Penetrative convection via internal heating in superposed
fluid and anisotropic porous layers with throughflow
Gangadharaiah, Y. H., Suma, S. P., Nagarathnamma. H
Abstract. The onset of thermal convection due to heating from below in a system consisting of
a fluid layer overlying a porous layer with anisotropic permeability and thermal diffusivity is
studied. The onset of penetrative convection simulated via internal heating in the presence of a
vertical throughflow. Flow in the porous medium is governed by Forchheimer-extended Darcy
equation and the Beavers–Joseph slip condition is applied at the interface between the fluid and
the porous layers. The boundaries are considered to be rigid, however permeable, and insulated
to temperature perturbations. The eigen value problem is solved using a regular perturbation
technique with wave number as a perturbation parameter. It is found that the depth of the relative
layers, the direction of throughflow, the presence of volumetric internal heat source in fluid
and/or porous layer and mechanical and thermal anisotropy parameters have a profound effect on
the stability of the system. Decreasing the mechanical anisotropy parameter and increasing the
thermal anisotropy parameter leads to stabilization of the system. Besides, the possibility of
control of thermal convection by suitable choice of physical parameters is discussed in detail.
Key words: composite layer; penetrative convection; throughflow.
Gangadharaiah, Y. H.
Department of Mathematics, New Horizon College of Engineering, Bangalore-560 103,
India
E-mail address: gangu.honnappa@gmail.com,
Suma, S. P
Department of Mathematics, Cambridge Institute of Technology,Bangalore-560 036,
India
Nagarathnamma. H
Department of Mathematics, Ambedkar Institute of Technology, Bangalore, India
1
Nomenclature
l 2  m2
a
horizontal wave number,
A
ratio of heat capacities
D
differential operator d dz
d
thickness of the fluid layer
dm
thickness of the porous layer
Da
Darcy number k d m2
g
acceleration due to gravity
k
permeability
l, m
wave number in x and y-directions respectively
R
Rayleigh number in the fluid layer  g T0  Tu  d 3 / 
Rm
Rayleigh number in a porous medium  g Tl  T0  dm k / m
p
pressure
Pr
Prandtl number for fluid layer, v 
Prm
porous medium Prandtl number, v  m
Pe
throughflow dependent Peclet number for fluid layer, W0 d 
Pem
through flow dependent Peclet number for a porous medium, W0 dm  m
T
temperature
T0
temperature at the interface
V
velocity vector (u, v, w)
W
amplitude of perturbed vertical velocity
qf
heat source in the fluid layer
qm
heat source in the porous layer
Ns f
dimensionless heat source in the fluid layer q f d 2 / 2 T0  Tu 
Nsm
dimensionless heat source in the porous layer qm dm2 / 2 m Tl  T0 
2
Greek symbols

slip parameter
 2h
horizontal Laplacian operator  2h   2 x 2   2 y 2
2
Laplacian operator
T
ratio of thermal diffusivities   m

