three – dimensional mhd free – convection...
TRANSCRIPT
124
6.1 INTRODUCTION
Free - convection flows past a vertical flat plate have been investigated
extensively. This configuration is relevant to solar energy collection and cooling of
modern electronic systems. In the latter application, electronic components are mounted
on circuit cards, an array of which is positioned vertically in a cabinet forming vertical
flat plates through which coolants are passed. The coolant may be propelled by free
convection, forced convection, or mixed convection, depending on the application.
Soundlgekar and Patil (1980) have studied Stokes problem for infinite vertical plate with
constant heat flux. They concluded that the velocity of the fluid increases with increasing
time and Grashof number. Acharya et al (2000) have analyzed the effect of magnetic
field on the free convection and mass transfer flow through a porous medium with
periodic suction and constant heat flux.
In low gravity or micro gravity environment, it can be expected that reduction or
elimination of natural convection may enhance the properties and performance of
materials such as crystals. Space related technology has demanded a profound knowledge
of forces involving vibrations that occur due to interaction of several phenomena. In
order to develop new innovative techniques it is important to understand the mechanism
of heat and mass transfer and flow characteristics. The idea of using mechanical vibration
as a mean of enhancing the heat transfer rate has received lot of attention. The influence
of vibration with zero mean on the heat transfer behavior of thermal system was
examined by Kamotami et al (1981) using small amplitude and frequency of vibration.
They analyzed that vibrations have no significant effect on heat transfer rate. Theoretical
efforts have been made by many researchers to explain the effects of the gravitational
field on material processing inside a space shuttle environment. Their analysis focused on
convection inside differentially heated enclosures. These studies show that gravitational
field can be resolved into a mean and fluctuating component. The fluctuating
accelerations act on density gradients in the fluid caused by heat and mass transfer
between the fluid and boundaries to produce convective motions. These motions may
lead to increase in heat transfer significantly beyond that of pure conduction.
125
Space laboratories offer microgravity environment, which makes it possible to
achieve somewhat true diffusion conditions. However, due to the existence of static and
oscillatory residual accelerations in space laboratories, diffusion experiments are affected
by the g-jitter induced convection. Although much weaker compared to that on the earth,
this convection has to be fully considered to ensure the accuracy of experiments. g-jitters
have a wide spectrum of amplitudes and frequencies. Their effect on diffusion varies
depending on both amplitude and frequency. Theoretical study of the g-jitter effect at
different conditions becomes essential in supporting space experiments and leads to
better understanding. Chen et al (1980) analyzed the combined heat and mass transfer
convection flow along vertical and inclined flat plates under the combined buoyancy
effects of thermal and mass diffusion, maintaining the plate either at a uniform
temperature/concentration or subjected to a uniform heat/mass flux. The analysis includes
the processes in which the diffusion-thermo and thermo-diffusion effects as well as the
interfacial velocities due to mass diffusion were negligibly small. Numerical results for
the local Nusselt number and the local Sherwood number were presented for diffusion of
common species into air and water.
Amin (1998) has investigated the heat transfer from a sphere immersed in an
infinite fluid medium in a zero gravity environment under the influence of g – jitter. She
has shown that heat transfer is negligibly small for high frequency g – jitter. But under
special circumstances, when Prandtl number is sufficiently high, low frequency g – jitter
may play an important role. Wadih and Roux (1998) proved that vibrations can either
substantially enhance or retard heat transfer and thus drastically affect the convection.
Christov and Homsy (2001) studied the convective flow in a vertical slot with
differentially heated walls and vertical temperature gradient with and without gravity
modulation. The time-dependent Boussinesq equations governing the two-dimensional
convection were solved numerically. The flow was investigated for a range of Prandtl
numbers from Pr = 1000, when fluid inertia was insignificant and only thermal inertia
played a role to Pr = 0.73, when both were significant and of the same order. The
presence of jitter adds two more parameters, the dimensionless jitter amplitude ε and
frequency ω, rendering the flow susceptible to new modes of parametric instability at a
126
critical amplitude c . They investigated the response of the system to jitter near the
neutral curves of the various instabilities modes.
Rees and Pop (2001) have investigated g – jitter induced from convection near a
stagnation point. They found that forcing amplitude and forcing frequency parameters
affect the shear stress and the rate of heat transfer. Rees and Pop (2003) investigated the
boundary-layer flow induced by a constant temperature vertical surface embedded in a
porous medium with time periodic variations in the gravitational acceleration. They
observed that the main effect of such g-jitter was confined mainly to the region near the
leading edge. For small g-jitter amplitudes the numerical results compare very well
indeed with their earlier analysis in Rees and Pop (2000).
Chacha et al (2002) investigated the thermal diffusion in a binary mixture of
methane and n-butane subject to g-jitters with moderately high frequency. For various
configurations, they noticed that the g-jitter causes mixing and overcomes the Soret effect
in the cavity. Yan et al (2005) have analyzed the effect of low frequency g-jitter on
thermal diffusion. Low frequency g-jitters have more effect on diffusion than high
frequency ones. The fluid flow, concentration and temperature distributions have been
thoroughly analyzed for different g-jitter scenarios.
Sharidan and Amin (2005) studied the generation of steady streaming induced by
g-jitter on double diffusion from a sphere immersed in a viscous and incompressible
fluid. The governing equations were solved analytically and numerically by introducing
the stream function. For small Reynolds number, the solution was obtained by asymptotic
method, while for large Reynolds number Keller-box method was used. They have
studied the effect of g-jitter on skin friction, heat and mass transfer from the sphere. They
noted that for opposing buoyant forces, the skin friction and heat and mass transfer rates
follow complex trends depending on the buoyant ratio parameter, Prandtl and Schmidt
numbers. Sharidan et al (2006b) investigated the effect of periodic gravity modulation, or
g – jitter induced mixed convection on the flow and heat transfer characteristic associated
with a stretching vertical surface in a viscous, incompressible fluid.
