Instability of Rayleigh â€“Benard -Marangoni convection in a ... - IJAMM

BENARD-MARANGONI CONVECTION IN A POROUS LAYER 

PERMEATED BY A NON-LINEAR MAGNETIC FLUID 

ABSTRACT 

Int. J. of Appl. Math and Mech. 8 (12): 13-41, 2012. 

A. A. Abdullah 1 and Z. Z.Rashed 2 

1 Department of Mathematics, Faculty of Science, Umm Al-Qura University, 

Makkah, Saudi Arabia 

2 Department of Mathematics, Faculty of Science, Northern Border University, 

Arar, Saudi Arabia 

Email: aamohammad@uqu.edu.sa 

Received 12 June 2011; accepted 27 September 2011 

The onset of Benard-Marangoni convection in a horizontal porous layer permeated by a 

magnetohydrodynamic fluid with a non-linear magnetic permeability has been examined. The 

porous layer is assumed to be governed by the Brinkman model, bounded from below by a 

rigid surface and from above by a non deformable free surface. The critical effective 

Marangoni number and the critical Rayleigh number are obtained for different values of the 

effective Darcy number, Biot number, Chandrasekhar number and nonlinear magnetic 

parameter for the cases of stationary convection and overstability. The related eigenvalue 

problem is solved using the first order Chebyshev polynomial method. 

Keywords: Marangoni convection, Porous layer, Brinkman model, Thermal instability, 

Magnetic field 

1 INTRODUCTION 

Thermal instability in a horizontal layer of a fluid heated from below has attracted 

considerable attention due to its relevance to a variety of applications such as geophysics, 

oceanography and atmospheric sciences as well as engineering. Systematic investigation of 

this topic began in the early last century with the experiments of (Benard1900;1901). 

(Rayleigh 1916) provided a theoretical explanation of Benard's experimental results by 

considering instability due to the action of buoyancy forces and he showed that the numerical 

value of the non-dimensional Rayleigh number, R, decides the stability or otherwise of a layer 

of fluid heated from below. Rayleigh-Benard convection driven by buoyancy has been a 

subject of considerable interest since the pioneering work of (Rayleigh 1916). Many of these 

studies are mentioned in (Chandrasekhar 1981). Numerous other papers exist on this topic; 

see for example (Abdullah and Lindsay 1990; 1991; Lombardo and Mulone 2005; Guray and 

Tarman 2007, Banjar and Abdullah 2011, Dubey et al. 2011). 

Apart from the buoyancy forces, convective instability can also occur due to temperature 

dependent surface tension forces (known as Marangoni convection) which was

14 


A. A. Abdullah and Z. Z. Rashed 

firstly studied by (Pearson 1958) who assumed infinitesimally small amplitude analysis with 

non-deformable free surface at the top and no slip boundary at the bottom and showed that the 

variations in the surface tension due to temperature gradients could induce motion within the 

fluid when the numerical value of the non-dimensional Marangoni number, M, exceeds a 

critical value. Marangoni instability has been studied widely for the cases of deformable and 

non-deformable free surface with various factors such as variable viscosity, magnetic field, 

rotation etc.(Takashima and Namikawa 1971; Takashima 1981a; 1981b; Friedrich and 

Rudraiah 1984; Hashim and Wilson 1999; Hashim and Arifin 2003; Awang and Hashim 

2008)). 

The theories of (Rayleigh 1916; Pearson 1958) were discussed by (Nield 1964) who 

combined both buoyancy and surface tension mechanisms into a single analysis (known as 

Benard-Marangoni) and showed that the two agencies causing instability reinforce one 

another and are tightly coupled. Some of the major features of the Benard-Marangoni 

convection problem are elucidated by ( Nield 1966; Garcia-Ybarra et al. 1987; Maekawa and 

Tanasawa 1989; Perez-Garcia and Carneiro 1991; Wilson 1993; Char and Chiang 1994; Jou 

et al. 1997; Chang and Chiang 1998; Douiebe et al. 2001) and others. 

Rayleigh-Benard convection in porous layers has been studied extensively since the 

pioneering work of (Horton & Rogers 1945; Lapwood 1948). The copious literature covering 

different developments in this field are well documented in (Kaviany 1995; Ingham 1998; 

Vafai 2005; Nield and Bejan 2006). The study of Marangoni and Benard-Marangoni 

convections in porous media has drawn little attention compared to the study of Rayleigh- 

Bénard convection in porous media in spite of their importance in material science 

processing, solidification of alloy, etc. 

Marangoni convection in a porous media has been studied using the Brinkman model by 

(Hennenberg et al. 1997; Rudraiah and Prasad 1998; Fadzillah et al. 2008). The onset of 

coupled Darcy-Benard-Marangoni convection in a liquid saturated porous layer has been 

investigated by (Shivakumara et al. 2009) by employing the Brinkman-Forchheimer- 

Lapwood-extended Darcy flow model. 

In the literature, so far no research have been conducting regarding the effect of magnetic 

field on Marangoni or Benard-Marangoni convections in a porous layer. The presence of 

magnetic field in an electrically conducting fluid usually has the effect of inhibiting the 

development of instabilities. (Thompson 1951; Chandrasekhar 1981) and others have 

examined Benard convection in the context of magnetohydrodynamic fluid with a linear 

constitutive relationship between the magnetic field H and the magnetic induction B. 

However a non-linear constitutive relationship between H and B may be appropriate for 

certain classes of materials. The relevance of this criterion to the configuration of a neutron 

star is discussed by (Roberts 1981; Muzikar and Pethick 1981).(Cowley and Rosensweig 

1967; Gailitis 1977) and others use non-linear magnetization laws to describe the properties 

of ferrofluids. (Abdullah and Lindsay 1990; 1991; Abdullah 2000; Jan and Abdullah 2000; 

Al-aidrous and Abdullah 2005) have examined Benard convection problems using the 

nonlinear relationship suggested by (Roberts 1981). 

The aim of this work is to study the effect of magnetic field on the stability of Benard- 

Marangoni convection in a horizontal porous layer using a linear and non-linear relationship 

between H and B. The non-linear relationship used by (Abdullah and Lindsay 1990;1991) has

Benard-Marangoni Convection In A Porous Layer Permeated By A Non-Linear Magnetic Fluid 

been adopted. The flow in the porous medium is controlled by the Brinkman model and the 

porous layer is bounded from below by a rigid surface and from above by a non deformable 

free surface. The first order Chebyshev polynomial method is used to obtain the numerical 

solutions of the corresponding eigenvalue problem (Bukhari 1997). 

