I ntnat. J. Math. & Math. Sci.
(1982)165-182
Vol. 5 No.
165
COMBINED EFFECT OF FREE AND FORCED CONVECTION ON
MHD FLOW IN A ROTATING POROUS CHANNEL
D.R.V. PRASADA RAO and D.V. KRISHNA
Department of Mathematics
Post Graduate Centre, Anantapur
Andhra Pradesh, India
and
LOKENATH DEBNATH*
Mathematical Institute
University of Oxford
Oxford, England
(Received September 2, 1979 and in revised form July 2, 1980)
ABSTRACT.
This paper gives a steady linear theory of the combined effect of the
free and forced convection in rotating hydromagnetic viscous fluid flows in a porous channel under the action of a uniform magnetic field.
The flow is governed by
the Grashof number G, the Hartmann number H, the Ekman number E, and the suction
Reynolds number S.
The solutions for the velocity field, temperature distribution,
magnetic field, mass rate of flow and the shear stresses on the channel boundaries
are obtained using a perturbation method with the small parameter S.
The nature of
the associated boundary layers is investigated for various values of the governing
flow parameters.
The velocity, the temperature, and the shear stresses are dis-
cussed numerically by drawing profiles with reference to the variations in the flow
parameters.
KEY WORDS AND PHRASES. Free and Forced Convection rn a rotating fluid and
Magnetohydrodynamic oundary Layers.
1980 ATHEATICS SUBTECT CLASSIFICATION CODES.
1.
76R05, 76R10, 76W05.
INTRODUCTION.
In the last several years considerable attention has been given to the study
166
D.R.V.P. RAO, D.V. KRISHNA, AND L. DEBNATH
of the hydrodynamic thermal convection due to its numerous applications in geo-
physics and astrophysics.
Several authors including Gupta
[I], Jana [2], Nanda and
Mohanty [3], Mishra and Mudali [4], Mohan [5], Soundalgekar [6], Yen [7], and Gill
and Casal
[8], have investigated the effects of the free and/or forced convection
on hydromagnetic fluid flows confined to non-porous boundaries under various geo-
metrical configurations.
Some of these authors have also considered the effects of
wall conductance on the heat transfer aspect of hydromagnetic channel flows.
has been shown that the wall
conductance
It
exerts a destabilizing influence on the
flow whereas the magnetic field stabilizes the flow.
In spite of these studies, relatively less attention has been given to the
simultaneous effects of the free and forced convection on the hydromagnetic rotat-
ing viscous flows confined to porous boundaries.
Such work seems to be important
and useful partly for gaining basic understanding of such flows, and partly for
possible applications to geophysical and astrophysical problems.
The present paper
deals with a steady linear theory of the combined effect of the free and forced
convection in rotating hydromagnetic viscous fluid flows in a porous channel under
the action of a uniform magnetic field.
The solutions for the velocity field,
temperature distribution, magnetic field, mass rate of the flow and the shear
stresses on the channel boundaries are investigated using a perturbation method
with the small suction Reynolds number S.
The structure of the associated boundary
layers is examined for various values of the governing flow parameters.
The velo-
city, the temperature, and the shear stresses are discussed numerically by drawing
profiles with reference to the variations in the flow parameters.
2.
MATHEMATICAL FORMULATION OF THE PROBLEM.
We consider an incompressible electrically conducting viscous steady fluid
flows in a channel bounded by two porous non-conducting parallel plates at z
+ L.
Both the fluid and the boundary plates are in a state of solid body rotation with
a uniform angular velocity
Cartesian coordinate system
about the z-axis normal to the plates.
Oxyz rotating with the angular velocity
We take a
with the x-
axis taken in the direction of pressure gradient and the z-axis positive upward.
FREE AND FORCED CONVECTION ON MHD FLOW
A uniform magnetic field of strength B
O
167
parallel to the z-axis is applied to the
fluid system, and the fluid is driven by a constant horizontal pressure gradient.
