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Physical and Numerical Simulation of Geotechnical Engineering
10th Issue, Mar. 2013
Stability Analysis of Thermosolutal Convection in a
Horizontal Porous Layer Using a Thermal Non-equilibrium
Model
CHEN Xi, WANG Shaowei, TAN Wenchang
Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, P.R.China
[email protected]
ABSTRACT: We use a linear stability analysis to investigate the onset of thermosolutal convection in
a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the boundaries’
surfaces. The critical Rayleigh number and the corresponding wave number for the exchange of
stability and overstability are obtained. The critical value of the parameters for oscillatory convection
occurs are obtained. Besides, asymptotic analysis for both small and large values of inter-phase heat
transfer coefficient H.
KEYWORDS: Double diffusive, Porous media, Non-equilibrium model, Critical rayleigh number
1 INTRODUCTION
In many geological systems, with the fluid convect in a
porous medium, the amount of solute in the fluid will be
dissolved or precipitated because of temperature, pressure
or chemical reactions, and it will also affect the convection
of the fluid. Then double-diffusive convection
(thermosolutal convection) occurs. The first linear stability
analysis of a double-diffusive system in a porous layer was
investigated by Nield [7], the critical Rayleigh numbers for
the onset of stationary and overstable convection are
obtained for different thermal and solutal boundary
conditions using a linear stability analysis. Recently,
Pritchard and Richardson [2] discussed how the dissolution
or precipitation of the solute affect the onset of convection.
The other emphasis of this paper is the thermal
non-equilibrium model. To be convenient, most of the
former works on convective heat transfer in porous medium
are under the assumption that the solid and fluid phases are
in local thermal equilibrium (LTE). But in fact, in many
applications, they are not in local thermal equilibrium.
Then Nield and Bejan [1] give us the equations of the
thermal non-equilibrium model. Recently, Malashetty et
al.[11] present the effect of thermal non-equilibrium on the
onset of convection in a porous layer using the
Lapwood-Brinkman model.
Nevertheless, the former studies focused on either
double-diffusive convection or non-equilibrium model,
none of them investigated the double-diffusive convection
using a non-equilibrium model in spite of it has many
important applications. Thus, the aim of this paper is to
make the linear stability analysis of double-diffusive
convection in a horizontal porous layer using a thermal
non-equilibrium model.
temperatures are T0 and T1 at the bottom and top,
respectively, and solutal mass concentrations are C0
and C1 . Taking a horizontal coordinate x and a vertical
coordinate z , assuming the fluid satisfies Darcy Law,
neglecting the small quantity of the second order, we get
the linear non-dimensional equations:
 xx  zz  C C x  T  x
(1)
 t  x   xx   zz  H (   )
(2)
t   xx   zz  H (   )
(3)
Ct   x 
(4)
Here,  is stream function,  、  are perturbations
of temperature of fluid and solid, C is perturbations of
solute.  T and C are related to the Rayleigh
numbers that emerge naturally from the pure thermal and
pure solutal problems. H is scaled inter-phase heat
transfer coefficient.  is porosity-modified conductivity
ratio.  is diffusivity ratio. k is the thermal
conductivity. Le is Lewis number.
3 LINEAR STABILITY ANALYSIS
We use Fourier-mode solutions to discuss the stability
(see Pritchard & Richardson [2]).
   0 et e imx sin(nz) ,
   0 et e imx sin( nz)
(5)
t imx
  0 e e sin(nz) ,
C   0 et e imx sin(nz)
(6)
Substituting (5) and (6) into (1)-(4), and eliminate 0 ,
0 ,
2
DOUBLEDIFFUSIVE
CONVECTION MODEL
1 2
 C  k (  C )
Le
0
and
 0 , we obtain:
A 3  B 2  C  D  0
REACTION-
(7)
Here, A, B, C and D are functions with respect to M, N,
2 2
 T and C , where M  m 2 , N   n . For simplicity, we
In a horizontal porous layer of depth d, we assume that
won’t give the expressions of the coefficients A, B, C and D
© ST. PLUM-BLOSSOM PRESS PTY LTD
Stability Analysis of Thermosolutal Convection in a Horizontal Porous Layer Using a Thermal Non-equilibrium Model
DOI: 10.5503/J.PNSGE.2013.10.007
in the paper.
It is easy to find that D=0 is the critical condition for the
exchange of stability and the critical condition for
oscillatory convection occurs is
BC  AD  0
and C  0 .
M T( c )  
N 2)
C
(M   2 ) 2
M  2
 C  (

