chapter 2 section 1 three dimensional couette...
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34
CHAPTER 2
SECTION 1
THREE DIMENSIONAL COUETTE FLOW THROUGH A POROUS
MEDIUM WITH HEAT AND MASS TRANSFER*
* Vidhya, M., Sundarammal Kesavan and Govindarajan, A., 2010, “Three Dimensional Couette Flow through a Porous Medium with Heat and Mass Transfer,” CiiT International Journal of Artificial Intelligent and Machine Learning, 2(11), pp. 352−358. 2.1.1 INTRODUCTION
In recent years, the flows of fluid through porous media attracted the attention
of a number of scholars because of their possible applications in many branches of science
and technology. Porous media are widely used in high temperature heat exchangers,
turbine blades, jet nozzles etc. In practice, cooling of porous structure is achieved by
forcing the liquid or gas through capillaries of solid. Actually, they are used to insulate a
heated body to maintain its temperature. Porous media are considered to be useful in
diminishing the natural free convection which would otherwise occur intensely on a
vertical heated surface. In order to make heat insulation of surface more effective, it is
necessary to study the free convection flow through a porous medium and to estimate its
effect in heat and mass transfer.
Mass transfer finds its application in ablative cooling (sudden decrease in the
temperature of space vehicles during their re-entry into the atmosphere, transpiration and
film cooling of rocket and jet engines). In fact, a porous material containing the fluid is a
non-homogeneous medium. But, it may be possible to treat it as a homogeneous one.
35
Thus, a complicated problem of the flow through a porous medium gets reduced to the
flow problem of a homogeneous fluid with some additional resistance.
When heat and mass transfer occurs simultaneously, it leads to a complex fluid
motion (the combination of temperature and concentration gradients in the fluid will lead
to buoyancy-driven flows). This problem arises in numerous engineering processes, for
example, biology and chemical processes, nuclear based repositories and the extraction of
geothermal energy.
Simultaneous heat and mass transfer from different geometries embedded in
porous media has many engineering and geophysical applications such as geothermal
reservoirs, drying of porous solids, thermal insulation and underground energy transport.
A series of investigations were made by different scholars where the porous
medium is either bounded by horizontal or vertical surfaces. Ram and Mishra [60] applied
the equations of motion derived by Ahmadi and Manvi [7] to study the unsteady MHD
flow of conducting fluid through porous medium. Varshney [8] analyzed the effect of
oscillatory two-dimensional flow through porous medium bounded by a horizontal porous
plate subjected to a variable suction velocity. Raptis [61] investigated the unsteady flow
through a porous medium bounded by an infinite porous plate subjected to a constant
suction and variable temperature. Raptis and Perdikis [9] studied the problem of free
convective flow through a porous medium bounded by a vertical porous plate with
constant suction when the free stream velocity oscillates in time about a constant mean
value. Singh et al. [62] analyzed the effect of periodic vibration of suction velocity used
by Gersten and Gross [63] on the three dimensional convective flow and heat transfer
through a porous medium. On the other hand, the channel flows through porous medium
have numerous engineering and geophysical applications, for example, in the fields of
chemical engineering for filtration and purification processes; in the fields of agriculture
engineering, to study the under ground water resources; in petroleum technology, to study
the movement of natural gas and oil and water through the oil channels/reservoirs. In view
of these applications, Singh and Sharma [54] studied couette flow with transpiration
cooling for ordinary medium. Singh and Sharma [15] also studied the effect of the
permeability of the porous medium on the three dimensional couette flow and heat transfer.
36
Singh [64] about free convection flow along a vertical wall in a porous medium with
periodic permeable variation. Combined Heat and Mass transfer effects on MHD free
convection-flow past an oscillating plate embedded in porous medium was discussed by
Chaudhary and Jain [10]. Ahmed and Ahmed [55] analyzed two-dimensional MHD
oscillatory flow along a uniformly moving infinite vertical porous plate bounded by porous
medium. Influence of moving magnetic field on three dimensional couette flow was
discussed by Singh [27]. Chemical Reaction effects on Infinite vertical plate with uniform
heat flux and variable mass diffusion was analyzed by Muthucumaraswamy et al. [35].
Three dimensional free convection couette flow with transpiration cooling was discussed
by Jain and Gupta [11]. Muthucumaraswamy and Meenakshisundaram [12] analyzed heat
transfer on vertical oscillating plate with mass flux in the presence of an optically thin
Gray Gas. Chamkha [13] discussed the unsteady MHD convective heat and mass transfer
past a semi-infinite vertical permeable moving plate with heat adsorption. Chamkha [14]
analyzed MHD flow of a uniformly stretched vertical permeable surface in the presence of
heat generation/adsorption and a chemical reaction. Chaudhary et al. [10] discussed
combined heat and mass transfer effects on MHD free convective flow past an oscillating
plate embedded in porous medium. Acharya et al. [23] analyzed magnetic field effects on
the free convective and mass transfer flow through porous medium with constant suction
and constant heat flux.
It is noted to the survey of literature no attempt has been made in the study of
heat and mass transfer with porous channel. Keeping this in view and the wide range of
applications in the branches of science and technology an attempt has been made to study
heat and mass transfer effects on three dimensional couette flow through a porous medium
where the upper plate is subjected to a constant suction and lower plate is subjected to a
transverse sinusoidal injection velocity in the absence of magnetic field in Chapter 2
section 1 and in the presence of magnetic field in Chapter 2 section 2.
This work is the extension of Singh et al. [15] for mass transfer.
The solutions for main flow velocity profiles u, cross flow velocity profiles w,
temperature field θ and concentration distribution φ are obtained using perturbation
technique. The skin friction coefficient in the main flow direction Tx and in cross flow
direction Tz, rate of heat transfer in terms of Nusselt number (Nu) and rate of mass transfer
37
are important physical parameter for this type of boundary layer flow which are defined
and determined. The effects of flow parameters such as injection/suction parameter Re,
Grashof number for mass transfer Gc, porosity parameter k0, Prandtl number Pr, Schmidt
number Sc on the velocity profiles in main flow direction, cross flow direction,
temperature and concentration distribution of the flow field have been studied analytically
and presented graphically. Further, the effects of flow parameters on skin friction in main
flow direction and cross flow direction, rate of heat transfer and rate of mass transfer have
been discussed with the help of graphs and tables.
2.1.2 FORMULATION OF THE PROBLEM
Consider couette flow of a viscous incompressible fluid through a porous
medium bounded between two infinite parallel flat porous plates with mass transfer. A
coordinate system is introduced with the origin at the lower stationary plate lying
horizontally x* − z* plane and the upper plate at a distance ‘d’ from it is subjected to a
uniform motion U. The y* axis is taken perpendicular to the plane of plates. The lower
and the upper plates are assumed to be at constant temperature T0 and T1 respectively with
T1 > T0.
It is assumed that the concentration C* of the diffusing species is very less in
comparison to other chemical species. The plate as well as fluid are assumed to be at the
same temperature T0 and the concentration of the species is very low, with concentration
level ∗∞C at all points. This leads to the assumption that Soret and Dufour effects are
negligible. The temperature of the plate is raised to T1 and the concentration of the species
is raised (or lowered) to ∗wC . It is also assumed that the effect of viscous dissipation is
negligible in the energy equation. The upper plate is subjected to a constant suction V0
whereas the lower plate to the transverse sinusoidal injection velocity distribution of the
form:
V*(z*) = ���
����
����
����
�+
d
z�cos�1V
*
0 (2.1.1)
38
Figure 2.1.1 Couette flow with periodic injection and constant suction at the porous plate
where ε (<<1) is a positive constant quantity. Without any loss of generality, the distance
‘d’ between the plates is taken equal to the wavelength of the injection velocity. Further,
all the fluid properties are assumed to be constant except that of the influence of the
density variation with temperature. All physical quantities are independent of x* for this
problem of fully developed laminar flow but the flow remains three dimensional due to the
periodic injection velocity given by equation (2.1.1). Denoting the velocity components
u*, v*, w* in the x*, y*, z* directions respectively and the temperature by T*. The
diagrammatic representation about the couette flow with periodic injection and constant
suction at the porous plate is given in Figure 2.1.1.
2.1.3 EQUATIONS OF MOTION FOR THE FLUID
The problem is governed by the following equations under usual Boussinesqu
approximation:
Continuity equation:
0z
w
y
v*
*
*
*
=∂
∂+
∂
∂ (2.1.2)
Momentum equations:
*
*
2*
*2
2*
*2***
0*
*
**
*
**
k
�uz
u
y
u�)C(Cg�)T�(Tgz
uw
y
uv −��
�
����
�∂
∂+
∂
∂+−+−=
∂
∂+
∂
∂∞ (2.1.3)
*
*
2*
*2
2*
*2
*
*
*
**
*
**
k
�vz
v
y
v�y
p
�1
z
vw
y
vv −��
�
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂ (2.1.4)
39
*
*
2*
*2
2*
*2
*
*
*
**
*
**
k
�wz
w
y
w�z
p
�1
z
ww
y
wv −��
�
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂ (2.1.5)
Energy equation:
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂2*
*2
2*
*2
*
**
*
**
z
T
y
T�z
Tw
y
Tv (2.1.6)
Mass concentration equation:
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂*2
*2
*2
*2
*
**
*
**
z
C
y
CD
z
Cw
y
Cv (2.1.7)
* stands for dimensional quantities. The last terms on the right hand side of the equations
(2.1.3), (2.1.4) and (2.1.5) account for the pressure drop across the porous material.