The present analysis is based on the Boussinesq approximation which implies that
e
<<
I, when the density
temperature T
is given by 0
0O
(1
e), 0 O is the density at the
is the temperature variation from T
O
of thermal expansion.
ties of the liquid,
O
and
is the coefficient
The Boussinesq approximation also implies that the proper-
,
the kinematic viscosity and < the thermal diffusivity, are
independent of temperature and hence can be taken as constants.
With the Cartesian coordinate system and the Boussinesq approximation, the
unsteady motion of a viscous conducting fluid in the presence of a magnetic field
-
B in this rotating coordinate system is governed by the Navier-Stokes equations,
the continuity equation and the energy equation with usual notations
u
+ (u_
V) u__ + 2 k
)
k_g(1
-VP*
u__
+--
j
B + v
V2u__
(2.1)
O
div u
8t
where the velocity u
+ (u
v)e
<
0,
(2.2)
v2e
(2.3)
is the pressure including the centrifugal
(u, v, w), P
force, j is the current density and g is the acceleration due to gravity.
further assume that j, B and the electric field
E_
We
satisfy the Maxwell equations
with usual notations.
The flow is assumed to be fully developed and steady so that all the physical
variables except the pressure, depends only on z.
tinuity equation (2.2) that w =-wO
w
O
< 0 for injection.
w
w
u
o
z
-
Clearly, w
O
> 0 for suction and
Thus the governing flow equations are
2v
+
a constant.
It then follows from the con-
P*
2u= v
z
+
+
22u
z
+
BO
o
ore
Bx.
( -i-!
(2.4)
(2.5)
oPe
Bx
___z + Bx --/
0oU e By
B
Z
g(1
13e),
(2.6)
168
D.R.V.P. RAO, D.V. KRISHNA, AND L. DEBNATH
(uT+o--)e
2Bx
8z
2
y
2
;)z
(Bx, By Bo)
where
u
2 0 v [(-,.---+.az
p
_
w
__o
q
(Bx)
z
B
w
q
B
+
\Sz
I
2
+
v
(-fT)
2-
+
i
2
(i
C o -x
o p
2
+J y ),
(2.7)
o
o
+--
z
u
=0
(2.8)
8v
0
(2.9)
’f
are the components of the magnetic field,
e
is the magnetic
permeability, o is the electrical conductivity, 8 is the coefficient of volume
expansion, C
p
(op)
is the specific heat and q
-i
e
is the magnetic viscosity.
The
last two terms of the energy equation (2.7) represent the viscous and ohmic dissipation respectively.
Integration of (2.6) gives that
P*
where
-gz
P1
20oP e
(B
2
x
+.B
2
y
+ Bo2
+ pl x +
8g
18 dz,
(2.10)
is the uniform pressure gradient with which the fluid is driven along the
x-axis.
It is assumed that the porous plates at z
heated by a constant temperature gradient A
ture varies linearly along the plates.
+ L
of thickness d are cooled or
along the x-axis so that the tempera-
1
Eliminating P* from (2.4) and (2.6), we
obtain
O
where O
l(z)
AlX
+
O
l(z),
(2.11)
is an arbitrary function of z.
We next take
To + AlX + @I0
and
To + AlX + @II
as the temperatures at the
lower and upper plates respectively.
It is convenient to write down the non-dimensional form of the basic field
equations (2.4)
(2.9) through the non-dimensional variables (starred) defined by
the relations
(x*, z*)
1
=’"
(x,z), (u*,v*)
L
(u,v), and (B x *
B *)
y
n
O
(Bx, By )’
(2.12 abc)
We next combine the non-dimensional equations (2.4) and (2.5) and use (2.10)
FREE AND FORCED CONVECTION ON MHD FLOW
to obtain the equation
d2F
dz
(after dropping the asterisks)
S
2
169
dF
z
F
2i
+ H 2 dB
z
Gz
+ R,
(2.13)
Similarly, it follows from (2.8) and (2.9) that
d2B
dz
where F
+
u
dF
dB
SPmz +z
2
- --
iv is the complex velocity field, B
wL
o__
tic field, S
H
Prandtl number
+
B
x
1/2
o
o
is the complex magne-
iB
y
is the cross-flow Reynolds number, Pm
B L
Grashof number, E
(2.14)
0,
n
is the magnetic
8g A
is the Hartmann number, G
L4
is the
o
L2
is the Ekman number and R
L
2
)v
P1
Substitution of (2.11) into (2.7) yields the energy equation in the nondimensional form
P
dz
S
2
x
Ec
2
+
2
Ec
dz
is the non-magnetic Prandtl number, Ec
where P
and
Gu
z
gL
3
(e
I
du
z
BgL/C
p
2
+
dv
z
2
(2 15)
is the Eckert number
010)/2.
