Mixing fluids with chaos: topology, ghost rods, and almost invariant sets

Transcription

1 Mixing fluids with chaos: topology, ghost rods, and almost invariant sets Mark A. Stremler Department of Engineering Science & Mechanics Virginia Polytechnic Institute & State University

2 Collaborators/Colleagues Hassan Aref DTU & Virginia Tech Philip Boyland U Florida Jean-Luc Thiffeault U Wisconsin Madison Shane Ross Virginia Tech Andrew Duggleby Texas A&M Jie Chen Mohsen Gheisarieha Piyush Grover Pankaj Kumar funded by the U.S. National Science Foundation

3 Fluid mixing is a common necessity Mixing is essential in: food & beverage production pulp & paper production chemical processing - polymers & polymer blends - personal care products biochemical processing - pharmaceuticals - waste water treatment petrochemical processing - fossil fuels - asphalt nanoscale structures in polymers: David Zumbrunnen Clemson University multi-layered polymer films electrically conducting plastics fibrous polymer composites

4 Mixing and Stirring Mixing is the process of increasing spatial homogeneity in a combination of two or more substances To obtain (uniform) texture it is necessary to mix up the material, and to accomplish this it is necessary to attenuate the material this attenuation is only the first step in the process of mixing all involve the second process, that of folding, piling, or wrapping, by which the attenuated layers are brought together in the case of some fluids the physical process of diffusion completes the admixture. For miscible fluid-fluid systems: O. Reynolds (1894) Nature 50, Stirring: Mixing: increasing mean values of gradients by stretching and folding through advection decreasing mean values of gradients through diffusion

5 Stirring in laminar and turbulent flows turbulent mixing spoon in coffee laminar mixing eccentric journal bearing flow

6 Stirring by chaotic advection in laminar flows chaotic advection: the occurrence of chaotic particle trajectories in a regular (non-chaotic) Eulerian velocity field chaotic advection Aref (1984), Ottino (1989) rapid stretching and folding of material interfaces Welander (1955) enhanced mixing chaotic advection can occur in unsteady 2D flow or any 3D flow

7 Stirring a viscous fluid with 3 rods experiment in glycerin at low Reynolds number Boyland, Aref & Stremler (2000) JFM 403, 277 R + first interchange L + L second interchange

8 finite order 3 R + 0 L σ 1 time σ 2 2 9

9 pseudoanosov R L 1 6 time σ σ 2

10 Topological chaos through stirring Complexity is built in the flow due to the topology of boundary motions Thurston-Nielsen classification theorem: a periodic stirring motion can be one of three topological types t finite order topologically trivial pseudo-anosov chaotic reducible both f.o. & pa

11 Thurston Nielsen Classification Theorem TN theorem gives a quantitative lower bound on the stretching of non-trivial material lines in the flow non-trivial material lines grow as l l 0 λ n bounded stretching rate λ λ TN also use topological entropy finite order pseudo-anosov n = 0 n = 2

12 Topological chaos in a viscous fluid finite order pseudo Anosov ( 3+ 5 ) λ TN = 1 2 h TN = log(λ TN )= finite order: chaos present due to fluid dynamics pseudo-anosov: chaos guaranteed by motion of stirrers

13 A taffy puller stirring device one fixed rod with two moving rods 3-rod taffy puller compute by measuring length of line

14 both pseudoanosov and finite order stirring motions are possible pseudo Anosov finite order λ TN = λ TN =1 h TN = log(λ TN ) 1.76 h TN =0 h f 1.94 h f 2.0

15 both pseudoanosov and finite order stirring motions are possible pseudo Anosov λ TN = h TN = log(λ TN ) 1.76 h f 1.94 stirring experiment with glycerin

16 finite order motions are topologically trivial, but the dynamics can produce significant levels of chaos why consider topological chaos? topology persistence of behavior under perturbation for finite order motions, perturbations can significantly reduce levels of mixing initial steady forcing periodic forcing lid-driven cavity flow Chien, Rising & Ottino (1986) final

17 Perturbing the system change relative positions of the rods change dynamics of the flow, preserve topology of the rod motions finite order: stretching depends on the fluid dynamics h f 2.0 h f 2.0 h f 1.5

18 Perturbing the system change relative positions of the rods change dynamics of the flow, preserve topology of the rod motions pseudoanosov: stretching depends on rod topology h f 1.9 h f 2.1 h f 2.0

19 Planetary mixers planetary gear J.-L. Thiffeault & M. Finn (2006) Phil. Trans. R. Soc. London A fixed stirrer moving stirrer fixed gear rods move on crossing paths

20 J.-L. Thiffeault and M. D. Finn, Topology, Braids, and Mixing in Fluids, Philosophical Transactions of the Royal Society A 364, (2006).