porosity of the porous medium

thermal diffusivity

amplitude of perturbed temperature

fluid viscosity
0
fluid density

mechanical anisotropy parameter K h / K v

thermal anisotropy parameter  mh /  mv

temperature dependent surface tension
v
kinematic viscosity  0

depth ratio d d m
 2   2 x 2   2 y 2   2 z 2
Subscripts
b
basic state
l
lower
m
porous medium
u
upper
1 Introduction
The problem of fluid flow over a porous medium is encountered in a wide range of industrial
and geophysical applications, such as flows in fuel cells, filtration processes, the extraction of oil
from underground reservoirs, ground-water pollution, the manufacture of composite materials,
3
and in flow in biological materials, see e.g. Allen (1984), Hill and Straughan (2009), Nield and
Bejan (2006).
A linear stability analysis of the convective instability in superposed horizontal fluid and
porous layers with throughflow in the vertical direction for isothermal boundaries has been
implemented by Chen (1990). It is found that in such a physical configuration both stabilizing
and destabilizing factors due to vertical throughflow can be enhanced so that a more precise
control of the buoyantly driven instability in either a fluid or a porous layer is possible. Khalili et
al. (2003) have investigated the effect of throughflow on thermal convective instability in
superposed fluid and porous layers system for the boundaries insulated to temperature
perturbations and an analytical expression for the critical Rayleigh number is obtained. Carr and
Straughan (2003) have considered penetrative convection in the porous-medium–fluid system by
employing a quadratic equation of state, while Carr (2004) has studied penetrative convection via
internal heating in a composite two-layer system in which a layer of fluid overlies and saturates a
layer of isotropic porous medium. Shivakumara et al. (2011)have investigated the criterion for
the onset of surface tension-driven convection in the presence of temperature gradients in a twolayer system comprising a fluid saturated anisotropic porous layer over which lies a layer of fluid
while the effect of internal heating on the problem has been considered by Shivakumara et al.
(2012).
The problem we wish to investigate is one of thermal convection in a system consisting
of a anisotropic porous layer underlying a fluid layer where there is throughflow through the
system. In addition, we intend focusing attention on the system heated internally in both fluid
and anisotropic porous layers, thereby incorporating penetrative convection. One way in which
penetrative convection with throughflow in composite fluid and porous layers system may be
important for industrial application is that it yields a potential method of controlling the onset
and behavior of convection. To be able to know how to control convection, one needs to
determine when the convection sets in.
The intent of the present study is to obtain the criterion for the onset of penetrative
convection via internal heating in a two-layer system in which a fluid layer overlies a layer of
fluid saturated anisotropic porous medium in the presence of vertical throughflow. The
4
boundaries are considered to be insulated to temperature perturbations. A regular perturbation
technique with wave number as a perturbation parameter is used to solve the eigen value
problem in a closed form .A wide-ranging parametric study is undertaken to explore their impact
on the stability characteristics of the system.
2 Mathematical Formulation
We consider penetrative convection via internal heating in a system consisting of an
infinite horizontal fluid layer of thickness d overlying a layer of Darcy porous medium of
thickness d m with throughflow of constant vertical velocity W0 as shown in Fig. 1. A Cartesian
coordinate system (x, y, z) is chosen with the origin at the interface and the z-axis vertically
upward. The gravity acts in the vertical direction with constant acceleration g . The top and
bottom boundaries are assumed to be rigid-permeable and are maintained at uniform but
different temperatures Tl and Tu ( Tl ) respectively.
Tu   Tl 
d
z
Fluid layer
g
x
dm
Anisotropic Porous
layer
Tl
W0
Fig. 1 Physical configuration
The governing equations for the fluid and the porous layers are:
Fluid layer:
 V  0
(1)
5
 V

 V  V   p  0 g 1   T  T0      2V
 t


0 

T
 V  T    2T  q .
t


(2)
(3)
Porous layer:
Vm  0
(4)
0 Vm
 pm  0 g 1   Tm  T0    K 1 Vm
 t
( 5)
A
Tm
 Vm  m Tm     m Tm   qm .
t


(6)
In the above equations, V  (u , v, w) is the velocity vector, p is the pressure, T is the temperature,
q is the constant heat source and  is the thermal diffusivity,  is the fluid viscosity,  is the
thermal expansion coefficient,  is the porosity of the porous medium, A is the ratio of heat
capacities and  0 is the reference fluid density, while K and  m are respectively the tensors of
ˆˆ ,
ˆˆ  ˆˆ
permeability and effective thermal diffusivity which are given by K  Kh (ii
jj )  Kv kk
ˆˆ , where the subscripts h and v refer to the quantities in the horizontal
 m   mh (iiˆˆ  ˆˆ
jj )   mv kk
and vertical directions respectively. Thus, the horizontal isotropy with coinciding principal axes
of permeability and effective thermal diffusivity to the coordinate axes are hereby assumed. The
subscript m refers to the value of the parameter in the porous region.
The basic state is quiescent and is of the form
 u, v, w, p, T   0, 0,W0 , pb  z  , Tb  z 
(7)
 um , vm , wm , pm , Tm   0, 0,W0 , pmb  z  , Tmb  z  .
(8)
The temperature distributions in the basic state are given by
qd
Tb  z   T0 
W0
 z 1  eW0 z   
 1  eW0 z  
  W0 d 
   T0  Tu   W0 d 
 ,0  z  d
 1 
1 
e
 d e
6
(9)
Tmb  zm   T0 
 1  eW0 zm  mv 
qm d m  zm eW0 zm  mv  1  