The boundary conditions on the velocity and temperature fields are
(u,v)
(0,0)
at z
0
at z
gBL3 (ell- I0 )/2
=_+
(2.16 ab)
(2.17)
o
(say) at z =-I,
(2.18)
The boundary conditions on the magnetic field are
-
dB
dz
+-B
dB
B
dz
d
B1
B2
at z
I,
(2.19)
at z
-i,
(2.20)
B1 Bxl + iByl B2 Bx2 + iBy2 l is the electrical conductivity of the plate,
Bxl and By are the values of Bx and By in the vacuum regiond
z =>
+d and Bx2 and By 2 are the values of Bx and By in the region z =< -I ---L
where
o
D.R.V.P. RAO, D.V. KRISHNA, AND L. DEBNATH
170
The magnetic field within ,the plates can be obtained using the Maxwell
equation J
(I
eJ (since u_
E and Curl B
0 on the plates).
If there is a net
current through the channel in the y-direction, the induced electric field exists
in the y-direction.
In view of this, it is observed that the magnetic field
components outside the boundary (in the vacuum) exist only in the x-direction and
The components
vanish in the y-direction.
Bxl
and
are fixed by knowing the
Bx2
net current in the y-direction through the channel and its return path.
3.
SOLUTION OF THE PROBLEM.
We shall adopt a small perturbation method with S, the cross-flow Reynolds
number as the perturbation parameter, in order to solve the differential equations
(2.13) and (2.14).
Thus we assume
Substituting (3.1)
+ SF 1 +
F
F
B
B3 + SB4
O
0 (S
2)
(3.1)
+ 0 (S 2),
(3.2)
(3.2) into (2.13) and (2.14) and equating the like
powers of S, we obtain
F
2i __qo
F"O
H2B,
+
Gz
+ R,
(3.4)
E
Fy
FI
2i
+ H2
E
B
(3.5)
F’
o
and
B’-’3 + F’O
0
B + F1
Pm B’
(3.6)
(3.7)
O
where the prime denotes the differentiation with respect to z.
(2.20) reduce to
The boundary conditions (2.16)
F
o
F
I
on z
0
+ 1,
(3.8 ab)
(3.9 ab)
-B
+
B
B
B
4
3
4
on z
I,
on z
-I,
0
B
+ 2
B--= 0
(3.10 ab)
0
FREE AND FORCED CONVECTION ON MHD FLOW
The solutions of (3.4)
H2B
2
where
a
G
3
H
+
12
al -i0 + S
2ia
2
a5
(I
+ cothl)
Gb
al0
212
R(I + )
b
b4-- 7
Qx
Qy
G
+
+ R,
al0
Bx2)
-
b4
(i + 3)
6i
b3
13
(g-
2
21 ), b
-
i
7
6
(i
(H2G213 Pm
(I + )
H2pm)
+
3
E
G
7
H2pm,
H2G
G
12
iS Pm b
+
tanhl,
3
E
21
H2S
+
i
El
2
I
--+ alob2b5
H
tanhl, a 8
iS Pm
2a10b2],
al0
2al0Sb2)
Pm] +
I), a II
2
i
+
(3.12)
+ S[ all
al0
2
+- (Bxl + Bx2
tanhl
2
asSz2 + aSz’
H2
+ 2)
(I
a
2t
b2b5G
13
(3.11)
Sb7z
3
+
GH2],
6
a
z} +
63
+
(a 2 +
cothl)
b
Sb z
al0
are given by
R), b 5
E
and
n
1/2(Bxl
2ial0
i
alO
+ H 2 (tanhl
(b2b5al0
And the constants b
2
1
sinhlz
sinhl
all
b2
+
GH 2
3
coshlCshlz
I
2
12
3(I + tanhl)
l0
GSb
7 (i + b5)
(1 + )
b
2a
+ S Pm b 4
A
+
a7
E
(I + cothl)
a
coshlz
coshl
sinhlsinhlz
2
9
If
a6
2Sb2G
13
G
GH
3
a
2i
