashof number,

aq - aq 2iR =1 (q - U ) = aU a q R

a q G T G C




at az

at az 2

c az 2 at r m

M : Magnetic parameter,
---------------------(1)
R : Rotation parameter

a 2T

aT aT

K : Permeability parameter,

az 2


Pr - Pr

az at

- a 0T = 0

p

a0 : heat source parameter,

a 2 C

---------------------(2)

aC aC

U : free steam velocity

az 2

Sc

az

- Sc = 0

at

Rc : non-Newtonian parameter
--------------------(3)
The non-dimensional boundary conditions are
q = 1 + Eeirot, (T = 1 + E eirot, ro<0), C = 1
+ E eirot at z = 0
q = (1 + Eeirot, ro>0), T 0, C = 0 at z �oo --------------------- (4)
The non-dimensional quantities
3. Solution of the equations:
In order to solve the equation [1] – [3] we assume velocity, temperature and concentration of the liquid in the neighborhood of the plate as
Q = (1 -q0) + E eirot (1 -q1)
---------------------(5) T = T0 + E T ' eirot
introduced in equations (1) – (3) are defined as
--------------------- (6)
C = C0 + E eirot C1

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Where
--------------------- (7)

C1'

S c C1' - iroS c C1 = 0

--------------------- (16)
(1 -q1) eirot = (Mr + I Mi) (Cos rot + i sin rot)
The boundary conditions are
T1 eirot = (Tr + iTi) (Cos rot + i sin rot)
q = 0, T
= 1, C
= 1, q
= 0, T =1, C =
C1 eirot = (Cr + iCi) (Cos rot + i sin rot)

0 0

0 at z = 0

0 1 1 1

For free stream (when E <<1)
U (t) = 1 + E eirot ------------------------(8)

The transient primary velocity and temperature profiles can be deduced from equations (5) and (6) for rot = n 2
Hence u (z, t) = u0 (z) - E Mi
--------------------- (9)
q0 0, T0 0, C0 0, q1 0, T1 0, C1 0 at z � oo --------------------- (17)
Solving equations (11) – (16) with boundary conditions (17) we obtain for Rc <<1 and ro small,
C0 = e- Sc z
C1 = e D1z

D2 z

T = T0 - E T1
--------------------- (10) Where u0 (z) + iv0 (z) = 1 - q0
Using equations (5) – (8) in equations (1),
T0 = e
T1 = e D3z
q0=

Gr(e D2 z - e D2 z )

Gm(e - scz - e D3 z )

(2) and (3) and equating harmonic and non- harmonic terms, we get

(D2 - D4 )(D2 - D5 )

q1=

(Sc

D4 )(Sc

D5 )

(1+iroRc)

Gr(e

D3 z

- e D7 z )

Gm(e

D3 z

- e D7 z )

T 1

q1' - Vq1' - \

(2iR

iro ) q1 = -Gr T1 - Gm C1

(D3 - D6 )(D3 - D7 )

(D1 - D6 )(D1 - D7 )

\L K p

1

--------------------- (11)
Putting values of T0, T1 in equation (6)
C0, C1 in equation (7) and q0, q1 in equation (5) we get

q1' - Vq0' -

J K p

2iR

q0 = -Gr T0 - Gm C0

T = e D2 z

e irot (e D3 z )

--------------------- (12)
C = e

- Sc z

eirot

(eD1z )

T0' - PrVT0' - a 0T0 = 0

q = 1 -

Gr(e

D2 z

- e D2 z )

Gm(e

- scz

- e D3 z )

--------------------- (13)
+Eeirot

(D2 - D4 )(D2 - D5 )

(Sc

D4 )(Sc

D5 )

T1' - PrVT1' - (iroPr

a 0 )T1 = 0

T Gr (e D3 z - e D7 z )

Gm(e D3 z - e D7 z )

--------------------- (14)

\1 - (D

- D )(D

- D ) - (D

- D )(D

- D )

C0'

S c C1' = 0

L 3 6 3 7

1 6 1 7

--------------------- (15)
Separating real and imaginary parts
u = 1 - Gr (B4 B12 - B5 B13) - Gm (B6 B14

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- B7 B13) + EH1 Cos rot -EH2 Sin rot
v = EH2 Cos rot + E H1 Sin rot -Gr (B5 B12
+ B4 B13) - Gm (B7 B14 + B6 B13) Where
H1 = 1 - (B8 B16 - B9 B17) Gr - (B10B18 -
B3 B11) Gm,
H2 = - Gr (B0 B16 + B8 B17) - Gm (B11 B18

+ B3 B10),

2

of view Gr<0 corresponds to cooling of the plate and Gr<0 corresponds to heating of the plate by free convection currents.
The transient primary velocity profiles are shown in Fig. 1. It is observed that the transient velocity u increases regressively and after attaining the maximum value decreases asymptotically and ultimately attains steady state.
Comparing the curves (4 and 5) of figures (1), it
D1 =
1 4

J K p

64R 2
is observed that the porosity parameter (Kp)
increases the primary velocity. The rotation

D2 = (P 2
4a 0 )

16ro 2 P 2 ,

parameter has negligible effect on the primary velocity while it decreases the secondary velocity
D3 =

