MS66: Topology and Mixing in Fluids

MS66: Topology and Mixing in Fluids Mark Stremler, Virginia Tech (replacing P. Boyland) Topological chaos in cavities and channels Matt Finn, University of Adelaide Topological entropy of braids on the torus Tsuyoshi Kobayashi, Nara Women s U., Japan Realizing topological chaos with simple mechanisms Kai de Lange Kristiansen, UCSB Braid theory and microparticle dynamics in ferrofluids

Topological chaos and fluid mixing in cavities and channels Mark A. Stremler and Jie Chen Department of Engineering Science & Mechanics Virginia Polytechnic Institute & State University SIAM Conference on Applications of Dynamical Systems

Topological chaos Complexity is built in the flow due to the topology of the boundary motions R N : 2D fluid region with N stirring rods stirrers move on periodic orbits stirrers = solid objects or fluid particles stirring with solid rods: stirring with point vortices: stirring with ghost rods : Boyland, Aref & Stremler (2000) J. Fluid Mech. Finn, Cox & Byrne (2003) J. Fluid Mech. Boyland, Stremler & Aref (2003) Physica D Gouillart, Thiffeault & Finn (2006) Phys. Rev. E

Topological chaos Complexity is built in the flow due to the topology of the boundary motions R N : 2D fluid region with N stirring rods stirrers move on periodic orbits stirrers = solid objects or fluid particles t stirrer motions generate a mapping f : R N R N (a diffeomorphism) stirrer trajectories generate braids in 2+1 dimensional space-time

Thurston-Nielsen Theory Thurston (1988) Bull. Am. Math. Soc. Casson & Bleiler (1988) Automorphisms... A stirrer motion f is isotopic to a stirrer motion g of one of three types:

the actual fluid motion Thurston-Nielsen Theory Thurston (1988) Bull. Am. Math. Soc. Casson & Bleiler (1988) Automorphisms... A stirrer motion f is isotopic to a stirrer motion g of one of three types: (i) finite order (f.o.): (ii) pseudo-anosov (pa): the nth iterate of g is the identity g has... an ideal motion: the TN representative dense orbits, Markov partition with transition matrix A λ > 1 : expansion or dilation = PF eigenvalue of A entropy equal to the topological entropy h top (g) = log(λ) (iii) reducible: g contains both f.o. and pa regions

Fixed stirrer motions: Isotopy all fluid motions with stirrers fixed in space, with rotations allowed Isotopy to the identity: a stirrer motion h is isotopic to the identity if the same result could have been obtained by a fixed stirrer motion for 0 or 1 stirrer, all motions are isotopic to the identity for 2 stirrers, all motions (or 2nd iterate) are isotopic to the identity Isotopic motions: motions f and g are isotopic if g = h f, with h isotopic to the identity Handel s isotopy stability theorem: the complex dynamics of the pa map remains under isotopy log(λ) provides a lower bound on the topological entropy

Braids on 3 strands or Stirring with 3 rods Finite Order: σ 1 σ 2 time σ 1 σ 2 fix the rods and unwind the outer boundary no lower bound for stretching in the flow pseudo Anosov: time σ 1 1 σ 2 σ 1 1 σ 2 complexity cannot be removed with rods fixed λ = 1 2 (3 + 5) every topologically non-trivial curve grows in length like λ n under iteration

Stirring experiments with 3 rods finite order 0 pseudo Anosov R + R + 1 L + L 2

Stirring with ghost rods Gouillart, Thiffeault & Finn (2006) Phys. Rev. E Finn, Thiffeault & Gouillart (2006) Physica D Topological chaos in doubly-periodic sine flow with no rods Can we generate topological chaos in a realistic flow without using any solid stirring rods?

Lid-driven cavity flow Chien, Rising & Ottino (1986) J. Fluid Mech. Leong & Ottino (1989) J. Fluid Mech. steady or time dependent 2D flow in a rectangular cavity steady forcing periodic forcing final initial

Our lid-driven cavity flow d U L 2c 2a U C design parameters: y x d U R 2b top boundary split into 3 segments other boundaries are fixed time-periodic operation of steady Stokes flow boundary velocities aspect ratio α = a/b boundary length ratio find periodic points that act as stirring rods β = 2c/d