The boundary conditions of the problem are:
y* = 0 : u* = 0, v* (z*) = ���
����
����
����
�+
d
z�cos�1V
*
0 , w* = 0, T* = T0, C* = *C∞
y* = d : u* = U, v* = V0, w* = 0, T* = T1, C
* = *wC (2.1.8)
Introducing the above non-dimensional quantities in eqns. (2.1.2) to (2.1.7),
�dV
Re,CCCC
,TT
TT
�Vp
p,Vw
w,Vv
v,Uu
u,dz
z,dy
y
0**
w
**
01
0*
20
*
0
*
0
****
=−
−=
−
−=
======
∞
∞ϕ
(2.1.9)
Substituting the above non dimensional quantities in equations (2.1.2) –
(2.1.7), the following equations are obtained.
Continuity equation:
0zw
yv
=∂
∂+
∂
∂ (2.1.10)
Momentum equations:
02
2
2
2
kReu
zu
yu
Re1
ReGcReGrzu
wyu
v −���
����
�∂
∂+
∂
∂++=
∂
∂+
∂
∂ϕ (2.1.11)
02
2
2
2
kRe
v
z
v
y
v
Re
1
y
p
z
vw
y
vv −��
�
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂ (2.1.12)
02
2
2
2
kRe
w
z
w
y
w
Re
1
z
p
z
ww
y
wv −��
�
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂ (2.1.13)
40
Energy equation:
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂2
2
2
2
z
y
PrRe
1
z
w
y
v (2.1.14)
Mass concentration equation:
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂2
2
2
2
z
y
ScRe1
z
wy
v (2.1.15)
The boundary conditions given by (2.1.8) reduces to
01,0,w1,v1,u:1y
10,0,wz,�cos�1v(z)0,u:0y
======
===+=== (2.1.16)
In order to solve these non-linear partial differential equations, it is assumed
that the solution of the following form because the amplitude ‘ε’, (<<1) of the injection
velocity is very small. Regular perturbation technique is used in this chapter.
Muthucumaraswamy used finite difference method using Crank Nicholson scheme. He
used numerical methods whereas the solution given in this chapter is a closed form
solution. The solution is defined for any region of the plate. But, when numerical methods
are employed stability and convergence of the (profiles) solution have to be checked
whereas that is not required in perturbation method.
f(y,z) = f0(y) + ε f1(y,z) + O(ε2) , (2.1.17)
Here f stands for u, v, w, θ and ϕ. When ε = 0, the problem reduces to the two
dimensional couette flow through porous medium with constant injection and suction at
the respective plates with heat and mass transfer. The solution of this two-dimensional
problem is
yPrRe13
yScRe2
ym2
ym10 eLLeLecec(y)u 21 ++++= , (2.1.18)
PrRe
yPrRe
0 e1
e1(y)
−
−= , (2.1.19)
ScRe
ScReyScRe
0 e1ee
(y)−
−= , (2.1.20)
with v0 = 1, w0 = 0, p0 = constant where
��
���
���
����
�++=
1/2
0
21 k
4ReRe
2
1m and
��
���
���
����
�+−=
1/2
0
22 k
4ReRe
2
1m , (2.1.21)
,)
k1
RePrPr)(Ree(1
ReGrL
0
22PrRe
2
1
−−−
=
41
)k1
ScReSc)(Ree(1b,bReGc
L0
222ScRe1
1
2
2 −−−=−= ,
ScRe0
ScRe2
e1keReGc
a−
−= , PrRe0
2
e1kReGr
b−
−= , 12
2
mm4
m3
1 ee
LeLc
−
−= ,
12
1
mm
m34
2 ee
eLLc
−
−=
baL3 += , 1234 LLLL ++= , 32ScRe
1PrRe
5 LLeLe1L −−−=
when ε ≠ 0, substituting (2.1.17) in equations (2.1.10) to (2.1.15) and comparing identical
power of ε, neglecting those of ε2, the following equations are obtained by equating the
coefficients of ε with the help of equations (2.1.18) to (2.1.21).
0z
w
y
v 11 =∂
∂+
∂
∂ (2.1.22)
0
121
2
21
2
1110
1 kReu
zu
yu
Re1ReGcGrRe
zu
yu
v −���
����
�∂
∂+
∂
∂++=
∂
∂+
∂
∂ (2.1.23)
0
121
2
21
211
kRe
v
z
v
y
v
Re
1
y
p
y
v−���
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂ (2.1.24)
0
12
12
21
211
kRe
w
z
w
y
w
Re
1
z
p
y
w−���
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂ (2.1.25)
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂21
2
21
210
1 z
y
PrRe
1
y
y
v (2.1.26)
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂21
2
21
210
1 z
y
ScRe
1
y
y
v (2.1.27)
The corresponding boundary conditions become
00,0,w0,v0,u:1y
00,0,wz,�cosv0,u:0y
11111
11111
======
====== (2.1.28)
These are the linear partial differential equations which describe the three
dimensional flow through porous medium with heat and mass transfer. To solve (2.1.23) –
(2.1.27) equations the equations (2.1.22), (2.1.24) and (2.1.25) which are independent of
the main flow component u1 and the temperature field θ1 are first considered for solving by
assuming them in the following form:
v1(y,z) = v11 (y) cos π z, (2.1.29)
zsin�(y)v�1
z)(y,w 1111
−= , (2.1.30)
p1(y, z) = p11(y) cos π z, (2.1.31)
42
where (y)v11′ denotes first differentiation of v11 with respect to y. Equations (2.1.29) and
(2.1.30) have been chosen such that the continuity equation (2.1.22) is satisfied.
Substituting (2.1.29)−(2.1.31) equations into equations (2.1.24) and (2.1.25) and applying
the corresponding transformed boundary conditions (2.1.28) the solutions of v1, w1 and p1
can be given as:
( ) �zcosecLecLececz)(y,v �y47
�y36
y�6
y�51
21 −−−+= (2.1.32)
z�cos)ece(cz)(y,p y�4
y�31
−+= (2.1.33)
where,
0
7
0
6
k
1�Re
�ReL;
k
1�Re
�ReL
−
=
+
= ,
�zsin]�ecLe�CLe�ce�[c�1
z)(y,w �y47
�y36
y�26
y�151
21 −+−+−= , (2.1.34)
where
���
����
�+++=
0
22
1 k
1�4
Re
2
Re� ���
����
�++−=
0
22
2 k
1�4
Re
2
Re� , (2.1.35)
c3, c4, c5, c6, are known constants but whose expressions are not given due to
the sake of brevity.
To solve (2.1.27), it is assumed that ϕ1(y,z) = ϕ11(y) cos πz in equation (2.1.27)
and applying the corresponding boundary conditions (2.1.28) the expression for ϕ1(y, z) is
obtained as
×−
++= ))De(1
ScRe(ececz)(y,
ScRe
22ym
8ym
7143
�zcosScRe�
eac
Sc�Re
eac
�ScRe��ec
�ScRe��ec
�)ySc(Re104
�)ySc(Re93
22
22
Sc)yRe(�6
21
21
Sc)yRe(�5
21
�
−
��
+
−++
−+
−
+++
(2.1.36)
where 222
42
22
3 �4
ScRe
2
ReScm,�
4
ScRe
2
ReScm +−=++= ,
34
3
34
4
43
mm21
m
8mm
m12
7m
8m
72871 ee
AAec,
ee
eAAc,ececA,ccA
−
−=
−
−=−−=−−=
1}){e��(�2
}e){e�)(��(�}e){e�)(��(�D21
1212
��12
����21
����21
+−−
+−+−++−=+
−++−
43
In order to solve the differential equations (2.1.23) and (2.1.26) for u1 and θ1
respectively, it is assumed that
u1 = u11 (y) cos πz, (2.1.37)
θ1 = θ11 (y) cos πz, (2.1.38)
Substituting the equations (2.1.36) and (2.1.37) in the equations (2.1.23) and
(2.1.26) the following equations are obtained
101111
211
22
011
111
1111 uvReGcReGrRe�
k
1uReuu +−−=��
�
����
�+−− , (2.1.39)
101111
2111
1111 vRePr�RePr =−− , (2.1.40)
with corresponding boundary conditions:
00
0,u
0,u
:1y
:0y
11
11
11
11
=
=
=
=
=
=, (2.1.41)
where the primes denote differentiation with respect to y. Solving equations (2.1.39) and
(2.1.40) under the boundary conditions (2.1.41) and using equations (2.1.33) and (2.1.36)
the following expressions for u1 and θ1 are obtained:
{ )ym(�5
ReSc)y(�4
RePr)y(�3
)ym(�2
)ym(�1
y12
y111
1221221121 etetetetetececz)(y,u +++++ ++++++= λλ
RePr)y(�11
)ym(�10
)ym(�9
ReSc)y(�8
RePr)y(�7
)ym(�6 etetetetetet 212222 ++++++ −−−+++
�)y(ReSc16
�)y(RePr15
�)y(m14
�)y(m13
ReSc)y(�12 etetetetet 21 −−−−+ −−−−−
]etetetetetet[GcRe �)y(ReSc22
�)y(ReSc21
ReSc)y(�20
ReSc)y(�19
ym18
ym17
2 2143 −+++ −++++−
�)y(RePr27
RePr)y(�26
RePr)y(�25
ym24
ym23
2 etetetete[tGrRe 2165 +++ −+−+−
} �zcos]et �)y(RePr28
−+ (2.1.42)
��
−
−++
−+
���
−−+=
++
22
22
RePr)y(�6
21
21
RePr)y(�5
RePr
22ym
10ym
91
�RePr��ec
�RePr��ec
)e(1PrRe
ececz)(y,
21
65
���
�
+
−+
�zcos�RePr
eac
�RePr
eac �)y(RePr104
�)y(RePr93 , (2.1.43)
222
5 �4
PrRe
2
RePrm ++=
222
6 �4
PrRe
2
RePrm +−=
44
,)e��(��)e�(���2c ��2
��225
21 −+ −−++−=
,�)e�(�)e��(���2c ��1
��126
11 +− −−+−=
,)e��(��)e(���)e(��ac 2112 ��12
��12
��2193
+−− −−+++−=−
,)e��(�)e�(���)e(�(�ac 2112 ��21
��12
��21104
+++ −+−+−−=
,RePr�
)e(eac)e(eac
��RePr�)e(ec
��RePr�)e(ec
)e1)(eD(ePrRe
c
RePr�m104
RePr�m93
22
22
RePr�m6
21
21
RePr�m5
mmPrRe