Deka and Soundalgekar (2006) have studied gravity modulation effect on transient
free – convection flow past an infinite vertical isothermal plate. They used Laplace
Transform technique for the solution of governing equations. They observed that
127
transient velocity decreases with increasing frequency of gravity modulation or Prandtl
number but increases with increasing time. Saeid (2006) investigated the effect of the
sinusoidal gravity modulation on the free convection from a vertical plate. The step
change in plate temperature has been assumed and the two dimensional laminar boundary
layer approximations is used in the formulation. Their results show the steady periodic
variation of Nusselts number and the friction coefficient with the amplitude and
frequency of the gravitational acceleration oscillation. The effects of gravity modulation
on convection in the annulus between two horizontal coaxial cylinders in a microgravity
environment have been analyzed by Dyko and Vafai (2007). To study the unsteady flow
structures in a large-gap annulus, the three-dimensional transient equations of fluid
motion and heat transfer were solved. Their work described convection in a cylindrical
annulus under microgravity and provided information on the influence of gravity
fluctuations on heat transfer in a space environment.
This chapter is divided into two parts. In the first part, we have studied the effect of
the periodic oscillation of the gravitational field on the free – convection from a vertical
plate. It is assumed that the gravity modulation is given by tggg cos10 .
The suction velocity on the vertical plate is imposed in the form
zvvv 0
0 cos1
The effect of gravity modulation on the velocity profiles and temperature profiles are
discussed. The transformed equations are solved by perturbation technique to investigate
the effects of Prandtl number Pr , and gravity modulation parameter. It has been noted
that these parameters affect considerably the shear stress and the rate of heat transfer.
In the second part, we investigate the effect of gravity modulation on free
convection unsteady laminar flow past an infinite vertical porous plate with constant heat
flux and varying suction velocity. Governing equations have been solved with regular
perturbation method. The variation in gravity modulation and magnetic parameter make
significant change in skin-friction.
128
PART I: THREE – DIMENSIONAL MHD FREE CONVECTION
FLOW PAST AN INFINITE VERTICAL POROUS PLATE
WITH PERIODIC SUCTION AND GRAVITY
MODULATION.
6.2 FORMULATION OF THE PROBLEM
We examine the fluid flow past an infinite porous plate placed vertically on zx -
plane such that x - axis is taken in the direction of the flow in upwards direction and
y - axis is taken perpendicular to the plate. The fluid is assumed to be incompressible,
viscous and electrically conducting. A uniform magnetic field is applied in the direction
equally inclined with the axes. The magnetic Reynolds number is taken to be very small
so that the induced magnetic field can be neglected in comparison to the applied magnetic
field. Hall effect, electrical and polarization effects are also neglected. The concentration
level of the foreign mass is considered to be very small.
Fig. 6.1 Physical configuration of the problem.
129
Let
wvu ,, be the velocity components in
zyx ,, direction respectively.
Since plate is infinite in x - direction, therefore all physical quantities are independent
of x . Under these assumptions, flow is governed by the following set of equations
0
z
w
y
v (6.2.1)
2
2
)(y
ug
z
uw
y
uv
t
u )(
2
02
2
tUuB
z
u (6.2.2)
2
2
2
2
z
v
y
v
y
p
z
vw
y
vv
t
v vB2
0 (6.2.3)
2
2
2
2
z
w
y
w
z
p
z
ww
y
wv
t
w wB2
0 (6.2.4)
22
2
2
2
2
2z
w
y
v
z
T
y
T
z
Tw
y
Tv
t
TC p
222
z
u
z
v
y
w
y
u (6.2.5)
2
2
2
2
*
*
*
*
*
*
z
C
y
CD
z
Cw
y
Cv
t
C (6.2.6)
where *T is the temperature, *C is the species concentration in the fluid, g is the
acceleration due to gravity, is the density of the fluid, D is the chemical molecular
diffusivity, is the thermal conductivity, 0B is the magnetic field intensity, is the
electric permeability and is the viscosity.
The suction velocity *
v on the vertical plate is imposed in the form
zvvv 0
0 cos1 (6.2.7)
where, 0v is the constant suction velocity.
This consists of a basic steady distribution with a superimposed weak transversally
varying distribution. This makes the flow three dimensional.
130
The relevant boundary conditions at 0y are
,0,cos1,0 0
0
w
zvvvu
w
tiCCeTTTT
,)(
**
00
(6.2.8a)
as y ,
,,0,1)( 0
ppweUtUuti
CCTT , (6.2.8b)
The time-dependent gravitational acceleration is assumed in the form
tggg cos10 , where 0g is the constant gravity level in the environment, 1g is the
amplitude of the oscillating component of acceleration and is the frequency of
gravitational oscillation.
The gravitational acceleration is rewritten in the form
tieggg 10 (6.2.9)
It is assumed that the real part is physically relevant.
6.3 METHOD OF SOLUTION
For 1T and 1 C , we can express )( in terms of )(
TT and
)(
CC as given below
,)()()(
CCgTTgg (6.3.1)
The following non-dimensional quantities are introduced
4
,,
2
000
tv
tzv
zyv
y , 000
,,v
ww
v
vv
U
uu
,
CC
CCC
w
*
,
TT
TT
0
*
, 0
0
U
v ,
2
0v
pp
, 2
0
4
v
,
Schmidt number Sc =D
, Magnetic parameter
2
0
0
v
BM
Eckert number)( 0
2
0
TTC
UEc
p
, Prandtl Number Pr =
pC,
131
Grashof Number Gr = 2
00
00 )(
vU
TTg ,
Modified Grashof Number Gc = 2
00
0 )(
vU
CCg w
, (6.3.2)
where, 0U is the free stream velocity, C is the dimensionless species concentration in the
fluid, is the coefficient of thermal expansion, is the coefficient of thermal
expansion with concentration, is the kinematic viscosity, is the dimensionless
temperature, is the dimensionless frequency of gravitational oscillation, 0T is the wall
temperature, p is the pressure, pC is the specific heat at constant pressure.