Our lid-driven cavity flow Numerical solution of the 2D biharmonic equation for the streamfunction in Stokes flow Meleshko & Gomilko (1997 ) PRSLA, (2004) PRSLA 2 2 ψ(x, y) = 0 d U L y 2c 2a U C x d U R 2b x = ±a : ψ = 0, ψ/ x = 0 y = b : ψ = 0, ψ/ y = V (x) y = b : ψ = 0, ψ/ y = 0 ψ(x, y) = ψ ee (x, y) + ψ eo (x, y) + ψ oe (x, y) + ψ oo (x, y)

Our lid-driven cavity flow ψ ee (x, y) = b m=1 ( 1) m P m p m (y) cos(α m x) a α m n=1 ( 1) n β n Q n q n (x) cos(β n y) x = ±a : ψ ee = 0, ψ ee / x = 0 y = ±b : ψ ee = 0, ψ ee / y = ±U ee (x) p m (y) = b tanh(α m b) cosh(α my) cosh(α m b) y sinh(α my) cosh(α m b) q n (x) = a tanh(β n a) cosh(β nx) cosh(β n a) x sinh(β nx) cosh(β n a) α m = β n = (2m 1)π 2a (2n 1)π 2b

Our lid-driven cavity flow ψ ee (x, y) = b m=1 ( 1) m P m p m (y) cos(α m x) a α m n=1 ( 1) n β n Q n q n (x) cos(β n y) key to the solution: assume infinite system of equations: P m = P 0 + ξ m Q n = Q 0 + η n ξ m b (α m b) = η n a (β n a) = η n n=1 ξ m m=1 4α 2 mβ n (α 2 m + β 2 n) 2 + F m(p 0, Q 0, α m, β n, U m ) 4β 2 nα m (α 2 m + β 2 n) 2 + H n(p 0, Q 0, α m, β n )

Our lid-driven cavity flow asymptotic approximation for solve finite system for P 0, Q 0 ξ m, η n ψ ee (x, y) = b m=1 ( 1) m P m p m (y) cos(α m x) a α m n=1 ( 1) n β n Q n q n (x) cos(β n y) = b M m=1 ( 1) m α m ξ m p m (y) cos(α m x) a N n=1 ( 1) n β n η n q n (x) cos(β n y) +P 0 b M+M m=1 ( 1) m α m p m (y) cos(α m x) Q 0 a N+N n=1 ( 1) n β n q n (x) cos(β n y)

Topological chaos in lid-driven cavity flow α = a/b = 3 β = 2c/d = 1 U 1.192 U R+ xc = (0, y0 ) xc xl xr = (x1, y1 ) xr change position after time τ 5.379 xl = ( x1, y1 ) is a stagnation point 1.192 U U xl L xc xr puncture the domain at the periodic orbits, examine the motion

Topological chaos in lid-driven cavity flow Poincaré section (a) design points are hyperbolic (b) xc elliptic point trajectories

3 1 5/2 2 3/2 elliptic orbits give a braid on 6 strands motion is reducible to a pa braid on 3 strands 1/2 1 [ 1 h top = log 2 (3 + ] 5) 0.96 1/2 t 0 1 2 3 4 5 6 x 1 2 3 4 5 6 0

Initial Stretching of non-trivial material lines 1/2 1 2 h flow 1.0 > h top Finn, Cox & Byrne (2003) JFM 3 h flow 1.92 2 h top h flow 0.80 < h top

other cases 6 aspect ratio α = a/b 5!=2 "=1 boundary length ratio d 2c U L y U C β = 2c/d d U R UC / UR 4 3 2!=3 "=1!=2 "=2 x 2b 1 2a!U 2.530 U U 0 0 1 2 3 4 5 6 U L / U R how does this complexity hold up under perturbation?

Extension to three dimensions Braided Pipe Mixer Finn, Cox & Byrne (2003) Phys. Fluids braided pipe inserts do not mix well

Lid-driven channel flow steady 3D flow in a rectangular channel surface grooves Stroock et al. (2002) Science electro-osmotic flow Qian & Bau (2002) Anal. Chem. lid-driven cavity flow + channel flow mixes well

Topological chaos in a lid-driven channel 1.231 U U lid-driven secondary flow + axial Poiseuille flow (V) α=2 Vmax = U l β=2 l 5.372b

Topological chaos in a lid-driven channel How does this compare with the Braided Pipe Mixer? How does the stretching compare with the 2D cases? Does the braiding matter in this flow? How is this affected by perturbations?