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Mass and heat transfer effect on MHD flow of a visco-elastic fluid through porous medium with oscillatory suction and heat source
International Journal of Heat and Mass Transfer 57 (2013) 433–438
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Mass and heat transfer effect on MHD flow of a visco-elastic fluid through porous medium with oscillatory suction and heat source S.R. Mishra ⇑, G.C. Dash, M. Acharya Department Of Mathematics, Institute of Technical Education and Research, Siksha ‘O’ Anusandhan University, Jagamohan Nagar, Khandagiri, Bhubaneswar-30, Orissa, India
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 4 April 2012 Received in revised form 9 October 2012 Accepted 16 October 2012 Available online 16 November 2012
The present study considers the effect of heat and mass transfer on free convective flow of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate through a porous medium with time dependent oscillatory permeability and suction in presence of a uniform transverse magnetic field and heat source. The novelty of the present study is to analyze the effect of time dependant fluctuative suction and permeability of the medium on a visco-elastic fluid flow. The necessity of the present analysis arises as the most industrial fluids exhibit the visco-elastic behavior. The coupled nonlinear partial differential equation are turned to ordinary by super imposing a solution with steady and time dependant transient part. Finally, the set of ordinary differential equations is solved with a perturbation scheme to meet the inadequacy of boundary condition. Elasticity of the fluid and the Lorentz force reduce the velocity and it is more pronounced in case of heavier species. Most interesting observation is the fluctuation of velocity appears near the plate due to the presence of sink and presence of elastic element as well as heat source reduce the skin friction. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: MHD flow Visco-elastic Mass and heat transfer Porous medium
Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . Formulation of the problem Method of solution. . . . . . . . Results and discussion . . . . . Conclusion . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . References . . . . . . . . . . . . . .
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1. Introduction An important class of two dimensional time dependent flow problem dealing with the response of boundary layer to external unsteady fluctuations of the free stream velocity about a mean value attracted the attention of many researchers. MHD flow with heat and mass transfer has been a subject of interest of many researchers because of its varied application in science and technology. Such phenomena are observed in buoyancy induced motions in the atmosphere, in water bodies, quasi-solid bodies such as earth, etc. In natural processes and industrial applications many transportation processes exist where transfer of heat and mass ⇑ Corresponding author. Tel.: +91 9937169245. E-mail addresses: [email protected] (S.R. Mishra), gcdash@india times.com (G.C. Dash), [email protected] (M. Acharya). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.10.053
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433 434 435 436 437 438 438
takes place simultaneously as a result of thermal diffusion and diffusion of chemical species. Several researchers have analyzed the free convective and mass transfer flow of a viscous fluid through porous medium. The permeability of the porous medium is assumed to be constant while the porosity of the medium may not be necessarily being constant. Kim [1] studied the unsteady MHD convective heat past a semiinfinite vertical porous moving plate with variable suction. The problem of three dimensional free convective flow and heat transfer through porous medium with periodic permeability has been discussed by Singh and Sharma [2]. Singh and Singh [3] have analyzed the heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity. Bathul [4] discussed the heat transfer in a three dimensional viscous flow over a porous plate moving with harmonic disturbance. Postelnicu [5] studied numerically the influence of a magnetic field on heat
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S.R. Mishra et al. / International Journal of Heat and Mass Transfer 57 (2013) 433–438
Nomenclature C0 D Gr K0 k Rc Nu S Sh T t u
v0 y
e b b0
t
species concentration molecular diffusivity Grashof number for heat transfer permeability of the medium thermal diffusivity elastic parameter Nusselt number heat source parameter Sherwood number non-dimensional temperature non-dimensional time non- dimensional velocity constant suction velocity non-dimensional distance along y-axis a small positive constant volumetric coefficient of expansion for heat transfer volumetric coefficient of expansion with species concentration kinematic coefficient of viscosity
and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Singh and Gupta [6] have investigated the MHD free convective flow of a viscous fluid through a porous medium bounded by oscillatory porous plate in slip flow regime with mass transfer. Ogulu and Prakash [7] considered heat transfer to unsteady magneto hydrodynamic flow past an infinite vertical moving plate with variable suction. Das et al. [8] analyzed the mass transfer effect on unsteady flow past an accelerated vertical porous plate with suction employing numerical methods. The basic equations of incompressible MHD flow are non-linear. But there are many interesting cases where the equations became linear in terms of the unknown quantities and may be solved easily. Linear MHD problems are accessible to exact solutions and adopt the approximations that the density and transport properties be constant. No fluid is incompressible but all may be treated as such whenever the pressure changes are small in comparison with the bulk modulus. Ferdows et al. [9] analyzed free convection flow with variable suction in presence of thermal radiation. Alam et al. [10] studied Dufour and Soret effect with variable suction on unsteady MHD free convection flow along a porous plate. Mishra et al. [16] have studied fluctuating flow of a non-Newtonian fluid past a porous flat plate with time varying suction. Majumdar and Deka [11] gave an exact solution for MHD flow past an impulsively started infinite vertical plate in the presence of thermal radiation. Muthucumaraswamy et al. [12] studied unsteady flow past an accelerated infinite vertical plate with variable temperature and uniform mass diffusion. Asghar et al. [13] have reported the flow of a non-Newtonian fluid induced due to oscillations of a porous plate. Moreover, Dash et al. [14] have studied free convective MHD flow of a visco-elastic fluid past an infinite vertical porous plate in a rotating frame of reference in the presence of chemical reaction. The specific area of application of heat and mass transfer phenomena is common in chemical process industries such as food processing and polymer production. Recently, Gireesh Kumar and Satyanarayana [17], Sivraj and Rushi Kumar [18] have considered MHD flow of visco-elastic fluid with short memory (Walters model). In the studies mentioned above, the oscillatory suction velocity in presence of oscillatory permeability of the medium under the
r x C Gc g KP M B0 Pr Sc T0 t0 u0 v(t0 ) y0
q s x0 w
electrical conductivity non-dimensional frequency of oscillation non-dimensional species concentration Grashof number for mass transfer acceleration due to gravity porosity parameter magnetic parameter magnetic field of uniform strength Prandtl number Schmidt number temperature of the field time velocity component along x-axis suction velocity distance along y-axis density of the fluid skin friction frequency of oscillation condition on porous plate
influence of uniform magnetic field with heat source has not been discussed. Therefore, the objective of the present study is to analyze the effect of permeability variation and oscillatory suction velocity in free convective mass transfer flow of a visco-elastic fluid (Walters B0 model) past an infinite vertical porous plate through a porous medium in presence of a uniform transverse magnetic field and heat source.