T

T




 ,  d m  zm  0
l
0  W0 d m  mv
W0  d m e W0 dm  mv  1  
1 
e
(10)
Where T0 is the interface temperature. It is observed that the effect of vertical throughflow is to
alter the basic temperature distribution from linear to exponential with respect to vertical
coordinate in both fluid and porous layers. To investigate the stability of the basic state,
infinitesimal disturbances are superimposed in the form
V  V , T  Tb  z    ,
p  pb ( z )  p , Vm  Vm
(11)
Tm  Tmb  zm   m , pm  pmb ( zm )  pm .
(12)
Following the standard linear stability analysis procedure and noting that the principle of
exchange of stability holds, we arrive at the following stability equations (for details see Chen et
al. 1990 and Shivakumara et al. 2011):
D

2
 a2
 D
2

  D  a 2 W  Ra 2 
(13)
 Ns
e Pe z
D 2  PeD  a 2   
 ( Pe  2 Ns )
1  e Pe
 Pe




W

(14)
1 2
2 
2
 Dm  am  Wm   Rm am m



D
2
m
(15)
 Ns
e Pem zm
 Pem Dm   am2  m   m  ( Pem  2 Nsm )
1  e  Pem
 Pem





 Wm .


(16)
Here, W is the amplitude of perturbed vertical velocity and  is the amplitude of perturbed
temperature, D  d / dz , Pr   /  is the Prandtl number, Pe  W0 d /  is the Peclet number,
R   g T0  Tu  d 3 /  is the Rayleigh number, Ns  q d 2 / 2 T0  Tu  is the dimensionless heat
source strength, a  l 2  m2 is the overall horizontal wave number and  2   2h   2 / z 2 is the
Laplacian operator with  2h   2 / x 2   2 / y 2 . The corresponding quantities for the porous
region are Wm ,  m , Dm  d / dzm , Prm   /  mv  Pr T , Pem  W0 d m /  mv  Pe T /   ,
7
Rm   g Tl  T0  dm Kv / mv  RDa 4T2 , Nsm  qmdm2 / 2 mv Tl  T0  , am  l 2  m 2 and  2m 
 2mh   2 / zm2 with  2mh   2 / xm2   2 / ym2 . Further, Da  K v / d m2
is
the
Darcy
number,
  K h / Kv is the mechanical anisotropy parameter,    mh /  mv is the thermal anisotropy
parameter,  T   /  mv is
the
ratio of
thermal
diffusivities and   Pe / Pr is a
nondimensional group.
The boundary conditions are:
W  D  DW  0
Wm  Dmm  0
at z  1
(17)
at zm  1 .
(18)
At the interface (i.e., z = 0) the continuity of velocity, temperature, heat flux, normal stress and
the Beavers and Joseph (1967) slip conditions are imposed. Accordingly, the conditions are:


Wm ,   T  m , D  Dm m ,
T

(19)
 4
 D 2   D  3a 2  DW 
DW
 T Da m m
(20)
 2


  3
DW 
DmWm
D 
Da 
 T Da

(21)
W
where   d dm is the ratio of fluid layer to porous layer thickness and  is the Beavers-Joseph
slip parameter. Thus, the problem is reduced to an eigen value problem consisting of a sixth
order ordinary differential equation in the fluid layer and a fourth order ordinary differential
equation in the porous layer, subject to 10 boundary conditions. If matching of the solutions in
the two layers is to be possible, the wave numbers must be the same for the fluid and porous
layers, so that we have a / d  am / d m and hence   a / am .
8
3 Solution by regular perturbation technique
Since the critical wave number is exceedingly small for the assumed temperature
boundary conditions (Nield and Bejan 2006), the eigen value problem is solved using a regular
perturbation technique with wave number 𝑎as a perturbation parameter. Accordingly, the
dependent variables are expanded in powers of 𝑎2 in the form
N
  W ,  
W ,     a 2
i 0
i
i
(22)
i
i
 a2 
W
,