The symbol * indicates dimensional quantities. The subscript denotes the free stream
condition.
By using (6.2.9), (6.3.1) and (6.3.2) into the equations (6.2.1) to (6.2.6) and (6.2.8), we
have
0
z
w
y
v (6.3.3)
)1)((4
1 tieGcCGrz
uw
y
uv
t
u
tieuMz
u
y
u
1
2
2
2
2
(6.3.4)
vMz
v
y
v
y
p
z
vw
y
vv
t
v
2
2
2
2
4
1 (6.3.5)
wMz
w
y
w
z
p
z
ww
y
wv
t
w
2
2
2
2
4
1 (6.3.6)
22
2
2
2
2
2
2Pr
1
4
1
z
v
y
vEc
zyzw
yv
t
2
2
22
z
v
y
wEc
z
u
y
uEc (6.3.7)
2
2
2
21
4
1
z
C
y
C
Scz
Cw
y
Cv
t
C (6.3.8)
where 01 gg
132
The corresponding boundary conditions reduce to
1,1,0,)cos1(,0;0 Cewzvuyti
.0,0,,0,1; Cppweuy ti (6.3.9)
Solution is assumed in the form
)(),,()(),,( 2
10 OtzyFyFtzyF (6.3.10)
where F stands for any of ,,,,, pwvu or C .
Substitute (6.3.10) into the equations (6.3.3) - (6.3.9), and equating the coefficient of like
powers of , we get
Zeroth – order equations
00 dy
dv (6.3.11)
MGcCGruMdy
du
dy
ud 000
0
2
0
2
(6.3.12)
)(0
0 xAyMpdy
dpM (6.3.13)
00
0
2
0
2
wMdy
dw
dy
wd (6.3.14)
2
0
2
02002
0
2
PrPr2Pr
dy
duEc
dy
dvEc
dy
dv
dy
d
0Pr
2
02
dy
dwEc (6.3.15)
00
2
0
2
dy
dCSc
dy
Cd (6.3.16)
where )(xA is unknown to be determined.
The modified boundary conditions are
1,1,0,1,0:0 00000 Cwvuy
0,0,,0,1: 00000 Cppwuy (6.3.17)
Equation (6.3.11) and the boundary conditions in (6.3.17) yield
10 v (6.3.18)
133
The equations (6.3.12) and (6.3.15) are ordinary coupled second order differential
equations which are solved under the boundary conditions (6.3.17). For incompressible
fluid flows, Eckert number Ec is very small, therefore )(0 yu and )(0 y can be expanded
in powers of Ec as given by
)()()()( 2
01000 EcOyuEcyuyu (6.3.19)
)()()()( 2
01000 EcOyEcyy (6.3.20)
Using equations (6.3.19) and (6.3.20) into the equations (6.3.12) and (6.3.15) and
equating the coefficient of like powers of Ec , we get
MGcCGruMuu 000000000 (6.3.21)
01010101 GruMuu (6.3.22)
0Pr 0000 (6.3.23)
2
000101 PrPr u (6.3.24)
where prime denote differentiation with respect to y .
Corresponding boundary conditions are reduced to
0,0,0,1:
,0,1,0,0:0
01000100
01000100
uuy
uuy (6.3.25)
The solution of ordinary differential equations (6.3.14), (6.3.16) and (6.3.21) – (6.3.24)
under the corresponding boundary conditions are as follows
1)( 2
Pr
13001 yScyym
eaeaeayu (6.3.26)
ymyScyymyym
eaeaeaeaeaeayuPr)(
14
2
13
Pr2
12
2
11
Pr
101701111)(
yScmySc
eaea)(
16
Pr)(
151
(6.3.27)
yey Pr
00 )( (6.3.28)
ymyScyymy
eaeaeaeaeayPr)(
7
2
6
Pr2
5
2
4
Pr
90111)(
yScmySc
eaea)(
9
Pr)(
81
(6.3.29)
ySceyC )(0 (6.3.30)
0)(0 yw (6.3.31)
where 1a to 17a and
9a are constants have been recorded in the APPENDIX - VII.
134
First – order equations
011
z
w
y
v (6.3.32)
11011
11
2
1
2
2
1
2
4
1GcCGruvuM
t
u
y
u
z
u
y
u
titi eMGcCGre ][ 00 (6.3.33)
2
1
2
2
1
2
11
11
4
1
z
v
y
v
y
pvM
y
v
t
v
(6.3.34)
2
1
2
2
1
2
11
11
4
1
z
w
y
w
z
pwM
y
w
t
w
(6.3.35)
2
1
2
2
1
2'
0111
Pr
1
4
1
zyv
yt
y
uuEc
1'
02 (6.3.36)
2
1
2
2
1
2'
0111 1
4
1
z
C
y
C
ScCv
y
C
t
C (6.3.37)
The corresponding boundary conditions are
0,,0,cos,0:0 11111 Cewzvuyti
0,0,0,: 1111 Cweuy ti (6.3.38)
To solve coupled equations separation of variable technique is used as
zyGeyGtzyG ti cos)()(),,( 21 (6.3.39)
zyv
eyzvtzyw ti
sin)('
)('),,( 12111 (6.3.40)
where G stands for any of 1111 ,,, pvu or 1C .
The equation of continuity (6.3.32) is identically satisfied.
