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修平㈻報 第㈩㆕期 民國㈨㈩㈥年㆔㈪
HSIUPING JOURNAL. VOL.14, pp.85-98 (March 2007) 85
Chung-Ting Hsu, Lecturer, Department of Mechanical Engineering, HIT.投稿日期:950906 接受刊登日期:950929
Vortex instability of MHD natural convection flow over an isothermal horizontal plate in a
saturated porous medium
Chung-Ting Hsu
Abstract
A numerical analysis is performed to study the vortex instability of a horizontal
magnetohydrodynamics (MHD) natural convection boundary layer flow in a saturated porous
medium with surface mass flux. The stability analysis is based on the linear stability theory.
The resulting eigenvalue problem is solved by the local similarity method. The velocity and
temperature profiles, local Nusselt number, as well as instability parameters for magnetic
parameter M ranging from 0 to 4 are presented. It is found that as magnetic parameter M
increases, the heat transfer rate and tangential velocity decrease. Furthermore, it is shown that as
the magnetic parameter M increases, the neutral stability curves shift to lower Rayleigh number
and lower wave number, indicating a destabilization of the flow to vortex instability. It is also
shown that suction stabilizes the flow, while blowing destabilizes the flow.
Keywords: MHD, porous medium, natural convection, horizontal.
86 修平㈻報 第㈩㆕期 民國㈨㈩㈥年㆔㈪
徐仲亭:修平技術學院機械工程系講師
磁場效應對多孔性介質㆗等溫㈬平板㉂然對流渦漩不穩定性之影響
徐仲亭
摘 要
本文以理論方法探討在多孔性介質中,一水平等溫板的自然對流磁液動流場,磁場
效應對其貫軸渦漩不穩定性(vortex instability)的影響。基本流場部份利用相似轉換理論,
同時考慮邊界質量通量及磁場強度等效應。擾動流場利用線性穩定理論(linear stability
theory),導出之特徵值常微分方程組利用局部相似(local similarity)方法求解。探討主題為
磁場參數M(0~4)對速度曲線、溫度曲線、熱傳係數及中性穩定曲線的影響。數值結果顯
示隨著磁場參數M增加,溫度邊界層變厚、熱傳率與切線速度降低、中性穩定曲線趨向
較低的雷利數及波數,並使流場趨於不穩定。數值結果亦顯示邊界具質量吸入(suction)能
使流場趨於穩定,相反的,質量噴出(blowing)則使流場趨於不穩定。
關鍵詞:磁液動、多孔性介質、水平板、自然對流。
87Vortex instability of MHD natural convection flow over an isothermal horizontal plate in a saturated porous medium:Chung-Ting Hsu
INTRODUCTIONLiterature on magnetohydrodynamics
(MHD) convective heat transfer is very
extensive due to its technical importance in
the scientific community. Recently, there has
been a renewed interest in studying MHD
flow and heat transfer in viscous fluid and
porous media due to the effect of magnetic
fields on the boundary layer flow control
and on the performance of many systems
using electrically conducting fluids. In
addition, the MHD flow has attracted the
interest of many investigators in view of
its applications in heat exchanger devices,
filtration, cooling of nuclear reactors by
liquid sodium, geothermal energy extractions
and novel power-generating systems. For
example, Sparrow and Cess [1] and Riley
[2] studied the effects of a transversely
applied magnetic field on free convection
flow of an electrically conducting fluid past
a semi-infinite hot vertical plate. Watanabe
and Pop [3] examined the simultaneous
occurrence of buoyancy and magnetic forces
in the flow of an electrically conducting fluid
over a wedge. Pop and Watanabe [4] studied
the Hall effects on MHD free convection
about a semi-infinite vertical flat plat.
For the studies of magnetic field effect
on the flow and heat transfer rate in porous
media, Kumari [5] examined the effects of
magnetic field on hydrodynamic Darcian
porous flow in various configurations.
Soundalegkar [6] obtained approximate
solutions for two-dimensional flow of
an incompressible, viscous fluid past an
infinite porous vertical plate with constant
suction velocity. Chamkha [7] analyzed the
non-Darcy hydromagnetic free convection
problem from a cone and a wedge in porous
media. Chamkha [8] and Kim [9] studied
the influence of a magnetic field upon the
unsteady convective flow past a semi-infinite
vertical porous moving plate with variable
suction or heat absorption. They presented
that the existence of magnetic field decreased
the velocity and Nusselt number. Both of
velocity and Nusselt number decreased as
increasing the heat absorption coefficient.
