vortex instability of mhd natural convection flow over an

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


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��

� � �


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~��


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 ]


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


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.

vortex instability of mhd natural convection flow over an

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