22
9
66
2616
65
�
−+−−
��
−+
−+
−+
−
−−=
+−+
++
.RePr�
)e(eac)e(eac
��RePr�)e(ec
��RePr�)e(ec
)e1)(eD(ePrRe
c
RePr�m104
RePr�m93
22
22
RePr�m6
21
21
RePr�m5
mmPrRe
22
10
55
2515
65
�
−+−−
��
−+
−+
−+
−
−−
−=
+−+
++
2.1.4 SKIN FRICTION IN MAIN FLOW DIRECTION
After knowing the velocity field, we can calculate the skin-friction components
Tx and Tz in the main flow and transverse directions respectively are calculated and it is
given by
�zcosy
u
y
u
y
uTx
0y
11
0y
0
==
���
����
�∂
∂+��
�
����
�∂
∂=
∂
∂= ε ,
[ ] { )m(�t�c�c�ReScLRePrLmcmc 111212111212211 +++++++=
)m(�t)m(�tReSc)(�tRePr)(�t)m(�t 2261252413222 ++++++++++
RePr)(�t)m(�t)m(�tReSc)(�tRePr)(�t 11210192827 +−+−+−++++
�)(ReSct�)(RePrt�)(mt�)(mtReSc)(�t 161521411312 −−−−−−−−+−
�)(ReSctReSc)(�tReSc)(�tmtm[tGcRe 212201194183172 +++++++−
RePr)(�tRePr)(�tmtm[tGrRe�)](ReSct 2261256245232
22 +++−+−−−
} �zcos�)](RePrtRePr)(�t 2827 −++− (2.1.44)
45
2.1.5 SKIN FRICTION IN CROSS FLOW DIRECTION
�zsiny
w�y
wTz
0y
1
=
���
����
�∂
∂=
∂
∂= ,
[ ] �zsin�ac�ac�c�c�� 2
1042
93226
215 −−+−= , (2.1.45)
where c1−c12, t1−t18 are known constants. The values of these constants are not given due
to the sake of brevity but, these values are taken into account while drawing the profiles.
2.1.6 RATE OF HEAT TRANSFER
The rate of heat transfer in terms of Nusselt number is given by
�zcosy
�y
y
Nu
0y
11
0y
0
0y ===
���
����
�∂
∂+��
�
����
�∂
∂=��
�
����
�∂
∂= , (2.1.46)
���
�
��
−
−++−
−= RePr
22
61059RePr e1
PrRemcmc�
e1
RePrNu
�z,cos�RePr
�)(RePrac
�RePr
�)(RePrac
�RePr��RePr)(�c
�RePr��RePr)(�c
10493
22
22
262
121
15
���
�−
++
−
��
−+
++
−+
+
(2.1.47)
2.1.7 RATE OF MASS TRANSFER
Rate of mass transfer at the plate y = 0 is given by
��
���
�−+
+
−+++
−=��
�
����
�∂
∂−
=
21
21
15ScRe
22
4837ReSc0y
�ReSc��ReSc)(�
ce1
ScRemcmc�
e1
ReSc
y
φ
z�cos�ReSc
�)(ReScac
�ReSc
a�)c(ReSc
�ReSc��ReSc)(�
c 104
932
222
26 �
���
�−+
+−
−+
++
(2.1.48)
46
2.1.8 RESULTS AND DISCUSSIONS
The problem of three dimensional couette flow with heat and mass transfer of a
viscous incompressible fluid bounded between two infinite parallel flat porous plate
through a porous medium has been considered. The solutions for velocity field in main
flow and cross flow direction, temperature field and concentration field are obtained using
perturbation technique. The effects of flow parameters such as injection/suction parameter
(Re), Grashof number for mass transfer (Gc), Schmidt number (Sc), permeability of the
porous medium (k0), Grashof number (Gr) on the main flow velocity field have been
studied analytically and presented with the help of Figures 2.1.2 − 2.1.6. The effect of
flow parameters Schmidt number Sc and injection/suction parameter on concentration
distribution have been presented in Figures 2.1.7 and 2.1.8 respectively. The effects of
injection/suction parameter Re and permeability of the porous medium k0 on cross flow
velocity profiles are given in Figure 2.1.9. The temperature profiles are drawn for various
injection/suction parameter Re in the case of both air (Pr = 0.71) and water (Pr = 7.0) in
Figure 2.1.10. Further, the effects of flow parameters on skin friction in both main flow
and cross flow direction have been discussed with the help of Figures 2.1.11 − 2.1.15. The
effect of rate of heat transfer for various values of k0 against Re in the case of both air
(Pr = 0.71) and water (Pr = 7.0) is given in Table 2.1.1. The effect of rate of mass transfer
for various values of Schmidt number Sc and injection/suction parameter Re are discussed
with the help of Table 2.1.2. The computational work of the problem is carried on using
MATLAB 6.5 programme.
2.1.8.1 Velocity field in the main flow direction u(y, z)
The main flow velocity profiles are shown graphically in Figure 2.1.2. It is
clear from Figure 2.1.2 that the velocity profile decrease with the increase of
injection/suction parameter Re. It is observed from Figure 2.1.3 that the main flow
velocity profiles decrease due to an increase in Grashof number for mass transfer.
47
Figure 2.1.2 Main flow velocity profiles u(y, z) against y for various
values of injection/suction parameter Re when k0 = 0.2, Gc = 2, Gr = 1,
Pr = 0.71, Sc = 0.22, ε = 0.02, z = 0
Figure 2.1.3 Main flow velocity profiles u(y, z) against y for various values of modified
Grashof number Gc when k0 = 0.2, Sc = 0.22, Gr = 1, Pr = 0.71, Re = 1, ε = 0.02, z = 0
The effect of Schmidt number on velocity profiles are shown graphically in
Figure 2.1.4 taking Re = 1, k0 = 0.2, Gc = 2, Gr = 1, Pr = 0.71 and z = 0. The value of
Schmidt number Sc are chosen in such a way that they represent the diffusing chemical
48
species of most common interest in air (for example the values of Schmidt number for H2
(Hydrogen), H2O (water), NH3 (Ammonia) and propyl benzene in air is 0.22, 0.6, 0.78 and
2.62 respectively, Perry [58]). It is noted from Figure 2.1.4 that the velocity profiles
increase due to an increase in Schmidt number.
Figure 2.1.4 Main flow velocity profiles u(y, z) against y for various values of Schmidt
number Sc when k0 = 0.2, Gc = 2, Gr = 1, Pr = 0.71, Re = 1, ε = 0.02, z = 0
The effect of permeability and Re on main flow profiles are drawn in
Figure 2.1.5. It is seen that an increase in the permeability of the porous medium leads to
an increase in the main flow profiles. An increasing permeability means reducing the drag
force and hence, causing the flow velocity to increase as shown in Figure 2.1.5. The
straight line in the Figure 2.1.5 represents couette flow in an ordinary medium when there
is neither injection nor suction in the plate. Here also, all the profiles increase steadily near
the lower plate and reach the maximum value at the other plate.
49
Figure 2.1.5 Main flow velocity profiles u(y, z) against y for various values
of injection/suction parameter Re and permeability of the porous medium k0
when Sc = 0.22, Gc = 2, Gr = 1, Pr = 0.71, ε = 0.02, z = 0
The effect of Grashof number Gr on main flow profiles is shown in Figure
2.1.6. It is observed that the increase in Grashof number leads to an increase in the main
flow profiles.
Figure 2.1.6 Main flow velocity profiles u(y, z) against y for various values of Grashof
number Gr when Sc = 0.22, Gc = 2, Pr = 0.71, Re = 1, k0 = 0.2, ε = 0.02, z = 0
50
2.1.8.2 Concentration profiles
The effect of Schmidt number on concentration species profiles are shown in
Figure 2.1.7. To be realistic, the values of Sc are chosen to represent the presence of
various species. For example, for hydrogen Sc = 0.22 (curve A), helium Sc = 0.30 (curve
B), oxygen Sc = 0.60 (curve C), ammonia Sc = 0.78 (curve D), carbon di oxide Sc = 0.94
(curve E), ethyl benzene Sc = 2.0 (curve F) and propyl benzene Sc = 2.62 (curve G). The
concentration profiles increase due to an increase in Schmidt number.
Figure 2.1.7 Concentration profiles ϕ(y, z) against y for various values of Schmidt number
Sc when Re = 1.5, ε = 0.02, z = 0
The effect of injection/suction parameter Re on concentration profiles are
given in Figure 2.1.8 for Pr = 0.71, Gc = 3, Sc = 2, ε = 0.02, k0 = 0.5 and z = 0. The values
of Re in Figure 2.1.8 for curve A = 0.2, curve B = 0.5 and curve C = 0.8 are taken. It is
noted from Figure 2.1.8 the species concentration profiles increase with an increase in
injection/suction parameter.