+--’
SPmb31
H2
a7
a9
2
(3.7) with the above conditions are
al + a2
F
171
(I + )
+ 2S)b 4,
H2G
Pm
3(1 + ) G b
2
2b2a I0
(R +
2ial0
E’
i b
and b
7
6
E
represent the dimensionless mass flow rates, then
Qx
+
2a2
iQy
tanhl
The non-dimensional shear stresses
2
+
u
Y
Sb
7
+ 2a
and
(3.13)
v
z are
y
obtained at the
upper and lower plates from (3.11), and are given by
(T
x
+
i
yz-- 1
I a
2
tanhl
a
a
(Tx
+
iy)z=_l
a21
tanhl
3
+7
3
(I cothl
+i (I
cothl
i) + 2S b 7
i)
2S b
7
(3.14)
(3.15)
172
D.R.V.P. RAO, D.V. KRISHNA, AND L. DEBNATH
+ i where
To solve the energy equation, it is convenient to write A
is a fictitious function which satisfies the differential equation
d27
dz
2
S P"
d
zz=P
Gv,
(3.16)
with the boundary conditions
o.
(+I)
(3.17)
We next combine (2.15) with (3.16) to obtain the equation for A
d2A
P
dz
S
2
_---dA +
dz
Ec H
2 dB dB
dz dz
dF dF
dz dz
+ Ec
(3.18)
GF.
The boundary conditions on & are then given by
&
0 on z
and A
80
on z
(3.19 ab)
-i
Substituting F and B in (3.18) and solving the resulting differential equation with the boundary conditions (3.19 ab), (z) can readily be determined.
How-
ever, since the solution for O(z) is quite complicated, we avoid writing the long
expression for it.
4.
FIELD
THE VELOCITY
MAGNETIC FIELD AND THE ASSOCIATED BOUNDARY LAYERS.
In order to investigate the salient feature of the boundary layers, we shall
consider the following special cases related to various magnitudes of the Ekman
and Hartmann numbers.
Case (i):
E <<
and H
2 >>
I.
To determine the flow field, the magnetic field and the associated boundary
layers on the plate z
I,
it is convenient to write
in (3.11)
z
and then take the limit of the resulting expressions as -/ 0
+
a
iv
0 such that
This leads to the following results:
is finite.
u
(3.12),
+
a
(a
+7)
exp (-) (cosB- i sin)
a3
-7
(i- ) +
S H
2
(I
4
)
2
G Pm,
(4.1)
H2(B x
i
+
iB
-R
i S H
Y
3 E
s
G b
2
2
4
)3
G Pm
)2
+ (I
2
a2 +
2blb2S
3
2
[G H 2 +
b b
(I + )} +
2iS Pm
2b5 (I
4
2ia
E
+ )
E
G
2
2ia
+ R)] + E
H
2
+7
(G
+
2S G
b2)
FREE AND FORCED CONVECTION ON MBD FLOW
+
-
2ial
S Pm
3
2
+ H
E
(Bxl
i S INn
+
El2
where
3’
Bx2)
+
b2(l
(I + )
2ialo
G
H2
2%
G
SH2
+
exD (-a) (cosB- i sinB)
b 5)
Pm
-a16+7(
%3
G H
2
2%
2
2ialo
(i + 2)
R)]
E
(4.2)
R),
E
2
are the real and imaginary parts of A, and
and
G
+
(I + 2)
at0 +
a
a
+
,]
b2b5G
3 alO
R) +
173
G H2 Pm
]’
4
2
[262a10- all
S
262a16
x--r-
b
2ial
+ S Pm
al
al0
S
I0 +
a
a2
R)
E
and
al0
%2(I
+ )
The solution (4.1) represents the steady hydromagnetic boundary layer flow which
consists of the Ekman-Hartmann layer on the boundary plate.
