Applied Mathematical Sciences, Vol. 3, 2009, no. 13, 629 - 651 Double-Diffusive Convection-Radiation Interaction on Unsteady MHD Flow over a Vertical Moving Porous Plate with Heat Generation and Soret Effects R. A. Mohamed Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt [email protected]Abstract The present work analyzes the influence of a first-order homgeneous chemical reaction and thermal radiation on hydromagnetic free convection heat and mass transfer for a viscous fluid past a semi-infinite vertical moving porous plate embedded in a porous medium in the presence of thermal diffusion and heat generation. The fluid is considered to be a gray, absorbing-emitting but non-scattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The plate moves with constant velocity in the direction of fluid flow while the free stream velocity is assumed to follow the exponentially increasing small perturbation law. A uniform magnetic field acts perpendicular to the porous surface, which absorbs the fluid with a suction velocity varying with time. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. The effects of various parameters on the velocity, temperature and concentration fields as well as the skin-friction coefficient, Nusselt number and the Sherwood number are presented graphically and in tabulated forms. Keywords: MHD; boundary layer; porous medium; heat and mass transfer; thermal radiation; chemical reaction; thermal diffusion; heat generation. 1 – Introduction Unsteady free convection flows in a porous medium have received much attention in recent time due to its wide applications in geothermal and oil reservoir engineering as well as other geophysical and astrophysical studies. Moreover, considerable interest has been shown in radiation interaction with convection for heat
The present work analyzes the influence of a first-order homgeneous chemical reaction and thermal radiation on hydromagnetic free convection heat and mass transfer for a viscous fluid past a semi-infinite vertical moving porous plate embedded in a porous medium in the presence of thermal diffusion and heat generation. The fluid is considered to be a gray, absorbing-emitting but non-scattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the energy equation. The plate moves with constant velocity in the direction of fluid flow while the free stream velocity is assumed to follow the exponentially increasing small perturbation law. A uniform magnetic field acts perpendicular to the porous surface, which absorbs the fluid with a suction velocity varying with time. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. The effects of various parameters on the velocity, temperature and concentration fields as well as the skin-friction coefficient, Nusselt number and the Sherwood number are presented graphically and in tabulated forms. Keywords: MHD; boundary layer; porous medium; heat and mass transfer; thermal radiation; chemical reaction; thermal diffusion; heat generation. 1 – Introduction
Unsteady free convection flows in a porous medium have received much attention in recent time due to its wide applications in geothermal and oil reservoir engineering as well as other geophysical and astrophysical studies. Moreover, considerable interest has been shown in radiation interaction with convection for heat
630 R. A. Mohamed and mass transfer in fluids. This is due to the significant role of thermal radiation in the surface heat transfer when convection heat transfer is small, particularly in free convection problems involving absorbing-emitting fluids. The unsteady fluid flow past a moving plate in the presence of free convection and radiation were studied by Monsure [21], Cogley et al. [1], Raptis and perdikis [6], Das et al. [34], Grief et al. [28], Ganeasan and Loganathan [26], Mbeledogu et al. [15], Makinde [25], and Abdus-Satter and Hamid Kalim [23]. All these studies have been confined to unsteady flow in a nonporous medium. From the previous literature survey about unsteady fluid flow, we observe that little papers were done in porous medium. The effect of radiation on MHD flow and heat transfer must be considered when high temperatures are reached. El-Hakiem [19] studied the unsteady MHD oscillatory flow on free convection-radiation through a porous medium with a vertical infinite surface that absorbs the fluid with a constant velocity. Ghaly [8] employed a symbolic computation software Mathematica to study the effect of radiation on heat and mass transfer over a stretching sheet in the presence of a magnetic field. Raptis et al. [5] studied the effect of radiation on 2D steady MHD optically thin gray gas flow along an inifinite vertical plates taking into account the induced magnetic field. Cookey et al. [9] researched the influence of viscous dissipation and radiation on unsteady MHD free-convection flow past on inifinite heated vertical plate in a porous medium with time-dependent suction. Abd El-Naby et al. [18] employed an implicit finite-difference methods to study the effect of radiation on MHD unsteady free-convection flow past a semi-infinite vertical porous plate but did not take into account the viscous dissipation. Singh and Dikshit [4] investigated the hydromagntic flow past a continuously moving semi-infinit plate at large suction. Takhar et al. [13] described the radiation effects on MHD free-convection flow past a semi-infinite vertical plate. Kim [35] studied unsteady MHD convective heat transfer past a semi-infinite vertical porous moving plate.