51
Figure 2.1.8 Concentration profiles ϕ(y, z) against y for various values of injection/suction
parameter Re when Sc = 2, ε = 0.02, z = 0
2.1.8.3 Cross flow velocity profiles w(y, z)
The cross flow velocity component w1 due to the transverse sinusoidal
injection velocity distribution applied through the porous plate at rest. This secondary
flow component is shown in Figure 2.1.9. It is interesting to note that in the lower half of
the channel, the cross flow component w1 decreases with the increase of the permeability
k0, of the porous medium or the injection/suction parameter Re, whereas in the upper half
of the channel, the effect of permeability or the injection/suction parameter on w1 is
reversed. This is due to the fact that there is injection at the stationary plate and suction at
the plate in uniform motion which are two exactly opposite processes.
52
Figure 2.1.9 Cross flow velocity profiles w(y, z) against y for various values of
injection/suction parameter Re and permeability of the porous medium k0
when z = 0.5, ε = 0.02
2.1.8.4 Temperature profiles
Variation of temperature profiles for different values of injection/suction
parameter in the case of both air and water are given in Figure 2.1.10. It is noted from
Figure 2.1.10 that the temperature profiles decrease steadily when there is an increase in
the injection/suction parameter Re in the case of both air and water. The temperature
profiles are much higher in water than in air. All the profiles increase steadily near the end
of the lower plate and reaches the maximum height at the upper plate.
53
Figure 2.1.10 Variation of temperature profiles θ(y, z) against y for various values of
injection/suction parameter Re in the case of both air (Pr = 0.71) and water (Pr = 7.0)
when z = 0, ε = 0.02
2.1.8.5 Skin friction component in main flow direction
The variation of skin friction component Tx in the main flow direction is shown
in Figures 2.1.11 − 2.1.14. It is evident from Figures 2.1.11 − 2.1.13 that the skin friction
component Tx increases with an increase of either permeability of the porous medium (or)
Grashof number (or) Schmidt number. It is also clear from Figures 2.1.11 and 2.1.13 that
skin friction component in the main flow direction decreases as Re (injection/suction
parameter) increases upto the point Re = 0.6 in Figure 2.1.11 and Re = 0.5 in Figure 2.1.13
and thereafter the profiles change their trend and start increasing and reaches the maximum
value at Re = 1. It is seen in Figure 2.1.14 that the skin friction component Tx decreases
due to an increase in modified Grashof number Gc. But from Figures 2.1.12 and 2.1.14 it
is observed that Tx (skin friction in main flow direction) decreases as Re increases.
54
Figure 2.1.11 Variations of skin friction component in main flow direction Tx against Re
for various values of permeability of the porous medium k0 when Sc = 0.22,
Gc = 1, Gr = 1, z = 0, ε = 0.02, Pr = 0.71
Figure 2.1.12 Variations of skin friction component in main flow direction Tx against Re
for various values of Grashof number Gr when Sc = 0.22, k0 = 0.2, Gc = 1,
z = 0, ε = 0.02, Pr = 0.71
55
Figure 2.1.13 Variations of skin friction component in main flow direction Tx against Re
for various values of Schmidt number Sc when k0 = 0.2, z = 0, ε = 0.02,
Gc = 1, Gr = 1, Pr = 0.71
Figure 2.1.14 Variations of skin friction component in main flow direction Tx against Re
for various values of modified Grashof number Gc when Sc = 0.22, k0 = 0.2, Gr = 1,
z = 0, ε = 0.02, Pr = 0.71
56
2.1.8.6 Skin friction component in cross flow direction
The variation of skin friction component Tz in the cross flow direction (or) in
the transverse direction is shown graphically in Figure 2.1.15. It is clear that the skin
friction component Tz decreases with an increase in permeability of the porous medium
(or) an increase in injection/suction parameter.
Figure 2.1.15 Variations of skin friction component in cross flow direction Tz against Re
for various values of permeability of the porous medium k0 when
z = 0.5, ε = 0.02,
2.1.8.7 Rate of heat transfer
Table 2.1.1 shows the variation of rate of heat transfer in terms of Nusselt
number with permeability of porous medium for the cases of air (Pr = 0.71) and water
(Pr = 7.0). It is found that the rate of heat transfer decreases with the increase of
permeability of the porous medium and injection/suction parameter in the case of both air
and water.
57
Table 2.1.1 Variations of rate of heat transfer in terms of Nusselt number Nu against Re for
various values of permeability of porous medium k0 in the case of both air (Pr = 0.71) and
water (Pr = 7.0) when ε = 0.02 and z = 0
Re (Pr = 0.71) k0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1 0.9619 0.9249 0.8889 0.8538 0.8197 0.7867 0.7545 0.7234 0.6932 0.6640
0.2 0.9616 0.9243 0.8879 0.8525 0.8181 0.7847 0.7522 0.7208 0.6902 0.6606
0.3 0.9610 0.9230 0.8859 0.8498 0.8145 0.7800 0.7463 0.7133 0.6808 0.6489
0.4 0.9600 0.9209 0.8825 0.8447 0.8070 0.7691 0.7296 0.6959 0.6285 0.6300
Re (Pr = 7.0) k0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1 0.6684 0.4271 0.2615 0.1539 0.0874 0.0481 0.0257 0.0134 0.0068 0.0034
0.2 0.6658 0.4230 0.2569 0.1495 0.0836 0.0450 0.0233 0.0117 0.0056 0.0025
0.3 0.6608 0.4149 0.2474 0.1399 0.0747 0.0373 0.0170 0.0096 0.0036 0.0017
0.4 0.6528 0.4017 0.2312 0.1223 0.0640 0.0310 0.0110 0.0066 0.0160 0.0010
2.1.8.8 Rate of mass transfer
The rate of mass transfer for various values of Sc and Re are given in
Table 2.1.2. It is noted that the rate of mass transfer decreases whenever there is either an
increase in injection/suction parameter Re (or) there is an increase in Schmidt number Sc.
In the case of Re, the rate of mass transfer decreases steadily whereas the significant
decrease is noted in the rate of mass transfer in the case of Sc.
Table 2.1.2 Variations of rate of mass transfer for different values of Schmidt number Sc
and injection/suction parameter Re when ε = 0.02, z = 0
Sc Mass Transfer Re Mass Transfer
0.22 (Hydrogen) 0.9299 0.10 0.9881
0.30 (Helium) 0.9053 0.20 0.9763
0.60 (Water) 0.8171 0.30 0.9645
0.78 (Ammonia) 0.7672 0.40 0.9529
0.94 (Carbon dioxide) 0.7248 0.50 0.9414
2.62 (Propyl Benzene) 0.3792 0.60 0.9299
58
2.1.9 CONCLUSIONS
The above analysis brings out the following results of physical interest on the
velocity both in main flow and cross flow direction, temperature, concentration distribution
of the flow field, skin friction component in the main flow and cross direction, rate of heat
transfer and mass transfer.
(1) The effect of Schmidt number Sc, permeability of the porous medium k0 and Grashof
number Gr on main flow velocity profiles and skin friction in the main flow direction
are same. They accelerate the velocity of the flow field as well as skin friction in the
main flow direction at all points. Injection/suction parameter Re and Grashof number
for mass transfer Gc reduces the velocity and skin friction in main flow direction.
(2) Permeability of the porous medium k0 has got the influence of increasing main flow
velocity profiles and skin friction in the main flow direction, whereas it has got an
opposite influence on skin friction in cross flow direction and heat transfer coefficient.
(3) Modified Grashof number Gc and injection/suction parameter Re show a significant
decrease in the profiles of main flow velocity, skin friction in main flow direction.
(4) As far as the cross flow velocity profiles are concerned, permeability parameter k0 and
injection/suction parameter Re have the same effect of increasing the profiles upto the
midpoint of the channel thereafter they have reverse effect on the cross flow profiles.
(5) The presence of foreign species increases the profiles of main flow velocity, species
concentration and shear stress.
(6) The porosity of the medium has considerable effect on velocity and skin friction in
main flow direction. Both the profiles increase with increases in permeability
parameter. But, it decelerate the rate of heat transfer and skin friction in cross flow
direction.
(7) The thickness of concentration layer decreases both in magnitude and extent in the
presence of thicker diffusing species.
(8) The rate of mass transfer decreases with an increase of Re and Sc.
(9) The concentration distribution of the flow field increases at all points as the Schmidt
number increases. This means that heavier diffusing species have a greater
accelerating effect on the concentration distribution of the flow field.
59
The results for the corresponding non mass transfer case may be
recovered as limiting case by allowing Gc →→→→ 0. When modified Grashof number Gc
is ignored, it is found that our results are in perfect agreement with the results
obtained by Singh [15] for non mass transfer case.
60
CHAPTER 2
SECTION 2
MHD-THREE DIMENSIONAL COUETTE FLOW THROUGH A
POROUS MEDIUM WITH HEAT AND MASS TRANSFER*
* Vidhya, M., Sundarammal Kesavan and Govindarajan, A., 2011, “MHD Three Dimensional Couette Flow through a Porous Medium with Heat and Mass Transfer,” IJMMSA, 4(1), pp. 80−95.
2.2.1 INTRODUCTION
The porous medium can be considered as a homogeneous medium by assuming
its dynamical properties to be equal to local averages of the original non-homogeneous
continuum. A lot of work has been done by various scholars on flow of fluid through
porous media due to its wide use in Science and Technology. The channel flows through
porous medium also have numerous applications in the field of Engineering and
Geophysics as, in Chemical Engineering for the filtration and purification processes, in the
fields of Agriculture Engineering to study the underground water resources, in the
Petroleum Technology to study the movement of natural gas oil and water through the oil
channels/reserviours.