thickness of this layer is of the order e
-I
4
2
[H + (H +
2
The non-dimensional
1/2]-1/2 which
is in
agreement with the result of Debnath [9] and also with that of Nanda and Mohanty
[3] for the case E <<
and H
2
0(i).
In particular, the Ekman-Hartmann layer
reduces to the Ekman layer of thickness O(E
0(
1
according as H
Case (ii):
2
<< or >>
1/2)
or the Hartmann layer of thickness
2
H2O(1)
E << i and
In this case, the solutions are obtained from (3.11)
(3.12), and have the
form
u
+
V
1
air + a3r
ali +
a3i
S(I
1/2 E
(a2i
+
)
2d7
+ 1/2 E
+
a3r)
a3i
d8(l
-- - - - -- -
H2E 22 + [(a2r + 1/2 E a3i)
1/2 E
[(a2i
(a2r
H2B x
S G Pm
sin{
a3r)
cos
sin{
)
EH
(I
{
(I
+
[d13
2
)}]
EH
(i
2
)}
/---
ep{-
(I +
H2E
)},
(4.3)
)}H2E
(i
EH
cos{
2
)}] exp{-- (I
/E
(i
sin
/E
EH2
)}
EH
2
+--- )} (4.4)
174
d
i.i
cos {
12
(I
2
+
1
8-
H2BY
S E
5
2
H2G
+
)2
sin {
d12
E G
4
(I
+
d
b
1
3
(5d14
+
11(1
) +
I
+-- )}
R)
+
[d13
H2E
5
3
+)
b
1+ 6
S E
G H
d5
d7
d9
d12
2
2
2E 3
8
2ali
(R
G Pm H
E
2
d
)’
d
S Pm
E
dl0
12
1/2
H2E [a2r
_H2
GE
4
1/2 H 2 E
[a2i
E
Sbl(d 5
a2i
alr
3d12
+
E
4E 3
6
8"5 (I’-+ "8)
8
R +
2
E
2H2
2
+
S G
G E
E
d5(l
a2r + Sbl(d 5 + d6)]
SGd6(l
+ 6) + S Pm4 E
3
2
d
bl
),
1
),
2ali
2alr +
E
S G E
2
5
3
1+6
4
2
2
E
dll
bl
+ 8) + S 4 em
Pm
4
+ (I + )d
+ d3
Pm-
S G Pm H 2 E
4
d6)]
(H2
16(1 + 6)
G H
2
3
H2[1/4
d15
_(d14
I respectively, and
_3
d13
)}
(4.6)
d4
+ H 2 Pro),
85(I + 8)
.2-
2
R)
+ H 2 Pro),
16(1 + 6)
Pm
(i + 6)
(4.5)
5
E
3d12
EH
(i
cos
5
d3
-
d15)
etc. denote the real and imaginary parts of a
alr ali
2
{--- (i +-- )}
)}] exp
(I + 26)
(I + )
EH
)}] exp {------ (I
EH2
H2 + 1/4 G E H 2
H2R
1/2 S E
2
(I + 6)
)3 + dl0( I
d9(l
EH
+ S 2 Pm (1 + )
+_ (Bxl Bx2)
where
---
D.R.V.P. RAO, D.V. KRI SHNA, AND L. DEBNATH
R} E
+ 1/2 GE
{1+8
3G
+ 8)d 5 + 2(1blE+ 8)’
R}] + --(I
FREE AND FORCED CONVECTION ON
175
FLOW
M}ID
5
H
d14
2
Pm
6
+E
and
3H2pmE3
2
d15
85
It is noted that the exponential terms in the above solution decays vewy
2
EH
In the limit
increases and exceeds the value /(i
rapidly as
-
+---
the flow field and the magnetic field assumes the form
H2Bx
S2Pm(
H2By
v
ali + a3i
2
3S G H
I +
d14
S2Pm
+
+ 5)
G
)
SH2R
d15
2
(I + 5)
(4.8)
5(1
/ E
)2 + dll( 1
)3 + dl0( I
(4.7)
b
1
E
+ 5)
+ 3d15
)’
E
H2
+ 1/4
4
5(1 + 5)
E
I
+
(5d14
2
S G H
R)
2E2
1 S G
P
a3r
H
+_
(Bxl + Bx2)
)
bl
+.2)
d3(l
+
a
)2 + dS(l
Sd7(l
Ir +
u
G
bl
1+(S
R)
(4.9)
H2E
(I +)
R).