The study of heat generation or absorption effects in moving fluids is important in view of several physical problems, such as fluids undergoing exothermic or endothermic chemical reactions. Vajravelu and Hadjinicolaou [16] studied the heat transfer charcteristics in the laminar boundary layer of a viscous fluid over a stretching sheet with viscous dissipation or frictional heating and internal heat generation. Hosssain et al. [20] studied the problem of natural convection flow along a vertical wavy surface with uniform surface temperature in the presence of heat generation / absorption. Alam et al. [24] studied the problem of free convection heat and mass transfer flow past an inclined semi-infinite heated surface of an electrically conducting and steady viscous incompressible fluid in the presence of a magnetic field and heat generation. Chamkha [3] investigated unsteady convective heat and mass transfer past a semi-infinite porous moving plate with heat absorption. Hady et al. [11] studied the problem of free convection flow alonge a vertical wavy surface embedded in electrically conducting fluid saturated porous media in the presence of internal heat generation or absorption effect.
Combined heat and mass transfer problems with chemical reaction are of importance in many processes and have, therefore, received a considerable amount of attention in recent years. In processes such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat and the mass transfer occur simultaneously. Possible applications of this type of
flow can be found in many industries, For example, in the power industry, among the methods of generating electric power is one in which electrical energy is extracted directly from a moving conducting fluid. Many practical diffusive operations involve the molecular diffusion of a species in the presence of chemical reaction within or at the boundary. There are two types of reactions. A homogeneous reaction is one that occurs uniformly throughout a give phase. The species generation in a homogeneous reaction is analogous to internal source of heat generation. In contrast, a heterogeneous reaction takes place in a restricted region or within the boundary of a phase. It can therefore be treated as a boundary condition similar to the constant heat flux condition in heat transfer. The study of heat and mass transfer with chemical reaction is of great practical importance to engineers and scientists because of its almost universal occurrence in many branches of science and engineering. Muthucumaraswamy and Ganesan [30] studied the effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate. Deka et al. [27] studied the effect of the first-order homogeneous chemical reaction on the process of an unsteady flow past an infinite vertical plate with a constant heat and mass transfer. Chamkha [2] studied the MHD flow of a numerical of uniformly stretched vertical permeable surface in the presence of heat generation / absorption and a chemical reaction. Muthucumaraswamy and Ganesan [29] investigated the effects of a chemical reaction on the unsteady flow past an impulsively started semi-infinite vertical plate which subjected to uniform heat flux. Muthucumaraswamy and Ganesan [31] analyzed the effect of a chemical reaction on the unsteady flow past an impulsively started vertical plate which is subjected to uniform mass flux and in the presence of heat transfer. Muthucumaraswamy [32] studied the effects of suction on heat and mass transfer along a moving vertical surface in the presence of a chemical reaction. Raptis and Perdikis [7] analyzed the effect of a chemical reaction of an electrically conducting viscous fluid on the flow over a non-linearly (quadratic) semi-infinite stretching sheat in the presence of a constant magnetic field which in normal to the sheet. Seddeek et al. [22] analyzed the effects of chemical reaction, radiation and variable viscosity on hydromagnetic mixed convection heat and mass transfer for Hiemenz flow through porous media. Ibrahim et al. [12] analyzed the effects of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi-infinite vertical permeable moving plate with heat source and suction.