2 MATHEMATICAL FORMULATION 

Consider a horizontal porous layer permeated by an incompressible electrically conducting 

magnetohydrodynamic viscous fluid. The porous layer is governed by Brinkman model and is 

subjected to a constant gravitational acceleration in the negative x 3 direction, and to a 

constant magnetic field H in the positive x 3 direction. The lower boundary, x3 � 0 , is 

assumed to be rigid while the upper boundary, x3 � d , is assumed to be free and subjected to 

temperature–dependant surface tension forces. The temperatures at x3 � 0 and x3 � d are 

T � and T� �Tu 

respectively. (see Figure.1) 


Figure 1: Schematic diagram of the problem 

The Boussinesq approximation is applied to the layer such that 

�1 � �T �T 

�� 

, 

� (1) 

� �0 

� 

� 

where � is the density of the fluid, � � is the density of the fluid at T � , T is the fluid 

temperature, and � is the coefficient of volume expansion. The relation between B i and i H 

is assumed to be non-linear(see Abdullah & Lindsay 1990) such that 

Hi 

� 

� 

�� (2) 

� 

�Bi 

where � �� , B� 

� � � is the internal energy function. In fact 

�� 

�B 

�� 

B 

H 

i 

i � �� 

� �� 

Bi 

(3) 

�B 

�B 

�B 

B 

i 

15

16 



� 1 

where � 

�B 

B 

� 

� is the magnetic susceptibility. We shall assume that the magnetic induction, 

B i , is constant of the form Bi � �0, 0, 

B�. 

Thus the governing field equations are 

�� 

�Vi 

� 

� �P, 

i � V 

� �t 

K 

V 

i, 

i 

� 0, 

�� B � 

�T 

�� c� 

� �� c � 

B 

� 

i, 

i 

�B 

i 

�t 

m 

�t 

� 0, 

� V 

i, 

j 

� 

B 

j 

� 

� 

�V 

j 

i , k 

p 

B 

i 

f 

i, 

j 

� � 

B 

i 

k 

� 

� 

� 

, 

� 

� � e 

� � 

2 

ijk 

V 

e 

i 

� � 

2 

V ��T 

� k� 

T , 

krs 

� 

�1 �� 

�T �T 

�� 

�B�� , 

s, 

rj 

where V is the velocity, P is the hydrostatic pressure, k is the thermal conductivity, � is the 

i 

dynamic viscosity, � is the porosity, �eff � � � is the effective viscosity, K is the 

permeability, p 

c is the specific heat at constant pressure, � � c p � f 

� 

g 

i 

� and � � c�m 

(4) 

� are the heat 

capacity per unit volume of the fluid and porous medium respectively and �� is the electrical 

conductivity. The fluid is confined between the planes 3 0 � x and x3 � d and on these 

planes we need to specify the following boundary conditions 

� �� 

At x3 � d : V3 � 0, 

k� T �n� 

qT � 0, 

B3, 3 � 0, 

2�� 

Dnt 

� �T 

�t, 

(5) 

� �T 

where q is the heat transfer coefficient, � is the kinematic viscosity, n , t denote the normal 

and tangential unit vectors at the upper free surface respectively, and � is the surface tension 

which assumed to be a linear function of temperature such that � � � � �� 

�T � T� 

�, where 

� � being the surface tension at temperature T � and � is the change of surface tension with 

temperature. 

At x3 � 0 : V3 � 0, 

V3, 

3 � 0, 

T � T� 

, B3 

� 0. 

(6) 

Equations (4) have a steady solution in which 

�B�, B �0, 0, 

B� 

B is constant 

V � 0, ��P 

� � g � 0, 

T � T� 

� � x3 

, � � � � , 

where � is the adverse temperature gradient. We shall suppose that the steady solution be 

perturbed by the following linear perturbation quantities 

V � 0 �V, 

P � P 

�x �� p, 

B � �0, 0, 

B�� 

�b , b , b � 

� � � � b � , T � T � � x �� 

. 

3 

, B 

3 

� 

3 

1 

2 

3 

,


Applying these perturbations, equations (4) become 

1 �Vi 

� p � � � 2 

� �� Vi 

Vi 

g i3 

t � 

� 

� 

� � � � � �� 

� � � �� 

� K � 

� � B�b 

i, 

3 � � b3, 

3 �i3 

�, 

Vi 

, i � 0, 

�� 

2 

Gm 

�V3 

� � �m� 

� , 

�t 

bi 

, i � 0, 

�bi 

� BVi 

, 3 � 

�t 

where 

Gm 

� 

2 2 

�� b � � �� b � � b ��. 

i 

�� c� 

� 

m 

��c p �f i3 

3, 

i3 


3 

� is the ratio of heat capacities, 

��c p � f 

(7) 

17 

k 

�m � is the thermal 

� � 

diffusivity, � � 

� � 

� 

B �, 

B 

is the electrical resistivity and � � is the non-linear magnetic 

� 

parameter. The boundary conditions (5) and(6) become 

�� 

V3 � 0, 

k � q� 

� 0, 

�x 

3 

�b 

�x 

3 � 

3 

2 

� � V � 

2 

0, 

� � 

� 3 � 

� �� 

�h� 

� 0 , (8) 

� � 2 

x � 

� � 3 � 

V3 � 0, 

V3, 

3 � 0, 

� � 0, 

b3 

� 0. 

(9) 

We now non-dimensionalize equations (7) using the dimensionless variables 

2 

d � 

� 

ˆ , , 

m ˆ μ 

x , 

m ˆ , ˆ , � ˆ 

i � d xi 

t � t Vi 

� Vi 

p � p bi 

� Bbi 

� T� 

�Tu 

� 

� d K 

m 

Thus equations (7) become 

Da ˆ 

eff �V 

i 

ˆ ˆ 

2 

� �� p �V 

ˆ ˆ 

i � Daeff 

� Vi 

� R� 

�i3 

Pr 

�t 

� Q Pm 

i, 

3 

Vˆ 

i, 

i � 0, 

� ˆ � ˆ 2 

G 

ˆ 

m � H �V3 

� � � , 

�t 

bˆ 

i, 

i � 0, 

�bˆ 

i 

�Vˆ 

i, 

3 � Pm 

�t 

�bˆ � � bˆ 

� � 

3, 

3 

i3 

2 � ˆ 2 

� b � ˆ ˆ 

i � � � b3�i3 

� b3, 

i3 

�� 

, 

, 

(10)

18 

where 



�� 

� � � gdKT� 

� Tu 

B K 

K 

Pr � , Pm 

� , R � 

, Q � � , Daeff 

� , 

� � 

� � 

� � � 

2 

m 

m 

m 

� 

d � 

�T � T � 

� � u 

H � � 

T� 

� Tu 

�1 

� � 

�� 

1 

Here P r Pm 

, R, 

Q , Daeff 

when heating from above, 

when heating 

from below. 