Using (6.3.39) and (6.3.40) into the equations (6.3.33) – (6.3.37), and equating the
coefficients of harmonic and non-harmonic terms, we get
11110111111114
GcCGruvuMi
uu
MGcCGr ][ 00 (6.3.41)
1201212
2
1212 GruvuMuu 12GcC (6.3.42)
135
111111114
pvMi
vv
(6.3.43)
1212
2
1212 pvMvv (6.3.44)
04
111111
vM
ivv
(6.3.45)
12
2
12
2
1212 pvMvv (6.3.46)
110011111111 Pr2Pr4
PrPr uuEcv
i
(6.3.47)
12001212
2
1212 Pr2PrPr uuEcv (6.3.48)
0111111114
CvScCSci
CScC
(6.3.49)
01212
2
1212 CvScCCScC (6.3.50)
The corresponding boundary conditions are reduced to
:0y
,0,0,0,0,0 1211111211 vvvuu
0,0,0,1,1 1211121112 CCv
:y
,0,0,0,0,0,1 121112111211 vvvvuu
.0,0,0,0,0,0 121112111211 ppCC (6.3.51)
The solutions of ordinary coupled differential equations (6.3.43) – (6.3.46), ( 6.3.49)
and (6.3.50) under the boundary condition (6.3.51) are given as
0)(11 yv (6.3.52)
yymeme
myv
3
3
123
1)( (6.3.53)
0)(11 yp (6.3.54)
yme
m
Mmyp 3
)(
)()(
3
312
(6.3.55)
0)(11 yC (6.3.56)
yScymScymeaeaeayC )(
22
)(
21231234)(
(6.3.57)
136
A perturbation technique is used to solve ordinary coupled second order differential
equation (6.3.41), (6.3.42), (6.3.47) and (6.3.48) under the boundary condition (6.3.51).
Eckert number Ec is very small, for incompressible fluid flows, therefore, we consider
)()()()(2
10 EcOyHEcyHyH (6.3.58)
where H stands for any of ),(),(),( 111211 yyuyu or )(12 y . Using (6.3.58) into the
equations (6.3.41), (6.3.42), (6.3.47) and (6.3.48) and equating the coefficients of like
power of Ec , we get
1111000111101101104
GcCGruvuMi
uu
MGcCGr ][ 000 (6.3.59)
121200012120
2
120120 GcCGruvuMuu (6.3.60)
0011110110110 Pr4
PrPr
v
i (6.3.61)
0012120
2
120120 PrPr v (6.3.62)
11101111111111114
GruvuMi
uu
01Gr (6.3.63)
1210112121
2
121121 GruvuMuu (6.3.64)
110000111111111111 Pr2Pr4
PrPr uuv
i
(6.3.65)
120000112121
2
121121 Pr2PrPr uuv (6.3.66)
The corresponding boundary conditions are reduced to
:0y ,1,0,,,,,, 110121120111121120111110 uuuu
:y 0,,,,,,,1 121120111110121120111110 uuuu (6.3.67)
Equations (6.3.59) – (6.3.67) yield
yymymeaeaeayu Pr
37364011057)( 3938 aea ySc (6.3.68)
ymymmymmymym
eaeaeaeaeayuPr)(
126
)(
125
)(
1241231371111517157)(
yScyymymyScm
eaeaeaeaea)(Pr
131
Pr2
130
)(Pr
129
)(Pr
128
)(
127571
ymyyScymScymSc
eaeaeaeaea 157 2
136
Pr
135
2
134
)(
133
)(
132
(6.3.69)
137
ymymScymymmym
eaeaeaeaeayu)(
30
)(
29
)(Pr
28
)(
2735120133313)(
ymymyScy
eaeaeaea 46
3433
)(
32
)(Pr
31
(6.3.70)
ymymmymymmym
eaeaeaeaeayuPr)2(
101
)2(
100
)(Pr
99
)(
981221211313313)(
yScmmymScPsymmymSc
eaeaeaea)(
105
)(
104
Pr)(
103
)2(
102313313
yScyymyym
eaeaeaeaea)2(
110
Pr)2(
109
)2(
108
Pr)(
107
)(
10611
ymmymyScmyScym
eaeaeaeaea)(
115114
)(
113
)Pr(
112
Pr)(
11161611
ymScymymymm
eaeaeaea)(
119
)(Pr
118
)(Pr
117
)(
11634641
ymScymSc
eaea)(
121
)(
12046
(6.3.71)
ymey 5)(110
(6.3.72)
yScmymymmymmym
eaeaeaeaeay)(
44
Pr)(
43
)(
42
)(
41521111151715)(
ymScyScyymym
eaeaeaeaea)(
49
Pr)(
48
Pr2
47
)(Pr
46
)(Pr
45757
ScyymSc
eaea2
51
)(
505
(6.3.73)
yymymeaeaeay Pr)(
25
Pr)(
242612036)(
(6.3.74)
ymScymymmymym
eaeaeaeaeay)2(
78
)Pr2(
77
)2(
76
)(Pr
7597121333136)(
ymm
eaPr)(
7931
ymyyScmmymSceaeaeaea
)2(
83
)(Pr
82
)(
81
)(Pr
801313
yScy eaea )2(
85
Pr)2(
84
yScmyScym
eaeaea)(
88
)Pr(
87
Pr)(
8611
ymmymmymm
eaeaea)(
91
)(
90
)(
89416131
ymymeaea
)(Pr
93
)(Pr
9246
ymScymScymSc
eaeaea)(
96
)(
95
)(
94463
(6.3.75)
where 24a to 137a , 1m , 3m to 7m are constants have been recorded in the APPENDIX -
VII.
Finally the expressions of ),,(),,,(),,,( tzywtzyvtzyu , ),,(),,,( tzyCtzyp and
),,( tzy are written as
ti
eyuEcyuyuEcyutzyu
)()()()(),,( 1111100100
zyuEcyu cos)()( 121120 (6.3.76)
)cos)()()(),,( 12110 zyveyvyvtzyv ti (6.3.77)
138
z
yveyvzywtzyw
ti
sin
)()()(),,( 12
110 (6.3.78)
)cos)()()(),,( 12110 zypeypyptzyp ti (6.3.79)
)cos)()()(),,( 12110 zyCeyCyCtzyC ti (6.3.80)
ti
eyEcyyEcytzy
)()()()(),,( 1111100100
zyEcy cos)()( 121120 (6.3.81)
6.4 RESULTS AND DISCUSSION
We discuss the effect of various parameters on velocity field, temperature field,
concentration field, skin-friction and heat transfer. Numerical values have been calculated
for Prandtl number Pr = 0.71(air) and 7.0 (water). The values of Schmidt number Sc are
taken for Hydrogen (Sc = 0.24), Ammonia (Sc = 0.78), Helium (Sc = 0.30), and Water-
vapor (Sc = 0.60). The value of Eckert number Ec is taken 0.01 and = 0.25. The values
of Grashof number Gr , Modified Grashof number Gc and frequency are selected
arbitrarily.