In spite of all these studies, the unsteady MHD double-diffusive free convection for a heat generating fluid with themal radiation and chemical reaction has recived little attention. Hence, the main object of the present investigation is to study the effect of a first-order homgeneous chemical reaction, thermal radiation, heat source and thermal diffusion on the unsteady MHD double-diffusive free convection fluid flow past a vertical porous plate in the presence of mass blowing or suction. It is assumed that the plate is embedded in a uniform porous medium and moves with a constant velocity in the flow direction in the presence of a transverse magnetic field. It is also assumed that the free stream to consist of a mean velocity and temperature over which are superimposed an exponentially varying with time.
632 R. A. Mohamed 2- Mathematical Formulation of the Problem Two dimensional unsteady flow of a laminar, incompressible, viscous, electrically conducting and heat generation/absorption fluid past a semi-infinite vertical moving porous plate embedded in a uniform porous medium and subjected to a uniform transverse magnetic field in the presence of a pressure gradient has been considered with double-diffusive free convection, thermal diffusion, chemical reaction, and thermal radiation effects. According to the coordinate system the *x - axis is taken along the porous plate in the upward direction and *y -axis normal to it. The fluid is assumed to be a gray, absorbing-emitting but non-scattering medium. The radiative heat flux in the *x -direction is considered negligible in comparison with that in the *y -direction [10]. It is assumed that there is no applied voltage of which implies the absence of an electric filed. The transversely applied magnetic field and magnetic Reynolds number are very small and hence the induced magnetic field is negligible [33]. Viscous and Darcy resistance terms are taken into account the constant permeability porous medium. The MHD term is derived form an order-of-magnitude analysis of the full Navier-stokes equation. It is assumed here that the hole size of the porous plate is significantly larger than a characteristic microscopic length scale of the porous medium. We regard the porous medium as an assemblage of small identical spherical particles fixed in space, following Yamamoto and Iwamura [17]. A homgeneous frist-order chemical reaction between the fluid and the species concentration. The chemical reactions are taking place in the flow and all thermophysical properties are assumed to be constant of the linear momentum equation which is approximation. The fluid properties are assumed to be constants except that the influence of density variation with temperature and concentration has been considered in the body-force term. Due to the semi - infinite place surface assumption furthermore, the flow variable are functions of *y and *t only. The governing equation for this investigation is based on the balances of mass, linear momentum, energy, and concentration species. Taking into consideration the assumptions made above, these equations can be written in Cartesian frame of reference, as follows:
where, *x , *y and *t are the dimensional distances along and perpendicular to the plate
and dimensional time, respectively. *u and *v are the components of dimensional velocities along *x and *y directions, ρ is the fluid density, µ is the viscosity, pc is the specific heat at constant pressure, σ is the fluid electrical conductivity, 0B is the magentic induction, *K is the permeability of the porous medium, *T is the dimensional temperature, MD is the coefficient of chemical molecular diffusivity, TD is the coefficient of thermal diffusivity, *c is the dimensional concentration, k is the thermal conductivity of the fluid, g is the acceleration due to gravity, and *
rq , *R are the local radiative heat flux, the reaction rate constant respectively. The term
)( **0 ∞−TTQ is assumed to be amount of heat generated or absorbed per unit volume
0Q is a constant, which may take on either positive or negative values. When the wall temperature *T exceeds the free stream temperature *
∞T , the source term 00 >Q and heat sink when 00 <Q . The magnetic and viscous dissipations are neglected in this study. It is assumed that the porous plate moves with a constant velocity *
pu in the
direction of fluid flow, and the free stream velocity *∞U follows the exponentially
increasing small perturbation law. In addition, it is assumed that the temperature and concentration at the wall as well as the suction velocity are exponentially varying with time.
The boundary conditions for the velocity, temperature, and concentration fields
are given as follows:
****
)(,)(, ********** tnww
tnwwp ecccceTTTTuu ∞∞ −+=−+== εε at 0=y , (5)
****
0** ,),1(
**
∞∞∞ →→+=→ ccTTeUUu tnε as ∞→y (6)
where, *wT and *
wc are the wall dimensional temerature and concentration, respectively. *
∞c is the free stream dimensional concentration. 0U and *n are constants.
