Arthur Straube

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PATTERNS IN CHAOTICALLY MIXING FLUID FLOWS
Arthur Straube
Department of Physics, University of Potsdam,
Germany
COLLABORATION A. Pikovsky, M. Abel
URL http//www.stat.physik.uni-potsdam.de
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• OUTLINE
• MOTIVATION
• - mixing in a relation to microfluidics
• - reaction-advection-diffusion systems variety
  of patterns
• TEMPORAL CHAOS VS SPATIAL MIXING
• - nontrivial transition spacially
  homogeneous/spacially
• inhomogeneous state (theory and numerical
  simulation)
• CHEMICAL INSTABILITY INDUCED BY MIXING FLOWS
• - mixing leads to homogenezation
• - can mixing destabilize homogeneous state?

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PATTERNS AND MIXING IN CLOSED FLOWS
Passive scalar advection-diffusion
D. Rothstein, E. Henry, J.P. Gollub, Nature,
401, 770 (1999)
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Blinking vortex with BZ reaction
C.R. Nugent, W.M. Quarles, T.H. Solomon, Phys.
Rev. Lett., 93, 218301 (2004)
advection-diffusion
reaction-diffusion
reaction-advection-diffusion
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TEMPORAL CHAOS VERSUS SPATIAL MIXING
A. Straube, M. Abel, A. Pikovsky, Phys. Rev.
Lett. 93, 174501 (2004)
Diffusion of a scalar
Advection and diffusion

• 2D time-dependent flow mixing

Reaction

• One or two species fixed point or oscillations.
• Final state is homogeneous (completely mixed)

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To allow for a nontrivial dynamics we consider
the dynamics of several species (three or more)
We start with the reaction equations
Additionally, each component is subjected to
advection and diffusion.
We also assume

• the velocity field is incompressible
• concentrations do not influence the flow
• no-flux/periodic boundary conditions

Reaction-Advection-Diffusion (RAD).
Spatio-temporal dynamics
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Spatially homogeneous chaos
Spatially homogeneous solution advection and
diffusion terms vanish
solution of chemistry
Problem of stability
Linearization near the homogeneous solution
Jacobi-matrix
Stability is defined by the largest transverse
Lyapunov exponent
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Analytical approach

• time-independent velocity field
• equal diffusion constants

Space eigenvalue problem
Time
Substitution
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Asymptotic solution
decay of the advection-diffusion mode
Lyapunov exponent due to reaction (chemistry)
Thus
The critical reaction rate
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Generalizations and further results

• time-dependent velocity field
• different diffusion constants

A simple 2D model (mixing) flow
T.M. Antonsen, et al., Phys. Fluids. 8, 3094
(1996)
Irregular (weakly turbulent)
is a - telegraph process with
independent exponentially distributed time
intervals and independent uniformly distributed
phases
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Chaos meets chaos
Decay of the advection-diffusion mode
Time-independent velocity
Time-dependent velocity
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Different diffuson constants. Connection with the
theory of synchronization
Numerical results
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Partial case
Chemical and biological applications

• surface reactions
• processes in marine sediments

Here, the mobility of one species is zero
Numerical simulation of a fully nonlinear model

• Near the threshold linear
• growth of variances (contrast)
• Strong intermittency beyond
• stability threshold

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Concentration patterns
Close to stability threshold
Far from stability threshold
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CHEMICAL INSTABILITY INDUCED BY A MIXING FLOW
A. Straube, A. Pikovsky, in progress
RAD with time-independent reaction
Classical Turing instability (no advection)
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Consider now the parameter region, where the
Turing patterns are not possible (without
advection) Influence of advection 1D shear flow
induces instability D.A. Vasquez, Phys. Rev.
Lett. 93, 104501, (2004).
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Question can mixing flow cause instability?

• Both species are davected
• Only one species is advected

1D problem with a steady sinus profile
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2D problem
Mixing flow can lead to instability for both
cases when one or two species are advected
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