) H  O( H 2 )
M
M
M  2  k
By setting T( c ) / M  0 , then we obtain
4
M T( c )   2 
H  O( H 2 )
2(2 2  k ) 2 ( 2  1)
Hence,
C
  C  (1 
) H  O( H 2 )
2 2  k
T( c )  4 2

M 1  ( C  4 2 ) /[ 2(1   ) /  2  2 C k /( 2 2  k ) 3 ]
and C must satisfy (14)
(13)
C k  0
(14)


5 DISCUSSION
(9)
Obviously, when N   2 ,  T take the lowest value. So
we assume that N   2 , then paint the plot of variation of
 T with M for specific values of k and H. The values
(10)
of the other parameters are H=5, Le  2 ,   1 , C  30 .
 (M   2  k )
C (M   )
(M   )
1
1

)
 O( 2 )
H
 2 (M   2  k )
M 2
H
2
(
(12)
(8)
(ii)For large values of H, we obtain (assumed that Le  1
and N   2 )
(M   2 ) 2 1  
M  2
 T( c ) 
  C (1 
)  (11)
M
1
1
 O(
)
H
H2
Therefore, we assume that k=0, than our result recovers
the result of non-reaction.
Thus,
1 
1
1
(15)
 T( c )  (4 2   C )
 1
 O( 2 )

H
H
Here,
2 2  C
8 4
(16)
1 

2
2
Now we discuss the asymptotic analysis for both small
and large values of H. For large H, LTE is recovered.
Following, we will find the asymptotic solutions, and in
section 5 we will compare it with numerical solutions.
(i)When H is small, we get (assumed that Le  1
 T( c ) 
 M1
and substitute it into T( c ) / M  0 , we get
4 RESULTS AND ASYMPTOTIC ANALYSIS
and
2
2
2
3
Assuming that,
Figure 1 Variation of  T with
values of k
M
Figure 2 Variation of  T with
values of H
for specific
M
for specific
compare  O(c ) and T(c) , and their corresponding wave
numbers. We take five virtual numbers. When k = 1.8708 ,
 O( c )  T( c ) . So we obtain the critical value of k (k=1.8708)
From Figure 1, we can see that the critical Rayleigh
number T decreases with k increases. So when k
becomes smaller, the system will be more stable. Thus, the
existent of reaction will make the system instable. Form
Figure 2 ( Le  2 ,   1 , k  2 , C  30 ), we know that
(c )
for the exchange of oscillatory convection and stationary
convection. So when k  1.8708 , over-stability will be
considered, when k  1.8708 , oscillatory convection won't
occur.
From Table 1, we can see when 0  k  2 , the
variation of O with M for k is very small. So the Figure
can't give a obviously variation. But from the numerical
solutions, it can be seen that the critical Rayleigh number
 O(c ) decreases with k increases. So when k becomes
the critical Rayleigh number T(c) increases with H
increases. So when H becomes bigger, the system will be
more stable. Thus, the existent of non-equilibrium between
fluid and solid will make the system instable.
When  O( c )   T( c ) , oscillatory convection can occur. In
Table 1 (( Le  2 ,   1 , H  5 ,   0.2 , C  30 )), we
30
Physical and Numerical Simulation of Geotechnical Engineering
10th Issue, Mar. 2013
smaller, the system will be more stable.
Table 1 Comparison of