, are non-dimensional numbers denote respectively the viscous 

Prandtl number, magnetic Prandtl number, Rayleigh number of porous layer, Chandrasekhar 

number and effective Darcy number. The non-dimensionalization of boundary conditions (8) 

and (9) yield 

Vˆ 

3 � 0, 

� ˆ � 

� Bi ˆ � � 0, 

�xˆ 

3 

2 

� 

Vˆ 

2 

M ˆ 

3 � eff �h� 

� 0, 

xˆ 

2 

� 3 

bˆ 

3, 

3 � 0, 

Vˆ 3 � 0, 

Vˆ 

3 3 � 0, 

ˆ � 0, 

bˆ 

, � 3 � 0 . (12) 

where Bi and M eff are the Biot and effective Marangoni numbers which are given by 

Bi � 

qd 

k 

, 

M 

eff 

� �� d 

� 

� �� 

� 

m 

2 

. 

Now we look for a normal mode solution of equations (10) of the from 

� t i 

� � � � �a1 x1 

�a 

2 x 

t x �� 

x e e 

2 � 

� , 

i 

3 

� � , 1 2 a , a are the wave numbers of the harmonic disturbance and � is the 

growth rate. Thus equations (10) become 

where �w ,� , b� 

Da 

G 

m 

eff 

P 

r 

� 

� b � Dw � P 

2 2 

2 2 

2 2 

�D � a �w �� 

Q Db � ��D 

� a �w � Da �D � a � 

�� H � w 

m 

� a 

2 2 

� �D � a �� 

, 

2 2 2 

��D � a �� 

� a � b. 

2 

R� 

� Q D 

2 

w, 

eff 

2 

w 

2 

(11) 

(13)


� 2 2 

where D � , a � a1 

� a2 

is the wave number. The boundary conditions (11) and (12) 

�x3 

become 

w � 0 

D� 

� Bi� 

� 0 

D 

2 

w � a 

Db � 0, 

2 

M 

eff 

� � 0 

w � 0, 

Dw � 0, 

� � 0, 

b � 0 . (15) 

3RESULTS AND DISCUSION 

The governing equations (13) together with the boundary conditions (14) and (15) are solved 

when the fluid layer is heated from below (i.e. H� � �1 

). The first order Chebyshev spectral 

method is used to obtain the numerical results. Here we shall consider the following cases. 

3.1 The effect of buoyancy forces only 

In the following analysis we shall study the special case of buoyancy effect only (Rayleigh- 

Benard instability). In this case we shall suppose that M eff � 0 in condition (14)3(i.e. no 

surface tension). We may eliminate b and � from (13)1 using equations (13)2,3 to obtain 

� 

Da � 

L� 

� Daeff 

L �� 

P � 

� 

� 

r � 

� Q 

�L �� 

G � 

2 2 

2 

�L ��G 

� �L � � a �D w � L �L �� 

G �� L � � � a �w 

� 0 

m 


m 

� � 2 � 2� 

� 2 � 

� 

� L � � � a w R a L a w 

P � 

� � 

� 

� � � � 

m 

P � 

� 

� 

� � m � 

m 

� 

� 

� 

where we have assumed that the porous medium layer is heated from below ( i.e. H � � �1). 

This equation can be expanded to yield 

Gm 

Daeff 

� 

3 

2 � Gm 

Daeff 

Daeff 

Gm 

Daeff 

� 2 � 2 

Gm 

Daeff 

Gm 

� � 

� 

Lw � � �� 

�L 

w � a 

�Lw� 

Pr 

P 

� 

� � 

� 

m 

Pr 

Pr 

Pm 

p � � 

� 

� 

m 

pr 

p � 

�� 

� 

� � 

m � �� 

�� 

Daeff 

Daeff 

� 3 � 2 

Daeff 

2 

1 � 2 

� � �� 

Gm 

Daeff 

�L 

w � a 

a Daeff 

GmQ 

Gm 

� 

� 

� � 

L w 

Pr 

p � 

� 

� 

� � � � � � 

m 

Pr 

p � 

�� 

� 

� � 

m � 

� 

� 

� 2 

Ra 

� � 

2 2 

2 

4 

4 2 

�� a G Q � a G Q � � a G �Lw� � � � a G Q �w� 

� Da L w � �� a Da � Q � 1� 

m 

� 

Pm 

2 2 

2 2 2 4 

4 

�a Q � � a G Q � � a �L w � �Ra � � a Q�Lw 

� Ra � w � 0 

m 

m 

m 

� 

� 

pm 

m 

� 

� 

�� 

eff 

� 

� 

� 

eff 

3 

L w 

19 

(14) 

(16) 

(17)

20 



which is an eighth order ordinary differential equation to be satisfied by w . Suppose that both 

w � A sin 

s� 

x where A is a constant and s is an integer. Thus 

boundaries are free and that � � 

2 

4 

6 

w � D w � D w � D w � 0, 

on x3 

� 0, 

1. 

2 

2 

3 

Let L � D � a , then Lw � �� 

w , where � � s � � a , thus equation (17) becomes 

3 Gm 

Daeff 

2 �Gm 

Daeff 

� 

� � � 

Pr 

Pm 

� Pr 

� � 

� � ��Daeff 

� 

�Gm 

� 

� � 

2 2 

� s � GmQ 

� � 

Pm 

� 

1 

Pr 

� 

� 

� 

� 

2 �� a 

� 

2 2 Daeff 

2 

�� a �� G �� a � 

2 

2 

Ra � � 

2 

Ra � 

2 

2 2 2 2 

�� a �� a � �Da 

� � � � Q� 

s � � � 0 . 