Velocity profiles
Fig. 6.2 represents the effect of Prandtl number Pr , magnetic parameter M , and
gravity modulation parameter for .2/ t It is observed from fig. 6.2 that
transient velocity increases in the neighborhood of the plate due to increase in gravity
modulation parameter , while it decreases with increase in Prandtl number Pr and
magnetic parameter M . Similar behavior is shown by Fig. 6.3 for .4/ t Fig. 6.4
exhibit the variation of velocity profiles versus y for different values of Grashof number
Gr , modified Grashof number Gc , Schmidt number Sc , Eckert number Ec and
frequency . It is seen that velocity increases with increase in Gr and Gc , while it
decreases with increase in Schmidt number Sc , Eckert number Ec and frequency .
139
Fig. 6.5 represents the velocity profiles versus z . It is seen that velocity profiles
decrease with increase in Prandtl number Pr and magnetic parameter M , while it
increase with increase in gravity modulation parameter . Fig. 6.6 depicts that fluid
velocity decreases with increase in Schmidt number Sc, Eckert number Ec and
frequency , while it increases with increase in Grashof number Gr and Modified
Grashof number Gc .
Temperature profiles
Fig. 6.7 exhibits the variation of temperature profiles for different values of Prandtl
number Pr , magnetic parameter M and gravity modulation parameter . With
increase in Prandtl number Pr , the temperature decreases more rapidly for water in
comparison to air. Fluid temperature increases with increase in magnetic parameter M
and gravity modulation parameter . Fig. 6.8 depicts that fluid temperature decreases
140
141
142
with increase in Grashof number Gr , Schmidt number Sc , frequency , and Eckert
number Ec . It also decreases with increase in modified Grashof number Gc for
6.00 y and then increases in the rest part.
Fig. 6.9 represents the variation of temperature profiles along z – axis. It is seen that
temperature decreases with increase in Schmidt number Sc , and magnetic parameter M ,
while it increases with increase in gravity modulation parameter .
Concentration profiles
Fig. 6.10 exhibits the variation of concentration profiles for different values of
Schmidt number Sc . It is observed that concentration decreases due to increase in
Schmidt number Sc . Gravity modulation parameter has no effect on concentration.
143
144
Skin friction
Skin friction coefficient at the plate is given by
0
yy
u
ti
emaScaamamaScaama
1145144143514271411401391138 PrPr
15014914811473146 Pr aScaamama zmama cos41526151 (6.4.1)
where 138a to 152a are constants and have been recorded in the APPENDIX - VII.
Fig. 6.11 depicts that skin-friction coefficient at the plate shows periodic behavior
along z - direction. It is noted that peaks of skin friction coefficient increase due to
increase in gravity modulation parameter and Eckert number Ec , while these
decrease due to increase in Schmidt number Sc and magnetic parameter M . It is
observed that gravity modulation parameter has more impact on skin friction as
compared to other parameters.
145
Heat transfer
The rate of heat transfer in terms of Nusselt number at the plate is given by
0000 )(
yyyy
T
TTvNu
where
51561551154153
0
Pr maScamaay
y
tieScaamama 16015971581157 Pr
116516416331626161 Pr maaamama zmaSca cos4167166 (6.4.2)
where 153a to 167a are constants recorded in the APPENDIX - VII.
It is seen from fig. 6.12 that the Nusselt number shows periodic behavior along
z-direction. Peaks of the Nusselt number increase due to increase in Schmidt number Sc ,
while these decrease due to increase in magnetic parameter M and gravity modulation
parameter . It is seen that gravity modulation parameter has more impact than
Schmidt number. Fig. 6.13 shows that peaks of Nusselt number increase due to increase
146
147
in Grashof number Gr, modified Grashof number Gc, frequency , Prandtl number Pr
and Eckert number Ec.
6.5 NUMERICAL SOLUTION AND COMPARISON
In order to validate the perturbation method, we have also done a numerical
treatment of the problem. We have used a bvp4c method which is inbuilt programme in
MATLAB software. We have compared the values of skin friction on the plate along z-
axis. Value of various parameters have been taken as ,5,78.0,71.0Pr GrSc
,3Gc 5.0,10,10 M and 4/ t . Result is shown in the Table-1.
Table-1 Comparison of Skin Friction.
z 0 1 2 3 4
Numerical
Solution
53.7086 52.8533 53.7086 52.8533 53.7086
Perturbation
Solution
53.4629 52.9891 53.4629 52.9891 53.4629
6.6 CONCLUSION
(i) It is observed that increase in gravity modulation parameter increases the
transient velocity in the neighborhood of the plate, while it decreases with
increase in Prandtl number and magnetic parameter.
(ii) Fluid temperature increases with increase in magnetic parameter and gravity
modulation parameter, while it decreases with increase in Grashof number and
Schmidt number.
148
(iii) It is observed that gravity modulation parameter has more impact on skin
friction as compared to other parameters. It is noted that peaks of skin friction
coefficient increase due to increase in gravity modulation parameter and
Eckert number, while these decrease due to increase in Schmidt number and
magnetic parameter.
(iv) Gravity modulation parameter has more impact on Nusselt number than
Schmidt number. It is observed that Nusselt number increases due to increase
in Schmidt number, while it decreases due to increase in magnetic parameter
and gravity modulation parameter.
149
PART II: THREE – DIMENSIONAL MHD FREE CONVECTION
FLOW PAST AN INFINITE VERTICAL POROUS PLATE
WITH GRAVITY MODULATION AND HEAT FLUX.