It is clear from equation (1) that the suction velocity at the plate surface is a function of time only. Assuming that it takes the following exponential form:
)1(**
0* tnAevv ε+−= (7)
where, A is a real positive constant, ε and Aε are small less than unity, and 0v is a scale of suction velocity which has non-zero positive constant.
In the free stream, from equation (2) we get
634 R. A. Mohamed
.*20
***
*
*
*
UBUK
gxp
dtdU
σµρρ −−−∂∂
−= ∞∞∞ (8)
Eliminating *
*
xp∂∂ between equation (2) and equation (8), we obtain
)()()()( **20
****
*2
*
*
*
**
*
*
2 uUBuUKy
udt
dUg
yuv
tu
−+−+∂
∂++−=
∂∂
+∂∂
∞∞∞
∞ σµµρρρρ , (9)
by making use the equation of state [14]
)()( *****∞∞∞ −+−=− ccTT ρβρβρρ (10)
where, β is the volumetric coefficent of thermal expansion, *β the volumetric coefficient of expansion with concentration, and ∞ρ the density of the fluid far away the surface. Then substituting from equation (10) into equation (9) we obtain
)()()( ***
******
*2
*
*
*
**
*
*
2 uUK
ccgTTgyu
dtdU
yuv
tu
−+−+−+∂
∂+=
∂∂
+∂∂
∞∞∞∞ υββυ
)( **20 uUB
−+ ∞ρσ (11)
where, ρµυ = is the coefficient of the kinematic viscosity. The third term on the
RHS of the momentum equation (11) denote body force due to nonuniform temperature, the fourth term denote body force due to nonuniform concentration.
The radiative heat flux term by using the Rosseland approximation is gived by
*
*
*1
**
4
34
yT
kqr ∂
∂−=
σ (12)
where, *σ and *
1k are respectively the Stefan-Boltzmann constant and the mean absorption coefficient. We assume that the temperature difference within the flow are sufficiently small such that
4*T may be expressed as a linear function of the temperature. This is accomplished by expanding in a Taylor series about *
By using equations (12) and (13), into equation (3) is reduced to
2
3
2 *
*2
*1
****0
*
*2
*
**
*
*
316
)(yT
kcT
TTc
QyT
ck
yTv
tT
ppp ∂
∂+−+
∂
∂=
∂∂
+∂∂ ∞
∞ ρσ
ρρ . (14)
We now introduce the dimensionaless variable, as follows:
,,,,,,,2
0*2
0
*0
*0
*
0
*0
*0
*
υυυ nV
nVttUUuUUU
VyyvVvuUu pp ======= ∞
).(),(, ********2
0
2*
∞∞∞∞ −+=−+== ccccTTTTVKK ww φθυ (15)
Then substituting from equation (15) into equations (11), (14) and (4) and taking
into acount equation (7) we obtain.
)()1( 2
2
uUNGrGryu
dtdU
yuAe
tu
cTnt −+++
∂∂
+=∂∂
+−∂∂
∞∞ φθε , (16)
ηθθθεθ+
∂∂
+=∂∂
+−∂∂
2
2
)3
41(Pr1)1(
yR
yAe
tnt , (17)
δφθφφεφ−
∂∂
+∂∂
=∂∂
+−∂∂
2
2
2
21)1(y
SoyScy
Aet
nt (18)
where,
02
0
** )(UV
TTgGr w
T∞−
=υβ
is the thermal Grashof number , 0
20
*** )(UV
ccgGr w
C∞−
=υβ
is the
solutal Grashof number, 20
20
VB
Mρ
υσ= is the magnetic field parameter ,
pcVQ
20
0
ρυ
η =
is the dimensionless heat generation / absorption coefficient , kc pυρ
=Pr is the
Prandtl number, kk
TR *1
** 3
4 ∞=σ is the thermal radiation parameter , 2
0
*
VR υδ = is
the chemical reaction parameter , MD
Sc υ= is the Schmidet number,
)()(
**
**
∞
∞
−−
=CCTTD
Sow
wT
υ is the Soret number,
and
KMN 1
+= . (19)
636 R. A. Mohamed
The dimensionless form of the boundary condition (5) and (6) become
ntntp eeUu εφεθ +=+== 1,1, at 0=y , (20)
0,0),1( →→+→= ∞ φθε nteUu as ∞→y . (21)
3- Analytical Approximate Solutions
In order to reduce the above system of partial differential equations to a system
of ordinary equations in dimensionless form, we may represent the velocity, temperature and concentration as
)()()( 2
10 εε Oyueyuu nt ++= , (22)
)()()( 210 εθεθθ Oyey nt ++= , (23)
)()()( 2
10 εφεφφ Oyey nt ++= . (24)
By substituting the above equations (22)-(24) into equations (16)-(18), equating the harmonic and non-harmonic terms and neglecting the higher-order terms of
)( 2εO , we obtain the following pairs of equations for ),,( 000 φθu and ),,( 111 φθu .
