Environmental Engineering Reference
In-Depth Information
Murray
,
2002
;
Henderson et al.
,
2004
):
∂
u
2
u
t
=
f
(
u
,v
)
+∇
,
∂
∂v
∂
2
t
=
g
(
u
,v
)
+
d
∇
v,
(B.1)
where
t
is time,
f
and
g
are the local reaction kinetics,
d
is the ratio
d
=
d
2
/
d
1
2
is the Laplace
between the two diffusivities
d
1
and
d
2
of
u
and
v
, respectively, and
∇
operator. All variables are in dimensionless units.
Turing
(
1952
) demonstrated that this diffusive system exhibits diffusion-driven
instability if (i) in the absence of diffusion the homogeneous steady state is linearly
stable (i.e., stable with respect to small perturbations) and (ii) when diffusion is present
the homogeneous steady state is linearly unstable. Thus we first need to determine
the homogeneous steady state (
u
0
,
2
u
v
0
) as the solution of Eqs. (
B.1
) with
∇
=
2
∇
v
=
0 (homogeneous state) and
∂
u
/∂
t
=
∂v/∂
t
=
0 (steady state):
f
(
u
0
,v
0
)
=
0
and
g
(
u
0
0. Then we need to impose the condition that this solution be stable
in the absence of diffusion. To this end, we can study the stability of (
u
0
,
,v
0
)
=
v
0
) with
respect to small perturbations,
u
−
u
0
v
−
v
0
w
=
,
(B.2)
around the steady state. For small perturbations of the steady homogeneous state
(i.e., for
|
w
|→
0) system (
B.1
) can be linearized around (
u
0
,
v
0
). Using a linear
Taylor expansion we have
u
0
,v
0
,
⎛
⎞
∂
f
∂
f
∂v
d
w
d
t
=
∂
u
⎝
⎠
J
w
,
J
=
(B.3)
∂
g
∂
g
∂v
∂
u
where
J
is the Jacobian of dynamical system (
B.1
).
The solutions of this set of equations are in exponential form and express the
temporal evolution of the perturbation of the homogeneous steady state, where
γ
is
an eigenvalue of system (
B.1
), i.e., a solution of the secular polynomial
|
J
−
γ
I
|=
0
,
(B.4)
where
I
is the identity matrix.
When the real part of
γ
γ
|
|
→∞
and the
steady homogeneous state (
u
0
,v
0
) is linearly stable with respect to small perturbations.
From the analysis of Eq. (
B.4
) we obtain that this condition is met when
∂
,Re[
], is negative,
w
tends to zero for
t
u
+
∂
f
g
∂v
<
∂
u
∂
f
∂v
−
∂
g
f
∂v
∂
g
0
,
>
0
,
(B.5)
∂
∂
∂
u
with all derivatives being calculated in (
u
0
,v
0
)(
Murray
,
2002
).
Search WWH ::
Custom Search