(4.10)
(4.4) shows the existence of the modified Ekman
The velocity field (4.3)
layer of thickness of the order /-(I
EH
2
+--- )-
1
It follows from (4.7)
(4.10)
that both components of the velocity field persist, and are functions of the vertiThis means the violation of the Taylor-Proudman theorem valid in
cal coordinate z.
an inviscid rotating fluid flows.
Thus the suction or injection at the porous
plates prevents the flow to reduce to two-dimensional.
However, in the non-porous
case the limiting flow satisfies the conditions of the Taylor-Proudman theorem.
Case (iii):
H
2 >>
1 and
E-
0(I).
-
In this case, the solutions are given by
+
(f2
u
fl
V
f4 + (f5
H2Bx
i
2
(GH
2
2H
f3
exp (-H)-
f6
exp (-H)
+7
+7
4
7
S Pm
f7
(I
f3
H-
+ SG Pm2
2H
2
(4.11)
-H-)2
+ [R
(4.12)
2f4
E
SG(I + Pm)
H
2
(I- )
176
D.R.V.P. RAO, D.V. KRISHNA, AND L. DEBNATH
2Sf7
[H(f 2 +
-
S Pm R
H
bl
H2
2G
H2
H2
G +
2f7
4"
H
(Bxl
H
H
(i
+ Pm)
+
Bx2)
+ 2SG(I + Pm)
2
bl
(I + )] exp (- H)
H2
SH2(I
+
2SblPm
R Pm
3
(I + 6)
G
3
"H
H
4
G
+ )
+ Pro) (i
(i
2H
2)
(I + 2)
(I + 6) E
(4.13)
2
and
H2BY
SGPm
3EH2
where
fl H2(I
f3
G
-
H3(1
2f
bl
+ S(
+ )
S R Pm,
f8
2s Pm b
3
EH
7
(I
+ G(I +4 Pm)
EH
2Pm b
EH5(I
f2
exp (- H)
+ )
H2(I
EH
3G
(1 + 3)
+ 26) }
+
+ )
S(f8
H
b
f
7
3
I(I
7
(4.14)
)’
+ Pro)
+
(I + Pm)
(I + 6)
f7
2
2f
1
2SblPm
Sf9’ f5 Sfs’ f6 E(I + 6.H
2)
H2
)
SR Pm (I
b
G Pm
2
2H
f8
36(2f 7 + G)
f9 :’HE
I
G(I + Pm)
+
4
+ 6)
2fl
)2 + T
(I
E(I + 6)H
EH
f4
G Pm
f4
H2E2
f7 +
+5
[Hf5
+ S
2S Pm
)3
(i
G)
3(-2
EH
G(I + 6)
G Pm
E
2H
2f
H
7
2
(4.14) represent the steady hydromagnetic boundary
-I
layers flow which consists of the Hartmann Layer of thickness of the order H
The solutions (4.11)
It is noted that the results for this case are completely independent of rotation.
5.
THE NUMERICAL RESULTS AND DISCUSSION.
In order to present numerical results and their discussion, it is necessary
to assume certain fixed values for
Bxl
Bx2
0.5, R
and
Bxl +__ Bx2,
R and 6.