2. Formulation of the problem The unsteady free convective flow of a visco-elastic (Walters B0 ) fluid past an infinite vertical porous plate in a porous medium with time dependent oscillatory suction as well as permeability in presence of a transverse magnetic field is considered. Let x0 -axis be along the plate in the direction of the flow and y0 -axis normal to it. Let us consider the magnetic Reynolds number is much less than unity so that the induced magnetic field is neglected in comparison with the applied transverse magnetic field. The basic flow in the medium is, therefore, entirely due to the buoyancy force caused by the temperature difference between the wall and the medium. It is assumed that initially, at t0 6 0, the plate as well as fluids are at the same temperature and also concentration of the species is very low so that the Soret and Dofour effects are neglected. When t0 > 0, the temperature of the plate is instantaneously raised to T 0w and the concentration of the species is set to C 0w . x
Gravity
Porous medium
B0
Flow geometry
y
Let the permeability of the porous medium and the suction velocity be of the form
435
S.R. Mishra et al. / International Journal of Heat and Mass Transfer 57 (2013) 433–438 0 0
K 0 ðt 0 Þ ¼ K 0p ð1 þ eeix t Þ
ð1Þ
and 0 0
v ðt0 Þ ¼ v 0 ð1 þ eeix t Þ
ð2Þ
where v0 > 0 and e 1 are positive constants. Under the above assumption with usual Boussinesq’s approximation, the governing equations and boundary conditions are given by
B20
2 0
@u0 @u0 @ u r 0 þ v 0 ¼ t 02 þ gbðT 0 T 1 Þ þ gb ðC 0 C 1 Þ u @t 0 @y @y q ! tu0 ko @ 3 u0 @ 3 u0 0 þ v 03 0 t0 02 i x @y K p ð1 þ ee Þ q @t@y
ð3Þ
ð4Þ
@C 0 @C 0 @2C0 ¼ D 02 0 þv 0 @t @y @y
ð5Þ
u ¼ 0; T 0 ¼ T 1 þ eðT w T 1 Þe
; C 0 ¼ C 1 þ eðC w C 1 Þe
ix0 t0
at y ¼ 0;
ð6Þ
Introducing the non-dimensional quantities,
y ¼ v 0t ;
t¼
0
1 C ¼ CCwC ; C 1
v 20 t0 4t
x¼
; 0
S ¼ tvS2 ;
0
Gc ¼ tgb ðCvw3C 1 Þ ; 0
4tx0
v 20 ;
KP ¼
v 20 K 0P
0
t2
0
u0
1 T ¼ TTwT ; T 1
u ¼ v0 ; ;
Gr ¼ tgbðTvw3T 1 Þ ; 0
M2 ¼
rB20 t q v 20 ;
Sc ¼ Dt ;
Pr ¼ tk
Rc ¼ k40tv20
9 > > > > > = > > > > > ;
ð7Þ
2
1 @u @u @ u u ð1 þ eeixt Þ ¼ þ Gr T þ Gc C 4 @t @y @y2 K p ð1 þ eeixt Þ M u Rc
! 3 @3u ix t @ u ð1 þ ee Þ 3 @t@y2 @y
1 @T @T 1 @2T ð1 þ eeixt Þ ¼ þ ST 4 @t @y P r @y2
ð8Þ
ð9Þ
2
1 @C @C 1 @ C ð1 þ eeixt Þ ¼ 4 @t @y Sc @y2 u ¼ 0;
T ¼ 1 þ eeixt ;
u ! 0;
T ! 0;
ð16Þ
T 000 þ Pr T 00 þ Pr ST 0 ¼ 0 ix T 1 ¼ Pr T 00 T 001 þ Pr T 01 þ Pr S 4
ð17Þ
C 000
ð19Þ
þ
Sc C 00
ð18Þ
¼0
ix C 001 þ Sc C 01 Sc C 1 ¼ Sc C 00 4
ð20Þ
C ¼ 1 þ eeixt
C ! 0 as y ! 1
ð10Þ at y ¼ 0
u0 ¼ u1 ¼ 0;
T 0 ¼ T 1 ¼ 0;
C 0 ¼ C 1 ¼ 0 at y ¼ 0
u0 ¼ u1 ! 0;
T 0 ¼ T 1 ¼ 0;
C 0 ¼ C 1 ! 0 as y ! 1
ð21Þ
u0 ¼ u00 ðyÞ þ Rc u01 ðyÞ þ o R2c u1 ¼ u10 ðyÞ þ Rc u11 ðyÞ þ o R2c
ð22Þ
Inserting Eq. (22) into (15) and (16) and equating the coefficients of R0c and Rc to zero we have following sets of ordinary differential equations; Zeroth order equations
The Eqs. (3)–(5) reduce to following non-dimensional form:
2
@ 3 u0 @u0 u0 Gr T 1 Gc C 1 @y3 @y Kp
Eqs. (15) and (16) are of third order but two boundary conditions are available. Therefore the perturbation method has been applied using Rc (Rc 1), the elastic parameter as the perturbation parameter.
)
u ! 0; T 0 ! T 1 ; C 0 ! C 1 ; as y ! 1
y0
¼ Rc
ð15Þ
The boundary conditions now reduce to
@T 0 @T 0 @2T 0 ¼ j 02 þ S0 ðT 0 T 1 Þ 0 þv 0 @t @y @y
ix0 t0
1 2 00 0 u0 ¼ Gr T 0 Gc C 0 Rc u000 þ u þ u M þ 0 0 0 KP 1 ix 2 00 0 u1 Rc u000 þ ð1 þ R i x Þu þ u M þ þ c 1 1 1 KP 4
)
1 u00 ¼ Gr T 0 GC C 0 u0000 þ u000 M 2 þ KP 1 u01 ¼ u000 u0001 þ u001 M 2 þ 00 KP
ð23Þ ð24Þ
First order equations 1 ix u00 u0010 þ u010 M2 þ þ ð25Þ u10 ¼ u000 Gr T 1 GC C 1 Kp 4 Kp 1 ix u01 00 000 u0011 þ u011 M 2 þ þ ð26Þ u11 ¼ u000 10 ixu10 u00 Kp KP 4 The corresponding boundary conditions are:
ð11Þ
u00 ¼ u01 ¼ 0;
as y ¼ 0
u10 ¼ u11 ! 0;
as y ! 1
ð27Þ
3. Method of solution
Solving these differential equations with the help of boundary conditions we get,
In view of periodic suction, temperature and concentration at the plate let us assume the velocity, temperature, concentration in the neighbourhood of the plate be
uðy; tÞ ¼ fA3 em4 y þ A1 em1 y þ A2 eSc y þ Rc ðA7 em4 y
uðy; tÞ ¼ u0 ðyÞ þ eu1 ðyÞeixt
ð12Þ
Tðy; tÞ ¼ T 0 ðyÞ þ eT 1 ðyÞeixt
ð13Þ
Cðy; tÞ ¼ C 0 ðyÞ þ eC 1 ðyÞeixt
ð14Þ
The above method of solution has been adopted by Mishra et al. [16], Das et al. [19] and Gireesh Kumar and Satyanarayana [17] to solve the periodically fluctuating flow problems. ubstituting Eqs. (12)–(14) into (8)–(10) and equating the nonharmonic (coefficient of e0) and harmonic (coefficient of e) terms, we get
þ A4 yem4 y þ A5 em1 y þ A6 eSc y Þg þ efðA13 em5 y þ A8 em1 y þ A9 em2 y þ A10 em3 y þ A11 em4 y þ A12 eSc y Þ þ Rc ðA21 em5 y þ A14 yem5 y þ A15 em1 y þ A16 em2 y þ A17 em3 y þ A18 em4 y þ A19 eSc y þ A20 yem4 y Þgeixt
ð28Þ
4im1 m1 y ðe em3 y Þ eixt Tðy; tÞ ¼ em1 y þ e em3 y þ x 4iS 4iSc Sc y ixt c em2 y þ e e Cðy; tÞ ¼ eSc y þ e 1
x
x
ð29Þ ð30Þ
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S.R. Mishra et al. / International Journal of Heat and Mass Transfer 57 (2013) 433–438
The skin friction at the plate in terms of amplitude and phase angle is given by
@u @u @u s ¼ 0 þ eeixt 1 ¼ 0 þ ejNj cosðxt þ aÞ @y y¼0 @y y¼0 @y y¼0 N ¼ Nr þ iN i ;
tan a ¼
1 0.9 0.8
ð31Þ
0.7
Ni Nr
ð32Þ
0.6
III
0.2 0.1
ð34Þ
0
0
1
2
3
4
5
6
y
7
8
9
10
Fig. 2. Effect of Pr on temperature profile with e = 0.002 and xt = p/2.