k
Table
114.8983
113.4234
112.3849
112.0339
2
(
Le  2
,
 1
M O(c )
12.859
12.796
12.070
11.973
112.5922
112.4721
112.3849
112.3548
15.293
15.181
15.098
15.070
will be considered, when H  4.8535 , oscillatory
,
convection won't occur. From Figure3, we can see,
 O( c )  T( c ) . So we obtain the critical value of H
with
 O(c )
M T(c )
and
with
M O(c )
for variation of
H
T(c)
M T(c )
 O(c )
4.0000
4.5000
4.8535
5.0000
109.1689
110.6322
111.6293
112.0339
11.553
11.773
11.916
11.973
107.3666
109.8712
111.6293
112.3548
Figure 3 Variation of
O
 O(c )
increases with H increases. Hence when H becomes larger,
the system will be more stable.
( H = 4.8535 ) for the exchange of oscillatory convection
and stationary convection. When H  4.8535 , over-stability
T(c)
k
 O(c )
k  2 ,   0.2 , C  30 ) we get, when H = 4.8535 ,
Table 2 Comparison of
M O(c ) for variation of
M T(c )
(c )
T
1.0000
1.2000
1.8708
2.0000
From
T(c) with  O(c ) and M T(c) with
H
M O(c )
14.092
14.582
14.927
15.070
with M for specific values of H
In Table 3, we compare the exact and asymptotic values
of the wave number and the critical Rayleigh number when
H is small. The other parameters are Le  1 ,
  1 , k  10 , C  30 . We can see that when H is small,
exact values are almost the same as the asymptotic values,
but when H grows bigger, the contrast of them are more
obviously.
Table 3 Comparison of exact and asymptotic values of M T(c ) and T(c) for small H. (E denotes the exact solution and
A denotes the asymptotic solution)
log10 H
M T(c ) ( E )
M T(c ) ( A)
T(c ) ( E)
T(c ) ( A)
-2.0
-1.5
-1.0
-0.5
0.0
0.5
9.86627
9.88063
9.92556
10.0630
10.4549
11.3640
9.85965
9.85976
9.86011
9.86120
9.86467
9.87562
69.4685
69.5334
69.7374
70.3708
72.2674
77.4214
69.4685
65.5019
69.6393
70.0738
71.4478
75.7923
H is large. The other parameters are Le  1 ,   1 ,
In Table (4), we compare the exact and asymptotic values
of the wave number and the critical Rayleigh number when
31
Stability Analysis of Thermosolutal Convection in a Horizontal Porous Layer Using a Thermal Non-equilibrium Model
DOI: 10.5503/J.PNSGE.2013.10.007
k  0 , C  30 . We can see when H is large, exact
values are almost the same as the asymptotic values.
Table 4 Comparison of exact and asymptotic values of
and
T(c)
for large H.
log10 H
M T(c ) ( E )
M T(c ) ( A)
T(c ) ( E)
T(c ) ( A)
2.5
3.0
3.5
4.0
4.5
5.0
10.1763
9.98205
9.91777
9.89715
9.89061
9.88853
9.56052
9.76503
9.82969
9.85014
9.85661
9.85865
74.9185
76.4605
76.9723
77.1367
77.1889
77.2055
74.7985
76.4497
76.9718
77.1369
77.1891
77.2056
[2].
6 CONCLUSIONS
Linear stability of double-diffusive convection in a
horizontal porous layer using a thermal non-equilibrium
model has been studied. The critical Rayleigh number and
the corresponding wave number for the exchange of
stability and over-stability are obtained. We find that the
critical Rayleigh number for stability increases when H
increases or k decreases. Differ from stability, the critical
Rayleigh number for over-stability increases when H
increases or k increases. We also find the critical values of
the parameters for exchange of stationary convection and
oscillatory convection. Then asymptotic solutions for both
small and large values of H are studied. We present
(c )
comparison of numerical and asymptotic values of M T
(c )
and T and for both small and large values of H . We
(c)
2
can see when H   or  0 , M T   and
(c )
T has a certain limit. For convenience, we assumed
that Le  1 , the errors of numerical and asymptotic values
are produced by the other parameters as k and C , and
we take the appropriate values to decreased the errors.
Although the variation of C is very small, it makes a
(c )
considerable affection of T , that's the affection of
double-diffusive convection.
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[4].
[5].
[6].
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32