J. Comp. & Math. Sci. Vol. 1(5), 528-536 (2010)
Unsteady MHD Flow Past a Vertical Oscillating Plate with Thermal Radiation and Variable Mass Diffusion G. V. RAMANA REDDY1, CH.V. RAMANA MURTHY1 and N. BHASKAR REDDY2 LakiReddy BaliReddy1 College of Engineering, Mylavaram-521230, (A.P), (India) S.V.University2, Tirupati, 517502.(A.P), (India) Email: gvrr2020@yahoo.co.in. Corresponding Author
G.V Ramana Reddy, Plot No. 337, Anil Castles, Gollapudi, Vijayawada, Andhra Pradesh. Pin-521225. E-mail: gvrr2020@yahoo.co.in
ABSTRACT The aim of the present analysis is to investigate the effects of magnetic field, frequency of excitation, time, Schmidt number and Prandtl numbers on the velocity field, temperature, concentration field and the skin friction. In this paper, the hydromagnetic flow of a viscous incompressible laminar fluid past an oscillating vertical plate with radiation and variable mass diffusion is considered. The governing equations of motion are solved for the velocity field, concentration, temperature and skin-friction. The results are presented in graphical format for different values physical parameters involved in the study. It is noticed that the critical parameters effect the flow pattern significantly.
Keywords: Thermal conductivity, MHD, Radiation effect, Skin-friction
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
G. V. Ramana Reddy et al., J. Comp. & Math. Sci. Vol. 1(5), 528-536 (2010)
INTRODUCTION The study of radiative heat and mass transfer in convective flow is found to be most important in industrial and technological applications. The applications are many and are often found in situations such as fiber and granular insulation, geo thermal systems in the heating and cooling chambers, fossil fuel combustion, energy process, and Astrophysical flows etc. When mass transfer takes places in a fluid at rest, the mass is transferred purely by molecular diffusion resulting from concentration gradients. For low concentration of the mass in the fluid and low mass transfer rate, the convective heat and mass transfer processes are similar in nature. A number of investigations have already been carried out with combined heat and mass transfer under the assumption of different physical situations. Radiation in free convection has also been made by several investigators due to several application areas of prime importance. Exact solutions of free convection flow past a vertical oscillation plate in the free convective flow was first obtained by Soundlgeskar9, and the same problem with mass transfer effect was considered by Soundlegekar and Akolkan7. The effects of mass transfer on free convection flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction was investigated by
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Das et al.2, subsequently Soundlgekar et al.10 examined the effects of mass transfer on the flow past an infinite vertical oscillating plate with constant heat flux. Gupta and Gupta6 studied the effect of radiation on the combined free and forced convection of an electrically conducting fluid flow inside and open ended vertical channel in the presence of a uniform transverse magnetic field. Using CoglyVincentime-Gilles1 equilibrium model, the radiation effects on free convection flow past a vertical plate was examined by Soundlegekar and Thakhar8. The effects of mass transfer on free convection flow past a vertical isothermal plate was examined by Gebhart and Pera5. The combined effects of thermal radiation and chemical reaction on free convection flow past a vertical plate in porous medium was studied by Deka and Neogy3. All of the considered the fact that free convection current caused by temperature differences is also caused by the differences in the concentration are material constitution as suggested by Gebhart6. Although different authors studied mass transfer with or without radiation effects on the flow past oscillating vertical plate by considering different surface conditions but the study on the effects of magnetic field on free convection heat and mass transfer with thermal radiation and variable mass diffusion in flow through an oscillating plate has not been found in literature and hence the motivation to
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
G. V. Ramana Reddy et al., J. Comp. & Math. Sci. Vol. 1(5), 528-536 (2010)
MATHEMATICAL FORMULATION We consider an unsteady natural convection flow of a viscous incompressible electrically conducting fluid past an infinite vertical plate. To visualize the flow pattern a Cartesian co-ordinate system considered where x’ - axis is taken along the infinite vertical plate, the y’ - axis is normal to the plate and fluid fills the region y’ > 0. Initially, the fluid and the plate are kept at the same constant temperature T’ and species concentration C’. At time t’ > 0, the plate is given an oscillatory motion in its own plane with a velocity U0 cos ‘ t’. At the same time the plate temperature is raised to T’w and concentration is raised linearly with time and a magnetic field of uniform strength B0 is applied normal to the plate. It is assumed that the magnetic Reynolds number is very small and the induced magnetic field is negligible in comparison to the transverse magnetic field. Further, it is also assumed that the effect of viscous dissipation is negligible in the energy equation and the level of species concentration is very low so that the Soret and Dufour effects are totally negligible.