" " # # @C 0 @C 0 ixt @C 1 ¼ Sh ¼ þ ee þ ejQ jcosðxt þ cÞ ð35Þ @y y¼0 @y y¼0 @y y¼0 Qi Qr
IV
0.3
The mass transfer coefficient, i.e, the Sherwood number (Sh) at the plate in terms of amplitude and phase is given by
Q ¼ Q r þ iQ i ; tan c ¼
Pr S 0.71 0 0.71 1 7.0 1 7.0 -1
II
0.4
" # " # @T 0 @T 0 ixt @T 1 Nu ¼ þ e e þ e jRj cosð x t þ dÞ ð33Þ ¼ @y y¼0 @y y¼0 @y y¼0 Ri Rr
I
T 0.5
The rate of heat transfer, i.e. heat flux at the (Nu) in terms of amplitude and phase is given by,
R ¼ Rr þ iRi ; tan d ¼
Curve I II III IV
ð36Þ
1 0.9
4. Results and discussion
0.8 0.7
The present study considers the effect of mass transfer on free convective flow and heat transfer of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate through a porous medium with time dependent oscillatory permeability and suction in presence of a constant transverse magnetic field and heat source. The significance of the present study is to bring out the effect of oscillatory characteristics of suction velocity, permeability of the medium, plate temperature and concentration. The common feature of the velocity profile is parabolic with picks near the plate. It is interesting to note that due to dominating effect of heat source no oscillatory behavior of the profile is exhibited but in the presence of sink an oscillation in the profile is well marked (curve X). The result of the present study is verified by comparing with that of Das et al. [19] for viscous case Rc = 0.0 i.e. (curve I). The effect of Rc is to thinning of velocity boundary layer which may be attributed due to elastic property of the fluid under consideration. Further, due to interaction of applied magnetic field with conducting fluid the current is generated. Consequently, it gives rise to an induced magnetic field and ponder motive force. That force
0.6
C 0.5
Sc=0.22,0.30,0.78
0.4 0.3 0.2 0.1 0
0
1
2
3
4
5
6
II
2.5
Curve I II III IV V VI VII VIII IX X
VII IV
2
u 1.5 1 0.5 0 -0.5
V
0
1
2
IX
3
4
5
6
VI
Gr Gc Sc M Kp S 10 5 0.22 1 1 0 10 5 0.22 1 1 0 10 5 0.22 1 1 1 10 5 0.22 1 10 1 10 5 0.22 3 1 1 10 5 0.6 1 1 1 10 10 0.22 1 1 1 20 5 0.22 1 1 1 10 5 0.22 1 1 1 10 5 0.22 1 1 -1
X
7
9
10
acts opposite to the direction of the flow and unless an additional electric field is applied to counter act. In the present study the Lorentz force decelerates the fluid flow consequently resulted in thinning of boundary layer. Our result is in conformity with the result of Das et al. [19].
I VIII III
8
Fig. 3. Effect of Sc on concentration profile with e = 0.002 and xt = p/2.
3.5 3
7
y
8
9
10
y Fig. 1. Velocity profile with e = 0.002, xt = p/2.
Rc 0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
Pr 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 7 0.71
437
S.R. Mishra et al. / International Journal of Heat and Mass Transfer 57 (2013) 433–438 Table 1 Values of skin friction, amplitude jNj and phase angle tana. Gr
Gc
Sc
M
Kp
S
Rc
Pr
jNj
tana
s
10 10 10 10 10 10 10 20 10 10 10 10 10
5 5 5 5 5 5 10 5 5 5 5 5 5
0.22 0.22 0.22 0.22 0.22 0.6 0.22 0.22 0.22 0.22 0.22 0.22 0.22
1 1 1 1 3 1 1 1 1 1 1 1 3
1 1 1 10 1 1 1 1 1 1 1 1 1
0 0 1 1 1 1 1 1 1 0.05 1 1 1
0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.05 0.2 0.05
0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 7 0.71 7 0.71 7
7.812374 18.86142 14.29378 16.56542 7.72E+00 2.031926 18.77444 24.14028 27.04664 18.51528 7.802468 22.5394 5.733448
0.21441 0.70396 0.72923 0.73339 0.42541 0.67949 0.66323 0.78356 0.165356 0.70575 0.01749 1.02549 0.624294
9.961591 6.139915 2.994724 1.313887 1.717626 2.031926 5.873305 3.112612 6.259483 5.922713 5.484785 6.87322 3.082736
Table 2 Presents the Nusselt number, Sherwood number and their amplitude and phase angle with S = 1, e = 0.002 and xt = p/2. Pr
x
jRj
Nu
tan d
Sc
x
jQj
Sh
tanc
0.71 0.025 7 0.71
5 5 5 10
2.215852 0.226676 20.00294 2.076165
1.266666 0.170728 7.864045 1.265907
0.75853 1.530945 0.722882 0.677664
0.22 0.3 0.66 0.78
5 5 5 10
0.714568 0.86793 1.45935 1.898976
0.219095 0.298966 0.658814 0.776574
0.818722 0.741898 0.444656 0.478527
The effect of source parameter is well marked from Fig. 1 (curve II and III). Presence of source reduces the velocity, significantly in all the layers. Most interesting case is due to the presence of sink (S = 1.0, curve X) which gives rise to a span wise oscillatory flow after a few layers from the plate. This is due to depletion of thermal energy causing a sudden reduction of velocity. Again it is compensated due to buoyancy effect. Now, the following lines reveal the effects of Gr, Gc, Sc, Kp and Pr. The free convection current due to thermal buoyancy, mass buoyancy and porosity of the medium enhances the fluid velocity whereas Sc and Pr reduce it. Gireesh kumar and Satyanaraya [17] observed that the velocity profile increases with increase of thermal Grashof number (Gr) and Solutal Grashof number (Gc) and they have also observed that the velocity decreases with increase in magnetic parameter. Hence our observation in respect of these parameters agree with [17] in the absence of porous medium. Thus, the above result indicates that heavier species with lower thermal conductivity reduces the fluid flow. Fig. 2 exhibits the effect of Prandtl number (Pr) and source/sink on temperature field.The variation is asymptotic in nature. Sharp fall of temperature is noticed in case of water (Pr = 7.0) for both source/sink. Absence of source contributes to increase the thickness of thermal boundary layer. Higher Prandtl number fluid causes lower thermal diffusivity and hence reduces the temperature at all points. It is interesting to note that the effect of source is to lower down the temperature than the effect of sink. From Fig. 3 it is observed that the concentration distribution asymptotically decreases with the heavier species. This result is in good agreement with Das et al. [19]. In Table 1 we present the values of skin friction, amplitude and phase angle in case of cooling of the plate. The increase of viscoelastic parameter leads to decrease the skin friction. This is due to the fact that, elastic property in visco-elastic fluid reduces the frictional drag. Thus the elastic property of the fluid may have great significance in polymer producing industry as the choice of higher order visco-elastic fluid would reduce the power consumption for