It is now important to calculate the physical quantities of primary interest, which are the local wall shear stress, the local surface heat, and mass flux. Given the velocity field in the boundary layer, we can now calculate the local wall shear stress ( i.e., skin- friction) is given by
by using equation (13), we can write equation (44) as follow
0*
*
*1
***
*316
=
∞⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛+−=
yw y
Tk
Tkq
σ, (45)
which is written in dimensionless form as;
( )0
0**
*
341
=
∞⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎠⎞
⎜⎝⎛ +
−−=
y
ww y
RVTTkq θ
υ. (46)
The dimensionless local surface heat flux (i.e., Nusselt number) is obtained as
( ) 0
1**
*
341Re
=
−
∞⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
⎟⎠⎞
⎜⎝⎛ +−=⇒
−=
yx
w
w
yRNu
TTkxq
Nu θ
( )( )[ ]13113341 HRHZeRR nt −−+−⎟
⎠⎞
⎜⎝⎛ +−= ε (47)
where, υ
xVx
0Re = is the Reynolds number.
The definition of the local mass flux and the local Sherwood number are
respectively given by
0*
*
*=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−=y
w ycDj , (48)
( )**∞−
=ccD
xjSh
w
wx , (49)
with the help of these equations, one can write
( )
⎟⎟⎠
⎞⎜⎜⎝
⎛−−+++−
+−+=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
−==
−
153431
54323121
0
1 1)(Re
HZRZZRZZZR
eRRZRy
Sh nt
yxx εφ
. (50)
It should be mentioned that in the absence of the chemical reaction, thermal radiation, heat source, and thermal diffusion effects, all of the flow and heat, mass transfer solutions reported above are consistent with those reported earlier by Chamkha [3].
640 R. A. Mohamed 4- Results and Discussions:
The formulation of the effects of chemical reaction, thermal diffusion, heat source and thermal radiation on MHD convective flow and mass transfer of an incopressible, viscous fluid along a semi infinite vertical porous moving plate in a porous medium has been performed in the preceding sections. This enables us to carry out the numercial calculations for the distribution of the velocity, temperature and concentration across the boundary layer for various values of the parameters. In the present study we have chosen A= 0.5, t =1.0, n = 0.1, 5.0=pU and 2.0=ε , while R, δ , η , So, CGr , Sc , TGr , M , Pr and K are varied over a range, which are listed in the figure legends. Also, the boundary condition for ∞→y is replaced by where
maxy is a sufficiently large value of y where the velocity profile u approaches to the relevant free stream velocity. We choose maxy =10.0 and a step size .001.0=∆y
0 2 4 6 8 10
0.6
0.8
1.0
1.2
1.4
1.6
1.8
R = 0.0,0.2,0.4,0.6,0.8,1.0
u
y
GrT=2.0, Grc=1.0, Pr=0.71,Sc = 0.6, M=1.0, η = 0.0,Up =0.5, ε = 0.2, t = 1.0,n =0.1, δ =0.0, K = 0.5,A =0.5, So=0.0.
For different values of radiation parameter R, the velocity profiles are plotted in
Fig.1. Here we find that, as the value of R increases the velocity increases, with an increasing in the flow boundary layer thickness. Thus, thermal radiation enhaces convective flow.