�pm 

�� 

Daeff 

� 

� � 

P � 

� 

m � 

Pm 

Now we shall consider two cases of instabilities. 

3.1.1 Stationary Convection 

m 

�� 

1 

Pr 

2 

eff 

2 

� Gm 

� 

� Gm 

� 

�� 

� 

� Pm 

� 

� �� 

To find the critical Rayleigh number for the onset of stationary convection we set � � 0 in 

equation (18) to obtain 

� 2 2 2 

R � �Daeff � � � � Q s � �. 

(19) 

2 

a 

Since this equation does not contain the non-linear parameter � then we deduce that the nonlinear 

magnetic permeability has no effect in the case of stationary convection. In fact in 

absence of magnetic field ( Q � 0 ) and � 0 , s � 1 we obtain 

2 2 �a � � 

2 

2 

� � 

R � � , 

2 2 

a a 

Da eff 

2 

and the critical Rayleigh number Rcrit 

by (Lapwood 1948). Clearly 

� 4� 

� 39. 

4784 ,which is the same result obtained 

dR 

dDaeff 

3 

6 � 2 

a � 

4 

� 

� 2 

�1 

� 

dR � � 

� � , � a 

� 1 � � 

2 2 

a 

� � 

2 

� � 

� 2 � 

� 

dQ a � � � 

which indicate that both the effective Darcy number and the magnetic field have a stabilizing 

effect on the system. 

(18)


3.1.2 Overstability Convection 

To obtain the critical Rayleigh number for the case of overstability we suppose that 

2 

a 

n1 

� , 

2 

� 

� 

� 

2 

� � � �1 � n1 

�, 

� i�1 

, 

2 

R 

� R1 

, 

2 

Q � Q1 

, 


� 

Daeff 

1 

� Da 

2 eff 

� 

, 

2 

s � 1, 

where � is complex and � 1 � 0 . Substitute into equation (18), we obtain 

� i� 

� 

�1 � n � 

� 

� i�1 

� 

� 

� 

��1 � n �� 

� n � 

�1 � n �Da �G 

� � ��1 � n �� 

�n 

� 

2 eff 1 

�1 � n � � G ��1 � n �� 

�n 

� 

�1 � n � 

� 

P 

3 

1 

m 

1 

G 

1 

1 

Da 

P P 

Da 

Da 

G 

P 

P 

Q 

2 �G 

�� 

1 � 

� 

1 

P 

� G 

Da 

� G 

� 

� � 

� P 

m 1 ��1 � n1 

�� 

�n1 

� 

�1 � n � 

n1R1 

p 

�1 � n � 

2 

��1 � n �� 

�n 

� Da �1 � n � � �1 � n � 

1 

m 

� 

r 

eff 1 

1 

eff 1 

m 

m 

m 

1 

1 

� 

� 

� 

� 

r 

eff 1 

� 

� 

� 

� 

� 

� 

m 

m 

eff 1 

m 

m 

P 

r 

eff 1 

1 � 

P � 

r � 

1 

1 

m 

m 

� 

� 

� 

� 

1 

1 

� 

1 

1 

1 

m 

1 

� 

� 

� 

� Q 

The real and imaginary parts of equation (20) are 

n 

1 

R 

1 

� Da 

1 

� Pm 

�1 � n1 

� � n1 

� 

�1 � n1 

� G G 

m m 

� 

�1 � n � � n � P 

1 

eff 1 

1 

3 2 

�1 � n � � �1 � n � � �1 � n � 

m 

1 

r 

1 

1 

Q 

1 

1 

1 

� 

n 

1 

R 

1 

�1 � n � 

1 

� 

� � 0, 

� 

2 � 

�� 

� 

Daeff 

1 � 1 

� 

� 

� Pr 

� 2 

� Gm 

� 

��1 

� n1 

� � 

� 

(21) 

Daeff 

1 

�1 � n1 

��, 

P 

21 

(20)

22 

n 

� 

� 

1 

R 

P 

1 

m 

2 �1 � n � �1 � n � � n � 

�1 � n � � n � 



�1 � n � � G Q �1 � n � � n � 

�1 � n � �1 � n ��G �1 � n � � n �� 

�� 

2 

1 

� 

Da 

1 

1 

eff 1 

m eff 1 

�1 � n � . 

1 

1 

1 

� 1 

� 

� Pm 

G 

P 

m 

Da 

r 

1 

P P 

m 

1 

1 

� 

� 

� 

m 

1 

� 

�Da 

� 

m 

1 

eff 1 

From equation (21) and (22) we obtain 

� 

p 

� 

2 

1 

2 

m 

� 

� 

� 

� 

� 

�1 � n � 

� 

� 

� 

�1 � n � � n � 

� 

� 

�G 

� 

1 

� 

� 

�G 

� 

�1 � � n � n ��1 � n � Da �G 

� � � G � . 

1 

1 

1 

Da 

eff 1 

1 

1 

m 

� 

1 

1 

P 

r 

eff 1 

� 

� 

� 

� 

� 

� � G 

� 

2 

From equation (23), � 1 � 0 provided 

P 

m 

and 

G 

m 

� 

� 

� 

�1 

� 

� 

1 

� n 

1 

� �� 1� 

n 

1 

�1 � n � � n ��1 � n � 

� 

m 

m 

1 

1 

m 

� 

� �� 

�Q1� 

�� 

� 

� 

� 

�� 

�� 

� 

� 

1 � 

P � 

r � 

1 

P 

m 

r 

1 

1 

P 

� 

� 

� 

� 

m 

�� 

�� 

�� 

� G 

m 

� � n 

� 

� � 

1G 

� 

m 

�1 � n � 

1 

� 

� 

� 

� 

(22) 

(23) 

, (24) 

� 

� 

� � � 

� 1 1 1 

1 

Q1 

�Daeff 

1�1 

� n1 

�� Gm 

� � 

� � Gm 

� . (25) 

� 1 � 

� � �� 

� 

� Pr 

� � 

� 

� � Gm 

� 

� 1 n1 

� n1Gm 

� Pm 

� 

In fact for overstability to be possible the conditions (24) and (25) must be satisfied. Note 

that when the relation between the magnetic field and the magnetic induction is linear ( i.e. 

� � 0 ) and using Darcy's model, the conditions (24) and (25) reduce to 

2 2 

P � � 

PmGm � 1 ,and 

m � � a 2 2 

Q1 � � � �� a �� 

Gm 

�, 

�1 � PmGm 

� 

Da 

where � � . These are the same conditions obtained by (Bhadauria and Sherani 2008). 