stretching the boundary sheet. This is in good agreement with Khan and Sanjayanand [15]. Further, it is to note that an increase in Gr, Gc, Pr and Kp leads to exert greater skin friction on the boundary whereas Sc, M, S and Rc reduce it. Thus, the flow under consideration is stable for heavier species with greater magnetic field strength. Presence of elastic elements and source reduces the drag producing a favorable condition for stretching. This coincides with the result of Khan and Sanjayanand [15]. Further, Table 1 presents the values of amplitude jNj and phase angle tana for various values of the parameters. The peculiarity is seen for high values of Pr (i.e. Pr = 7.0). When magnetic parameter increases from 1.0 to 3.0 and elastic parameter increases from 0.05 to 0.2 then there is a change in phase i.e. from phase lag to phase lead. Thus, the present flow consideration is quite sensitive to the high Prandtl number fluid in the presence of elasticity and magnetic field. For low Prandtl number fluid i.e. Pr = 0.71, an increase in phase angle is noticed due to increase in Gc, Sc, M and Pr but reverse effect is observed in case of Gr, Kp, Rc and S. From the Table 2 it is seen that an increase in Pr leads to an increase in Nusselt number and amplitude but decreases the phase angle. The Prandtl number is a measure of the relative importance of heat conduction and viscosity of a flow. The careful study of the table reveals that for high values of Pr = 7.0 (i.e. for water) the Nusselt number increases significantly along with its amplitude. This shows that when viscosity of a fluid dominates over conductivity then the rate of heat transfer increases significantly. The Schmidt number i.e. mass transfer coefficient affects Sherwood number, phase and amplitude in a similar manner as that of Prandtl number in heat transfer.
5. Conclusion Presence of heat source prevents the oscillatory flow whereas the sink favors it. Elasticity of the fluid is responsible for thinning of the boundary layer.
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Application of magnetic field decelerates the fluid flow. The heavier species with low conductivity reduces the flow within the boundary layer. An increase in elasticity of the fluid leads to decrease the velocity which is an established result. Elasticities of the fluid and heat source reduce the skin friction providing a favorable condition for stretching.
Appendix A
Pr þ
m1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2r þ 4Pr S
Sc þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2c þ ixSc
; m2 ¼ ; 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pr þ P 2r þ 4S þ ixPr ; m3 ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ix 1 þ 1 þ 4a1 ; þ ; m4 ¼ a1 ¼ M 2 þ ; a2 ¼ M2 þ Kp Kp 2 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ 4a2 Gr ; m5 ¼ ; A1 ¼ 2 2 m1 m1 a1 A2 ¼
Gc S2c
Sc a1
; A3 ¼ A1 A2 ; A4 ¼
m34 A3 ; 1 2m4
m31 A1 S3 A 2 ; A6 ¼ 2 c ; A7 ¼ A5 A6 ; m1 a1 Sc Sc a1 m1 A1 4im1 Gr =x A1 =K p 4iSc =x 1 ; A9 ¼ 2 ; A8 ¼ m2 m2 a2 m21 m1 a2 4iSc =x 1 A3 m4 A3 =K p A10 ¼ 2 ; A11 ¼ 2 ; m2 m2 a2 m3 m3 a2 m1 A1 4im1 Gr =x A1 =K p ; A12 ¼ m24 m4 a2 A5 ¼
m21
A13 ¼ A8 A9 A10 A11 A12 ; A14 ¼ 3 m1 m21 ix A8 þ m31 A1 A1 =K p A15 ¼ ; m21 m1 a1
A13 m35 ixm5 A13 ; 1 2m5
A14 m32 ixm2 A14 A15 m33 ixm3 A15 ; A17 ¼ ; m22 m2 a2 m23 m3 a2 3 4 ð2m4 1Þ m4 m24 ix A16 þ m34 A3 KA7p þ K Am 2 p ð 4 m4 a2 Þ ¼ ; 2 m4 m4 a2 S3c S2c ix A17 þ S3c A2 A6 =K p A4 =K p ¼ ; A20 ¼ 2 ; m4 m4 a2 S2c Sc a2
A16 ¼ A18 A19
A21 ¼ ðA14 þ A15 þ A16 þ A17 þ A18 þ A19 Þ:
References [1] Y.J. Kim, Unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate with variable suction, Int. J. Eng. Sci. 38 (8) (2000) 833–845. [2] K.D. Singh, R. Sharma, Three dimensional free convective flow and heat transfer through a porous medium with periodic permeability, Indian J. Pure Appl. Math. 33 (6) (2002) 941–949. [3] A.K. Singh, N.P. Singh, Heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity, Indian J. Pure Appl.Math. 34 (3) (2003) 429–442. [4] S. Bathul, Heat transfer in three dimensional viscous flow over a porous plate moving with harmonic disturbances, Bull. Pure Appl. Sci. E 24 (1) (2005) 69–97. [5] A. Postelnicu, Influence of a magnetic field on heat and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects, Int. J. Heat Mass Transfer 47 (2004) 1467–1472. [6] P. Singh, C.B. Gupta, MHD free convective flow of a viscous fluid through a porous medium bounded by an oscillating porous plate in slip flow regime with mass transfer, Indian J. Theor. Phys. 53 (2) (2005) 111–120. [7] A. Ogulu, J. Prakash, Heat transfer to unsteady magneto hydrodynamics flow past an infinite moving vertical plate with variable suction, Phys. Scr. 74 (2006) 232–239. [8] S.S. Das, S.K. Sahoo, G.C. Dash, Numerical solution of mass transfer effects on unsteady flow past an accelerated vertical porous plate with suction, Bull. Malays. Math. Sci. Soc. 29 (1) (2006) 33–42. [9] M. Ferdows, M.A. Sattar, M.N.A. Siddiqui, Numerical approach on parameters of the thermal radiation interaction with convection in boundary layer flow at a vertical plate with variable suction, Thammasat Int. J. Sci. Technol. 9 (2004) 19–28. [10] M.S. Alam, M.M. Rahman, M.A. Samad, Dufour and Soret effects on unsteady MHD free convection and mass transfer flow past a vertical plate in a porous medium, Nonlinear Analysis: Modeling and Control 11 (2006) 211–226. [11] M.K. Majumder, R.K. Deka, MHD flow past an impulsively started infinite vertical plate in the presence of thermal radiation, Rom. J. Phys. 52 (5–7) (2007) 565–573. [12] R. Muthucumaraswamy, M. Sundarraj, V. Subhramanian, Unsteady flow past an accelerated infinite vertical plate with variable temperature and uniform mass diffusion, Int. J. Appl. Math. Mech. 5 (6) (2009) 51–56. [13] S. Asghar, M.R. Mohyuddin, T. Hayat, A.M. Siddiqui, The flow of a nonNewtonian fluid induced due to the oscillation of a porous plate, Math. Probl. Eng. 2 (2004) 133–143. [14] G.C. Dash, P.K. Rath, N. Mohapatra, P.K. Dash, Free convective MHD flow through porous media of a rotating visco-elastic fluid past an infinite vertical porous plate with heat and mass transfer in the presence of a chemical reaction, AMSE Model. B 78 (4) (2009) 21–37. [15] S.K. Khan, E. Sanjayanand, Visco-elastic boundary layer flow and heat transfer over an exponential stretching sheet, Int. J. Heat Mass Transfer 48 (2005) 1534–1542. [16] S.P. Mishra, G.C. Dash, J.S. Roy, Fluctuating flow of a non-Newtonian fluid past a porous flat plate with time varying suction, Indian J. Phys. 48 (1974) 23–36. [17] J. Gireesh Kumar, P.V. Satyanarayana, Mass transfer effect on MHD unsteady free convective Walters memory flow with constant suction and heat sink, Int. J. Appl. Math Mech. 7 (19) (2011) 97–109. [18] R. Sivraj, B. Rushi Kumar, Unsteady MHD dusty visco-elastic fluid Couette flow in an irregular channel with varying mass diffusion, Int. J. Heat Mass Transfer 55 (2012) 3076–3089. [19] S.S. Das, A. Satapathy, J.K. Das, J.P. Panda, Mass transfer effects on MHD flow and heat transfer past a vertical porous plate through a porous medium under oscillatory suction and heat source, Int. J. Heat Mass Transfer 52 (2009) 5962–5969.