Isreal-Cookey et al. [10] investigated
the influence of viscous dissipation and
radiation on the problem of unsteady MHD
free-convection flow past an infinite heated
vertical plate in a porous medium with
time-dependent suction.
The problem of the vortex mode
of instability in natural convection flow
over a heated plate has received much
attention in the heat transfer literature.
The instability mechanism is due to the
88 修平㈻報 第㈩㆕期 民國㈨㈩㈥年㆔㈪
presence of buoyancy force component in
the direction normal to the plate surface.
The buoyancy force gives rise to vortex
instability under critical conditions. It is
very important to predict the onset of vortex
instability in convective flow because of
their industrial applications such as chemical
vapor deposition and cooling of electronic
packages. Cheng and Chang [11], Hsu et
al [12] and Hsu and Cheng [13] analyzed
the vortex mode of instability of horizontal
and inclined natural convection flows in a
saturated porous medium. Jang and Chang
[14] reexamined the same problem for an
inclined plate, where both the streamwise
and normal components of the buoyancy
force are retained in the momentum
equations. Therefore, ref. [14] provided
new vortex instability results for small
angles of inclination from the horizontal (030�� ) than the previous study [12]. Later,
Jang and Chang [15, 16] investigated the
non-Darcy effects and combined heat and
mass buoyancy effects on vortex instability
of a horizontal natural convection flows in
a saturated porous medium. Jang and Lie
[17] studied the non-Darcy effects on vortex
instability of a horizontal natural convection
flows with mass flux surface. Hassanien et
al [18] follows the investigation of Jang and
Chang [15] studied the effect of surface mass
flux on vortex instability of non-Darcian
natural convection flow.
In conclusion to the above review,
the literature concerned the magnetic field
effect on the natural convection flow and
vortex instability in a saturated medium
has been mostly limited to the vertical and
inclined plates. The MHD free convection
on horizontal plates has received relative
less attention. Furthermore, the extended
study to the vortex instability of natural
convection flow with magnetic effect has
never been investigated. It is motivated the
present study. The purpose of this paper
is to examine the magnetic effect on the
vortex instability of a horizontal Darcian
natural convection boundary layer flow in a
saturated porous medium with surface mass
flux. This is accomplished by considering
Darcy equation of motion. In the main
flow, the boundary layer approximations
are invoked. The stability analysis is based
on the linear stability theory incorporated
with non-parallel flow model. The resulting
eigenvalue problem is solved using a
variable step-size sixth-order Runge-Kutta
integration scheme in conjunction with the
Gram-Schmidt orthogonalization procedure
to maintain the linear independence of
89Vortex instability of MHD natural convection flow over an isothermal horizontal plate in a saturated porous medium:Chung-Ting Hsu
the eigenfunction. A parametric analysis
to the flow, heat transfer characteristic,
and stability to vortex mode with respect
to system parameters such as M ( M=
0,0.5,1,1.5,2,4 ), fw ( fw = -1,0,1 ) will be
discussed in detail.
MaTHEMATICAL ANALYSIS
Before proceeding to the instability
problem, consideration is given first to
the basic natural convection flow along a
horizontal surface, since the computation
of instability criteria requires knowledge
of the velocity and temperature profiles for
the main flow and the solution has not been
investigated before.
Figure 1 The physical model and
system coordinate
1.The base flow
We cons ider the s teady na tura l
convection flow of an electrically conducting
fluid on a heated semi-infinite, horizontal
plate(Tw), embedded in a porous medium(T∞),
as shown in Fig.1, where x represents the
distance along the plate from its leading
edge, and y is the distance normal to the
surface. The wall temperature is assumed
to be a constant Tw (Tw>T∞). The uniform
magnetic field (B0) is applied in the y
direction, normal to the surface which is
electrically non-conducting( 0y BBB ���
). The electrical field is assumed to be zero (
0E ��
). Then, in the equation of motion,
the extra body force (also called Lorentz
force) becomes B)BV(BJ�����
����� . It is
assumed that the magnetic Reynolds number
Rem=µ0σV L<<1, where µ0 and σ are
the magnetic permeability and electrical
conductivity, respectively. The V and L
are the characteristic velocity and length,
respectively. Under these conditions, it is
possible to neglect the induced magnetic
field as compare to the applied magnetic
field. The Lorentz force can be simplified
as uBFx20��� .Darcy’s model is used for
the momentum equation and the Boussinesq
approximation is applied. The governing
equations are given by
0u vx y
� �� �� �
(1)
90 修平㈻報 第㈩㆕期 民國㈨㈩㈥年㆔㈪
Where K is the permeability of the
porous medium;β the coefficient for thermal
expansion;αe represents the equivalent
thermal diffusivity.