With
Bxl + Bx2
3.0,
0.i, the velocity, the temperature and the shear
stresses are discussed numerically.
The pressure gradient is taken to be positive
FREE AND FORCED CONVECTION ON MKD FLOW
177
along the direction of x increasing, and hence the flow in the opposite sense
indicated by negative u is the actual flow whereas a positive u corresponds to a
reversed velocity.
The velocity profiles for u and v, the secondary flow, are
presented for different values of the Hartmann number H, and the Grashof number G
for fixed K (=
i
and S as well as for different values of the suction parameter
S fixing H, G and K.
Figure I shows that in the absence of free convection (G --_ 0)
for fixed K and small S, as H increases the velocity profiles for u become more and
more asymmetric with the point of maximum velocity shifting towards the upper half
of the region.
When G
0, even for moderate values of G, u is found to change its
direction as we move from the upper plate to the lower one.
The maximum value of
u is attained in the vicinity of the upper plate, and for large values of G, there
is a sharp increase in the magnitude of u both at the upper and the lower plates
irrespective of the magnitude of H.
The region of reversed velocity near the lower
plate gradually grows with the increase in G.
However, for a fixed G, as H
increases the velocity near the upper plate enormously increases while the velocity
at the lower plate decreases in magnitude.
The secondary velocity normal to the direction of the pressure gradient, as
shown in Figure 2, behaves in the same manner as the primary velocity except that
the regions of adverse velocity zones grow in size almost comparable to each other
with increase in G for a fixed H.
However, for a fixed G the reversed secondary
velocity zone near the lower plate shrinks with the increase in H.
It has been
pointed out earlier that the presence of suction eliminated the possibility of flow
being two-dimensional in the central core for large K:
It is interesting to
observe the behavior of u and v for increase in the suction parameter S for fixed
H, G and K.
From Figure 3 it follows that while the magnitude of u increases
rapidly with increase in S there is a steady fall of v, although the maximum velocity of either of them tilt towards the upper plate similar to the H increasing
case.
For fixed values of H, G and S, both u and v decrease as K increases and
for large K, a reversed flow develops near the lower plate as shown in Figure 4.
D.R.V.P. RAO, D.V. KRISHNA, AND L. DEBNATH
178
C
-’3
.0
Veiocity profile for
lg. 2:
-2
-I
Fzg. i:
C
,
2,
0.2
o
when K
Velocity proflle for
i,
0.2
G
i0
-, Z
0.0
0
-1
when
i0
o.0
0.
0.
0.0
0.0
-0.2
..4
Fig. 4:
and
i0
i0
,OJ
Fig. 3:
when K
and
when H
i, G
a’
0.0
0.2
0.4
2,
0.2
2,
i,
a’
b’
c’
0.0
0.2
0.4
K
b’
c’
FREE AND FORCED CONVECTION ON MHD FLOW
179
Figure 5 indicates the behavior of the dimensionless mass flow rates compared
The mass flow rate
to H.
Qx
along the x-axis rapidly increases with increase in
H for a fixed K as well as increases with increase in K for a fixed H.
other
hand,Qy,
On the
for moderate values of K retains the same nature but almost remains
uniform for large K.
The non-dimensional shear stresses
x
and
y
at the upper and lower plates
are plotted in Figures 6-9 for various values of the governing flow parameters S,
H,G and K.
It is to be noted that the reversed flow occurs whenever the shear
No such separation
stress on the upper and the lower plates are of the same sign.
in the flow arises if the shear stress on the upper plate is opposite to that of
the lower plate.
Figure 6 shows that
x
remains positive and decreases with in-
crease in H for a fixed G, although increases for increase in G for a fixed H.
At the lower plate, it is negative and almost uniform for G
0.
But as G in-
creases it remains negative for large values of H (say greater than 4) but becomes
positive as H approches to zero.
This shows that for G z 0 and H
velocities appear in the vicinity of the lower plate.
0, reversed
However, this region of
reversed velocities which grows with increase in G can be made to shrink by choosing sufficiently large H.