 m m    2  Wmi , mi 
i 0  

N
(23)
Substitution of Eqs. ( 22 ) and ( 23) into Eqs. (13)  16  and the boundary conditions (17 )   21
yields a sequence of equations for the unknown functions Wi ( z ), i ( z ) , Wmi ( zm ) and mi ( zm ) for
i  0,1, 2,3...... .
At the leading order in 𝑎2 Eqs. (13)  16  become, respectively,
D 4W0   D 3W0  0
(24)
D 2 0  PeD0  W0 f ( z )
(25)
Dm2Wm 0  0
(26)
Dm2  m 0  Pem Dm  m 0  Wm 0 g ( zm )
(27)
where
f ( z) 
2 Ns  Pe  2 Ns  Pe z

e
Pe
1  e Pe


(28)
2 Nsm  Pem  2 Nsm  Pem zm
g ( zm ) 

e
Pem
1  e  Pem


and the boundary conditions (17 )   21 become
W0  0,
Wm0  0,
D0  0,
DW0  0
Dmm0  0,
at z  1
(29)
at zm  1.
(30)
And at the interface( 𝑖. 𝑒 𝑧 = 0)
W0 


Wm 0 , 0  T m 0 , D0  Dm m 0
T

(31)
9
 4
D W0   D W0 
DmWm 0
Da T
3
2
D2W0 

Da
DW0 
 3
 T Da
(32)
DmWm0 .
(33)
The solution to the zeroth order Eqs. ( 24 )  ( 27 ) is given by
W0  0, 0 
T
, Wn 0  0, m0  1 .

(34)
At the first order in a 2 Eqs. (13)  (16 ) then reduce to
D 4W1   D 3W1  R
T

(35)
D 2 1  Pe D1  W1 f ( z ) 
T

(36)
Dm2 Wm1   Rm
(37)
Dm2 m1  Pem Dm m1  Wm1 g ( zm )  
(38)
and the boundary conditions (17 )   21 become
W1  0,
Wm1  0,
D1  0, DW1  0
Dmm1  0,
at z  1
(39)
at zm  1.
(40)
And at the interface( 𝑖. 𝑒 𝑧 = 0)
W1 
1
 T
Wm1 , 1 
D3W1   D 2W1 
D 2W1 

Da
T
1
 m1 , D1  2 Dm m1
3


(41)
 2
DmWm1
Da T
DW1 
 
 T Da
(42)
DmWm1 .
(43)
The general solutions of Eq. (25) and (27) are respectively given by

Da  T3 3 
2
z
W1  R C1  C2 z  C3 z  C4e 
z 
6


(44)
10

 Da  T2 2 
Wm1  R C5  C6 zm 
zm 
2 4


(45)
where
C

b  b C 
 b     T  C3  C3 
C
 

C1   2  C4  , C2   T  1  C3  e C4  , C3  10 9 6 , C4  6 T
b4
 C1
2
 

 T 

 2 2 Da

C1
 C6 , b1  
  3  b2 
2 (1  2 )
2


  T  3  3 Da

 3 
b3    2 Da 
,
b




T 
 T   4  6

C5 
 2
Da
b6   T 
 6
T 

b10 
 b1b7  b8b6 

b4 
 b1b7 

T  



b2
 , b7   b1b3 
T 


 
2

Da   3   3
2


Da  e ,


 , b5    1 e  Da ,



b2
 , b8   b3b5 
T 



 , b9   b4 b5  2b7 b6  ,

,
Equations (36) and (38) involving D 2 1 and Dm2  m1 respectively provide the solvability
requirement which is given by
1