� P 

r


The governing equations (13) together with the boundary conditions (14) and (15) are 

solved numerically in absence of surface tension when the upper boundary is free and the 

lower boundary is rigid. We suppose first that the effect of magnetic field is ignored. For the 

stationary convection case the relation between the Biot number, Bi , and the critical Rayleigh 

number, R , is displayed in figure (2) for different values of the effective Darcy number, 

Da . As Bi increases R increases which indicates that the Biot number delays the onset of 

eff 

convection. Moreover as Da eff decreases R decreases. 

In the presence of magnetic field, the relation between Q and the critical R for both 

stationary convection and overstability cases is displayed in figures (3) when � 0. 

001. 

In 


Da eff 

both stationary and overstability cases as Q increases the critical R increases which indicates 

that the magnetic field has a stabilizing effect. For the overstability case we noticed that in the 

case of nonlinear magnetic permeability ( � � 0 ), the critical R increases as � increases 

which indicates that the nonlinearity has a stabilizing effect. However this nonlinearity has no 

effect in the case of stationary convection which is the preferred mechanism. Moreover as 

Da decreases R decreases for stationary convection and overstability cases and for the 

eff 

linear and non-linear cases. The numerical results related to figure (3) are listed in tables (1) 

and (2). 

The critical Rayleigh, R , is obtained when Da � � (fluid layer ) and M � 0 for 

different values of Q . The numerical results of this case are listed in table (3) which are in 

excellent agreement with those of (Biswal and Rao 1999). Note that the relation between Q f 

(in fluid layer) and Q ( in porous layer) is Q � Q f Daeff 

. 

3.1.3 Special Cases 

Here we shall discuss the same problem (the effect of buoyancy forces) using Darcy's model 

when 

(i) The boundaries are mixed. 

(ii) Both boundaries are free. 

( i ) Mixed boundaries 

In this case we shall assume that the upper boundary is free and the lower boundary is rigid. 

Figure (4) shows the relation between Q and critical R for the stationary convection and 

�6 overstability cases when Da � 10 , � � 0. 

3 . Clearly as Q increases R increases for both 

cases and for linear and non-linear cases. The numerical results related to this figure are listed 

in table (4). We note that in absence of magnetic field �Q � 0�the 

critical Rayleigh number 

R � 27. 

098 which is in excellent agreement with the same result mentioned in (Nield and 

Bejan 2006). 

eff 

23

24 

(ii) Both boundaries are free 



In this case we shall assume that both boundaries are free. The critical Rayleigh numbers are 

obtained for different values of Q for the stationary convection case. The numerical results 

are listed in table (5). These results are in excellent agreement with those of (Bhadauria & 

Sherani 2008). 

3.2 The effect of surface tension force only 

In this case the effect of surface tension is studied in absence of buoyancy forces. First we 

suppose that the effect of magnetic field is ignored. For the stationary convection case the 

relation between the Biot number, Bi, and the critical effective Marangoni number, M eff , is 

displayed in figure (5) for different values of the effective Darcy number, Da eff . Clearly as 

Bi increases M eff increases which indicates that the effect of the Biot number is to delay the 

onset of Marangoni convection. Moreover as Da eff decreases M eff increases. Similar 

relations are presented in figure (6) between Bi and the critical effective Darcy-Marangoni 

number 

M ( � Daeff 

M eff ) for different values of eff 

Deff 

�5 

Daeff 

� 10 then D � 2. 

0532602 and 4. 

2875 

M eff 

result obtained by (Hennenberg et al. 1997). 

Da . We note that if Bi � 0 and 

a � which is in excellent agreement with the 

In the presence of magnetic field, the relation between, Q , and the critical M eff for both 

stationary convection and overstability cases is displayed in figure (7) when � 0. 

001. 

In 

Da eff 

both stationary convection and overstability cases the magnetic field has a stabilizing effect. 

For the overstability case when � 0 

M increases as � increases which 

� the critical eff 

indicates that the nonlinear magnetic permeability has a stabilizing effect. However this 

nonlinearity has no effect in the case of stationary convection. In fact we noticed that as 

M increases for stationary convection and overstability cases and for 

Da eff decreases eff 

linear and non-linear cases. The numerical results related to this figure are listed in tables (6) 

and (7). Similar relations between M and Q are displayed in figure (8). 

Deff 

Figure (9) shows the effect of Bi on M eff for different values of Q for the stationary 

convection case. Clearly as Bi increases M eff increases. In table (8) the critical values of 

M eff are shown for different values of Da eff , Bi , � and Q when the rigid boundary is 

thermally insulated ( D � 0 

excellent agreement with those of (Shivakumara et al. 2009). 

� ). In absence of magnetic field � 0� 

Q � the results are in 

The critical Marangoni number, M , is obtained when Da � � (fluid layer) and 

R � 0 for different values of Q . As Q increases, M increases which indicates that the 

magnetic field has a stabilizing effect in this case also. The numerical results obtained 

coincide with those of (Biswal and Rao 1999). 

eff


3.3 The effect of both surface tension and buoyancy forces 

In this case the effect of both surface tension and buoyancy forces is studied in the presence 

of magnetic field. The relation between Q and the critical M eff for both stationary 

convection and overstability cases is displayed in figure (10) when � 0. 

01 for different 


Da eff 

values of R . In both stationary convection and overstability cases as Q increases the critical 

M eff increases for linear and non-linear cases which indicates that the magnetic field has a 

stabilizing effect. Moreover as � increases M eff increases which indicates that the nonlinear 

magnetic permeability has a stabilizing effect in the overstability case. We also note from the 

figure that as R increases M eff decreases. The numerical results related to this figure are 

listed in tables (9a) when � � 0 and table (9b) when � � 0. 

5, 

1. 

Similar relations between 

M D and Q are displayed in figure (11). 

eff 

Figures (12) and (13) show the relation between R and M eff for different values of � , 

when Q � 50, 

Da � 0. 

01and 

Bi � 0 . Clearly as the buoyancy forces increases, the surface 

tension effect decreases and vice versa for the stationary convection and over stability cases 

and for linear and non-linear cases. 