The phenomenon of MHD flow with heat and mass transfer has been a subject
of interest of many researchers because of its varied applications in science and
technology. Such phenomena is observed in buoyancy induced motions in the atmosphere,
in bodies of water, quasi-solid bodies such as earth etc. Three dimensional couette flow
through porous medium with heat and mass transfer play an important role in chemical
engineering, turbo machinery and in aerospace technology. In natural processes and
industrial applications many transport processes exist where transfer of heat and mass
61
takes place simultaneously as a result of combined buoyancy effects of thermal diffusion
and diffusion of chemical species. The phenomenon of heat and mass transfer is also very
common in chemical process industries such as food processing and polymer production.
Several researchers have analyzed the free convection and mass transfer flow
of a viscous fluid through porous medium. In their studies, the permeability of the porous
medium is assumed to be constant while the porosity of the medium may not necessarily
be constant because the porous material containing the fluid is a non homogeneous
medium. Therefore, the permeability of the porous medium may not necessarily be a
constant.
The analysis of the flow through porous medium has become the basis of
several scientific and engineering applications. Flow and heat transfer phenomena over a
moving flat surface are important in many technological processes, such as the
aerodynamic extrusion of plastic sheet, rolling, purification of molten metals from non
metallic inclusion by applying magnetic field and extrusion in manufacturing processes. In
continuous casting, i.e., the process consist of pouring molten metal into a short vertical
metal die or mould (at a controlled rate) which is open at both ends, cooling the mould
rapidly and withdrawing the solidified product in a continuous length from the bottom of
the mould at a rate consistent with that of pouring, the casting solidified before leaving the
mould. The mould is cooled by circulating water around it. This process is used for
producing blooms, pillets, slabs for rolling structural shaped, it is mainly employed for
copper, brass, bronze and aluminium and also increasingly with cast iron (C, I) and steel.
In light of these facts, Gebhart and Pera [65] showed the nature of vertical
natural convection flows resulting from the combined buoyancy effects of thermal and
mass diffusion. Gerston and Gross [66] have discussed the flow and heat transfer along
the plane wall with periodic suction. Soundalgekar and Gupta [67] investigated the effect
of free convection on oscillatory flow past an infinite vertical plate with variable suction
and constant heat flux. Georgantopolous et al. [68] have estimated the effect of mass
transfer on free convective hydro magnetic oscillatory flow past an infinite vertical porous
plate. Hayat et al. [69] have reported the periodic unsteady flows of non Newtonian fluid.
Kim [70] studied the unsteady MHD convective heat transfer past a semi infinite vertical
porous moving plate with variable suction. The problem of three dimensional free
62
convective flow and heat transfer through a porous medium with periodic permeability has
been discussed by Singh and Sharma [24]. Govindarajulu and Thangarj [71] studied the
effect of variable suction on free convection on a vertical plate in a porous medium. Singh
and his co workers [72] have analyzed the heat and mass transfer in MHD flow of a
viscous fluid past a vertical plate under oscillatory suction velocity. Asghar et al. [73]
have reported the flow of a non Newtonian fluid induced to the oscillations of a porous
plate. Bathul [74] discussed the heat transfer in a three dimensional viscous flow over a
porous plate moving with a harmonic disturbance. Singh and Gupta [75] have investigated
the MHD free convective flow of a viscous fluid through a porous medium bounded by an
oscillatory porous plate in the slip flow regime with mass transfer. Das and his co workers
[76] analyzed the mass transfer effects on unsteady flow past an accelerated vertical porous
plate with suction employing numerous methods. Ogulu and Prakash [77] considered heat
transfer to unsteady MHD flow past an infinite vertical moving plate with variable suction.
Das et al. [78] discussed the free convective and mass transfer flow of a viscous fluid past
an infinite vertical porous plate through a porous medium in presence of source/sink with
constant suction and heat flux.
Andreas Raptis [79] discussed the two dimensional free convective oscillatory
flow and mass transfer past a porous plate in the presence of radiation for an optically thin
fluid. The fluid is a gray, absorbing-emitting radiation but non-scattering medium.
Chauhan and Kumar [80] studied the heat transfer effects in three dimensional couette flow
of a viscous fluid through a channel partly filled with a porous medium and partly filled
with a clear fluid. Rajeswari [81] investigated the effect of suction on the MHD forced and
free convection flow past a vertical porous plate. Dulal Pal and Babulal [82] analysed the
study of combined effects of buoyancy force and first order chemical reaction in two
dimensional MHD flow, heat and mass transfer of a viscous incompressible fluid past a
permeable vertical plate embedded in a porous medium in the presence of viscous
dissipation, ohmic dissipation and thermal radiation. Khalek [36] discussed the effect of
heat and mass transfer in a hydromagnetic flow of a moving permeable vertical surface.
Murali Gundagani [83] studied about a finite element solution of thermal radiation effect
on unsteady MHD flow past a vertical porous plate with variable suction. Dulal Pal and
Babulal [84] studied the combined effect of MHD and ohmic heating in unsteady two
dimensional boundary layer slipflow, heat and mass transfer of a viscous incompressible
63
fluid past a vertical permeable plate with the diffusion of species in the presence of thermal
radiation incorporating first order chemical reaction.
Upto the knowledge of author, three dimensional couette flow through a
porous medium with heat and mass transfer in the presence of a transverse magnetic field
where the upper plate is subjected to a constant suction and lower plate is subjected to a
transverse sinusoidal injection velocity has not been discussed so far using perturbation
technique.
The objective of the present chapter is to analyze the effects of permeability
variation k0, injection/suction parameter Re, modified Grashof number Gm, Grashof
number Gr, Schmidt number Sc and magnetic parameter M on three dimensional couette
flow through a porous medium where the upper plate is subjected to a constant suction and
lower plate is subjected to the transverse sinusoidal injection velocity in the presence of a
transverse magnetic field. This work in this chapter is the extension of Singh et al. [15]
and Sarangi et al. [108] for mass transfer in the presence of transverse magnetic field.
The solutions for main flow velocity profiles u, cross flow velocity profiles w,
temperature field θ and concentration distribution φ are obtained using perturbation
technique. The skin friction coefficient in the main flow direction Tx and in cross flow
direction Tz, rate of heat transfer in terms of Nusselt number Nu and rate of mass transfer
are important physical parameter for this type of boundary layer flow which are defined
and determined. The effects of flow parameters such as injection/suction parameter Re,
Grashof number for mass transfer Gm, Grashof number Gr, porosity parameter k0, Prandtl
number Pr, Schmidt number Sc and magnetic parameter M on the velocity profiles in main
flow direction, cross flow direction, temperature and concentration distribution of the flow
field have been studied analytically and presented graphically. Further, the effects of flow
parameters on skin friction in main flow direction and cross flow direction, rate of heat
transfer and rate of mass transfer have been discussed with the help of graphs and tables.
2.2.2 FORMULATION OF THE PROBLEM
Consider MHD Couette flow of a viscous incompressible electrically
conducting fluid through a porous medium bounded between two infinite parallel non-
64
conducting porous plates in a porous medium in the presence of a transverse magnetic field
with mass transfer. It is also assumed that the magnetic Reynolds number is much less
than unity. So that induced magnetic field is neglected in comparison to the applied
magnetic field.
A co-ordinate system is introduced with the origin at the lower stationary plate
lying horizontally on x* − z* plane and upper plate at a distance ‘d’ apart from it is
subjected to a uniform motion U. The y* - axis is taken perpendicular to the plane of
plates. The lower and upper plates are assumed to be at constant temperatures T0 and T1
respectively, with T1 > T0. The upper plate is subjected to a constant suction “V0” whereas
the lower plate to a transverse sinusoidal injection velocity distribution of the form
V* (z*) = V0 ���
����
����
����
�+
d
z�cos�1
*
(2.2.1)
where ( )1� << is a positive constant quantity. It is assumed that the concentration C* of
the diffusing species in the binary mixture is very less in comparison to the other chemical
species which are present. The plate as well as fluid are assumed to be at the same
temperature T0 and the concentration of the species is very low, with concentration level
∗∞C at all points. This leads to the assumption that the Soret and Dufour effects are
negligible. The temperature of the plate is raised to T1 and the concentration of the species
is raised (or lowered) to ∗wC . It is also assumed that the effects of viscous dissipation is
negligible in the energy equation. Without any loss of generality, the distance ‘d’ between
the plates is taken equal to the wavelengths of the injection velocity. Further, all the fluid
properties are assumed to be constant except that of the influence of the density variation
with temperature. All physical quantities are independent of x* for this problem of fully
developed laminar flow, but the flow remains three-dimensional due to the periodic
injection velocity given by equation (2.2.1). Denoting the velocity components u*, v*, w*
along the x*, y*, z* directions respectively and the temperature by T*. The schematic
diagram flow of the above problem is same as Figure 2.1.1.