0

f ( z ) W1 dz  2

T 
1 g ( zm )Wm1 dz      2
0
.
(46)
In the absence of throughflow (i.e.,Pe→ 0), both the functions f ( z ) and g ( zm ) take the value
−1 and the above condition reduces to that of Nield (1977).
The expressions for W1 and Wm1 are back substituted into Eq. (53) and integrated to yield an
expression for the critical Rayleigh number Rmc , which is given by
11
Rmc 

   Da  T2 
 T  2 

4
    

(47)

1C1   2 C2   3C3   4 C4   5   2  C5   6C6   3 7 

where

 2 Ns  Pe  2 Ns   e Pe 2e Pe 2 1  e Pe
 4 Ns





1  
 1 ,  3  
2
Pe3
1  e Pe  Pe Pe
 3Pe
 Pe










 
Pe
3e Pe 6e Pe 6 e  1
 2 

Pe
Pe3
Pe4




 
  Ns  Pe  2 Ns   e Pem
e Pem  1
m
1
1

6  


Pem2
 Pem
1  e Pem  Pem





 
 Pe  

1
2 Ns  Pe  2 Ns  e

4 

 Pe
1  e Pe
 Pe   

  2 Ns  Pe  2 Ns   e Pe

5   T 

6  Pe
1  e Pe   Pe



 
  .
 2 Ns e  Pem 2e  Pe 2 e  Pem  1
1



, 7  

3Pem Pem
Pem2
Pem3




The expression for Rmc is evaluated for different values of various physical parameters and the
results are discussed in detail in the next section
4 Results and Discussion
The onset of penetrative convection via internal heating in the presence of a vertical
throughflow is considered in a system consisting of a fluid layer overlying a anisotropic porous
layer. In the calculation, we have chosen the value of   0.389, Cb =209.25 and
Da  3.04  103 which correspond to 3 cm deep porous layer consisting of 3mm diameter glass
beads (Chen, 1990). The results are discussed for different values of
depth ratios  .The
following three different cases of internal heating pattern is considered for discussion namely,
Case (i): internal heat source in the porous layer alone (i.e., Ns f  0 , Nsm  5)
Case (ii): internal heat sources in both fluid and porous layers(i.e., Ns f  5 , Nsm  5) and
Case (iii): internal heat source in the fluid layer alone (i.e., Ns f  5 , Nsm  0 ).
12
2000
Nsf = 5
(a)
5
2
Pr = 1
1500
2
10
Rc
0
0
1
10
1000
1
10
500
-10
-5
0
5
10
Pe
3000
(b)
Pe = 4
2000
c
R
3
1000
-4
-3
0
0.1
1
Nsf
10
100
Fig. 2. Critical Rayleigh number versus (a) Pe for two values of Pr and Ns f , (b) Ns f for different
values of Pe with Pr  10 (single fluid layer case,   1 )
13
150
Nsm=0
100
c
Rm
2
5
50
10
-10
-5
0
5
10
Pem
Fig. 3. Critical Rayleigh number versus Pem for different values of Nsm when   0.1 and  =0.5
,(single fluid layer case,   1 )
12
Nsm=0
c
Rm
8
2
5
4
10
-10
-5
0
5
10
Pem
Fig. 4. Critical Rayleigh number versus Pem for different values of Nsm when   1and =0.5 (single
fluid layer case,   1 )
14
300
Nsm=0
200
c
Rm
2
5
100
10
-10
-5
0
5
10
Pem
Fig. 5. Critical Rayleigh number versus Pem for different values of Nsm when   0.1and =1 (single
fluid layer case,   1 )
140
(c)
105
c
Rm
70
Pem=4
3
35
-4
-3
0
1
Nsm
10
100
Fig. 6. Critical Rayleigh number versus Nsm for different values of Pem with Ns f  0 when
  0.5and =0.5 (single fluid layer case,   1 )
15
4.1 Depth ratio   1
This is the case of a pure fluid layer and the stability characteristic of the system is
measured by the Rayleigh number R . When Ns f  0 and Pe  0 , the known exact value
R c  720 (Sparrow et al.1964) is retrieved. The direction of throughflow has no influence on the
stability of the system in the absence of internal heating and throughflow is to delay the onset of
convection. This may be attributed to the fact that the primary effect of throughflow is to confine
significant thermal gradients to a thermal boundary layer at the boundary toward which the
throughflow is directed. The effective length scale is thus smaller than the layer thickness d and
hence the effective Rayleigh number which will be depending on d 3 will be much less than the
actual Rayleigh number. A larger value of Rayleigh number is thus necessary to initiate the
convection. Nonetheless, the simultaneous presence of throughflow and internal heating alters
the basic temperature distribution so that the direction of throughflow also affects the stability of
the system. As a result of this some unusual behaviors are observed namely, (i) downward