6.7 FORMULATION OF THE PROBLEM
This part examines the previous problem with change in boundary condition of
temperature. We consider a constant heat flux on the vertical porous plate.
The corresponding boundary conditions at 0y are
,0,cos1,0 00
wzv
vvu
wCC
q
y
T
,
(6.7.1a)
as y ,
,,0,1)( 0
ppweUtUu ti
CCTT , (6.7.1b)
where q denotes heat transferred and the other physical variables used herein are defined
in part-I.
6.8 METHOD OF SOLUTION
For 1T and 1 C , we can express )( in terms of )(
TT and
)(
CC as given below
,)()()(
CCgTTgg (6.8.1)
The following non-dimensional quantities are introduced
,, 00
zv
zyv
y 2
0
4
v
, 000
,,v
ww
v
vv
U
uu
,
CC
CCC
w
*
,
q
KvTT 0
* )( ,
0
0
U
v ,
4,
2
0
2
0
tv
tv
pp ,
Prandtl Number Pr =
pC, Eckert number
qC
vUEc
p
0
2
0
150
Schmidt number Sc = D
, Magnetic parameter
2
0
0
v
BM
Grashof Number Gr =
3
00
2
0
vU
qg ,
Modified Grashof Number Gc = 2
00
0 )(
vU
CCg w
, (6.8.2)
The physical variables used herein are defined in Part-I.
By using (6.2.9), (6.8.1) and (6.8.2) into the equations (6.2.1) to (6.2.6), (6.7.1a) and
(6.7.1b) we have
0
z
w
y
v (6.8.3)
)1)((4
1 tieGcCGrz
uw
y
uv
t
u
tieuMz
u
y
u
1
2
2
2
2
(6.8.4)
vMz
v
y
v
y
p
z
vw
y
vv
t
v
2
2
2
2
4
1 (6.8.5)
wMz
w
y
w
z
p
z
ww
y
wv
t
w
2
2
2
2
4
1 (6.8.6)
2
2
2
2
2
2
2Pr
1
4
1
y
vEc
zyzw
yv
t
2
2
222
z
v
y
wEc
z
u
y
uEc
z
v (6.8.7)
2
2
2
21
4
1
z
C
y
C
Scz
Cw
y
Cv
t
C (6.8.8)
where 01 gg
The corresponding boundary conditions (6.7.1a) and (6.7.1b) reduce to
1,1,0,)cos1(,0:0
C
ywzvuy
.0,0,,0,1: Cppweuy ti (6.8.9)
151
Solution near the plate is taken as
)(),,()(),,( 2
10 OtzyFyFtzyF (6.8.10)
where F stands for ,,,,, pwvu or C .
Substituting (6.8.10) into the equations (6.8.3) - (6.8.9), and equating the coefficient of
like powers of , we get
Zeroth – order equations
00 dy
dv (6.8.11)
MGcCGruMdy
du
dy
ud 000
0
2
0
2
(6.8.12)
)(0
0 xAyMpdy
dpM (6.8.13)
00
0
2
0
2
wMdy
dw
dy
wd (6.8.14)
2
0
2
02002
0
2
PrPr2Pr
dy
duEc
dy
dvEc
dy
dv
dy
d
0Pr
2
02
dy
dwEc (6.8.15)
00
2
0
2
dy
dCSc
dy
Cd (6.8.16)
where )(xA is unknown to be determined.
The modified boundary conditions are
1,1,0,1,0:0 00000 Cwvuy
0,0,,0,1: 00000 Cppwuy (6.8.17)
Equation (6.8.11) and the boundary conditions (6.8.17) yield
10 v (6.8.18)
The equations (6.8.12) and (6.8.15) are ordinary coupled second order differential
equations which are solved under the boundary conditions (6.8.17). For incompressible
152
fluid flows, Eckert number Ec is very small, therefore )(0 yu and )(0 y can be expanded
in powers of Ec as given by
)()()()( 2
01000 EcOyuEcyuyu (6.8.19)
)()()()( 2
01000 EcOyEcyy (6.8.20)
Using equations (6.8.19) and (6.8.20) into the equations (6.8.12) and (6.8.15), and
equating the coefficient of like powers of Ec , we get
MCGcGruMuu 000000000 (6.8.21)
01010101 GruMuu (6.8.22)
0Pr 0000 (6.8.23)
2000101 PrPr u (6.8.24)
where prime denotes differentiation with respect to y .
Corresponding boundary conditions are reduced to
0,0,0,1:
,0,1,0,0:0
01000100
01000100
uuy
uuy (6.8.25)
The solution of ordinary differential equations (6.8.14), (6.8.16) and (6.8.21) – (6.8.24)
under the corresponding boundary conditions are as follows
1)( 2
Pr
13001 yScyym
eaeaeayu (6.8.26)
ymyScyymyym
eaeaeaeaeaeayuPr)(
14
2
13
Pr2
12
2
11
Pr
101701111)(
yScmySc
eaea)(
16
Pr)(
151
(6.8.27)
yey Pr
00Pr
1)( (6.8.28)
ymyScyymy
eaeaeaeaeayPr)(
7
2
6
Pr2
5
2
4
Pr
90111)(
yScmySc
eaea)(
9
Pr)(
81
(6.8.29)
ySceyC )(0 (6.8.30)
0)(0 yw (6.8.31)
where 1a to 17a and
9a are constants and have been recorded in the APPENDIX- VIII.