Contents lists available at SciVerse ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Mass and heat transfer effect on MHD flow of a visco-elastic fluid through porous medium with oscillatory suction and heat source S.R. Mishra ⇑, G.C. Dash, M. Acharya Department Of Mathematics, Institute of Technical Education and Research, Siksha ‘O’ Anusandhan University, Jagamohan Nagar, Khandagiri, Bhubaneswar-30, Orissa, India
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 4 April 2012 Received in revised form 9 October 2012 Accepted 16 October 2012 Available online 16 November 2012
The present study considers the effect of heat and mass transfer on free convective flow of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate through a porous medium with time dependent oscillatory permeability and suction in presence of a uniform transverse magnetic field and heat source. The novelty of the present study is to analyze the effect of time dependant fluctuative suction and permeability of the medium on a visco-elastic fluid flow. The necessity of the present analysis arises as the most industrial fluids exhibit the visco-elastic behavior. The coupled nonlinear partial differential equation are turned to ordinary by super imposing a solution with steady and time dependant transient part. Finally, the set of ordinary differential equations is solved with a perturbation scheme to meet the inadequacy of boundary condition. Elasticity of the fluid and the Lorentz force reduce the velocity and it is more pronounced in case of heavier species. Most interesting observation is the fluctuation of velocity appears near the plate due to the presence of sink and presence of elastic element as well as heat source reduce the skin friction. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: MHD flow Visco-elastic Mass and heat transfer Porous medium
Contents 1. 2. 3. 4. 5.
Introduction . . . . . . . . . . . . . Formulation of the problem Method of solution. . . . . . . . Results and discussion . . . . . Conclusion . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . References . . . . . . . . . . . . . .
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1. Introduction An important class of two dimensional time dependent flow problem dealing with the response of boundary layer to external unsteady fluctuations of the free stream velocity about a mean value attracted the attention of many researchers. MHD flow with heat and mass transfer has been a subject of interest of many researchers because of its varied application in science and technology. Such phenomena are observed in buoyancy induced motions in the atmosphere, in water bodies, quasi-solid bodies such as earth, etc. In natural processes and industrial applications many transportation processes exist where transfer of heat and mass ⇑ Corresponding author. Tel.: +91 9937169245. E-mail addresses: [email protected] (S.R. Mishra), gcdash@india times.com (G.C. Dash), [email protected] (M. Acharya). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.10.053
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433 434 435 436 437 438 438
takes place simultaneously as a result of thermal diffusion and diffusion of chemical species. Several researchers have analyzed the free convective and mass transfer flow of a viscous fluid through porous medium. The permeability of the porous medium is assumed to be constant while the porosity of the medium may not be necessarily being constant. Kim [1] studied the unsteady MHD convective heat past a semiinfinite vertical porous moving plate with variable suction. The problem of three dimensional free convective flow and heat transfer through porous medium with periodic permeability has been discussed by Singh and Sharma [2]. Singh and Singh [3] have analyzed the heat and mass transfer in MHD flow of a viscous fluid past a vertical plate under oscillatory suction velocity. Bathul [4] discussed the heat transfer in a three dimensional viscous flow over a porous plate moving with harmonic disturbance. Postelnicu [5] studied numerically the influence of a magnetic field on heat
434
S.R. Mishra et al. / International Journal of Heat and Mass Transfer 57 (2013) 433–438
Nomenclature C0 D Gr K0 k Rc Nu S Sh T t u
v0 y
e b b0
t
species concentration molecular diffusivity Grashof number for heat transfer permeability of the medium thermal diffusivity elastic parameter Nusselt number heat source parameter Sherwood number non-dimensional temperature non-dimensional time non- dimensional velocity constant suction velocity non-dimensional distance along y-axis a small positive constant volumetric coefficient of expansion for heat transfer volumetric coefficient of expansion with species concentration kinematic coefficient of viscosity
and mass transfer by natural convection from vertical surfaces in porous media considering Soret and Dufour effects. Singh and Gupta [6] have investigated the MHD free convective flow of a viscous fluid through a porous medium bounded by oscillatory porous plate in slip flow regime with mass transfer. Ogulu and Prakash [7] considered heat transfer to unsteady magneto hydrodynamic flow past an infinite vertical moving plate with variable suction. Das et al. [8] analyzed the mass transfer effect on unsteady flow past an accelerated vertical porous plate with suction employing numerical methods. The basic equations of incompressible MHD flow are non-linear. But there are many interesting cases where the equations became linear in terms of the unknown quantities and may be solved easily. Linear MHD problems are accessible to exact solutions and adopt the approximations that the density and transport properties be constant. No fluid is incompressible but all may be treated as such whenever the pressure changes are small in comparison with the bulk modulus. Ferdows et al. [9] analyzed free convection flow with variable suction in presence of thermal radiation. Alam et al. [10] studied Dufour and Soret effect with variable suction on unsteady MHD free convection flow along a porous plate. Mishra et al. [16] have studied fluctuating flow of a non-Newtonian fluid past a porous flat plate with time varying suction. Majumdar and Deka [11] gave an exact solution for MHD flow past an impulsively started infinite vertical plate in the presence of thermal radiation. Muthucumaraswamy et al. [12] studied unsteady flow past an accelerated infinite vertical plate with variable temperature and uniform mass diffusion. Asghar et al. [13] have reported the flow of a non-Newtonian fluid induced due to oscillations of a porous plate. Moreover, Dash et al. [14] have studied free convective MHD flow of a visco-elastic fluid past an infinite vertical porous plate in a rotating frame of reference in the presence of chemical reaction. The specific area of application of heat and mass transfer phenomena is common in chemical process industries such as food processing and polymer production. Recently, Gireesh Kumar and Satyanarayana [17], Sivraj and Rushi Kumar [18] have considered MHD flow of visco-elastic fluid with short memory (Walters model). In the studies mentioned above, the oscillatory suction velocity in presence of oscillatory permeability of the medium under the
r x C Gc g KP M B0 Pr Sc T0 t0 u0 v(t0 ) y0
q s x0 w
electrical conductivity non-dimensional frequency of oscillation non-dimensional species concentration Grashof number for mass transfer acceleration due to gravity porosity parameter magnetic parameter magnetic field of uniform strength Prandtl number Schmidt number temperature of the field time velocity component along x-axis suction velocity distance along y-axis density of the fluid skin friction frequency of oscillation condition on porous plate
influence of uniform magnetic field with heat source has not been discussed. Therefore, the objective of the present study is to analyze the effect of permeability variation and oscillatory suction velocity in free convective mass transfer flow of a visco-elastic fluid (Walters B0 model) past an infinite vertical porous plate through a porous medium in presence of a uniform transverse magnetic field and heat source.