65
2.2.3 EQUATIONS OF MOTION FOR THE FLUID
The problem is governed by the following equations in the presence of
transverse magnetic field of the constant intensity B0, under the usual Boussinesqu
approximation
0z
w
y
v*
*
*
*
=∂
∂+
∂
∂ (2.2.2)
)C(Cg�)T�(Tgz
uw
y
uv ***
0*
*
**
*
**
∞−+−=∂
∂+
∂
∂
�
u�B
k
�uz
u
y
u�*2
0*
*
2*
*2
2*
*2
−−�
��
∂
∂+
∂
∂+ (2.2.3)
�v�B
k
�vz
v
y
v�y
p
�1
z
vw
y
vv
*20
*
*
2*
*2
2*
*2
*
*
*
**
*
** −−�
��
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂ (2.2.4)
�w�B
k
�wz
w
y
w�z
p
�1
z
ww
y
wv
*20
*
*
2*
*2
2*
*2
*
*
*
**
*
** −−�
��
∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂ (2.2.5)
�
��
∂
∂+
∂
∂=
∂
∂+
∂
∂2*
*2
2*
*2
*
**
*
**
zT
yT�
zT
wyT
v (2.2.6)
�
��
∂
∂+
∂
∂=
∂
∂+
∂
∂2*
*2
2*
*2
*
**
*
**
z
C
y
CD
z
Cw
y
Cv (2.2.7)
Here * stands for dimensional quantities. Here, the viscous dissipation term is neglected in
the energy equation.
The boundary conditions are:
y* = 0 : u* = 0 , v* (z*) = *w
*0
***
0 CC,TT0,w,dz��cos1V ===��
�
����
����
����
�+
y* = d : u* = U , v* (z*) = V0, w* = 0, T* = T1,
** CC ∞= (2.2.8)
Introducing the following non-dimensional quantities
20
*
0
*
0
****
�Vp
p,V
ww,
V
vv,
U
uu,
d
zz,
d
yy ======
,�dV
Re,��
Pr,dk
k,��
d�BM,
TTTT 0
2
*
0
2202
01
0*
====−
−=
20
**w
*
**w
**
UV)C(C�g�
Gm,CCCC ∞
∞
∞ −=
−
−=ϕ (2.2.9)
66
into the equations (2.2.2) to (2.2.7) the following equations are obtained
0zw
yv
=∂
∂+
∂
∂ (2.2.10)
uRe
M
Rek
u
z
u
y
u
Re
1GmReGrRe
z
uw
y
uv
2
02
2
2
2
−−���
����
�∂
∂+
∂
∂++=
∂
∂+
∂
∂ϕ (2.2.11)
Re
vM
Rek
v
z
v
y
v
Re
1
y
p
z
vw
y
vv
2
02
2
2
2
−−���
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂ (2.2.12)
Re
wM
Rek
w
z
w
y
w
Re
1
z
p
z
ww
y
wv
2
02
2
2
2
−−���
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂+
∂
∂ (2.2.13)
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂2
2
2
2
z
y
RePr1
z
wy
v (2.2.14)
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂2
2
2
2
z
y
ReSc
1
z
w
y
v (2.2.15)
The corresponding boundary conditions are
y = 0 : u = 0 ; v = 1 + ε cos πz; w = 0 ; θ = 0, φ = 1
y = 1 : u = 1 ; v = 1 ; w = 0 ; θ = 1, φ = 0 (2.2.16)
In view of the boundary conditions, it is assumed that
f(y,z) = f0(y) + ε f1(y,z) + O(ε2) (2.2.17)
where f stands for u, v, w, p, θ and φ. Using equation (2.2.17) into the equations (2.2.10)
to (2.2.15) and equating the coefficients of like powers of ε zeroth-order equations are
obtained. These equations represent MHD two dimensional couette flow through porous
medium with heat and mass transfer. The solution of this two dimensional problem is,
,e1
eeconstant,p0,w1,vReSc
ReScReScy
0000−
−====
,eLeLececu ReScy2
RePry1
ym2
ym10
21 +++=
RePr
RePry
0 e1
e1−
−= (2.2.18)
where 2
0
2
22
0
2
1 Mk
1
4
Re
2
Rem,M
k
1
4
Re
2
Rem ++−=+++=
,)M
k
1PrRePr)(Ree(1
GrReL
2
0
222RePr
2
1
−−−−
−=
67
)Mk
1ScReSc)(Ree(1
GmReL
2
0
222ReSc
2
2
−−−−
−= , baL3 +=
,)
k
1)(Me(1
GrRea
0
2RePr
2
+−
= )
k
1)(Me(1
eGmReb
0
2ReSc
ReSc2
+−
=
,ee
)Le(Lc
12
2
mm5
m4
1−
+−=
12
1
mm
m45
2 ee
eLLc
−
+= , ,LLLL 1234 ++=
3ReSc
21RePr
5 LeLLe1L −−−= (2.2.19)
When ε ≠ 0, substituting (2.2.17) into equations (2.2.10) to (2.2.15) and comparing the
coefficients of identical powers of ε, neglecting those of ε2, the following equations which
are the coefficients of ε with the help of equation (2.2.18) are obtained
0z
w
y
v 11 =∂
∂+
∂
∂ (2.2.20)
Re
vM
Rek
v
z
v
y
v
Re
1
y
p
y
vv 1
2
0
121
2
21
211
0 −−���
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂ (2.2.21)
Re
wM
Rek
w
z
w
y
w
Re
1
z
p
y
w 12
0
12
12
21
211 −−��
�
����
�∂
∂+
∂
∂+
∂
∂−=
∂
∂ (2.2.22)
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂21
2
21
210
1 z
y
ReSc
1
y
y
v (2.2.23)
Re
uM
Rek
u
z
u
y
u
Re
1GmReGrRey
u
y
uv 1
2
0
121
2
21
2
1110
1 −−���
����
�∂
∂+
∂
∂++=
∂
∂+
∂
∂ (2.2.24)
���
����
�∂
∂+
∂
∂=
∂
∂+
∂
∂21
2
21
210
1 z
y
RePr1
y
y
v (2.2.25)
The corresponding boundary conditions become
y = 0 : u1 = 0 , v1 = cos πz, w1 = 0, θ1 = 0, φ1= 0
y = 1 : u1 = 0 , v1 = 0, w1 = 0, θ1 = 0, φ1= 0 (2.2.26)
These are the linear partial differential equations which describe the three-dimensional
flow through porous medium with heat and mass transfer.
When ε ≠ 0, to find the solution of first-order equations, it is assumed that
v1(y, z) = v11 (y) cos πz (2.2.27)
z�sin(y)v�1
z)(y,w 1111
−= (2.2.28)
68
p1(y,z) = p11 (y) cos π z (2.2.29)
where (y)v11′ denotes the first derivative of v11 with respect to y. Equations (2.2.27) and
(2.2.28) have been chosen so that the continuity equations (2.2.20) is satisfied.
Substituting (2.2.27)−(2.2.29) equations into equations (2.2.21) and (2.2.22) and applying
the corresponding transformed boundary conditions (2.2.26) the solutions of v1, w1 and p1
are obtained as follows:
{ } �zcosecLceLececz)(y,v �y675
�y6
ym8
ym71
43 −−−+= (2.2.30)
{ } �zsine�cL�ecLemcemc�1
z)(y,w �y67
�y56
ym48
ym371
43 −+−+−
= (2.2.31)
�zcos)ece(cz)(y,p �y6
�y51
−+= (2.2.32)
where
;k
1�M4
Re
2
Rem;
k
1�M4
Re
2
Rem
0
222
40
222
3 +++−=++++=
;�4
ScRe
2
ReScm;�
4
ScRe
2
ReScm 2
22
62
22
5 +−=++=
;�4
PrRe
2
RePrm;�
4
PrRe
2
RePrm 2
22
82
22
7 +−=++=
1098765
0
27
0
26 c,c,c,c,c,c,
k
1M�Re
�ReL;
k
1�ReM
�ReL
−−
=
++
=
are known constants but whose expressions are not given due to the sake of brevity but the
values of these constants are taken into account while drawing the profiles.
To solve (2.2.23), it is assumed that ϕ1(y,z) = ϕ11(y) cos πz in equation (2.2.23)
and applying the corresponding boundary conditions (2.2.26), the solution ϕ1(y, z) is given
by
��
���
���
�−+
+−+−
++=++
24
24
Resc)y(m8
23
23
Resc)y(m7
ReSc
22ym
10ym
91 �ReScmm
ec
�ReScmm
ec
e1
ScReececz)(y,
43
65
�zcose�ReSc
cae
ReSc�ca �)y(Resc639�)y(Resc537
������
+− −+ (2.2.33)
In order to solve equations (2.2.24) and (2.2.25), it is assumed that
69
u1(y,z) = u11 (y) cos πz , θ1(y,z) = θ11 (y) cos π z (2.2.34)
substituting these equations in (2.2.24) and (2.2.25) the following equations are obtained.
011112
11222
0111111 uvReGmReGrRe�M
k
1uuReu ′+−−=��
�
����
�++−′−′ (2.2.35)
0112
111111 RePrv�RePr ′=−′−′′ (2.2.36)
with corresponding boundary conditions
y = 0 : u11 = 0 , θ11 = 0
y = 1 : u11 = 0, θ11 = 0 (2.2.37)
where 11u′ and 11u ′′ denotes the first and second derivatives of u11 with respect to y.