throughflow initially shows some destabilizing effect (Fig. 2a) and (ii) increasing internal heat
source strength causes stabilizing effect for upward throughflow initially (Fig. 2b).
The presence of throughflow brings in the effect of Prandtl number Pr on the
characteristics of stability of the system. Increasing the value of Prandtl number is to hasten the
onset of convection in the absence of internal heating but the variation in the critical Rayleigh
number R c for Pr =1 and 10 is found to be not so significant. In the presence of internal heating,
however, the effect of increasing Pr is destabilizing for only downward throughflow and an
opposite behavior is noticed for upward throughflow. Besides, R c for Pr =1 and 10 are not close
in the presence internal heating. The upward throughflow is found to be more stabilizing than
downward throughflow.
4.2 Depth ratio   1
In the absence
of
internal
heating ( Nsm  0 ) and
throughflow ( Pem  0 ), we
recover the known exact value Rmc  12 (Nield and Bejan 2006) for pure isotropic porous layer (
  1   ).
16
In the case anisotropic porous layer it is observed that the direction of throughflow
matters only in the presence of internal heating. Figures 3 - 6 it is noted that Rmc attains higher
values at lower values of  for the absence and presence of internal heating. That is, decrease in
the mechanical anisotropy parameter is to delay the onset of convection. This is because,
decrease in  corresponds to smaller horizontal permeability which in turn hinder the motion of
fluid in the horizontal direction. As a consequence, the conduction process in the porous medium
becomes more stable and hence higher values of Rmc are needed for the onset of convection and
Figures 3 - 6 illustrate that the results are qualitatively similar to those of   1 .
4.3 Depth ratio   1
The stability of the system is characterized by Rmc . Figure 7 exhibit plots of Rmc as a
function of P em respectively for the above mentioned three cases of internal heating pattern for
  0.5 ,   0.5 . The results are presented for two values of Prandtl number Pr  1 and 100.
From Fig. 7 it is seen that for all values of Pr for the case of internal heating porous layer
stabilizes the system both upward and downward throughflow. And upward throughflow shows
stabilizing effect for other two combination heating layer ( Ns f  5 , Ns f  5 and Ns f  5 ,
Ns f  0 ) and for downward through flow the system is destabilizing when 3.5  Pem  0 ,and
4.2  Pem  0 for Pr = 1 and 100,respectively and for higher values of Pem the system is
stabilizing. For Pr =0.1, the system is always stabilizing as the value of Rmc increases with Pem .
Figure 8 exhibit plots of Rmc as a function of P em for different values of  . The results
are presented for   0.5 , Pr  10 for the above mentioned three cases of internal heating
pattern. From Fig. 8 it is seen that for all values of  for the case of internal heating porous layer
stabilizes the system both upward and downward throughflow and for other two combination
heating layer ( Ns f  5 , Ns f  5 and Ns f  5 , Ns f  0 ) and for downward through flow the
17
system is destabilizing when 3.8  Pem  0 for all values of  and stabilizes the system for
upward through flow. Hence the system is more stabilizing for decreasing the values of  .
Figure 9 exhibit plots of Rmc as a function of P em for different values of  . The results
are presented for   0.5 , Pr  10 for the above mentioned three cases of internal heating pattern.
From Fig. 9 it is seen that for all values of  for the case of internal heating porous layer
stabilizes the system both upward and downward throughflow and for other two combination
heating layer ( Ns f  5 , Ns f  5 and Ns f  5 , Ns f  0 ) and for downward through flow the
system is destabilizing when 3.4  Pem  0 for all values of  and stabilizes the system for
upward through flow. Hence the system is more stabilizing for increasing the values of  .
0.02
Nsf=0, Nsm=5
Nsf=5, Nsm=5