By applying the boundary layer
assumptions, equations(1)-(4) become:
where �� /20BKM � is the dimensionless
magnetic parameter expressing the relative
importance of the MHD effects.
The boundary conditions are defined as
follow:
O n i n t r o d u c i n g t h e f o l l o w i n g
transformations:
uBK
xPK
u��
���
���
20
� ���
���
�����
��
��� �� TTg
yPK
v
2
2e
T T Tu v
x y y�� � �� �
� � �
(2)
(3)
(4)
ww TTxavvy ���� � ,;0 1
����� TTuy ,0; (7)
� �13, x
yx y R a
x� �
31
)(
xeRaf
�
�� �
� ��
��
��
TxT
TT
w )(��
(8)
Where Ψ is the stream function,
ewx xTTKgRa ���� /)( �� �� i s t h e
modified local Rayleigh number.
E q u a t i o n ( 5 ) a n d ( 6 ) c a n b e
nondimensionlized as follows:
T h e b o u n d a r y c o n d i t i o n s a r e
transformed as follows
Where
is the surface mass flux. It is suction for
fw>0 , blowing for fw<0 and fw=0 for
impermeable surface. It is noted that for
M=0 and fw=0 corresponds the case (m=0)for
the base flow without MHD effect , which
was investigated by Hsu et al[12].
In terms of the dimensionless variables, it
can be shown that the local Nusselt number
is given by
2.The disturbance flow
In the usual manner for stability
�������32
)1( fM
031 ����� �� f
(9)
(10)
0,0;
1,;0
��������
����
f
ff w (11)
3113
/
w
e
ew )TT(KAg
af ��
�
����
����
�
����
���
����
�
(�=-2/3) (12)
)0(/ 3/1 � ���xx RaNu (13)
(5)
(6)
� �xTKg
yu
M��
���
����� �1
2
2e
T T Tu v
x y y�� � �� �
� � �
91Vortex instability of MHD natural convection flow over an isothermal horizontal plate in a saturated porous medium:Chung-Ting Hsu
analys is , the veloci ty , pressure and
temperature are assumed to be the sum of
a mean and fluctuating component, here
designed as barred and subscript 1 quantities,
respectively. The perturbed flow can be
represented as
After substituting equations (16)
into the governing equations for the three-
dimensional convective flow in a porous
medium, subtracting the parts satisfied
by the base quantities, and linearizing the
disturbance quantities, we arrive at the
disturbances
),,,(),(),,,(
),,,(),(),,,(
),,,(),,,(
),,,(),(),,,(
),,,(),(),,,(
1
1
1
1
1
tzyxTyxTtzyxT
tzyxPyxPtzyxP
tzyxwtzyxw
tzyxvyxvtzyxv
tzyxuyxutzyxu
����
�����
(14)
0111 ����
���
��
zw
yv
xu
xPK
u)M(����� 1
11�
���
����
��
���� � 1
11 Tg
y
PKv ��
�
zPK
w���� 1
1 �
���
����
�
���
���
���
����
���
���
���
��
21
2
21
2
21
2
11111
z
T
y
T
x
T
yyT
vxT
uy
Tv
x
Tu
t
T
e�
�
0111 ����
���
��
zw
yv
xu
xPK
u)M(����� 1
11�
���
����
��
���� � 1
11 Tg
y
PKv ��
�
zPK
w���� 1
1 �
���
����
�
���
���
���
����
���
���
���
��
21
2
21
2
21
2
11111
z
T
y
T
x
T
yyT
vxT
uy
Tv
x
Tu
t
T
e�
�
(15)
(16)
(17)
(18)
(19)
Where ζ is the capacity ratio of porous
medium.
Following the method of order-of-
magnitude analysis prescribed in detail
by Hsu and Cheng[12], the terms xu �� /1
21
2 / xT ��,
xu �� /1
21
2 / xT �� in equation (15) and (19) can
be neglected. The omission of xu �� /1
21
2 / xT ��
in equation (15) implies the existence of
disturbance stream function 1� such that
At neutral stability, the vortex mode of
the three dimensional disturbances are of the
form
where a is the spanwise periodic wave
number which is real.