2. Formulation of the problem The unsteady free convective flow of a visco-elastic (Walters B0 ) fluid past an infinite vertical porous plate in a porous medium with time dependent oscillatory suction as well as permeability in presence of a transverse magnetic field is considered. Let x0 -axis be along the plate in the direction of the flow and y0 -axis normal to it. Let us consider the magnetic Reynolds number is much less than unity so that the induced magnetic field is neglected in comparison with the applied transverse magnetic field. The basic flow in the medium is, therefore, entirely due to the buoyancy force caused by the temperature difference between the wall and the medium. It is assumed that initially, at t0 6 0, the plate as well as fluids are at the same temperature and also concentration of the species is very low so that the Soret and Dofour effects are neglected. When t0 > 0, the temperature of the plate is instantaneously raised to T 0w and the concentration of the species is set to C 0w . x
Gravity
Porous medium
B0
Flow geometry
y
Let the permeability of the porous medium and the suction velocity be of the form
435
S.R. Mishra et al. / International Journal of Heat and Mass Transfer 57 (2013) 433–438 0 0
K 0 ðt 0 Þ ¼ K 0p ð1 þ eeix t Þ
ð1Þ
and 0 0
v ðt0 Þ ¼ v 0 ð1 þ eeix t Þ
ð2Þ
where v0 > 0 and e 1 are positive constants. Under the above assumption with usual Boussinesq’s approximation, the governing equations and boundary conditions are given by
B20
2 0
@u0 @u0 @ u r 0 þ v 0 ¼ t 02 þ gbðT 0 T 1 Þ þ gb ðC 0 C 1 Þ u @t 0 @y @y q ! tu0 ko @ 3 u0 @ 3 u0 0 þ v 03 0 t0 02 i x @y K p ð1 þ ee Þ q @t@y
ð3Þ
ð4Þ
@C 0 @C 0 @2C0 ¼ D 02 0 þv 0 @t @y @y
ð5Þ
u ¼ 0; T 0 ¼ T 1 þ eðT w T 1 Þe
; C 0 ¼ C 1 þ eðC w C 1 Þe
ix0 t0
at y ¼ 0;
ð6Þ
Introducing the non-dimensional quantities,
y ¼ v 0t ;
t¼
0
1 C ¼ CCwC ; C 1
v 20 t0 4t
x¼
; 0
S ¼ tvS2 ;
0
Gc ¼ tgb ðCvw3C 1 Þ ; 0
4tx0
v 20 ;
KP ¼
v 20 K 0P
0
t2
0
u0
1 T ¼ TTwT ; T 1
u ¼ v0 ; ;
Gr ¼ tgbðTvw3T 1 Þ ; 0
M2 ¼
rB20 t q v 20 ;
Sc ¼ Dt ;
Pr ¼ tk
Rc ¼ k40tv20
9 > > > > > = > > > > > ;
ð7Þ
2
1 @u @u @ u u ð1 þ eeixt Þ ¼ þ Gr T þ Gc C 4 @t @y @y2 K p ð1 þ eeixt Þ M u Rc
! 3 @3u ix t @ u ð1 þ ee Þ 3 @t@y2 @y
1 @T @T 1 @2T ð1 þ eeixt Þ ¼ þ ST 4 @t @y P r @y2
ð8Þ
ð9Þ
2
1 @C @C 1 @ C ð1 þ eeixt Þ ¼ 4 @t @y Sc @y2 u ¼ 0;
T ¼ 1 þ eeixt ;
u ! 0;
T ! 0;
ð16Þ
T 000 þ Pr T 00 þ Pr ST 0 ¼ 0 ix T 1 ¼ Pr T 00 T 001 þ Pr T 01 þ Pr S 4
ð17Þ
C 000
ð19Þ
þ
Sc C 00
ð18Þ
¼0
ix C 001 þ Sc C 01 Sc C 1 ¼ Sc C 00 4
ð20Þ
C ¼ 1 þ eeixt
C ! 0 as y ! 1
ð10Þ at y ¼ 0
u0 ¼ u1 ¼ 0;
T 0 ¼ T 1 ¼ 0;
C 0 ¼ C 1 ¼ 0 at y ¼ 0
u0 ¼ u1 ! 0;
T 0 ¼ T 1 ¼ 0;
C 0 ¼ C 1 ! 0 as y ! 1
ð21Þ
u0 ¼ u00 ðyÞ þ Rc u01 ðyÞ þ o R2c u1 ¼ u10 ðyÞ þ Rc u11 ðyÞ þ o R2c
ð22Þ
Inserting Eq. (22) into (15) and (16) and equating the coefficients of R0c and Rc to zero we have following sets of ordinary differential equations; Zeroth order equations
The Eqs. (3)–(5) reduce to following non-dimensional form:
2
@ 3 u0 @u0 u0 Gr T 1 Gc C 1 @y3 @y Kp
Eqs. (15) and (16) are of third order but two boundary conditions are available. Therefore the perturbation method has been applied using Rc (Rc 1), the elastic parameter as the perturbation parameter.
