1. Arthur Straube PATTERNS IN CHAOTICALLY MIXING FLUID FLOWS Department of Physics, University of Potsdam, Germany COLLABORATION: A. Pikovsky, M. Abel URL: http://www.stat.physik.uni-potsdam.de

2. 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?

3. PATTERNS AND MIXING IN CLOSED FLOWS Passive scalar advection-diffusion [D. Rothstein, E. Henry, J.P. Gollub, Nature, 401, 770 (1999)]

4. Blinking vortex with BZ reaction [C.R. Nugent, W.M. Quarles, T.H. Solomon, Phys. Rev. Lett., 93, 218301 (2004)] advection-diffusion reaction-diffusionreaction-advection-diffusion

5. TEMPORAL CHAOS VERSUS SPATIAL MIXING Advection and diffusion  2D time-dependent flow: mixing [A. Straube, M. Abel, A. Pikovsky, Phys. Rev. Lett. 93, 174501 (2004)] Diffusion of a scalar Reaction  One or two species: fixed point or oscillations. Final state is homogeneous (completely mixed)

6. To allow for a nontrivial dynamics we consider the dynamics of several species (three or more) solution 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

7. 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

8. Analytical approach  time-independent velocity field  equal diffusion constants Separation ansatz: Space: eigenvalue problem Smallest nonzero eigenvalue Time: Substitution:

9. Asymptotic solution: decay of the advection-diffusion mode Lyapunov exponent due to reaction (“chemistry”)  spacially homogeneous regime is stable if  a regime with regular (reaction) oscillations is always stable  a nontrivial transition is possible only if and large enough Thus: The critical reaction rate

10. 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)] Time-periodic: Irregular (weakly turbulent): is a - telegraph process with independent exponentially distributed time intervals and independent uniformly distributed phases

11. Chaos meets chaos Decay of the advection-diffusion mode Time-independent velocity: Time-dependent velocity:

12. Numerical results Da Different diffuson constants. Connection with the theory of synchronization

13. Partial case Here, the mobility of one species is zero:  surface reactions  processes in marine sediments Chemical and biological applications: Numerical simulation of a fully nonlinear model  Near the threshold linear growth of variances (contrast)  Strong intermittency beyond stability threshold

14. Close to stability threshold: Concentration patterns Far from stability threshold:

15. [A. Straube, A. Pikovsky, in progress] CHEMICAL INSTABILITY INDUCED BY A MIXING FLOW RAD with time-independent reaction Classical Turing instability (no advection)

16. U 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).

17. Question: can mixing flow cause instability? U  Both species are davected  Only one species is advected 1D problem with a steady sinus profile

18. Typical patterns 2D problem Mixing flow can lead to instability for both cases: when one or two species are advected

19. CONCLUSIONS  Advection-diffusion: decaying, long-living (persistent) patterns  Chaotic reaction leads to nontrivial patterns  Characterization of transition can be done by generalized transversal LE  Mixing flows can induce instability