Substituting equation (20) and (21) into
equation (15)-(19) yields
yw
��� 1
1�
, z
v�
��� 11
�(20)
� � � � � � � �� � � �iazyxTyxuyxTu exp,~
,,~,,~,, 111 �� ��(21)
� �yx
~u~iaM
����� �2
1
T~Kgia~a
y
~
���� ����
�� 2
2
2
yT~ia
xT
u~yT~
v
xT~
uT~
ay
T~
e
���
���
��
�����
��
����
��
��
�
� 22
2
(22)
(23)
(24)
92 修平㈻報 第㈩㆕期 民國㈨㈩㈥年㆔㈪
Equations (22)-(24) are solved based on the
local similarity approximation [12]. Letting
We obtain the fol lowing system
of equations for the local s imilari ty
approximations:
With the boundary conditions
I t i s no ted tha t fo r M=0 and fw=0
co r re sponds the ca se (m=0) fo r t he
disturbance flow without MHD effect, which
was investigated by Hsu et al[12].
Numerical Method of SolutionThe equations (9)-(11) for the base
3/1
~
xRa
axk �
� �3/1
~
xe RaiF
��� �
� �T
T�
��~
� (25)
������ 312 /xRak
~Fk
~F
FRak~
)FF(
Rak~
)M(k~
B
/x
/x
31
312
1
32
31
13
2
��
��
�������
��
������� ��
(26)
(27)
0)()()0()0( �������� FF
Where
fB21
1 ��
(28)
(29)
flow constitute a system of linear ordinary
differential equations and are solved by
the sixth-order Runge-Kutta, variable step
size integration routine. The results are
stored for a fixed step size, Δη=0.02,
which is small enough to predict accurate
linear interpolation between mesh points.
In the stability calculations, the disturbance
equations (26)-(28) are solved by separately
integrating two linearly independent
integrals. The full equations may be written
as the sum of two linearly independent
solutions
Two independent integrals (Fi, i� ), with
i=1, 2 may be chosen so that their asymptotic
solutions are
Where
The calculating procedure for disturbance
equations (26)-(28) are then solved as
21 EFFF ��
21 ����� E (30)
� ��� ��exp1 NF , � ���� ��exp1
� ���� �exp2F , 02 �� (31)
)k~
/(Rak~
N /x
2231 ��� �
� � 2/~
42/122
11 ���
��� ��� kBB�
k~���
93Vortex instability of MHD natural convection flow over an isothermal horizontal plate in a saturated porous medium:Chung-Ting Hsu
follows. For specified M and k~
, Rax is
guessed. Using equations (31) as starting
values, the two integrals are integrated
routine separately from the outer edge of
the boundary layer to the wall using the
sixth-order Runge-Kutta variable size
integrating routine incorporated with the
Gram-Schmidt orthogonalization procedure
[19] to maintain the linear independence of
the eigenfunctions. The required input of
the base flow to the disturbance equations
is calculated, as necessary, by linear
interpolation of the stored base flow. From
the values of the integrals at the wall, E is
determined using the boundary conditions
i�2=0. The second boundary condition
F(0)=0 is satisfied only for appropriate
values of the eigenvalue Rax. A Taylor series
expansion of the second condition provides
a correction scheme for the initial guess
of Rax. Iterations continue until the second
boundary conditions is sufficiently close to
zero (typically<10-6)
Results and DiscussionThe formulation of the effects of
magnetic field and blowing/suction on the
flow and vortex instability of a horizontal
Darcian free convection in saturated
porous medium has been carried out in the
0 1 2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
1.2
�
f '
M = 0, 1, 2, 4
fw = 0
Fig. 2 Tangential velocity profiles at fw = 0.
0 1 2 3 4 5 6 7 8 90
0.2
0.4
0.6
0.8
1
�
�
M = 0, 1, 2, 4
fw = 0
Fig.3 The effect of M on the temperature profiles
preceding sections. Numerical results for the
velocity and temperature profiles, the critical
Rayleigh number and wave number at the
onset of vortex instability are presented for
a range of magnetic parameter M (M=0~4)
and suction parameter (fw = -1, 0, 1).
94 修平㈻報 第㈩㆕期 民國㈨㈩㈥年㆔㈪
Figures 2-3 depict the magnetic effect
on the tangential velocity and temperature
profiles, respectively, for fw=0. It is seen
that the magnetic effect markedly affects the
velocity and temperature fields. The velocity
profiles decrease and the thermal boundary
layer thickness increase as increase of M .
The tangential velocity f'(0) decreases about
67% at M=4 relative to the results without
magnetic field (M=0) for fw=0.