Solving equations (2.2.34) and (2.2.35) under the boundary conditions (2.2.26) the
solutions for u1(y, z) and θ1(y, z) are obtained as
{ )ym(m5
�)y(m4
�)y(m3
)ym(m2
)ym(m1
ym12
ym111
3211143143 esesesesesececz)(y,u +−+++ +−−+++=
�)y(RePr11
RePr)y(m10
RePr)y(m9
�)y(m8
�)y(m7
)ym(m6 eseseseseses 432242 +++−++ −++−−+
�)y(ReSc16
ReSc)y(�15
ReSc)y(m14
ReSc)y(m13
�)y(RePr12 eseseseses 43 −+++− −−++−
ReSc)y(m20
ReSc)y(m19
ym18
ym17
2 4365 esesese(sGmRe ++ +++−
ym24
ym23
2�)y(ReSc22
�)y(ReSc21
87 ese(sGrRe)eses +−+− −+
)} �zcoseseseses �)y(RePr28
�)y(RePr27
RePr)y(m26
RePr)y(m25
43 −+++ −+−− (2.2.38)
��
−
−++
−+
���
−−+=
++
22
22
RePr)y(�6
21
21
RePr)y(�5
RePr
22ym
10ym
91
�RePr��ec
�RePr��ec
)e(1
PrReececz)(y,
21
65
���
�
+
−+
�zcos�RePr
eac
�RePr
eac �)y(RePr104
�)y(RePr93 , (2.2.39)
where constants s1−s28 are known constants but not presented here due to the sake of
brevity but the values of these constants are taken into account while drawing the profiles.
2.2.4 SKIN FRICTION IN MAIN FLOW DIRECTION
After knowing the velocity field, the skin-friction components Tx and Tz in the
main flow and transverse directions are calculated and given below:
70
�zcosy
u�y
uTx
0y
11
0y
0
==
���
����
�∂
∂+��
�
����
�∂
∂=
)m(ms)m(msmcmc{�ReSc)LRePrLmcm(c 142311412311212211 +++++++++=
�)(ms�)(ms)m(ms)m(ms�)(ms�)(ms 28274263251413 −−+−++++−−+−
�)(RePrs�)(RePrsRePr)(msRePr)(ms 121141039 −−+−++++
�)(ReScs�)(ReScsReSc)(msReSc)(ms 1615414313 −−+−++++
ReSc)(msReSc)(msmsm[sGmRe 4203196185172 +++++−
] RePr)(msmsm[sGrRe�)(ReScs�)(ReScs 3258247232
2221 +−+−−++−
]} z�cos�)(RePrs�)(RePrs)m(RePrs 2827426 −−+++− (2.2.40)
2.2.5 SKIN FRICTION IN CROSS FLOW DIRECTION
z�siny
��
Tz 11���
����
�∂
∂=
z�sin)�cL�cLmcm(c�� 2
672
56248
237 −−+
−= (2.2.41)
2.2.6 RATE OF HEAT TRANSFER
From the temperature field, the heat transfer coefficient in terms of the Nusselt
number is obtained as
z�cosy
�y
Nu
0y
11
0y
0
==
���
����
�∂
∂+��
�
����
�∂
∂=
���
�
��
−
−++−
−= RePr
22
61059RePr e1PrRe
mcmc�e1
RePr
��
−+
++
−+
+2
222
262
121
15
�RePr��RePr)(�c
�RePr��RePr)(�c
�z,cos�RePr
�)(RePrac�RePr
�)(RePrac 10493
���
�−
++
− (2.2.42)
71
2.2.7 RATE OF MASS TRANSFER
Rate of mass transfer at the plate y = 0 is given by
z�cos�ReSc
�)(ReScca
�ReSc
�)(ReScca
�ReScmm
ReSc)(mc
�ReScmm
ReSc)(mc
e1ScRe
mcmcy
639537
24
24
482
323
37ReSc
22
61059
0y
�
−+
+−
��
−+
++
−+
+
−−−−=��
�
����
�∂
∂−
=
φ
(2.2.43)
2.2.8 RESULTS AND DISCUSSIONS
The problem of three dimensional couette flow with heat and mass transfer in
the presence of transverse magnetic field of a viscous incompressible fluid bounded
between two infinite parallel flat porous plate through a porous medium has been
considered. The solutions for velocity field in main flow and cross flow direction,
temperature field and concentration field are obtained using perturbation technique. The
effects of flow parameters such as injection/suction parameter (Re), magnetic parameter
M, Schmidt number (Sc), Grashof number for mass transfer (Gm), permeability of the
porous medium (k0), Grashof number (Gr) on the main flow velocity field, have been
studied analytically and presented with the help of Figures 2.2.1 − 2.2.6. The effects of
flow parameters injection/suction parameter Re and Schmidt number Sc on concentration
distribution have been presented in Figures 2.2.7 and 2.2.8 respectively. The effects of
injection/suction parameter Re, magnetic parameter M and permeability of the porous
medium k0 on cross flow velocity profiles are given in Figure 2.2.9. The temperature
profiles are drawn for various injection/suction parameter Re in the case of both air
(Pr = 0.71) and water (Pr = 7.0) in Figure 2.2.10. Further, the effects of flow parameters
on skin friction in both main flow and cross flow direction have been discussed with the
help of Figures 2.2.11 − 2.2.16 and Table 2.2.1 and 2.2.2. The effect of rate of heat
transfer for various values of k0 against Re in the case of both air (Pr = 0.71) and water
(Pr = 7.0) is given in Table 2.2.3. The effect of rate of mass transfer for various values of
Schmidt number Sc and injection/suction parameter Re are discussed with the help of
Table 2.2.4. The computational work of the problem is carried on using MATLAB 6.5
programme.
72
2.2.8.1 Velocity field in main flow direction
Figure 2.2.1 Main flow velocity profiles u(y, z) for various values of injection/suction
parameter Re when k0 = 0.2, Gm = 1, Gr = 1, Sc = 0.22, ε = 0.02, z = 0, M = 1, Pr = 0.71
Figure 2.2.2 Main flow velocity profiles u(y, z) for various values of magnetic parameter
M when k0 = 0.2, Gm = 1, Gr = 1, Sc = 0.22, ε = 0.02, z = 0, Re = 0.2, Pr = 0.71
73
The main flow velocity profiles are shown graphically in Figures 2.2.1 to 2.2.6.
It is clear from Figure 2.2.1 that the velocity profiles decrease with the increase of
injection/suction parameter Re. It is noted from Figure 2.2.2 that the velocity profiles
decrease with the increase of the Hartmann number M. Physically speaking, the effect of
increasing magnetic field strength dampens the velocity. It is because that the application
of transverse magnetic field will result a resistitive type force (Lorentz force) similar to
drag force which tends to resist the fluid flow and thus reducing its velocity. All the
profiles increase steadily near the lower plate and reach the highest value 1 at the other
plate.
Figure 2.2.3 Main flow velocity profiles u(y, z) for various values of Schmidt number Sc
when k0 = 0.2, Gm = 1, Gr = 1, ε = 0.02, z = 0, M = 1, Re = 0.2, Pr = 0.71
The effect of Schmidt number on velocity profiles are shown graphically in
Figure 2.2.3 taking Re = 0.2, k0 = 0.2, Gm = 2, Gr = 1, M = 1, Pr = 0.71. The values of
Schmidt number Sc are chosen in such a way that they represent the diffusing chemical
species of most common interest in air (for eg the values of Schmidt number for H2
(Hydrogen), H2O (water), NH3 (Ammonia) and (Propyl benzene in air is) 0.22, 0.6, 0.78
and 2.62 respectively, Perry [58]). It is noted from Figure 2.2.3 that the velocity profiles
decrease due to an increase in Schmidt number. In chapter 2 section 1, the main flow
velocity profiles increase in the case of increasing Schmidt number. But here, they behave
exactly in opposite manner. This is due to the presence of magnetic field.
74
Figure 2.2.4 Main flow velocity profiles u(y, z) for various values of modified Grashof
number Gm when k0 = 0.2, Sc = 0.22, Gr = 1, ε = 0.02, z = 0, M = 1, Re = 0.2, Pr = 0.71
It is observed from Figure 2.2.4 that the main flow velocity profiles decrease
due to an increase in Grashof number for mass transfer. The profiles are drawn taking
Re = 0.2, Pr = 0.71, M = 1, Sc = 0.22, Gr = 1, k0 = 0.2 and ε = 0.02. All the profiles
decrease steadily near the lower plate upto a little away from the centre of the plate and
thereafter a reverse trend is noticed.
Figure 2.2.5 Main flow velocity profiles u(y, z) against y for various values of Re and k0
when Gm = 1, Gr = 1, Sc = 0.22, ε = 0.02, z = 0, M = 1, Pr = 0.71
75
The effect of permeability parameter k0 and injection/suction parameter Re on
main flow profiles are drawn in Figure 2.2.5. It is seen that an increase in the permeability
of the porous medium leads to an increase in the main flow profiles. The straight line in the
Figure 2.2.5 represents couette flow in an ordinary medium when there is neither injection
nor suction in the plate. Here also, all the profiles increase steadily near the lower plate
and reach the maximum value at the other plate. The effect of permeability on main flow
profiles in the presence of Re (injection/suction parameter) and in the absence of Re
(injection/suction parameter) are drawn in Figure 2.2.5. It is seen that an increase in k0
(permeability parameter) leads to an increase in the main flow profiles in the presence or
absence of Re (injection/suction parameter).
Figure 2.2.6 Main flow velocity profiles u(y, z) against y for various values of Gr
when k0 = 0.2, Gm = 1, Sc = 0.22, ε = 0.02, Re = 0.2, z = 0, M = 1, Pr = 0.71
The effect of Grashof number Gr on main flow velocity profiles are shown
graphically in Figure 2.2.6. It is seen that the velocity profiles increase for increasing
values of Gr.
76
2.2.8.2 Concentration profiles
Figure 2.2.7 Concentration profiles against y for various values of Schmidt number Sc
when Re = 0.2, ε = 0.02, z = 0
The effect of Schmidt number on Species Concentration profiles are shown
graphically in Figure 2.2.7. To be realistic, the values of Sc are chosen to represent the
presence of various species. For example, for hydrogen Sc = 0.22 (curve A), helium
Sc = 0.30 (curve B), oxygen Sc = 0.60 (curve C), ammonia Sc = 0.78 (curve D), carbon di
oxide Sc = 0.74 (curve E) ethyl benzene Sc = 2.0 (curve F) and propyl benzene Sc = 2.62
(curve G). The concentration profiles increase due to an increase in Schmidt number.