21 Stirring with fluid particles TN theorem can be based on period motions of fluid particles one rod moving on an epicyclic trajectory Gouillart, Thiffeault & Finn (2006) Phys. Rev. E ghost rods solid rods Fluid is wrapped around ghost rods in the fluid f.o. rod motion + ghost rod motion = pa stirring

22 Designing a system with ghost rods design a flow with ghost rods that appear when and where we want them: a lid-driven cavity flow u = ψ y = ± N n=1 U n sin(nx/2) ψ =0 y x 2! 2b ψ =0 u = ψ y = N n=1 U n sin(nx/2)

23 Stokes flow in a lid-driven cavity u = ψ y = ± N n=1 U n sin(nx/2) u wall x Exact solution for the stream function f n (y) =y cosh(nb/2) sinh(ny/2) b sinh(nb/2) cosh(ny/2) C n = 2 [sinh(nb)+nb] 1

24 u top x generating pseudoanosov periodic orbits = U [ 1 βψ1 + βψ 2 ] u bottom β = U = x L =(π x 0, 0) is a stagnation point R + x C =(π, 0) x R =(π + x 0, 0) change position after time τ

25 u bottom x generating pseudoanosov periodic orbits = U [ 1 βψ1 + βψ 2 ] u top β = U = instantaneous switching every τ =0.5 x L =(π x 0, 0) is a stagnation point R + x C =(π, 0) x R =(π + x 0, 0) change position after time τ L {x L, x C, x R } trajectories produce a pseudoanosov braid

26 α =1/3 β = U = Poincaré section

27 α = 1/3 β = U = htn = ln λtn = hflow = ln λ the TN theorem gives an excellent estimate of the flow behavior

28 elliptic points: Periodic points as ghost rods hyperbolic points: parabolic points: material lines Gouillart, Thiffeault & Finn (2006) Phys. Rev. E

29 elliptic points: Periodic points as ghost rods hyperbolic points: parabolic points: parabolic points act most like flow around a solid rod however they are structurally unstable Gouillart, Thiffeault & Finn (2006) Phys. Rev. E

30 β =0.415 Perturbing the system change relative strengths of ψ 1, ψ 2 (vary β )

31 β =0.415 Perturbing the system change relative strengths of ψ 1, ψ 2 (vary β ) cylinder in Stokes flow this ghost rod structure acts like a semi-permeable rod

32 β = β = β = 0.430

33 Perturbing β : ψ = U [ 1 βψ1 + βψ 2 ] h h TN β β

34 Perturbing β : ψ = U [ 1 βψ1 + βψ 2 ] h h TN β β

35 Perturbing β : ψ = U [ 1 βψ1 + βψ 2 ] h h TN β β

36 Perturbing β : ψ = U [ 1 βψ1 + βψ 2 ] h h TN β β

37 Can we analyze stirring with ghost rods using topological chaos when there are no (obvious) low-order periodic points? use a set-oriented approach and consider almost invariant or almost cyclic sets Dellnitz, Froyland, et al. almost invariant sets (AIS) regions of fluid that stick together for a significant length of time

38 Set oriented approach t, τ

39 Almost invariant sets The eigenvectors of P show the almost invariant sets present in the phase space. The eigenvalues of P give a measure of how ʻleakyʼ the regions are. The zero contour of the second eigenvector in phase space gives the partition into two almost invariant sets with (almost) minimum transport between them Zero contours of lower eigenvectors give different partitions, which have a higher transport rates among themselves, and hence they are less invariant.

40 Almost invariant sets in the critical case almost invariant sets surround the periodic parabolic points

41 Almost invariant sets below the critical case almost invariant sets persist even without periodic points

42 Time dependent motion of AIS AIS exhibit pa braiding motion below the critical case system behavior remains close to that with periodic orbits for reasonable perturbations TN theorem gives accurate estimate of line stretching even though it does not formally apply

43 Summary/Conclusions Thurston-Nielsen classification theorem can be used to analyze stretching and folding in a fluid predictions of chaos can be based on limited data regarding the topology of boundary motions periodic points can act as ghost rods that assist in (cause?) stretching and folding the surrounding fluid almost invariant sets appear to explain topological entropy of the flow almost invariant sets can act as ghost rods