Figure 4 shows the magnetic effect on
the heat transfer rate for selected values of
fw. It is shown that the heat transfer rates
decrease with increasing value of M. And
there are larger heat transfer rates for suction
surface (fw>0)).
Fig.4 Heat transfer rate as a function of M
The simultaneous effects of magnetic
field and blowing/suction condition on
the vortex instability of horizontal free
convection flow are reported graphically
in Figures 5-8. Figure 5 shows the neutral
stability curves, in terms of the flow vigour
parameter Ra and the dimensionless wave
number k~
for selected values of M (M=0,
1, 2) at fw=0. The curve of M=0, which
represents the case without magnetic field, is
agreed quantitatively with the earlier results
of Hsu et al. [12]. It is also shown that as the
magnetic parameter M increases, the neutral
stability curves shift to lower Rayleigh
number and lower wave number, indicating
a destabilization of the flow to vortex
instability.
Fig.5 Neutral stability curves for selected values
of M
Figure 6 shows the neutral stability
curves, in terms of the flow vigour parameter
0 0.5 1 1.5 20.1
0.2
0.3
0.4
0.5
0.6
0.7
M
-�'(0)fw = 1
fw = 0
fw = -1
0 0.2 0.4 0.6 0.8 1 1.210
100
Rax
k~
M = 0
M = 1
M = 2
fw = 0Hsu et al.[12 ]
95Vortex instability of MHD natural convection flow over an isothermal horizontal plate in a saturated porous medium:Chung-Ting Hsu
Ra and the dimensionless wave number for
selected values of fw (fw=-1, 0, 1). It is seen
that the blowing surface (fw=-1) shifts the
neutral stability curves to lower Rayleigh
number and lower wave number. It indicates
the blowing flow motion destabilizes the
flow to vortex instability. As opposite to
blowing flow, the suction flow motion
stabilizes the flow to vortex instability.
Fig.6 Neutral stability curves for selected values
of fw
The corresponding critical Rayleigh
number Ra* and critical wave number k*,
which marks the onset of longitudinal
vortices, as functions of magnetic parameter
M are plotted in Figure 7 and 8. It is clearly
indicated that the critical Rayleigh number
and wave number decrease as M increases
and fw decreases. For fw=-1, the critical
Rayleigh number of M=2 is reduced by
about 66% relative to the result of M=0,
while for fw=1, the critical Rayleigh number
of M=2 is reduced by about 53%.
Fig.7 The critical Rayleigh number as a function
of M
Fig.8 The critical wave number as a function of M
Figures 9(a)-(c) show the streamlines
(Ψ*) ( solid lines ) and isotherms ( i�*)
0 0.2 0.4 0.6 0.8 1 1.210
100
Rax
k~
M = 1
fw = -1, 0, 1
0 0.5 1 1.5 215
20
25
30
35
40
45
M
Ra*
fw = 1
fw = 0
fw = -1
0 0.4 0.8 1.2 1.6 20.45
0.5
0.55
0.6
0.65
0.7
0.75
M
k*
fw = 1
0-1
96 修平㈻報 第㈩㆕期 民國㈨㈩㈥年㆔㈪
(a)M=0
(b)M=1
(c)M=2
Fig.9 The streamlines (solid lines)and isotherms.
(dashed lines) for the secondary flow at
the onset of instability for M=0 (without
magnetic field) and M=1,2 (with magnetic
field), respectively. It is shown that the value
of η, at which Ψ*max and i�*
max occur, for
M=2 is larger than that of M=0 and M=1.
We obtained that magnetic effect enlarges
the region of vortex and destabilizes the
flow.
Conclusion The coupled effects of magnetic
field and blowing/suction on the vortex
instability of horizontal free convection
flow in a saturated porous medium have
been examined by a linear stability theory.
The numerical results demonstrate that
the magnetic field produces a significant
retarding force on the flow and thermal
fields. Thus, it reduces the velocity and heat
transfer rate. For the disturbance flow, the
presence of magnetic field destabilizes the
flow to vortex instability. As the magnetic
parameter M increases, the critical Rayleigh
number and the associated wave number are
decreased. The numerical results also show
that suction surface stabilizes the flow and
blowing surface destabilizes the flow. For
selected values of blowing parameter fw,
the critical Rayleigh numbers of M=2 are
0
1
2
3
4
5
6
7
8
9
�
�� 0az �
2.0*��2.0*��
4.04.0
8.0 8.0
0
1
2
3
4
5
6
7
8
9
�
�� 0az �
2.0*��2.0*��
4.04.0
8.0 8.0
0
1
2
3
4
5
6
7
8
9
�
�� 0az �
2.0*��2.0*��
4.04.0
8.0 8.0
97Vortex instability of MHD natural convection flow over an isothermal horizontal plate in a saturated porous medium:Chung-Ting Hsu
reduced by about 53%-66% relative to the
results of M=0.