As the plate is infinite in the extent so the derivatives of all the flow variables with respect to x’ vanish and they can be assumed to be functions of y’ and t’ only. Thus the motion is one dimensional with only non-zero vertical velocity component u’ varying with y’ and t’ only. Due to one dimensional nature, the equation of continuity is trivially satisfied. Under the above assumptions and following the Oberbeck-Boussinesq approximation, the unsteady flow field is governed by the following set of equations:
u' g T ' T' t ' g C ' C'
CP
2u ' B02 u' y ' 2
T ' 2T ' q k 2 r t ' y ' y '
(1)
(2)
C ' 2C ' D 2 t ' y '
(3)
Along with the following initial and boundary conditions: For t 0 : u ' 0, T ' T ' , C ' C ' , y' For t 0 , u ' U 0 cos ' t ' , T ' T ' w ,
C ' C ' ( C ' w C ' ) At,
A
undertake this study. It is therefore proposed to study the effects of thermal radiation and variable mass diffusion on hydrom-agnetic flow past an oscillating vertical plate.
A
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y ' 0 (4) and u' 0,T ' T ' , C ' C ' ,as y' Where A
U0
We assume that the medium is optically thin with relatively low density. Thus
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
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G. V. Ramana Reddy et al., J. Comp. & Math. Sci. Vol. 1(5), 528-536 (2010)
qr e 4(T ' T ' ) K w b d 4 I T 'T ' y T ' w 0
(5)
Using equation (5), equation (2) can be written as:
T ' 2T ' CP k 2 4 I (T ' T ' ) t ' y '
(6)
To reduce the governing equations in dimensionless form the following quantities are introduced:
u' t 'U 02 y 'U 0 u ,t ,y U0
g (T ' w T ' ) , U 03
Gm
g (C ' w C ' ) U03
Pr F
are: For t 0, u 0 , 0 , 0 ,
y
For t 0, u cost, 1 1, at y 0 (11) 1) and u 0, 0, 0, as y Solution of the Problem The solution of the equations (8), (9) and (10) are assumed as:
u ( y , t ) u 0 ( y ) e i t
(12)
i t
(13) (14)
u 0 e i t cos t ,
(7)
4 I 2 , B02 M kU 02 U02
0 e i t , 0 te i t , at y 0 u0 0 , 0 0, 0 0, as y (15) Using equations (12), (13) , (14) and (15), the solution of equations (8),(9) and (10), are expressed as:
e Ry cos t Gr By u ( y , t ) 2 e e Ry (16) 2 B R Gm Dy Ry e e t 2 2 D R
Thus the equations (1), (6) and (3) reduce to:
1 2 F t Pr y 2 Pr
and the initial and boundary conditions (4)
With the modified boundary conditions are
C p , Sc , k D
u 2u Gr Gm Mu t y 2
(10)
( y , t ) 0 ( y )e ( y , t ) 0 ( y )e i t
T ' T ' C ' C ' , T 'w T ' C 'w C '
Gr
1 2 t Sc y 2
A
f o l l o w i n g C o g l y - Vi n c e n t i n e - G i l l e s equilibrium model we have
(8)
(9)
( y, t ) exp( By ) ( y, t ) t exp( Dy ) The Skin-friction on the wall is:
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
(17) (18)
G. V. Ramana Reddy et al., J. Comp. & Math. Sci. Vol. 1(5), 528-536 (2010)
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u y
Gr R cos t B 2 R 2 R B t Gm R D (19) D 2 R 2
y 0
Where
D
F i Pr, i Sc ,
R
M i
B
2. Fig 2 illustrates the combined effect of magnetic field, and dimensionless thermal radiation parameter. It is noticed that as the magnetic intensity is increased the velocity increases. Further, it is noticed that as the thermal radiation also contributes for the increases in the velocity field.
RESULTS AND DISCUSSION 1. The effect of magnetic field on velocity profiles is illustrated in fig 1. When all physical parameters in the system are held constant and applied magnetic field is increased, it is seen that the velocity increases. Also at times it is seen that there is a backward flow which can be attributed to the effect of magnetic intensity. Further, in general it is seen that as we move far away from the boundary, the velocity decreases.
Fig.2
Fig.1.
3. The eff ect of Schmidt Number on velocity field is illustrated in figure 3. It is seen that the velocity decreases as we mover far away from the plate. Further, as the Schmidt Number increases the f luid velocity also increases. But in general, the velocity decreases as we move away from the plate.
