Non-Turing stationary patterns in flow-distributed oscillators
with general diffusion and flow rates
Razvan A. Satnoianu
1
and Michael Menzinger
2
1
Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom
2
Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 3H6
Received 15 October 1999
An analytical prediction P. Andresen et al., Phys. Rev. E 60, 297 1999 and its experimental confirmation
M. Kaern et al., Phys. Rev. E 60, 3471 1999 establish a mechanism for forming stationary, space-periodic
structures in a reactive flow reaction-diffusion-convection system with equal diffusion and flow rates. In this
paper we generalize the analysis to systems with unequal diffusion and flow rates. Interestingly, stationary
waves also exist outside the oscillatory Hopf domain of the batch system—hence the parameter space in which
these structures exist is bigger than that initially predicted P. Andresen et al., Phys. Rev. E. 60, 297 1999
for equal diffusion and flow rates. On the other hand, we find that these stationary waves exist only for
parameter values outside of and up to the Turing regime. We clarify the nature of the instability in terms of a
boundary-forcing problem, whereby a time-periodic pattern is carried over the whole domain by the flow while
the phase is fixed at the inflow boundary.
PACS numbers: 82.20.-w
I. INTRODUCTION
Andresen et al. 1 proposed an alternative spatial insta-
bility mechanism that gives rise to stationary, space-periodic
patterns in a time-oscillating reaction-diffusion-convection
system with equal diffusion and flow rates through the con-
stant forcing at the inflow boundary of the flow domain see
also the earlier contribution 2. This mechanism is interest-
ing since it differs fundamentally from the Turing 4 and
differential flow instabilities DIFI3, which require differ-
ential transport of the key species, and since the conditions
for the alternative instability may be readily realized in
chemical flow reactors, using active media in the oscillatory
Hopf domain. This prediction was confirmed experimen-
tally by Kaern and Menzinger 5. From their results it is
clear that the patterns arise by a mechanism that is essentially
kinematic. Accordingly, the flow carries temporal oscilla-
tions into space, while the oscillation phase is locked at the
inflow boundary through a constant boundary condition.
Therefore we refer to these stationary structures as flow-
distributed oscillations FDO.
In this work we generalize the above results to the case of
a reaction-diffusion-convection system with differential
transport, i.e., with different diffusion and flow rates. To dis-
tinguish the resulting waves for this more general case
from the FDO waves, we refer to them here as flow-
distributed structures or FDS. The analysis provides the
boundary of the FDS instability in the parameter domain and
clarifies its relation to Turing and DIFI patterns. We analyze
how this instability connects with the Turing and DIFI do-
mains and discuss the interface with, and difference from,
the two classical, differential transport-induced instability
mechanisms. This paper sheds light on how the region,
where stationary Turing patterns are observed in the absence
of a bulk flow, may be extended by the presence of a flow.
We find that an unstable Hopf reference state is not neces-
sary for FDS to occur, and that their domain extends into the
region where the reference state is stable. Our analytical re-
sults confirm the kinematic interpretation 5 that the FDO
instability is being driven by boundary forcing that freezes
the temporal oscillation phase of the species traveling along
the flow domain until the other boundary is encountered.
II. A REACTION-DIFFUSION-CONVECTION SYSTEM
WITH UNEQUAL DIFFUSION AND FLOW RATES:
AN IONIC CHEMICAL SYSTEM WITH CUBIC
AUTOCATALYTIC STEP
We generalize the problem considered in 1 by employ-
ing a model for a differential-flow reactor based on applying
an electric field to a reacting medium with an ionic version
of cubic autocatalator or Gray-Scott kinetics 6. The dif-
ferential flow or migration of the reacting species arises
since substrate A
+
and autocatalyst B
+
have different drift
velocities due to their different diffusion coefficients. By us-
ing the cubic autocatalator model as opposed to the Bruss-
elator model in 1 it simplifies considerably the details of
the subsequent calculations while still preserving all the fea-
tures of the general case.
The model assumes that we have a precursor P
+
present
in excess. To maintain electroneutrality we also require a
further species Q
-
to be present in the reactor at a concen-
tration similar to that of P
+
, though this species does not
take part in the reaction. We assume that P
+
decays at a
constant rate to form the substrate A
+
via
P
+
→A
+
, rate=k
0
p
0
, 2.1
where p
0
is the initial concentration of the reservoir species
P
+
. The substrate A
+
and autocatalyst B
+
subsequently re-
act according to the scheme
A
+
+2 B
+
→3 B
+
, rate=k
1
ab
2
, 2.2
B
+
→C
+
, rate=k
2
b , 2.3
PHYSICAL REVIEW E JULY 2000 VOLUME 62, NUMBER 1
PRE 62 1063-651X/2000/621/1137/$15.00 113 ©2000 The American Physical Society