The effects of radiaton parameter R on the temperature profiles are presented in Fig.2. From this figure we observe that, as the value of R increasses the temperature profiles increases, with an increasing in the themal boundary layer thickness.
The influences of chemical reaction parameter δ on the velocity profiles across the boundary layre are presnted in Fig.3. We see that the velocity distribution across the boundary layer decreases with increasing of δ .
For different values of the chemical reaction parameter δ , the concentration profiels plotted in Fig.4. It is obvious that the influence of increasing values of δ , the concentration distribution across the boundary layer decreases.
The effect of heat generation η on the velocity profiles is shown in Fig.5. From this figure we see that the heat is generated the buoyancy force increases which induces the flow rate to increase giving rise to the increase in the velocity profiles.
Fig.6 shows the variation of temperature profiles for different values of η . It is seen from this figure that temperature profiles increase with an increasing of heat generation parameter η . The effects of Soret number So on the velocity profiles is shown in Fig.7. From this figure we see that velocity profiles increase with an increasing of So from which we concloude that the fluid velocity rises due to greater thermal diffusion.
0 2 4 6 8 10
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
δ = 3.0,2.0,1.0,0.5,0.2,0.0
u
y
GrT=2.0, GrC=1.0, Pr= 0.71, Sc = 0.6,
M = 1.0 , η = 0.0, Up= 0.5 , n = 0.1,
T = 1.0 , ε = 0.2, R = 1.0 , K = 0.5,So= 0.0, A = 0.5.
GrT=2.0, GcC= 1.0,M =1.0, δ =0.5 T =1.0, n =0.1So=0.0, R =0.0 K =0.5, A =0.5Pr =0.71, Sc =0.6Up = 0.5, ε = 0.2
Fig. 6. Effects of η on temperature profiles.
Fig.8 represents the concentration profiles for different values of So. From this figure we observe that the concentration profiles increase significantly with an increasing of Soret number.
The velocity profiles for different values of solutal Grashof number CGr are discribed in Fig.9. It is observed that an increasing in CGr leads to a rise in the values of velocity. In addition, the curves show that the peak value of velocity increases rapidly near the wall of the porous plate as soultal Grashof number increases, and then decays to the relevant free stream velocity.
642 R. A. Mohamed
0 2 4 6 8 10
0.8
1.2
1.6
2.0
2.4
2.8
So = 0.0,0.5,1.0,2.0,5.0,10.0.
u
y
K=0.0, GrT=2.0, GrC=1.0M=1.0, Pr=0.71, Sc=0.22n =0.1, t =1.0, Up=0.5ε =0.2, η =0.0, δ =0.0R=0.0, A=0.5
0 2 4 6 8 100.0
0.3
0.6
0.9
1.2
1.5
1.8
2.1
2.4
So = 0.0, 0.5, 1.0, 2.0, 5.0, 10.0.
φ
y
K=0.5, GrT=2.0, GrC=1.0, M=1.0, Pr=0.71Sc=0.22, n=0.1, t = 1.0, U
p=0.5, ε=0.2η=0.0, δ=0.0, R=0.0, A=0.5
Fig. 7. Effects of So on velocity pofiles. Fig. 8. Effects of So on concentration pofiles.
Fig. 9. Effects of Grc on velocity profiles. Fig. 10. Effects of Sc on velocity profiles.
For different values of the Schmidt number Sc, the velocity profiles are plotted in Fig.10. It is obvious that the effect of increasing values of Sc results in a decreasing velocity distribution across the boundary layer. Fig.11 shows the concentration profiles across the boundary layer for various values of Schmidt number Sc. The figure shows that an increasing in Sc results in a decreasing the concentration distribution, because the smaller values of Sc are equivalent to increasing the chemical molecular diffusivity.
The velocity profiles for different values of Grashof number TGr are described in Fig.12. It is observed that an increasing in TGr leads to a rise in the values of velocity. Here the Grashof number represents the effects of the free convection currents. Physically, 0>TGr means heating of the fluid of cooling of the boundary surface, 0<TGr means cooling of the fluid of heating of the
boundary surface and 0=TGr corresponds to the absence of free convection current. In addition, the curves show that the peak value of velocity increases rapidly near the wall of the porous plate as Grashof number increases, and then decays to the relevant free stream velocity.