Effect of Magnetic field on the velocity profiles
Effect of thermal radiation (F) on the velocity profiles
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
G. V. Ramana Reddy et al., J. Comp. & Math. Sci. Vol. 1(5), 528-536 (2010)
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5. When the frequency of excitation of the plate is held constant and the dimensionless thermal radiation parameter is increased, it is noticed that temperature goes down. Further it is noticed that as we move far away from the plate the temperature distribution significantly decreases.
Fig.3.
Effect of 'Sc' on the velocity profiles
4. The effect of time on the velocity field is shown in figure 4. For a constant value of the Prandtl Number Pr = 7), as t increases, an interesting point that is noticed in the graphical represent is that at a certain distance far away from the plate the effect of all fluid particles is found to be identical and subsequently a reverse trend is noticed. Fig.5.
Fig.4.
Effect of time (t) on the velocity profiles.
Effect of thermal radiation (F) on the temperature profiles.
6. The effect of frequency of excitation for a fixed Prandtl Number as been illustrated in fig 6. It is observed that as the frequency of excitation increases, in general it is observed that the temperature dissipates quiet fast in the fluid medium. It is noticed that as the frequency of concentration increases at a fixed distance, away from the plate the temperature in the fluid medium decreas
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Fig.6.
G. V. Ramana Reddy et al., J. Comp. & Math. Sci. Vol. 1(5), 528-536 (2010)
Effectof ‘‘on the temperature profiles.
Fig.8.
7. Figure7 illustrates the effect of
Effect of time (t) on the concentration field.
Schmidt Number on the concen-tration field. It is observed that as the Schmidt Number increases the concentration of the fluid medium decreases and is found to be almost linear.
Fig7.
9. The effect of Schmidt Number on the skin friction for a constant value of the other flow entities has shown in fig 9. It is noticed that, the skin -friction decreases in the presence of the Schmidt Number. Further it is seen as Schmidt Number increases the skin friction is found to be increasing.
Effect of 'Sc' on the concentration field
8. The effect of time ( t ) for a fixed value of Schmidt Number on the concentrated field as been shown in figure 6. It is seen that as 't' decreases the concentration on the fluid medium also decreases. However, the schematic representation illustrates that the rate at which the concentration decreases is found to be less as 't' increases.
Fig.9.
Effect of 'Sc' on Skin-Friction.
10. The effect of frequency of excitation, while maintaining the constant magnetic field has been examined on
Journal of Computer and Mathematical Sciences Vol. 1, Issue 5, 31 August, 2010 Pages (528-635)
G. V. Ramana Reddy et al., J. Comp. & Math. Sci. Vol. 1(5), 528-536 (2010)
skin friction in figure 10. It is seen that the skin friction decreases when the frequency of excitation is unity. Further, as the frequency of excitation is doubled the profile for the skin friction is found to be approximately parabolic and thereafter a wave pattern is noticed. Such a wave pattern could be attributed to be fact that, the frequency of excitation of the plate. In general, it can be concluded that, more the frequency of excitation a wave pattern could be seen increasing of skin friction which appears to be more interesting.
Fig.10. Effect of 'ď ˇ' on Skin-Friction.
11. Figure 11 shows the effect of small values of excitation on skin friction. When the frequency of excitation is low, the skin friction on the plate appears to be sinusoidal. Further, it is seen that when the frequency of excitation increases and is relatively small, the profiles for the skin friction are linear.
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Fig.11. Effect of 'ď ˇ' on Skin-Friction.
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pp 2025 - 2050(1971). 5. Gebhart.B, Heat Transfer, Tata McGraw Hill, 1971. 6. Gupta P.S. and Gupta A.S., Radiation effect on hydromagnetic convection in a vertical channel, Int. J. Heat Mass Transfer, vol 17, pp 1437 - 1442 (1974). 7. Soundlgekar V.M. and Akolkar S.P., Effects of free convection currents and mass transfer on flow past a vertical oscillating plate, Astrophysics and Space Science, vol 89, pp 241 - 254 (1983). 8. Soundlgekar V.M. and Thakhar H.S.,
Radiative connective flow past a semi infinite vertical plate, Modelling Measure and Cont., vol 51, pp 31 - 40 (1992). 9. Soundlgekar V.M. free convection effects on the flow past a vertical oscillating plate Astrophysics and Space Science, vol 66, pp 165 - 172(1979). 10. Soundlgekar V.M., Lahurikar R.M., Pohanerkar S.G. and Birajdar N.S., Effects of mass transfer on flow past an oscillating infinite vertical plate with constant heat flux, Thermo physics and Aeromechanics, vol 1, pp 119 - 124 (1994).
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