)
u ! 0; T 0 ! T 1 ; C 0 ! C 1 ; as y ! 1
y0
¼ Rc
ð15Þ
The boundary conditions now reduce to
@T 0 @T 0 @2T 0 ¼ j 02 þ S0 ðT 0 T 1 Þ 0 þv 0 @t @y @y
ix0 t0
1 2 00 0 u0 ¼ Gr T 0 Gc C 0 Rc u000 þ u þ u M þ 0 0 0 KP 1 ix 2 00 0 u1 Rc u000 þ ð1 þ R i x Þu þ u M þ þ c 1 1 1 KP 4
)
1 u00 ¼ Gr T 0 GC C 0 u0000 þ u000 M 2 þ KP 1 u01 ¼ u000 u0001 þ u001 M 2 þ 00 KP
ð23Þ ð24Þ
First order equations 1 ix u00 u0010 þ u010 M2 þ þ ð25Þ u10 ¼ u000 Gr T 1 GC C 1 Kp 4 Kp 1 ix u01 00 000 u0011 þ u011 M 2 þ þ ð26Þ u11 ¼ u000 10 ixu10 u00 Kp KP 4 The corresponding boundary conditions are:
ð11Þ
u00 ¼ u01 ¼ 0;
as y ¼ 0
u10 ¼ u11 ! 0;
as y ! 1
ð27Þ
3. Method of solution
Solving these differential equations with the help of boundary conditions we get,
In view of periodic suction, temperature and concentration at the plate let us assume the velocity, temperature, concentration in the neighbourhood of the plate be
uðy; tÞ ¼ fA3 em4 y þ A1 em1 y þ A2 eSc y þ Rc ðA7 em4 y
uðy; tÞ ¼ u0 ðyÞ þ eu1 ðyÞeixt
ð12Þ
Tðy; tÞ ¼ T 0 ðyÞ þ eT 1 ðyÞeixt
ð13Þ
Cðy; tÞ ¼ C 0 ðyÞ þ eC 1 ðyÞeixt
ð14Þ
The above method of solution has been adopted by Mishra et al. [16], Das et al. [19] and Gireesh Kumar and Satyanarayana [17] to solve the periodically fluctuating flow problems. ubstituting Eqs. (12)–(14) into (8)–(10) and equating the nonharmonic (coefficient of e0) and harmonic (coefficient of e) terms, we get
þ A4 yem4 y þ A5 em1 y þ A6 eSc y Þg þ efðA13 em5 y þ A8 em1 y þ A9 em2 y þ A10 em3 y þ A11 em4 y þ A12 eSc y Þ þ Rc ðA21 em5 y þ A14 yem5 y þ A15 em1 y þ A16 em2 y þ A17 em3 y þ A18 em4 y þ A19 eSc y þ A20 yem4 y Þgeixt
ð28Þ
4im1 m1 y ðe em3 y Þ eixt Tðy; tÞ ¼ em1 y þ e em3 y þ x 4iS 4iSc Sc y ixt c em2 y þ e e Cðy; tÞ ¼ eSc y þ e 1
x
x
ð29Þ ð30Þ
436
S.R. Mishra et al. / International Journal of Heat and Mass Transfer 57 (2013) 433–438
The skin friction at the plate in terms of amplitude and phase angle is given by
@u @u @u s ¼ 0 þ eeixt 1 ¼ 0 þ ejNj cosðxt þ aÞ @y y¼0 @y y¼0 @y y¼0 N ¼ Nr þ iN i ;
tan a ¼
1 0.9 0.8
ð31Þ
0.7
Ni Nr
ð32Þ
0.6
III
0.2 0.1
ð34Þ
0
0
1
2
3
4
5
6
y
7
8
9
10
Fig. 2. Effect of Pr on temperature profile with e = 0.002 and xt = p/2.
" " # # @C 0 @C 0 ixt @C 1 ¼ Sh ¼ þ ee þ ejQ jcosðxt þ cÞ ð35Þ @y y¼0 @y y¼0 @y y¼0 Qi Qr
IV
0.3
The mass transfer coefficient, i.e, the Sherwood number (Sh) at the plate in terms of amplitude and phase is given by
Q ¼ Q r þ iQ i ; tan c ¼
Pr S 0.71 0 0.71 1 7.0 1 7.0 -1
II
0.4
" # " # @T 0 @T 0 ixt @T 1 Nu ¼ þ e e þ e jRj cosð x t þ dÞ ð33Þ ¼ @y y¼0 @y y¼0 @y y¼0 Ri Rr
I
T 0.5
The rate of heat transfer, i.e. heat flux at the (Nu) in terms of amplitude and phase is given by,
R ¼ Rr þ iRi ; tan d ¼
Curve I II III IV
ð36Þ
1 0.9
4. Results and discussion
0.8 0.7
The present study considers the effect of mass transfer on free convective flow and heat transfer of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate through a porous medium with time dependent oscillatory permeability and suction in presence of a constant transverse magnetic field and heat source. The significance of the present study is to bring out the effect of oscillatory characteristics of suction velocity, permeability of the medium, plate temperature and concentration. The common feature of the velocity profile is parabolic with picks near the plate. It is interesting to note that due to dominating effect of heat source no oscillatory behavior of the profile is exhibited but in the presence of sink an oscillation in the profile is well marked (curve X). The result of the present study is verified by comparing with that of Das et al. [19] for viscous case Rc = 0.0 i.e. (curve I). The effect of Rc is to thinning of velocity boundary layer which may be attributed due to elastic property of the fluid under consideration. Further, due to interaction of applied magnetic field with conducting fluid the current is generated. Consequently, it gives rise to an induced magnetic field and ponder motive force. That force
0.6
C 0.5
Sc=0.22,0.30,0.78
0.4 0.3 0.2 0.1 0
0
1
2
3
4
5
6
II
2.5
Curve I II III IV V VI VII VIII IX X
VII IV
2
u 1.5 1 0.5 0 -0.5
V
0
1
2
IX
3
4
5
6
VI
Gr Gc Sc M Kp S 10 5 0.22 1 1 0 10 5 0.22 1 1 0 10 5 0.22 1 1 1 10 5 0.22 1 10 1 10 5 0.22 3 1 1 10 5 0.6 1 1 1 10 10 0.22 1 1 1 20 5 0.22 1 1 1 10 5 0.22 1 1 1 10 5 0.22 1 1 -1
X
7
9
10
acts opposite to the direction of the flow and unless an additional electric field is applied to counter act. In the present study the Lorentz force decelerates the fluid flow consequently resulted in thinning of boundary layer. Our result is in conformity with the result of Das et al. [19].
I VIII III
8
Fig. 3. Effect of Sc on concentration profile with e = 0.002 and xt = p/2.
3.5 3
7
y
8
9
10
y Fig. 1. Velocity profile with e = 0.002, xt = p/2.
