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
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
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
Fluid mixing is ubiquituous in the industry... Mixing step in many processes Low Reynolds number : no turbulence Closed (batch) or open flows
... or in the environment
What we would like to know about fluid mixing
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 ?
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
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
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
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
Chaotic advection : chaos in the fluid space ! t 0 l 0 l ( t ) = l 0 e λ t Successive shears in different Neighbouring particles separate directions → rapid stretching exponentially with time. and fluid dispersion. Enough flow complexity ⇒ chaotic trajectories.
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 � ˙ = v x ( t ) = − ψ , y x ˙ = v x x y ˙ = v y ( t ) = ψ , x y ˙ = v y ˙ = v z z Need for time dependency. Phase space = real fluid space Fountain et al., 2000 Aref 1984, Ottino 1989 chaotic advection
Poincaré sections reveal mixed and unmixed regions Chaiken et al., 1986
Lyapunov exponents measure stretching t 0 l 0 l ( t ) = l 0 e λ t Muzzio et al., PoF 1991 Lyapunov exponent : mean stretching rate A measure of mixing efficiency
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 � = metric description
� 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 + smart protocol ⇒ entangled braid → The motion of rods (its braid) gives a lower bound on the stretching rate of filaments.
� Topological chaos Topological entropy of the Topological entropy h braid of a flow : rate of exponential stretching of material lines braid : index of entanglement t L ( t ) = L 0 exp ( h flow t ) h braid ≤ h flow
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
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.
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.
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 h braid 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.
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 h braid 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.
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]
Designing mixers for efficient topological chaos Finn and Thiffeault, 2010 √ Maximize topological entropy per rods exchange. Best configurations reach the silver ratio for the entropy. Easy implementation with planetary gears. Same idea in some industrial planetary mixers !
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. 2009
Statistics of concentration fluctuations Evolution according to the advection-diffusion equation ∂ C ∂ t + v · ∇ C = D ∆ C One fluid particle is stretched and thinned up to the � Batchelor scale w B = D /λ
Statistics of concentration fluctuations One fluid particle is stretched and thinned up to the � Batchelor scale w B = D /λ The concentration levels of several fluid particles are averaged inside boxes of size w B → fluctuations of C decay.
Statistics of concentration fluctuations The concentration levels of several fluid particles are averaged inside boxes of size w B → fluctuations of C decay. A particle (concentration fluctuation) is mixed once it has reached w B and is averaged with other particles. Finite-size Lyapunov exponents [Boffetta et al. 2000, 2001]
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
What is the speed of homogenization ? Eulerian description Pierrehumbert 1994, Rothstein et al. 1999 Jullien et al. 2000 self-similar concentration field : strange eigenmode The variance σ 2 ( C ) decays exponentially (at long times)
What is the speed of homogenization ? Lagrangian description Eulerian description Antonsen et al. 1996 Exponential stretching should yield an exponential decay. Villermaux and Duplat, 2003 Pierrehumbert 1994, Rothstein et al. 1999 Jullien et al. 2000 self-similar concentration field : strange eigenmode The concentration PDF evolves The variance σ 2 ( C ) decays by auto-convolution due to exponentially (at long times) (random) self-averaging by diffusion [Turbulence].
(Non-asymptotic) homogenization mechanisms Stretching of filaments
(Non-asymptotic) homogenization mechanisms Stretching of filaments Self-averaging by diffusion
(Non-asymptotic) homogenization mechanisms Stretching of filaments Self-averaging by diffusion Persistent contrast in low-stretching regions
(Non-asymptotic) homogenization mechanisms Stretching of filaments Self-averaging by diffusion Persistent contrast in low-stretching regions Chaotic region ⇒ propagation of poorly stretched patches
(Non-asymptotic) homogenization mechanisms Stretching of filaments Self-averaging by diffusion Persistent contrast in low-stretching regions Chaotic region ⇒ propagation of poorly stretched patches Mixing is controlled by the worst-stretched elements !
Wall-controlled slow mixing 1 rod, figure-eight periodic protocol ( ∞ ). Chaotic advection : stretching and folding.
Wall-controlled slow mixing
Wall-controlled slow mixing Slow (algebraic) mixing
Wall-controlled slow mixing No-slip condition ⇒ unmixed fluid at the wall leaks into the mixed region ( d ( t ) ∼ 1 / t ) ⇒ contamination of the mixing pattern
Wall-controlled slow mixing Model : modified baker’s map Accounts for algebraic dynamics. No-slip condition ⇒ unmixed fluid at the wall leaks into the mixed region ( d ( t ) ∼ 1 / t ) ⇒ contamination of the mixing pattern
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