The shear stress
y
behaves similar to
ence of the reversed velocities near the lower plate for small H.
indicate the behavior of r
also observe that for K
or v for all values of S.
x
and r
y
x
with occur-
Figures 8-9
with respect to the variation of S and K.
We
i no reversed flow appears with reference to either u
When K is large for sufficiently small S, we find its
growth near the lower plate.
For a fixed S,
decreasing for an increase in K.
x
and r
y
at both the plates go on
Also the shear stresses increase with S for a
fixed K.
The non-dimensional temperature profiles are plotted in Figures 10-12 for
different sets of parameters.
In all these cases the profiles are almost similar
except that the concavity increases with either increase in S or G.
in either
For increase
S,G and H it can be noted that the profiles become more and more sharp
contributing to the increase in the maximum temperature attained.
180
D.R.V.P. RAO, D.V. KRISHNA, AND L. DEBNATH
-3
0
0
Fzg. 6:
Fig. 5:
Mass rate of flow
2,
when
Qx
and
Qy
S
2.5
0
against H
a’
=0.2
I,
when
against
b’
10
a’
b’
0.2
c’
10
c’
K
2,0
10
0 .g,
0.
O.S
0.25
0
Fzg. 8:
against
when H
a’
Fig. 7:
against
when
0.2
a’
i0
b’
c’
10
K
i,
b’
c’
FREE AND FORCED CONVECTION ON MHD FLOW
18!
"--Z
0
-I
0
Fxg. I0:
Temperature distribution when
0.2, K
1.5, Ec
1,
0
0.5, Pm
i0
1.50
3.5
0 -7’5
2.0
Fig. ii:
TemPerature
Ec
distrxbution when
0.2,
1.5
0.5, Pm
i, K
4,
0.0
Fig. 12:
Temperature dstributions when H
4, K=
1.5, Ec
0.5, Pm
H
0.0
0.2
0.4
1,
D.R.V.P. RAO, D.V. KRISHNA, AND L. DEBNATH
182
ACKNOWLEDGEMENT.
This work was partially supported by East Carolina Univ-
ersity and the University of Oxford.
REFERENCES
Combined free and forced convection effects on the magnetohydrodynamic flow through a channel, Z.Angew. Math. Phys Vol. 20 (1969)
pp. 506-513.
i.
Gupta, A.S.
2.
Jana, R.N.
3.
Nanda, R.S. and Mohanty, H.K. Hydromagnetic Flow in a Rotating Channel,
Appl. Sci. Res. Vol. 24 (1970) pp. 65-78.
4.
Mishra, S.P. and Mudali, J.G.
Wall Conductance Effects on Convective Horizontal Channel Flow,
Z.Angew. Math. and Phys. Vol. 26 (1975) pp. 315-324.
the
Sci.
Combined free and forced convection effects on
magnetohydrodynamic flow through a porous channel, Proc. Ind. Acad.
Vol. 84 (1976) pp. 257-272.
5.
Mohan, M.
6.
Soundalgekar, V.M.
7.
Yen, J.T. Effect of Wall Conductance on Magnetohydrodynamic Heat Transfer in
a Channel, J. Heat Transfer Transac. ASME. Vol. 85C (1963) pp. 371-
Combined effects of free and forced convection on magnetohydrodynamic flow in a rotating channel, Proc. Indian Acad. Sci. Vol 85
(1977) pp. 383-401.
On Heat Transfer in crossed fluids magnetohydrodynamic
channel flow between conducting walls, Proc. Nat. Inst. Sci. India
Vol. A35 (1969) pp. 329-342.
377.
8.
Gill, W.M. and Casal, A.D. A Theoretical Investigation of Natural Convection
Effects in Forced Horizontal Flows, Amer. Inst. Chem. Engg.. Jour. Vol.
8 (1962) pp. 513-520.
9.
Debnath, L.
Mech.
On Ekman and Hartmann Boundary Layers in a Rotating Fluid, Acta.
Vol. 18 (1973) pp. 333-361.
*Present address:
Mathematics Department
East Carolina University
Greenville, North Carolina 27834 U.S.A.