Rc 0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
Pr 0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 7 0.71
437
S.R. Mishra et al. / International Journal of Heat and Mass Transfer 57 (2013) 433–438 Table 1 Values of skin friction, amplitude jNj and phase angle tana. Gr
Gc
Sc
M
Kp
S
Rc
Pr
jNj
tana
s
10 10 10 10 10 10 10 20 10 10 10 10 10
5 5 5 5 5 5 10 5 5 5 5 5 5
0.22 0.22 0.22 0.22 0.22 0.6 0.22 0.22 0.22 0.22 0.22 0.22 0.22
1 1 1 1 3 1 1 1 1 1 1 1 3
1 1 1 10 1 1 1 1 1 1 1 1 1
0 0 1 1 1 1 1 1 1 0.05 1 1 1
0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.05 0.2 0.05
0.71 0.71 0.71 0.71 0.71 0.71 0.71 0.71 7 0.71 7 0.71 7
7.812374 18.86142 14.29378 16.56542 7.72E+00 2.031926 18.77444 24.14028 27.04664 18.51528 7.802468 22.5394 5.733448
0.21441 0.70396 0.72923 0.73339 0.42541 0.67949 0.66323 0.78356 0.165356 0.70575 0.01749 1.02549 0.624294
9.961591 6.139915 2.994724 1.313887 1.717626 2.031926 5.873305 3.112612 6.259483 5.922713 5.484785 6.87322 3.082736
Table 2 Presents the Nusselt number, Sherwood number and their amplitude and phase angle with S = 1, e = 0.002 and xt = p/2. Pr
x
jRj
Nu
tan d
Sc
x
jQj
Sh
tanc
0.71 0.025 7 0.71
5 5 5 10
2.215852 0.226676 20.00294 2.076165
1.266666 0.170728 7.864045 1.265907
0.75853 1.530945 0.722882 0.677664
0.22 0.3 0.66 0.78
5 5 5 10
0.714568 0.86793 1.45935 1.898976
0.219095 0.298966 0.658814 0.776574
0.818722 0.741898 0.444656 0.478527
The effect of source parameter is well marked from Fig. 1 (curve II and III). Presence of source reduces the velocity, significantly in all the layers. Most interesting case is due to the presence of sink (S = 1.0, curve X) which gives rise to a span wise oscillatory flow after a few layers from the plate. This is due to depletion of thermal energy causing a sudden reduction of velocity. Again it is compensated due to buoyancy effect. Now, the following lines reveal the effects of Gr, Gc, Sc, Kp and Pr. The free convection current due to thermal buoyancy, mass buoyancy and porosity of the medium enhances the fluid velocity whereas Sc and Pr reduce it. Gireesh kumar and Satyanaraya [17] observed that the velocity profile increases with increase of thermal Grashof number (Gr) and Solutal Grashof number (Gc) and they have also observed that the velocity decreases with increase in magnetic parameter. Hence our observation in respect of these parameters agree with [17] in the absence of porous medium. Thus, the above result indicates that heavier species with lower thermal conductivity reduces the fluid flow. Fig. 2 exhibits the effect of Prandtl number (Pr) and source/sink on temperature field.The variation is asymptotic in nature. Sharp fall of temperature is noticed in case of water (Pr = 7.0) for both source/sink. Absence of source contributes to increase the thickness of thermal boundary layer. Higher Prandtl number fluid causes lower thermal diffusivity and hence reduces the temperature at all points. It is interesting to note that the effect of source is to lower down the temperature than the effect of sink. From Fig. 3 it is observed that the concentration distribution asymptotically decreases with the heavier species. This result is in good agreement with Das et al. [19]. In Table 1 we present the values of skin friction, amplitude and phase angle in case of cooling of the plate. The increase of viscoelastic parameter leads to decrease the skin friction. This is due to the fact that, elastic property in visco-elastic fluid reduces the frictional drag. Thus the elastic property of the fluid may have great significance in polymer producing industry as the choice of higher order visco-elastic fluid would reduce the power consumption for
stretching the boundary sheet. This is in good agreement with Khan and Sanjayanand [15]. Further, it is to note that an increase in Gr, Gc, Pr and Kp leads to exert greater skin friction on the boundary whereas Sc, M, S and Rc reduce it. Thus, the flow under consideration is stable for heavier species with greater magnetic field strength. Presence of elastic elements and source reduces the drag producing a favorable condition for stretching. This coincides with the result of Khan and Sanjayanand [15]. Further, Table 1 presents the values of amplitude jNj and phase angle tana for various values of the parameters. The peculiarity is seen for high values of Pr (i.e. Pr = 7.0). When magnetic parameter increases from 1.0 to 3.0 and elastic parameter increases from 0.05 to 0.2 then there is a change in phase i.e. from phase lag to phase lead. Thus, the present flow consideration is quite sensitive to the high Prandtl number fluid in the presence of elasticity and magnetic field. For low Prandtl number fluid i.e. Pr = 0.71, an increase in phase angle is noticed due to increase in Gc, Sc, M and Pr but reverse effect is observed in case of Gr, Kp, Rc and S. From the Table 2 it is seen that an increase in Pr leads to an increase in Nusselt number and amplitude but decreases the phase angle. The Prandtl number is a measure of the relative importance of heat conduction and viscosity of a flow. The careful study of the table reveals that for high values of Pr = 7.0 (i.e. for water) the Nusselt number increases significantly along with its amplitude. This shows that when viscosity of a fluid dominates over conductivity then the rate of heat transfer increases significantly. The Schmidt number i.e. mass transfer coefficient affects Sherwood number, phase and amplitude in a similar manner as that of Prandtl number in heat transfer.
5. Conclusion Presence of heat source prevents the oscillatory flow whereas the sink favors it. Elasticity of the fluid is responsible for thinning of the boundary layer.
438
S.R. Mishra et al. / International Journal of Heat and Mass Transfer 57 (2013) 433–438
Application of magnetic field decelerates the fluid flow. The heavier species with low conductivity reduces the flow within the boundary layer. An increase in elasticity of the fluid leads to decrease the velocity which is an established result. Elasticities of the fluid and heat source reduce the skin friction providing a favorable condition for stretching.
Appendix A
Pr þ
m1 ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2r þ 4Pr S
Sc þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2c þ ixSc
; m2 ¼ ; 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pr þ P 2r þ 4S þ ixPr ; m3 ¼ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ix 1 þ 1 þ 4a1 ; þ ; m4 ¼ a1 ¼ M 2 þ ; a2 ¼ M2 þ Kp Kp 2 4 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ 4a2 Gr ; m5 ¼ ; A1 ¼ 2 2 m1 m1 a1 A2 ¼
Gc S2c
Sc a1
; A3 ¼ A1 A2 ; A4 ¼
m34 A3 ; 1 2m4
m31 A1 S3 A 2 ; A6 ¼ 2 c ; A7 ¼ A5 A6 ; m1 a1 Sc Sc a1 m1 A1 4im1 Gr =x A1 =K p 4iSc =x 1 ; A9 ¼ 2 ; A8 ¼ m2 m2 a2 m21 m1 a2 4iSc =x 1 A3 m4 A3 =K p A10 ¼ 2 ; A11 ¼ 2 ; m2 m2 a2 m3 m3 a2 m1 A1 4im1 Gr =x A1 =K p ; A12 ¼ m24 m4 a2 A5 ¼
m21
A13 ¼ A8 A9 A10 A11 A12 ; A14 ¼ 3 m1 m21 ix A8 þ m31 A1 A1 =K p A15 ¼ ; m21 m1 a1
A13 m35 ixm5 A13 ; 1 2m5
A14 m32 ixm2 A14 A15 m33 ixm3 A15 ; A17 ¼ ; m22 m2 a2 m23 m3 a2 3 4 ð2m4 1Þ m4 m24 ix A16 þ m34 A3 KA7p þ K Am 2 p ð 4 m4 a2 Þ ¼ ; 2 m4 m4 a2 S3c S2c ix A17 þ S3c A2 A6 =K p A4 =K p ¼ ; A20 ¼ 2 ; m4 m4 a2 S2c Sc a2
A16 ¼ A18 A19
A21 ¼ ðA14 þ A15 þ A16 þ A17 þ A18 þ A19 Þ:
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