For different values of the magnetic field parameter M, the velocity profiles are plotted in Fig.13. It is abvious that the effect of increasing values of M-parameter results in decreasing velocity distribution across the boundary layer because of the application of transfer magnetic field will result a restrictive type force (Lorentz force) similar to drag force which tends to resist the fluid flow and thus reducing its velocity.
Fig.14 shows the velocity profiles across the boundary layer for different values of Prandtl number Pr. The results show that the effect of increasing values of Pr results in a decreasing the velocity.
Typical variation of the temperature profiles along the spanwise coordinate y are shown in Fig. 15 for different values of Prandtl number Pr. The results show that an increase of Prandtl number results in a decreasing the thermal boundary layer thickness and more uniform temperature distribution across the boundary layer. The reason is that smaller values of Pr are equivalent to increasing the thermal conductivities, and therefore, heat is able to differ away from the heated surface more rapidly than for higher values of Pr. Hence, the boundary layer is thicker and the rate of heat transfer is reduced, for gradient have been reduced.
Fig.16 illustrates the concentration profiles for various values of Pr. We see that the concentration profiles increases (decreases) near (far) from vertical porous plate as parameter Pr increases.
Fig. 17 shows the velocity profiles for different values of the permeability K. Cleary as K increases the peak value of velocity tends to increase. These results could be very useful in deciding the applicability of enhanced oil recovery in reservoir engineering.
646 R. A. Mohamed Table(1): Numerical values of fC , XNu Re/ and xxSh Re/ for 71.0Pr = , ,0.1,20.0,5.0,0.0,60.0,0.1,0.2 ======= MUSoScGrGr pCT ε 1.0,0.1 == nt , 5.0=K and .5.0=A
Table (1) depict the effects of the radiation parameter R, the chemical reaction parameter δ , and the heat generation coefficients η , on the skin-friction coefficients fC , Nusselt number Nu, and the Sherwood number xSh , respectivily. It is observed from this table that as R increases, the skin- friction coefficients and the Nusselt number increase whereas the Sherwood number remain unchanged. However, as the chemical reaction patameter effects increase, the skin-friction coefficient decreases and Sherwood number increases whereas the Nusselt number remains unaffected. Also, increases in the heat generation effects increase, both the skin-friction coefficient and Nusselt number increase whereas the Sherwood number remains unchanged.
Double-diffusive convection-radiation interaction 647 Table (2): Numerical values of fC , 1Re−
The effects of the Soret number So , solutal Grashof number CGr and Schmidet number Sc , on the skin-friction coefficients fC , Nusselt number Nu and the Sherwood number xSh are given in table (2). It is seen from this table that as So increases, the skin-friction coefficient and Sherood number decreases whears the Nusselt number remains unchanged. However, as the CGr increase, the skin- fricition coefficients increase wheras the Nusselt number and Sherwood number remains unaffected. Also, an increaseing in the Sc effects, the skin-friction coefficient decreases and Sherwood number increases whereas the Nusselt number remains unchanged.
648 R. A. Mohamed
The numercial values of the skin-friction coefficients fC , Nusselt number Nu, and the Sherwood number xSh for different values of the thermal Grashof number TGr , the magnetic field parameter M and the permeability K are entered in table (3). It can be noted from this table that an increaseing in TGr and M lead to an increasing in the value of the skin- friction coefficients, wheras the Nusselt number and Sherwood number remains unchanged. However, as K increases, the skin-friction coefficient decreases wheras the Nusselt number and Sherwood number remains unchanged.
Table (3): Numercial values of fC , 1Re−
xNu and 1Re−xxSh for
,2.0,5.0,5.0,0.1,6.0,0.1,71.0Pr ======= εPC URSoScGr 1.0,5.0,1.0,0.1 ==== ηAnt and .0.1=δ
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