     Ns =5, Ns =0
f
m
c
Rm
Pr =1
100
1
0.01
100
0.00
-10
-5
0
5
10
Pem
Fig. 7. Critical Rayleigh number versus Peclet number for different values of Pr and
Ns f , Nsm for   0.5 ,   0.5 and   1.
18
0.02
Nsf=0, Nsm=5
Nsf=5, Nsm=5
     Ns =5, Ns =0
f
m

  0.1
c
Rm
1
0.01
10
0.1
1
10
0.00
-10
-5
0
5
10
Pem
Fig. 8. Critical Rayleigh number versus Peclet number for different values of  and
Ns f , Nsm for   0.5, Pr  10,   1.
0.03
Nsf=0, Nsm=5
Nsf=5, Nsm=5
 Ns =5, Ns =0
f
m

0.02
 2
c
Rm
2
0.01
1
0.1
1
0.1
0.00
-10
-5
0
5
10
Pem
Fig. 9. Critical Rayleigh number versus Peclet number for different values of  and
Ns f , Nsm for   0.5, Pr  10,   1.
19
0.018
 1
0.015
Rmc
0.5
0.01
0.1
0.005
0

0.5
1
1.5
Fig. 10. Critical Rayleigh number versus  for different values of 
for
Ns f  0, Nsm  2, Pr  1,   1.
0.02


  0.5


0.0175
Rmc
0.015
0.0125

Ns f  5, Nsm  5
 
0.3

 
0.1

 Ns f  5, Nsm  0
0.01
0

0.5
1




1.5
Fig. 11. Critical Rayleigh number versus  for different values of  and
Ns f , Nsm for Pr  1,   1.
20
The effect of mechanical and thermal anisotropic parameters on the onset of convection
is emphasized by depicting the variation of Rmc and over a range of mechanical anisotropy
parameter  for different values of thermal anisotropy parameter  in Figs. 10 and 11,
respectively for a fixed value of Pr  1,   1. It is observed that Rmc
increases with the
decreasing  . Physically, this means that the conduction solution in the porous medium becomes
more stable and the critical wavelength decreases as the horizontal permeability decreases.
Smaller horizontal permeability inhibits horizontal motion, and the conduction solution is thus
stabilized. The larger resistance to horizontal flow also leads to a shortening of the horizontal
wavelength at onset. In the same figure, it is seen that for a given and smaller values of the
horizontal thermal diffusivity correspond to destabilization of the basic state, and the onset of
convection at a smaller wavelength, This can be explained by the fact that as  decreases, a
heated fluid parcel loses less heat in the horizontal directions, and hence retains its buoyancy
better. Therefore, the base state becomes less stable, and the wavelength is reduced.
5 Conclusions
The criterion for the onset of penetrative convection in an anisotropic porous layer
underlain by a fluid layer is investigated theoretically to understand control of penetrative
convection. The Forchheimer-extended Darcy equation is used in the porous medium and the
Beavers–Joseph slip condition is applied at the interface between the fluid and the porous layers.
The boundaries are considered to be rigid, however permeable, and insulated to temperature
perturbations. The resulting eigen value problem is solved by regular perturbation technique.
From the foregoing analysis, it is observed that the stability characteristics of the configuration
depend crucially on (i) the presence of internal heating in fluid and/or porous layer, (ii) depth
ratio ζ, (iii) mechanical anisotropy parameter  , (v) thermal anisotropy parameter  , and (vi)
throughflow direction. For ζ =1, the system is more stabilizing if the porous layer alone is heated
internally
as compared to other two types of heating pattern. Decreasing the mechanical
anisotropy parameter 
and increasing the thermal anisotropy parameter  , the system is
stabilizing.
21
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22