4 CONCLUSION 

The onset of Benard-Marangoni convection in a horizontal porous layer permeated by a 

magnetohydrodynamic fluid with a non-linear magnetic permeability has been examined. The 

porous layer is assumed to be governed by the Brinkman model, bounded from below by a 

rigid surface and from above by a non deformable free surface. This problem is examined 

under the effect of surface tension only, the effect of buoyancy effect only and the effect of 

both surface tension and buoyancy force. 

The critical effective Marangoni number and the critical Rayleigh number are obtained for 

different values of the effective Darcy number, Biot number and Chandrasekhar number. The 

non-linear magnetic permeability has no effect on the development of instabilities through the 

mechanism of stationary convection and from the viewpoint of terrestrial applications, this is 

frequently the preferred process and so we should not expect the non-linear magnetic 

permeability to manifest itself under terrestrial circumstances. However in non-terrestrial 

applications overstability is the preferred mechanism and in this situation, the presence of the 

non-linear magnetic permeability influences the onset of overstable convection. 

Numerical results are obtained for the critical effective Marangoni and Rayleigh numbers 

under the effects of magnetic field in the cases of stationary convection and overstability and 

some special cases are produced. It is shown that the results of Darcy model case can be 

recovered in the limit as Darcy number Daeff � 0 , while the results of fluid layer can be 

recovered in the limit as Darcy number Da � � . 

eff 

25

26 

R 

R 

150 

140 

130 

120 

110 

100 

90 

80 

70 

60 

50 

40 

30 

20 

10 



0 

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 

Figure 2: Variation of R with Bi for different values of Da eff 

1400 

1200 

1000 

800 

600 

400 

200 

0 

0 10 20 30 40 50 60 70 80 90 100 110 

Figure 3: Variation of R with Q when � 0. 

001. 

Q 

Bi 

Stationary Convection 

Da eff 

Daeff � 0. 

1 

Daeff � 0. 

01 

Daeff � 0. 

001 

� 

� 1 

� � 0. 5 

� � 0


Table 1: The critical values of the Rayleigh number R for different values of Q and Da eff 

for the cases of stationary convection and overstability. Here M eff � 0 , � � 0 , Bi � 0 , 

Gm � 1. 

Q 

0 

1 

5 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

stationary convection 

Daeff � 0. 

01 Daeff � 0. 

001 

R a R a 

35.364 2.304 28.466 2.342 

53.225 2.684 45.345 2.808 

115.015 3.505 103.569 3.817 

184.723 4.091 169.423 4.557 

315.033 4.83 292.937 5.512 

439.561 5.331 411.334 6.173 

560.877 5.716 526.936 6.687 

680.060 6.031 640.706 7.111 

797.690 6.298 753.158 7.475 

914.118 6.532 864.599 7.794 

1029.582 6.739 975.236 8.079 

1144.246 6.926 1085.21 8.338 

1258.234 7.096 1194.63 8.574 


overstability 

Daeff � 0. 

01 Daeff � 0. 

001 

R a R a 

- - - - 

- - - - 

105.016 3.711 88.819 4.153 

140.113 4.319 120.14 4.862 

195.204 4.963 171.52 5.684 

242.738 5.386 216.87 6.243 

286.314 5.709 259.06 6.681 

327.317 5.974 299.20 7.046 

366.466 6.200 337.87 7.363 

404.193 6.397 375.40 7.644 

440.783 6.573 412.02 7.899 

476.434 6.732 447.89 8.133 

511.294 6.878 483.12 8.350 

Table 2: The critical values of the Rayleigh number R for different values of Q , Da eff and 

� for the case of overstability. Here M eff � 0 , Gm � 1, 

0 

Q 

5 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

Daeff � 0. 

01 

R 

114.951 

160.569 

232.639 

295.701 

354.201 

409.760 

463.203 

515.028 

565.559 

615.022 

663.584 

� � 0. 

5 

a 

3.518 

4.196 

4.861 

5.285 

5.607 

5.87 

6.094 

6.290 

6.465 

6.624 

6.769 

Daeff � 0. 

001 

R 

98.482 

138.89 

205.89 

266.00 

322.59 

376.91 

429.58 

481.01 

531.43 

581.01 

629.88 

a 

3.972 

4.748 

5.581 

6.139 

6.575 

6.938 

7.254 

7.535 

7.790 

8.025 

8.242 

Bi � , � 1, 

P � 0. 

25 . 

Daeff � 0. 

01 

R 

- 

179.81 

270.35 

349.96 

424.29 

495.3 

563.93 

630.75 

696.13 

760.32 

823.51 

a 

- 

4.752 

4.752 

5.181 

5.501 

5.761 

5.982 

6.174 

6.346 

6.502 

6.644 

Pr m 

� � 1 

Daeff � 0. 

001 

R 

- 

156.811 

239.888 

314.964 

386.098 

454.726 

521.568 

587.043 

651.426 

714.905 

777.618 

a 

- 

4.612 

5.483 

6.042 

6.474 

6.833 

7.145 

7.422 

7.673 

7.904 

8.118 

27

28 



Table 3: The critical values of Rayleigh number R for different values of Q for the case of 

stationary convection. Here Bi � 0 , 

R 

1400 

1200 

1000 

800 

600 

400 

200 

0 

Q f 

0 

4 

10 � 

10 � 

10 

3 

2 

10 

6 

Daeff � 10 , Gm � 1, 

M � 0 (fluid layer). 

(Biswal and Rao 1999) 

R a 

668.998 2.086 

669.000 

669.020 

874.862 

2424.90 

2.086 

2.086 

2.288 

3.128 

Q 

0 

10 

10 

10 

2 

3 

7 

8 

10 

Present 

R a 

668.998 2.0856 

669.000 

669.020 

874.862 

2424.903 


2.0856 

2.0856 

2.2884 

3.1281 

� � 1 

� � 0. 

5 

� � 0 

0 10 20 30 40 50 60 70 80 90 100 110 

Q 

�6 Figure 4: Variation of R with Q when Da � 10 , � � 0. 

3 (Darcy model)


Table 4: The critical values of the Rayleigh number R for different values of Q and � for 

�6 

the cases of stationary convection and overstability. Here M � 0 , Da � 10 , Bi � 0 , 

� � 0. 3 (Darcy's Model). 


overstability 

Q � � 0 � � 0. 