The effect of injection/suction parameter on concentration profiles are plotted
in Figure 2.2.8. The values of Re (injection/suction) parameter in Figure 2.2.8 are chosen
as (Re = 0.2 for curve A, Re = 0.5 for curve B and Re = 0.8 for curve C). It is observed that
the concentration profiles increase for an increase in injection/suction parameter.
77
Figure 2.2.8 Concentration profiles against y for various values of injection/suction
parameter Re when ε = 0.02, z = 0
2.2.8.3 Cross flow velocity profiles
Figure 2.2.9 Cross flow velocity profiles against y for various values of injection/suction
parameter Re, permeability of the porous medium k0 and magnetic parameter M when
ε = 0.02, z = 0.5
78
The cross flow velocity component W1 due to the transverse sinusoidal
injection velocity distribution applied through a porous plate at rest. This secondary flow
component is shown in Figure 2.2.9. It is clear that the cross flow velocity w(y, z)
increases due to the increase in Hartmann number M and also increases due to decrease in
permeability of the porous medium. The cross flow velocity component w(y, z) increases
due to an increase in injection/suction parameter upto the midpoint of the channel but, it is
interesting to note that the behaviour of w(y, z) is reversed, when y ≥ 0.4. This is due to
the fact that there is injection at the stationary plate and suction at the plate in uniform
motion which are two exactly opposite processes.
2.2.8.4 Temperature profiles
Figure 2.2.10 Variations of temperature profile θ(y, z) for various values of
injection/suction parameter Re in the case of both air (Pr = 0.71) and water (Pr = 7.0)
when ε = 0.02, z = 0
From the Figure 2.2.10, it is noted that the temperature profiles decrease steadily when
there is an increase in the injection/suction parameter, Re in the case of air and water. The
temperature profiles are much higher in water than in air. All the profiles increase steadily
near the ends of the plate x = 0 and reaches the maximum height x = 1 at the other end.
79
2.2.8.5 Skin friction in main flow direction
Figure 2.2.11 Variations of skin friction component in main flow direction Tx for
various values of magnetic parameter M against Re when Sc =0.22, Gm = 1, Gr = 1,
ε = 0.02, z = 0, Pr = 0.71, k0 = 0.2
Figure 2.2.12 Variations of skin friction component in main flow direction Tx for
various values of permeability parameter k0 against Re when Sc = 0.22, Gr = 1, Gm = 1,
M = 1, ε = 0.02, z = 0, Pr = 0.71
80
Figure 2.2.13 Variations of skin friction component in main flow direction Tx for
various values of modified Grashof number Gm against Re when Sc = 0.22, M = 1,
k0 = 0.2, Gr = 1, ε = 0.02, z = 0, Pr = 0.71
Figure 2.2.14 Variations of skin friction component in main flow direction Tx for
various values of Grashof number Gr against Re when Sc = 0.22, Gm = 1, M = 1,
k0 = 0.2, ε = 0.02, z = 0, Pr = 0.71
The variation of skin friction component Tx in main flow direction is shown in
Figures 2.2.11 − 2.2.14. It is evident from the Figures 2.2.11 − 2.2.14 that the skin friction
component decrease due to either an increase in Hartmann number M (or) modified
81
Grashof number Gm while it increases due to either an increase in permeability of the
porous medium k0 (or) an increase in Grashof number Gr. It is also noted from Figures
2.2.11 − 2.2.14 that skin friction component Tx decreases whenever there is an increase in
injection / suction parameter Re.
2.2.8.6 Skin friction in cross flow direction
The variation of Skin friction component Tz in the transverse direction is shown in
Figures 2.2.15 and 2.2.16. It is clear from Figures 2.2.15 and 2.2.16 that the skin friction
component Tz increases due to an increase in Hartmann number M whereas reverse trend is
seen in the case of an increase in the permeability of the porous medium.
Figure 2.2.15 Variations of skin friction component in cross flow direction Tz for
various values of magnetic parameter M when k0 = 0.2, ε = 0.02, z = 0.5
Figure 2.2.16 Variations of skin friction component in cross flow direction Tz for
various values of permeability of the porous medium k0 against Re when
M = 1, ε = 0.02, z = 0.5
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2.2.8.7 Skin friction in both main flow and cross flow direction for various values
of Re
Table 2.2.1 Variation of skin friction in main flow and cross flow direction against Re
Re TX TZ
0.1 0.0321 4.2918
0.2 0.0361 4.2600
0.3 0.0436 4.2285
0.4 0.0545 4.1972
0.5 0.0687 4.1661
It is noted from the Table 2.2.1 that the skin friction component in the main
flow direction increases whenever there is a small increase in injection/suction parameter.
But skin friction component in the cross flow direction has the reverse effect when
injection/suction parameter increase.
2.2.8.8 Variation of skin friction in main flow direction Tx for various values of
Schmidt number Sc against Re
Table 2.2.2 Variation of skin friction in main flow direction Tx against Re for various
values of Schmidt number Sc
Re
Sc 0.3 0.5 0.7 0.9
0.22 0.3785 0.3258 0.2527 0.1564
0.60 0.3783 0.3248 0.2503 0.1520
0.78 0.3781 0.3243 0.2491 0.1497
2.62 0.3770 0.3194 0.2359 0.1186
It is noted from Table 2.2.2 that the skin friction component in main flow
direction decrease due to either an increase in Schmidt number Sc or injection / suction
parameter Re.
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2.2.8.9 Variation of Nusselt number against Re in case of both air (Pr = 0.71) and
water (Pr = 7.0)
Table 2.2.3 Variation of Nusselt number against Re in case of both air (Pr = 0.71) and
water (Pr = 7.0)
Pr = 0.7 Pr = 7.0 Re
Nu Nu
0.1 0.9678 0.7085
0.3 0.9009 0.3015
0.5 0.8372 0.1126
0.7 0.7768 0.0382
0.9 0.7196 0.0122
It is noted from the Table 2.2.3 that the rate of heat transfer in terms of Nusselt number
decreases whenever the injection/suction parameter Re increase in the case of both air and
water. The rate of heat transfer is higher in the case of air than in the case of water. The
results of rate of heat transfer with respect to Re do not change irrespective of the fact that
magnetic field is present or absent in case of both air and water.
2.2.8.10 Rate of mass transfer
Table 2.2.4 Rate of mass transfer for various values of Sc and Re
Sc Mass
Transfer Re
Mass
Transfer
Hydrogen (0.22) 0.0045 0.1 0.0120
Helium (0.30) 0.0061 0.2 0.0235
Water (0.60) 0.0120 0.3 0.0343
Ammonia (0.78) 0.155 0.4 0.0447
Carbon di oxide (0.94) 0.0186 0.5 0.0545
Propyl benzene (2.62) 0.0481 0.6 0.0637
The rate of mass transfer for various values of Sc and Re are given by
Table 2.2.4. It is noted that the rate of mass transfer increases whenever there is either an
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increase in injection/suction parameter Re or there is an increase in Schmidt number Sc.
Significant difference is noted in rate of mass transfer whenever Schmidt number Sc,
injection/suction parameter Re increase.
2.2.9 CONCLUSIONS
The above analysis brings out the following results of physical interest on the
velocity both in main flow and cross flow direction, temperature, concentration distribution
of the flow field, skin friction component in the main flow and cross direction, rate of heat
transfer and mass transfer.
(1) Presence of foreign species reduces the velocity as well as thermal boundary layer
and further reduction occurs with increase in Schmidt number.
(2) Application of magnetic field causes decrease in main flow velocity. The magnetic
parameter M retards the velocity of the flow field at all points due to the magnetic
pull of the Lorentz force acting on the flow field.
(3) Magnetic parameter changes (reduces) the velocity characteristics in main flow
direction. But, it does not change skin friction in cross flow direction.
(4) Magnetic parameter has a remarkable feature of changing the effect of Schmidt
number on main flow velocity profiles.
(5) Grashof number for mass transfer decelerate the velocity of the flow field
irrespective of whether the transverse magnetic field is present or not.
(6) Injection/suction parameter increases the rate of mass transfer in the presence of
magnetic field. But, it has reverse effect on rate of mass transfer in the absence of
magnetic field.
(7) Schmidt number decreases the velocity profiles and skin friction in the main flow
direction in the presence of magnetic field. It increases rate of mass transfer in the
presence of magnetic field. But, Schmidt number decreases rate of mass transfer in
the absence of magnetic field. In the absence of magnetic field, Schmidt number
increases skin friction and velocity profiles in main flow direction.
(8) The permeability of the porous medium increases rate of heat transfer in the presence
of magnetic field. But, it decreases rate of mass transfer in the absence of magnetic
field.
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(9) Grashof number Gr accelerates the skin friction component in main flow direction in
the abscence of magnetic field while it shows a reverse effect in the presence of it.
The results for the corresponding non mass transfer in the presence of
tranverse magnetic field case may be recovered as limiting case by allowing Gm →→→→ 0
(when modified Grashof number Gm is ignored). When M = 0, Gm = 0 it is found
that the results of this chapter are in perfect agreement with the results obtained by
Singh [15] for non mass transfer case and non magnetic case. When M = 0, the
results coincides with the results of chapter 2 section 1 for non magnetic case.