153
First – order equations
011
z
w
y
v (6.8.32)
11011
11
2
1
2
2
1
2
4
1GcCGruvuM
t
u
y
u
z
u
y
u
titi eMGcCGre ][ 00 (6.8.33)
2
1
2
2
1
2
11
11
4
1
z
v
y
v
y
pvM
y
v
t
v
(6.8.34)
2
1
2
2
1
2
11
11
4
1
z
w
y
w
z
pwM
y
w
t
w
(6.8.35)
2
1
2
2
1
2
01
11
Pr
1
4
1
zyv
yt
y
uuEc
1
02 (6.8.36)
2
1
2
2
1
2
01
11 1
4
1
z
C
y
C
ScCv
y
C
t
C (6.8.37)
The corresponding boundary conditions are
;0,0,0,cos,0:0 11
111
C
ywzvuy
0,0,0,: 1111 Cweuy ti (6.8.38)
To solve coupled equations separation of variable technique is used.
zyGeyGtzyG ti cos)()(),,( 21 (6.8.39)
zyv
eyzvtzyw ti
sin)('
)('),,( 12111 (6.8.40)
where G stands for 1111 ,,, pvu or 1C .
The equation of continuity (6.8.32) is identically satisfied.
Using (6.8.39) and (6.8.40) into the equations (6.8.33) – (6.8.37), and equating the
coefficients of harmonic and non-harmonic terms, we get
11110111111114
CGcGruvuMi
uu
MGcCGr ][ 00 (6.8.41)
154
1201212
2
1212 GruvuMuu 12GcC (6.8.42)
111111114
pvMi
vv
(6.8.43)
1212
2
1212 pvMvv (6.8.44)
04
111111
vM
ivv
(6.8.45)
12
2
12
2
1212 pvMvv (6.8.46)
110011111111 Pr2Pr4
PrPr uuEcv
i
(6.8.47)
12001212
2
1212 Pr2PrPr uuEcv (6.8.48)
0111111114
CvScCSci
CScC
(6.8.49)
01212
2
1212 CvScCCScC (6.8.50)
The modified boundary conditions are
:0y ,0,0,0,0,0 1211111211 vvvuu
0,0,0,0,1 1211121112 CCv
,0,0,0,0,0,1: 121112111211 vvvvuuy
.0,0,0,0,0,0 121112111211 ppCC (6.8.51)
The solutions of ordinary differential equations (6.8.43) – (6.8.46), (6.8.49) and
(6.8.50) under the boundary condition (6.8.51) are given as
0)(11 yv (6.8.52)
yymeme
myv
3
3
123
1)( (6.8.53)
0)(11 yp (6.8.54)
yme
m
Mmyp 3
)(
)()(
3
312
(6.8.55)
0)(11 yC (6.8.56)
yScymScymeaeaeayC )(
22
)(
21231234)(
(6.8.57)
155
A perturbation technique is used to solve ordinary coupled second order differential
equation (6.8.41), (6.8.42), (6.8.47) and (6.8.48) under the boundary condition (6.8.51).
Eckert number Ec is very small, for incompressible fluid flows, therefore, we consider
)()()()( 2
10 EcOyHEcyHyH (6.8.58)
where H stands for ),(),(),( 111211 yyuyu or )(12 y .Using (6.8.58) into the equations
(6.8.41), (6.8.42), (6.8.47) and (6.8.48) and equating the coefficients of like power of
Ec , we get
1111000111101101104
GcCGruvuMi
uu
MGcCGr ][ 000 (6.8.59)
121200012120
2
120120 GcCGruvuMuu (6.8.60)
0011110110110 Pr4
PrPr
v
i (6.8.61)
0012120
2
120120 PrPr v (6.8.62)
11101111111111114
GruvuMi
uu
01Gr (6.8.63)
1210112121
2
121121 GruvuMuu (6.8.64)
110000111111111111 Pr2Pr4
PrPr uuv
i
(6.8.65)
120000112121
2
121121 Pr2PrPr uuv (6.8.66)
The corresponding boundary conditions reduce to
,0,,,,,,,:0 110121120111121120111110 uuuuy
0,,,,,,,1: 121120111110121120111110 uuuuy (6.8.67)
Equations (6.8.59) – (6.8.67) yield
3837
Pr
36391107)( aeaeaeayu yScyym
(6.8.68)
ymymymmymym
eaeaeaeaeayu)(Pr
122
Pr)(
121
)(
120119130111717157)(
ymScymyyScyySc
eaeaeaeaeaea)(
128
2
127
Pr
126
2
125
Pr2
124
)(Pr
12371
yScm
ea)(
1291
(6.8.69)
156
ymymScymymmym
eaeaeaeaeayu)(
30
)(
29
)(Pr
28
)(
2735120133313)(
y
ea)(Pr
31
ymymySc eaeaea 46
3433
)(
32
(6.8.70)
ymymmymymmym
eaeaeaeaeayuPr)2(
97
)2(
96
)(Pr
95
)(
941181211313313)(
ymSc
ea)2(
983
yScmmymScPsymmeaeaea
)(
101
)(
100
Pr)(
9931331
yScyymyym
eaeaeaeaea)2(
106
Pr)2(
105
)2(
104
Pr)(
103
)(
10211
yScym
eaea)Pr(
108
Pr)(
1071
ymyScm
eaea 61
110
)(
109
ymymymmymm
eaeaeaea)(Pr
114
)(Pr
113
)(
112
)(
111464161
ymScymSc
eaea)(
116
)(
11563 ymSc
ea)(
1174
(6.8.71)
0)(110 y (6.8.72)
ymmym
eaeay)(
4048111715)(
ymyScmymeaeaea
)(Pr
43
)(
42
Pr)(
41711
yScymScy eaeaea Pr)(
46
)(
45
Pr2
447 Scyea 2
47
(6.8.73)
yymymeaeaeay Pr)(
25
Pr)(
242612036)(
(6.8.74)
ymScymymmymym
eaeaeaeaeay)2(
74
)Pr2(
73
)2(
72
)(Pr
7193121333136)(
yymyyScmmymmeaeaeaeaea Pr)2(
79
)2(
78
)(Pr
77
)(
76
Pr)(
7513131
ymmyScmyScymySc eaeaeaeaea
)(
84
)(
83
)Pr(
82
Pr)(
81
)2(
803111
ymmymm
eaea)(
86
)(
854161
ymScymymeaeaea
)(
89
)(Pr
88
)(Pr
87346
ymScymScymSc
eaeaea)(Pr
92
)(
91
)(
90346
(6.8.75)
where 24a to 137a , 1m , 3m to 7m are constants and have been recorded in the
APPENDIX- VIII.