2

1 4M 2 4

J K p

(8R

4ro )2 ,
appreciably (curves 4 and 6). Comparing the curves (4, 8) and (2, 3), it is evident that the
Taking the value of V=1.
The values of the other constants involved are omitted here to save space.
4. Results and Discussions
The problem of hydrodynamic free convection and mass transfer flow of a rotating viscous fluid past an isothermal vertical porous plate has been considered. The effects of porosity parameter (Kp), rotation parameter (R), Grashof
numbers (Gr, Gm), frequency parameter (ro) and
the heat source parameter (a0) on the transient primary velocity, secondary velocity, temperature distribution profiles and the rate of heat transfer have been discussed to observe the physical significance and the mystery of the problem. To be realistic Prandtl number is chosen as 0.71 (the water vapour) and 7.0 (for water) approximately
at 1 atmosphere and 250 C). From physical point
Grashof number for heat transfer (Gr) and the Grashof number for mass transfer (Gm) have accelerating effect on both the components of the velocity field. On careful observation of curves (4, 9) and (2, 4), it is observed that Prandtl number (Pr) and heat source parameter (a0) have decelerating effect on the transient velocity. The frequency parameter has negligible effect on the velocity field.

Fig 1 : Effect of different parameters on transient

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primary velocity profile when E= 0.02
of the tempe rature field. An
increa

F amete


se in frequency parameter leads to an oscillatory dary temperature d for the transient ves 3 and 4). The ates the transient
oes not affect the

Fig 2: Effect of different parameters on transient

secondary velocity profile when E=0.02
The effects of different parameter on the transient secondary velocity have been shown in Fig.2. It is marked that the velocity V decreases with modified Grashof number Gm (curves 1, 2 and
3). This is valid for water vapour or air. But, the
velocity ‘V’ rises for liquid sodium (Pr=1). As the value of rotation parameter R increase, the transient secondary velocity also rises (curve 5). The variation of frequency parameter ro does not produce any appreciable variation in the velocity
‘V’. An increase in Grashof number Gr reduces

Fig 3 : Effect of different parameters on transient primary temperature profiles when E= 0.02, rot = n 2
Table (1 and 2) show the effects of a0, Pr and ro
on the transient primary rate of heat transfer

(Nu ) and transient secondary rate of heat

Curves Kp R Gr a0 ro Pr Gm

the velocity ‘V’. For water (Pr = 7.0), the velocity
‘V’ falls in comparison to that of liquid sodium
(Pr = 1.0)

1 1 .2 2 1 5 .71 0

2 1 .2 2 1 5 .71 2

3 1 .2 2 1 5 .71 3

4 1 .2 2 1 5 1 2

5 1 .4 2 1 5 1 2

6 1 .2 2 1 5 1 2

7 1 .2 4 1 5 1 2

8 1 .2 2 1 5 7 2

1 and rot =
reveal that
eter (a0>0),

a0 increase Nup and Nus and reverse effect is

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noted for generation type that source parameter (a0>0). It is interesting to note that an increase in frequency parameter leads to an oscillatory effect on both components of transient heat transfer.

Fig. 4 : Effect of different parameters on transient secondary temperature profiles


when E = 0.02, rot = n 2

Table: 2


Variation of transient primary (Nup) and secondary (Nus) heat transfer at E = 0.01 and rot = n 2 (generation type)

Table: 1

Variation of transient primary (Nup) and secondary (Nus) heat transfer at E = 0.01 and rot =

n 2 (absorption type)

5. Conclusion
The following conclusions are drawn for the transient primary and secondary velocity field, temperature field and the rate of heat transfer.
1. Heat source parameter (a0) and Prandtl
number (Pr) have decelerating effect on both components of the transient velocity field. The effect of the heat source parameter (a0) is more pronounced than the Prandtl number (Pr). The Prandtl number also increases the temperature field.
2. The Grashof number for heat transfer

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(Gr) and Grash of number for mass transfer (Gm) enhance both components of the velocity of the flow field.
3. The porosity parameter (Kp) accelerates the transient primary velocity field while it does not affect the transient secondary velocity field.
4. Both components of the velocity field decrease with the increase in rotation parameter but the effect is more pronounced in the case of the secondary velocity. The effect of frequency parameter(ro) is significant for both the components of temperature field and the rate of heat transfer and it has a negligible effect on the velocity field.
5. The heat source parameter enhances both components of rate of heat transfer for heat sink and the reverse effect is observed for heat source.

References

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Res. Comm., Vol. 12, pp 31-41 (1985)
2. Singh, A.K., Proc. Ind. Nat. Sci. Acad., Vol. 52 A, pp 221-226 (1986)
3. Dash, G.C. and Biswal, S., AMSE,
Modelling, Simulation and Control, B, Vol. 21, No.4 pp 13-24 (1989)
4. Dash, G.C. and Ojha, B.K., AMSE, Modelling, Simulation and Control, B, Vol 21 No. 1, pp 1-12 (1989)
5. Sacheti, N.C. and Singh, A.K., Int.

Comm. Heat and Mass transfer, Vol. 19,

423-433 (1992)
6. Sattar, M.A., Indian J. Pure Appl. Math., Vol 23, No. 1, 759-766 (1994)
7. Rath, R.S. and Bastica, S.N., Proc.
Indian Acad. Sci., 87A, No.9, 227-236 (1978)
8. Mukherje, S. and Mukherjee, Srikumar,
Indian J. pure appl. Math., 14 (12), 1534-
1541 (1983).
9. Datta, N. and Jana, R.N., Istanbul Univ.
Fen Fak. Mec. Seri A, 43, 121-130 (1978)

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