Reference1.E. M. Sparrow and R. D. Cess, Effect of
magnetic field on free convection heat
transfer, Int. J. Heat and Mass Transfer 3
(1961) 267-274.
2.N Riley, Magnetohydrodynamic free
convection, J. of Fluid Mechanics 18
(1964) 577-586.
3.T. Watanabe and I. Pop, Magnetohy-
drodynamic free convection flow over
a wedge in the presence of a transverse
magnetic field, Int. Comm. Heat Mass
Transfer 20 (1993) 871-881.
4.I. Pop and T. Watanabe, Hall effects on
magnetohydrdynamic free convection
about a semi-infinite vertical flat plate Int. I.
Engng. Sci. 32 (1994) 1903-1911.
5.M. Kumari, 1998, MHD flow over a wedge
with large blowing rates, Int. J. Eng. Sci.
36 (1998) 299-314.
6.V. M. Soundalgekar, Free convection
effects on the oscillatory flow past an
infinite, vertical, porous plate with constant
suction, Proc. R. Soc. London A 333 (1973)
25-36.
7.A. J. Chamkha, Non-Darcy Hydromagnetic
free convection from a cone and a wedge
in porous media, Int. Comm. Heat Mas
Transfer 23 (1996) 875-887.
8 . A l i J . C h a m k h a , U n s t e a d y M H D
convect ive heat and mass t ransfer
past a semi-infinite vertical permeable
moving plate with heat absorption, Int. J.
Engineering Science 42 (2004) 17-230.
9.Y. J. Kim, Unsteady MHD convective heat
transfer past a semi-infinite vertical porous
moving plate with variable suction, Int. J.
Engineering Science 38 (2000) 833-845.
10.C. Isreal-Cookey, A. Ogulu, V.B. Omubo-
Pepple, Influence of viscous dissipation
and radiation on the problem of unsteady
MHD free-convection flow past an infinite
heated vertical plate in a porous medium
with time-dependent suction., Int. J.
of Heat and Mass Transfer 46 (2003)
2305-2311.
11.P. Cheng and I. D. Chang, Buoyancy-
induced flows in a saturated porous
medium adjacent to impermeable
horizontal surfaces, Int. J. Heat and Mass
Transfer 19 (1976) 1267-1272.
12.C. T. Hsu , P. Cheng and G. M. Homsy,
Vortex instability in buoyancy induced
flow over horizontal heated surfaces
in porous media, Int. J. Heat and Mass
Transfer 21 (1978) 1221-1228.
13.C. T. Hsu and P. Cheng, Vortex instability
98 修平㈻報 第㈩㆕期 民國㈨㈩㈥年㆔㈪
in buoyancy-induced flow over inclined
heated surfaces in porous media, ASME
J. Heat Transfer 101 (1979) 660-665.
14.J.Y.Jang and W.J.Chang, Vortex instability
in buoyancy inclined boundary layer flow
in a saturated porous medium, Int. J. Heat
and Mass Transfer 31 (1988) 759-767.
15.J.Y.Jang and W.J.Chang, Inertia effects
on vortex instability of a horizontal
natural convection flow in a saturated
porous medium, Int. J. Heat and Mass
Transfer 32 (1989) 541-550.
16.J. Y. Jang and W. J. Chang, The flow and
vortex instability of horizontal natural
convection in a porous medium resulting
from combined heat and mass buoyancy
effects, Int. J. Heat and Mass Transfer 31
(1988) 769-777.
17.J. Y. Jang and K. N. Lie, Vortex
instability of free convection with surface
mass flux over a horizontal surface,
AIAA Journal of Thermophysics and
Heat Transfer 7 (1993) 749-751.
18.I .A. Hassanien, A.A. Salama and
N.M. Moursy, Inertia effect on vortex
i n s t ab i l i t y o f ho r i zon t a l na tu ra l
convection flow in a saturated porous
medium with surface mass flux. Int.
Comm. Heat Mass Transfer 31 (2004)
741-750.
19.A. R. Wazzan, T. T. Okamura and H. M.
O. Smith, Stability of laminar boundary
layer at separation, Physics Fluids 190
(1967) 2540-2545.