Piecewise isometries and mixing in granular tumblersrsturman/talks/banff.pdf · Piecewise...

Piecewise isometries and mixing in granulartumblers

Rob Sturman

Department of MathematicsUniversity of Leeds

BIRS Workshop on Low Complexity Dynamics, 28 May 2008Banff

Joint work with Steve Meier, Julio Ottino, NorthwesternSteve Wiggins, University of Bristol

Rob Sturman Granular mixing

Mixing

Mixing of granular materials:is important — Science 125th anniversary identifiedgranular flow as one of the 125 big questions in Scienceis ubiquitous — pharmaceuticals, food industry, ceramics,metallurgy, constructionwas initially explained by analogies with fluid mixing —hence terms like granular shear and granular diffusion

But the big difference is that granular materials tend tosegregate


Mixing

Mixing of granular materials:is important — Science 125th anniversary identifiedgranular flow as one of the 125 big questions in Scienceis ubiquitous — pharmaceuticals, food industry, ceramics,metallurgy, constructionwas initially explained by analogies with fluid mixing —hence terms like granular shear and granular diffusion

But the big difference is that granular materials tend tosegregate


Segregation

Granular materials segregate by (at least) 2 mechanisms:

Percolation — little particles fall through the gaps of bigparticlesBuoyancy — less dense particles tend to rise

The Brazil Nut effect


Flow regimes

[from S. W. Meier et al., 2007]


2D circular tumblers

In the bulk

r = 0, θ = ω

In the flowing layer

x = γ(δ(x)+y), y = ωxy/δ(x)

The flowing layer hasshape

δ(x) = δ0

√1− x2/L2


Constant rotation rate

At constant rotation rateparticle streamlines form closed loops passing throughflowing layersteady, divergence-free, integrablecan transform to action–angle coordinates ρ, φtrajectories in action–angle coordinates given by:

ρ = 0φ = 2π/T (ρ)

taking a time τ -map gives a twist map

P(ρ, φ) = (ρ, φ+ 2πτ/T (ρ))


Variable rotation rate

Break the integrability by varying the rate of angular rotationSinusoidal forcing has been well-studied.

[Fiedor and Ottino, JFM 255 2005]






Key idea is that streamlines changes and cross



Piecewise constant rotation rate

Simplify the forcing by using a blinking flow

ω =

ωb = ω + ω for iτ < t < (i + 1/4)τωa = ω − ω for (i + 1/4)τ < t < (i + 3/4)τωb = ω + ω for (i + 3/4)τ < t < (i + 1)τ

Alternate the angular velocity between ωa and ωb.


Streamline crossing structure

Dynamicalbehaviour stemsfrom intersectingstreamlinesConstant rotationrate gives analogywith blinking flowsThis can bemathematicallyformalised usinglinked twist maps


Linked Twist Maps on the plane

../FIGURES/annuli.jpg

Domain is two intersect-ing annuli with two dis-tinct regions of intersec-tion.



../FIGURES/big_blobs1_small.pdf

A twist map takes pointsin an annulus...




... and performs a shear,wrapping this initial setaround the annulus

F (r , θ) = (r , θ + f (r))

(centred at the centre ofleft annulus)




A linked twist map is thecomposition G◦F of suchmaps on a pair of annuli.

F (r , θ) = (r , θ + f (r))

(centre left annulus)

G(ρ, φ) = (ρ, φ+ g(ρ))

(centre right annulus)

Proof of ergodic mixing due to Burton & Easton (1980),Devaney (1980), Wojtkowski (1980), Przytycki (1983)



../FIGURES/planar_co_valid.jpg

Linked twist maps aremixing, in the sense that

limn→∞

µ(f n(A)∩B) = µ(A)µ(B)

providing:intersections aretransversetwists are monotonic


The Blinking Vortex


The Blinking Vortex


The Blinking Vortex


The Blinking Vortex


Microfluidics — patterned walls

from [Stroock, A. D. et al., Science 295, 647–651 (2002)]


Microfluidics — electroosmotic flow

from [Qian, S. & Bau, H. H., Anal. Chem., 74, 3616–3625 (2002)]


Streamline crossing structure

Dynamicalbehaviour stemsfrom intersectingstreamlinesConstant rotationrate gives analogywith blinking flowsThis can bemathematicallyformalised usinglinked twist maps


../FIGURES/2d_pics.pdf


Blinking experiments

In the 2d tumbler, shears are monotonic, but streamlines donot cross transversely.

θ

r

Parabolic islands

However the size and position of the islands can be predictedby a linked twist map analysis.


Three dimensional blinking system

../FIGURES/3d_a.jpg ../FIGURES/3d_c.jpg

../FIGURES/3d_d.jpg

../FIGURES/3d_e.jpg


Rotation about the z-axis

Solid body rotation in the bulk:

x = ωyy = −ωxz = 0

Shear in the flowing layer:

x = γ1(δ1(x , z) + y)

y = ω1xy/δ1(x , z)

z = 0

Boundary of flowing layer and bulk:

δ1(x , z) = δ0

√1− x2/L2

=√ω1/γ1

√R2 − x2 − z2


Rotation about the x-axis

Solid body rotation in the bulk:

x = 0y = −ωzz = ωy

Shear in the flowing layer:

x = 0y = ω2zy/δ2(x , z)

z = γ2(δ2(x , z) + y)

Boundary of flowing layer and bulk:

δ2(x , z) = δ0

√1− z2/L2

=√ω2/γ2

√R2 − x2 − z2


The δ → 0 limit

Consider a mathematical limit as the depth of flowing layer→ 0,and speed across flowing layer→∞.

F (r , θ, x) = (r , θ + ω1[π], x)

G(ρ, φ, z) = (ρ, φ+ ω2[π], z)

W (r , θ, x) = (ρ, φ, z)

../FIGURES/3d_b.jpg

H = W−1GWF (r , θ, x)


Dynamics of piecewise isometries


Dynamics of piecewise isometries

Piecewise isometries can possess efficient mixing behaviour inthe absence of any stretching and folding.


Comparison with non-zero flowing layer

Piecewiseisometry

../FIGURES/good_2_1000.jpg../FIGURES/good_5_1000.jpg../FIGURES/good_100_1000.jpgFast flowinglayer

../FIGURES/good_2_100.jpg../FIGURES/good_5_100.jpg

../FIGURES/good_100_100.jpg

Realisticflowing layer




Conclusions and questions

Segregation frequently dominates granular media, androtations about different axes offers an opportunity toproduce mixing in the absence of stretching.

What significant (robust) features of PWIs can we expect toobserve in granular experiments?

Is there a systematic understanding of PWIs as a limitingbehaviour of a continuous system?

How do the PWI dynamics compete with shearing from theflowing layer, and with segregation effects?


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