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Chaotic Mixing of Viscous Fluids Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Chaotic mixing of viscous fluids Topological entanglement and transport barriers Emmanuelle Gouillart Joint Unit CNRS/Saint-Gobain, Aubervilliers Olivier Dauchot, CEA Saclay Jean-Luc Thiffeault, University of Wisconsin Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Outline 1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Plan 1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Fluid mixing is ubiquituous in the industry... Mixing step in many processes Low Reynolds number : no turbulence Closed (batch) or open flows Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows ... or in the environment Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows What we would like to know about fluid mixing Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows What we would like to know about fluid mixing Can we avoid unmixed regions ? Homogenization mechanisms : black/white → gray ? Mixing speed ? Quality of mixing ? How to define it and how to measure it ? Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Plan 1 Why study fluid mixing - Context 2 Mechanisms of mixing 3 What is the speed of dye mixing in closed flows ? 4 Mixing in open flows Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Mixing = transport, stretching and diffusion Large-scale mixing : particles of a fluid patch should disperse (fast) in the whole domain. This imposes to stretch the initial patch Diffusion is more efficient for thin filaments Arratia et al. 2004 Successive shears in different directions → rapid stretching and fluid dispersion. Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Chaotic advection : chaos in the fluid space ! How can one disperse and stretch fluid patches ? Slow (linear with time) stretching. No mixing across streamlines. Meunier and Villermaux 2003 Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Chaotic advection : chaos in the fluid space ! How can one disperse and stretch fluid patches ? Slow (linear with time) Successive shears in different stretching. directions → rapid stretching No mixing across streamlines. and fluid dispersion. Meunier and Villermaux 2003 Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Chaotic advection : chaos in the fluid space ! t 0 l0 λt l(t) = l0e Successive shears in different Neighbouring particles separate directions → rapid stretching exponentially with time. and fluid dispersion. Enough flow complexity ⇒ chaotic trajectories. Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Chaotic advection : chaos in the fluid space ! Lagrangian trajectories are determined by a dynamical system, that can be chaotic if it has enough degrees of freedom (≥ 3). 2-D incompressible flows : Hamiltonian dynamics. 3-D flows ( x_ = vx (t) = −ψ,y x_ = vx y_ = vy (t) = ψ,x y_ = vy Need for time dependency. z_ = vz Phase space = real fluid space Fountain et al., 2000 Aref 1984, Ottino 1989 chaotic advection Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Poincaré sections reveal mixed and unmixed regions Chaiken et al., 1986 Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Lyapunov exponents measure stretching t 0 l0 λt l(t) = l0e Muzzio et al., PoF 1991 Lyapunov exponent : mean stretching rate A measure of mixing efficiency Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Topological mixing A new description of chaotic advection Principle : characterize chaotic advection by the braiding/entanglement of fluid particles (as opposed to the stretching of fluid particles). Topological description 6= metric description Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Topological chaos imposed by stirrers Boyland, Aref, Stremler JFM 2000 Clever motion of rods ⇒ entangled trajectories of the rods, on which material lines are stretched exponentially with time. The topological complexity is characterized by the topological entropy (the entanglement) of the braid traced by the rods. t σ 2 + PSfrag replacements smart protocol σ1 ⇒ entangled braid → PSfrag replacements The motion of rods (its braid) gives a lower bound on the stretching rate of filaments. Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Topological chaos Topological entropy of the Topological entropy hbraid of a flow : rate of exponential braid : index of entanglement stretching of material lines t σ 2 PSfrag replacements σ1 PSfrag replacements L(t) = L0 exp(hflowt) hbraid ≤ hflow Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Topological chaos with ghost rods Periodic structures of the flow cannot be crossed by material lines. They may also braid and stretch filaments, like "ghost rods" Elliptical islands are ghost rods Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Topological chaos with ghost rods Periodic structures of the flow cannot be crossed by material lines. They may also braid and stretch filaments, like "ghost rods" Elliptical islands are ghost rods Material line Poincaré section Rod + islands ⇒ stretched on the displays many islands entangled braid. braid. Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Topological chaos with ghost rods Material line Poincaré section Rod + islands ⇒ stretched on the displays many islands entangled braid. braid. Elliptical islands account for topological chaos and behave like ghost rods. A simple glimpse at the Poincaré section can give a lower bound on the stretching rate of material lines. Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Topological chaos with ghost rods Periodic structures of the flow cannot be crossed by material lines. They may also braid and stretch filaments, like "ghost rods" Unstable (point-like) periodic orbits are ghost rods as well 2.5 2 1.5 braid h 1 0.5 0 0 5 10 15 20 25 30 number of strands in braid A small number of orbits can account for the topological entropy of the flow. Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Topological chaos with ghost rods Periodic structures of the flow cannot be crossed by material lines. They may also braid and stretch filaments, like "ghost rods" Unstable (point-like) periodic orbits are ghost rods as well 2.5 2 1.5 braid h 1 0.5 0 0 5 10 15 20 25 30 number of strands in braid A small number of orbits can account for the topological entropy of the flow. Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows The complexity of chaotic advection arises from the entanglement of ghost rods’ trajectories. Different characterization of fluid mixing in terms of braiding and topological entanglement, as opposed to metric stretching (Lyapunov exponents). [E. Gouillart, J.-L. Thiffeault, M. Finn, Phys. Rev. E, 73, 036311, 2006] [M. Finn, J.-L. Thiffeault, E. Gouillart, Physica D, 221, 92, 2006] Mixing and pAs Train tracks Automata Optimization Conclusions References Why study fluid mixing - ContextOptimization Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows •DesigningConsider periodic latticemixers of rods. for efficient topological chaos −1 • Move all the rods such that they execute σ1 σ2 with their Finnneighbor and (Boyland Thiffeault, et al., 2000; 2010 Thiffeault & Finn, 2006). p • The dilatation per period is χ2, where χ = 1 + 2 is the Silver Ratio! • ThisMaximize is optimal for topological a periodic lattice entropy of two rods (Follows fromper D'Alessandro rods exchange. et al. (1999)). Best configurations reach the 19 / 24 silver ratio for the entropy. Easy implementation with planetary gears. Same idea in some industrial planetary mixers ! Why study fluid mixing - Context Mechanisms of mixing What is the speed of dye mixing in closed flows ? Mixing in open flows Applications of ghost rods Extension to non-periodic orbits Oceanography Dynamics of granular media Floats in the Labrador sea Thiffeault, Chaos 2010 Puckett et al.
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