Finally we have
)()()(),,( 1100100 yuyuEcyutzyu
zyuEcyueyuEc ti cos)()()( 121120111 (6.8.76)
zyveyvyvtzyv ti cos)()()(),,( 12110 (6.8.77)
z
yveyzvywtzyw ti
sin
)()()(),,(
'
12'
110 (6.8.78)
157
zypeypyptzyp ti cos)()()(),,( 12110 (6.8.79)
zyCeyCyCtzyC ti cos)()()(),,( 12110 (6.8.80)
)()()(),,( 1100100 yyEcytzy
zyEcyeyEc ti cos)()()( 121120111 (6.8.81)
6.9 RESULTS AND DISCUSSION
In order to understand the physical solution, velocity field, temperature field,
concentration field and skin-friction coefficient at the plate are discussed by assigning
numerical values to various parameters appearing in the solution. The values of Prandtl
number Pr for air and water are 0.71 and 7.0 respectively. The values of Schmidt number
Sc for Hydrogen, Ammonia, Helium and Water-vapor are 0.24, 0.78, 0.30, and 0.60
respectively. The value of Eckert number Ec is taken 0.01 and = 0.25. The values of
Grashof number Gr , Modified Grashof number Gc and frequency are selected
arbitrarily.
Velocity profiles
The variation of velocity profiles versus y for different values of Grashof number
Gr, modified Grashof number Gc , Schmidt number Sc and Eckert number Ec are
shown in fig. 6.14. It is observed that velocity increases with increase in Grashof number
Gr , Modified Grashof number Gc , while it decreases with increase in Schmidt number
Sc, and Eckert number Ec .
Fig. 6.15 depicts the effect of Prandtl number Pr , magnetic parameter M , and
gravity modulation parameter for .2/ t It is seen from Fig. 6.15 that transient
velocity increases in the neighborhood of the plate due to increase in gravity modulation
parameter , while it decreases with increase in Prandtl number Pr and magnetic
parameter M. Similar behavior is shown by fig. 6.16 for .4/ t
158
159
Fig. 6.17 represents the velocity profiles versus z. It is seen that velocity profiles
decreases with increase in Prandtl number Pr and magnetic parameter M , while it
increases with increase in gravity modulation parameter . Fig. 6.18 depicts that fluid
velocity decreases with increase in Schmidt number Sc , Eckert number Ec and
frequency , while it increases with increase in Grashof number Gr , Modified Grashof
number Gc .
Temperature profiles
Fig. 6.19 exhibits the variation of temperature profiles for different values of Prandtl
number Pr , magnetic parameter M and gravity modulation parameter . With
increase in Prandtl number Pr , the temperature decreases more rapidly for Pr = 7.0 in
comparison to Pr = 0.71. Fluid temperature decreases with increase in magnetic
parameter M , while it increases with increase in gravity modulation parameter .
160
161
Fig. 6.20 depicts that fluid temperature increases with increase in Grashof number
Gr , modified Grashof number Gc and Eckert number Ec, while it decreases with
increase in Schmidt number Sc . Fig. 6.21 represents the variation of temperature profiles
along z – axis. It is seen that temperature decreases with increase in Schmidt number Sc ,
and magnetic parameter M , while it increases with increase in gravity modulation
parameter .
Concentration profiles
Fig. 6.22 exhibits the variation of concentration profiles for different values of
Schmidt number Sc . It is observed that concentration decreases due to increase in
Schmidt number Sc . Gravity modulation parameter has no effect on concentration.
162
163
Skin friction
Skin friction coefficient in the non-dimensional form on the plate is given by
000
y
yx
y
u
Uv
Scaama 1331321131 Pr tieEcmamaScaama 511911371361357134 Pr
14214114011393138 Pr aScaamama zmama cos61444143 (6.9.1)
where 131a to 144a are constants have been recorded in the APPENDIX-VIII.
Fig. 6.23 depicts that skin-friction coefficient at the plate shows periodic behavior
along z - direction. It is noted that peaks of skin friction coefficient increase due to
increase in gravity modulation parameter , while it decreases due to increase in
Prandtl number Pr , and magnetic parameter M . It is observed that gravity modulation
parameter has significant impact on skin friction. Fig. 6.24 show that peaks of skin
friction coefficient increases with increase in modified Grashof number Gc and
frequency , while it decreases with increase in Grashof number Gr , Schmidt number
Sc and Eckert number Ec .
164
165
6.10 NUMERICAL SOLUTION AND COMPARISON
Perturbation solution has been compared with numerical solution. Numerical
solution has been obtained by using bvp4c. We have compared the values of skin friction
on the plate along z-axis. Solution has been done for 71.0Pr , 10,78.0 Sc and
5M . Results are shown in the Table-2.
Table-2 Comparison of Skin Friction.
z 0 1 2 3 4
Numerical
Solution
80.6109 79.4018 80.6109 79.4018 80.6109
Perturbation
Solution
80.5911 79.5321 80.5911 79.5321 80.5911
6.11 CONCLUSION
(i) It is noted that fluid velocity along y-axis increases due to increase in gravity
modulation parameter, while it decreases with increase in Prandtl number and
magnetic parameter. A similar behavior is seen along z-axis.
(ii) Fluid temperature along y-axis decreases with increase in magnetic parameter
and Schmidt number, while it increases with increase in gravity modulation
parameter. These parameters show similar behavior on fluid temperature
along z-axis.
(iii) It is observed that skin friction coefficient increases due to increase in gravity
modulation parameter, while it decreases due to increase in Prandtl number,
Schmidt number and magnetic parameter. Skin-friction coefficient at the plate
shows periodic behavior along z -direction.