5 � � 1 

R a R a R a R a 

0 27.098 2.326 - - - - - - 

1 43.651 2.810 - - - - - - 

5 100.903 3.857 86.541 4.204 95.837 4.014 - - 

10 165.800 4.630 117.759 4.950 136.112 4.834 153.493 4.697 

20 287.701 5.639 169.050 5.810 202.934 5.708 236.180 5.615 

30 404.671 6.343 214.577 6.393 263.166 6.291 311.134 6.204 

40 518.949 6.896 257.150 6.848 320.111 6.745 382.389 6.659 

50 631.464 7.356 297.838 7.228 374.975 7.123 451.340 7.036 

60 742.709 7.753 337.186 7.556 428.369 7.451 518.674 7.363 

70 852.982 8.103 375.520 7.848 480.649 7.741 584.790 7.653 

80 962.481 8.417 413.047 8.111 532.048 8.004 649.943 7.915 

90 1071.344 8.704 449.915 8.352 582.724 8.245 714.311 8.155 

100 1179.673 8.967 486.229 8.575 632.795 8.468 778.019 8.378 


29 

Gm � 1, 

Table 5: The critical values of the Rayleigh number R for different values of Q for the case 

�6 

of stationary convection. Here M � 0 , Da � 10 , � � 

model). 

Q 

0 

1 

5 

10 

25 

50 

100 

150 

200 

500 

(Bhadauria & Sherani 2008) 

R a 

39.4784 3.142 

57.5243 3.736 

117.4382 4.917 

183.9028 5.721 

367.12992 7.094 

654.1856 8.395 

1205.0762 9.959 

1742.7393 11.013 

2273.5117 11.829 

5396.3447 14.863 

Bi , Gm � 1, 

� � 0. 3, 

(Darcy's 

Present 

R a 

39.4784 3.1416 

57.5242 3.7360 

117.4382 4.9169 

183.9027 5.7213 

367.1298 7.0940 

654.1854 8.3954 

1205.0760 9.9593 

1742.7391 11.0127 

2273.5114 11.8290 

5396.3642 14.8631

30 

M eff 



Figure 5: Variation of M eff with Bi for different values of Da eff for the case of stationary 

convection. 

M 

Deff 

8000 

7000 

6000 

5000 

4000 

3000 

2000 

1000 

0 

-1 0 1 2 3 4 5 6 7 8 9 10 11 12 

60 

50 

40 

30 

20 

10 

Daeff � 0. 

001 

Daeff � 0. 

01 

Daeff � 0. 

1 

0 

-1 0 1 2 3 4 Bi5 

6 7 8 9 10 11 

Figure 6: Variation of M D with Bi for different values of Da eff 

eff for the case of stationary 


Bi 

Daeff � 0. 

1 

Daeff � 0. 

01 

Daeff � 0. 

001


Figure 7: Variation of M eff with Q when 0. 

001 

and overstability. 


Da eff � for the cases of stationary convection 

Table 6: The critical values of the effective Marangoni number M for different values of eff 

Q 

and Da eff for the cases of stationary convection and overstability. Here R � 0 , � � 0 , Gm � 1 

Q 

0 

1 

5 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

M eff 

140000 

120000 

100000 

80000 

60000 

40000 

20000 


Daeff � 0. 

01 

Daeff � 0. 

001 

M eff 

0 

372.210 

555.645 

1192.774 

1916.229 

3276.515 

4581.733 

5855.973 

7109.293 

8347.082 

9572.645 

10788.193 

11995.302 

13195.157 

0 10 20 30 40 50 60 70 80 90 100 110 

a 

2.423 

2.831 

3.701 

4.320 

5.110 

5.659 

6.088 

6.446 

6.754 

7.026 

7.270 

7.492 

7.696 

M eff 

2510.291 

4066.627 

9576.203 

15941.503 

28058.077 

39792.908 

51325.706 

62732.865 

74054.876 

85315.803 

96531.023 

107710.865 

118862.548 

Q 

a 

2.978 

3.616 

5.044 

6.100 

7.444 

8.363 

9.085 

9.695 

10.232 

10.720 

11.170 

11.590 

11.984 


overstability 

Daeff � 0. 

01 Daeff � 0. 

001 

M eff 

- 

- 

- 

1786.695 

2704.793 

3550.948 

4361.087 

5148.130 

5918.691 

6676.712 

7424.767 

8164.650 

8897.668 

a 

- 

- 

- 

4.547 

5.274 

5.754 

6.131 

6.448 

6.724 

6.971 

7.194 

7.399 

7.589 

� 

� 1 

� � 0. 5 

M eff 

� � 0 

- 

- 

9528.054 

14200.385 

22553.054 

30367.372 

37898.299 

45244.811 

52458.225 

59569.405 

66598.709 

73560.439 

80465.111 

31 

a 

- 

- 

5.234 

6.022 

6.912 

7.574 

8.126 

8.607 

9.040 

9.435 

9.799 

10.141 

10.461

32 



Table 7: The critical values of effective Marangoni number M for different values of eff 

Q , 

Da eff and � for the case of overstability. Here R � 0 , Gm � 1, 

Bi � 0 , Pr � 1 , Pm 

� 0. 

25 . 

Q 

20 

30 

40 

50 

60 

70 

80 

90 

100 

M Deff 

Daeff � 0. 

01 

M eff 

3145.795 

4195.587 

5205.536 

6190.848 

7158.764 

8113.486 

9057.731 

9993.375 

10921.786 

140 

120 

100 

80 

60 

40 

20 

0 

a 

5.141 

5.659 

6.055 

6.386 

6.672 

6.928 

7.159 

7.371 

7.568 

� � 0. 

5 

Daeff � 0. 

001 

M eff 

26132.837 

35676.511 

44942.342 

54027.250 

62981.200 

71834.278 

80606.382 

89311.527 

97960.038 

a 

7.123 

7.882 

8.501 

9.037 

9.517 

9.948 

10.346 

10.716 

11.062 

Daeff � 0. 

01 

M eff 

- 

- 

- 

- 

8295.793 

9429.100 

10550.448 

11662.067 

12765.521 

Q 

Figure 8: Variation of M Deff 

with Q when 0. 

001 


a 

- 

- 

- 

- 

6.491 

6.748 

6.978 

7.187 

7.380 


� � 1 

Daeff � 0. 

