Environmental Engineering Reference
InDepth Information
γ
Re( )
k
1
k
2
0
k
max
k
Figure B.1. Example of dispersion relation for a twodiffusivespecies monodimen
sional system.
k
1
and
k
2
are the extremes of the range of unstable Fourier modes, and
k
max
represents the most unstable Fourier mode.
0
), we consider full sys
tem (
B.1
) and use a Taylor expansion to linearize this set of equations around the
homogeneous steady state, (
u
0
To study the effect of diffusion on the stability of (
u
0
,v
,v
0
);
⎛
⎞
10
∂
w
∂
⎝
⎠
.
2
w
t
=
J
w
+
D
∇
,
D
=
(B.6)
0
d
The solution of Eqs. (
B.6
) can be written in the form of a sum of Fourier modes:
W
k
e
γ
t
+
i
k
·
r
w
(
r
,
t
)
=
,
(B.7)
where
k
=
(
k
x
,
k
y
) is the wavenumber vector,
r
=
(
x
,
y
) is the coordinate vector, and
W
k
are the Fourier coefficients (
Murray
,
2002
).
We can obtain the relation between eigenvalues and wave numbers (known as the
dispersion relation, see Fig.
B.1
) b
y insert
ing Eq. (
B.7
)into(
B.6
) and searching for
nontrivial solutions. Setting
k
k
x
+
=
k
y
, we obtain
Dk
2

γ
I
−
J
+
=
0
.
(B.8)
For the state (
u
0
,v
0
) to be unstable with respect to small perturbations, the solution
of dispersion relation (
B.8
) should exhibit positive values of Re[
γ
(
k
)] for some wave
number
k
=
0. Using Eq. (
B.8
), we find that this condition is met when
d
∂
f
∂
2
4
d
∂
f
∂
d
∂
f
∂
u
+
∂
g
u
+
∂
g
u
∂
g
∂v
−
∂
f
∂
g
∂
∂v
>
0
,
−
>
0
.
(B.9)
∂v
∂v
u
The first of conditions (
B.5
) combined with the first of conditions (
B.9
) implies that
d
=
1, indicating that the systemcannot be unstable with respect to small perturbations
Search WWH ::
Custom Search