001 

M eff 

- 

- 

- 

61996.042 

72501.979 

82903.700 

93221.716 

103470.448 

113660.512 

� 

� 1 

� � 0. 5 

� � 0 

0 10 20 30 40 50 60 70 80 90 100 110 

a 

- 

- 

- 

9.196 

9.690 

10.134 

10.535 

10.906 

11.252 

Da eff � for the cases of stationary convection


Figure 9: Variation of M eff with Q for different values of Bi for the case of stationary 


Table 8: The critical values of the effective Marangoni number M for different values of 

eff 

Da , Q , � and Bi when the rigid boundary is thermally insulated ( D� � 0). 

Here R � 0 . 

eff 

Stationary 

convection 

(Shivakumara 

et al. 2009) 

Stationary 

convection 

overstability 

M eff 

� 

0 

0 

0 

0.5 

110000 

100000 

90000 

80000 

70000 

60000 

50000 

40000 

30000 

20000 

10000 

Q 

0 

0 

50 

50 

Bi � 0. 

5 

Bi � 0 

0 

0 10 20 30 40 50 60 70 80 90 100 110 

Da eff 

0.1 

0.01 

0.1 

0.01 

0.1 

0.01 

0.1 

0.01 

0.1 

0.01 


M 

eff 

71.974 

281.212 

71.975 

281.212 

934.985 

7105.323 

837.180 

5150.602 

- 

6195.369 

Bi � 0 

Q 

a 

0.000 

0.000 

0.0009 

0.001 

3.6778 

6.3872 

4.2934 

6.4645 

- 

6.4142 

M 

Bi � 1 

eff 

138.862 

479.214 

138.862 

479.213 

1159.569 

8092.848 

955.449 

5636.163 

1110.327 

6760.099 

Daeff � 0. 

001 

Daeff � 0. 

01 

a 

1.856 

2.330 

1.8555 

2.3297 

4.6088 

7.9840 

4.3950 

6.6876 

4.2907 

6.7381 

M 

eff 

318.279 

978.212 

318.279 

978.207 

1899.077 

Bi � 5 

11159.449 

1418.54 

7521.53 

1638.310 

8887.256 

a 

33 

2.467 

3.467 

2.4672 

3.4677 

5.9456 

11.2008 

4.5593 

7.2806 

4.5294 

7.5529

34 

M eff 



Figure 10: Variation of M eff with Q when 0. 

01 

convection and overstability. 

M Deff 

12000 

11000 

10000 

9000 

8000 

7000 

6000 

5000 

4000 

3000 

2000 

1000 

Figure 11: Variation of 


0 

0 10 20 30 40 50 60 70 80 90 100 110 

120 

110 

100 

90 

80 

70 

60 

50 

40 

30 

20 

10 

0 

Q 

M with Q when 0. 

01 

Deff 


Q 

R � 30 

Da eff � for the cases of stationary 


R � 10 

R � 30 

R � 10 

� � 1 

� 

� 1 

0 10 20 30 40 50 60 70 80 90 100 110 

� � 0. 5 

� � 0 

� � 0. 5 

� � 0 

Da eff � for the cases of stationary convection


Table 9a: The critical values of effective Marangoni number M eff for different values of Q 

and R when � � 0 for the cases of stationary convection and overstability. Here Daeff � 0. 

01 , 

Bi � 0 , Gm � 1 


R 

10 

30 

Q 

1 

5 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

1 

5 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 


� � 0 

overstability 

M eff a M eff a 

472.997 2.710 - - 

1122.869 3.580 - - 

1853.524 4.209 1709.164 4.472 

3221.299 5.015 2626.743 5.209 

4530.832 5.574 3472.709 5.695 

5808.045 6.012 4282.908 6.077 

7063.604 6.377 5070.153 6.397 

8303.170 6.690 5840.994 6.676 

9530.192 6.967 6599.341 6.925 

10746.973 7.214 7347.747 7.150 

11955.143 7.439 8087.994 7.356 

13155.920 7.645 8821.382 7.548 

280.728 

967.507 

1717.467 

3104.218 

4424.188 

5708.393 

6969.108 

8212.704 

9443.001 

10662.517 

11873.022 

13075.823 

2.593 

3.393 

4.013 

4.834 

5.410 

5.863 

6.239 

6.562 

6.847 

7.101 

7.332 

7.543 

- 

- 

1539.181 

2459.244 

3306.751 

4118.330 

4906.881 

5678.987 

6438.544 

7188.113 

7929.477 

8663.937 

- 

- 

4.343 

5.092 

5.588 

5.976 

6.302 

6.585 

6.837 

7.065 

7.275 

7.469 

35

36 



Table 9b: The critical values of effective Marangoni number M eff for different values of 

Q , R and � for the case of overstability. Here Daeff � 0. 

01, 

Bi � 0 , Gm � 1. 

R 

10 

30 

Q 

20 

30 

40 

50 

60 

70 

80 

90 

100 

20 

30 

40 

50 

60 

70 

80 

90 

100 

� � 0. 

5 

M eff a 

3073.969 5.079 

4125.166 5.602 

5136.297 6.002 

6122.658 6.336 

7091.526 6.625 

8047.122 6.883 

8992.175 7.116 

9928.571 7.330 

10857.685 7.528 

2920.928 

3976.764 

4991.435 

5980.727 

6952.126 

7909.962 

8857.029 

9795.259 

10726.058 

4.964 

5.495 

5.902 

6.241 

6.535 

6.796 

7.032 

7.249 

7.450 

M eff 

� � 1 

- 

- 

- 

- 

8235.576 

9370.016 

10492.372 

11604.892 

12709.178 

- 

- 

- 

6958.300 

8111.313 

9248.440 

10373.148 

11487.757 

12593.924 

a 

- 

- 

- 

- 

6.447 

6.706 

6.938 

7.149 

7.344 

- 

- 

- 

6.062 

6.362 

6.625 

6.860 

7.075 

7.272

8000 

7000 

6000 

5000 

M 

4000 

3000 

2000 

1000 

0 


eff 

0 50 100 150 200 250 300 350 

Figure 12: Variation of M efff with R when Q � 50, 

0. 

01 

R 

800 

700 

600 

500 

400 

300 

200 

100 

0 


Bi � . 

Da eff � , 0 

� � 0. 5 

0 50 100 150 200 250 300 350 

Figure 13: Variation of R with M efff when Q � 50, 

0. 

01 

M 

R 


eff 


� � 1 

� � 0. 5 

� � 0 

� � 1 

� � 0 

Overstability 

Bi � . 

Da eff � , 0 

Overstability 

37

38 

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