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
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
Rob Sturman Granular mixing
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
Rob Sturman Granular mixing
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
Rob Sturman Granular mixing
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))
Rob Sturman Granular mixing
Variable rotation rate
Break the integrability by varying the rate of angular rotationSinusoidal forcing has been well-studied.
[Fiedor and Ottino, JFM 255 2005]
Rob Sturman Granular mixing
Variable rotation rate
Key idea is that streamlines changes and cross
[Fiedor and Ottino, JFM 255 2005]
Rob Sturman Granular mixing
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.
Rob Sturman Granular mixing
Streamline crossing structure
Dynamicalbehaviour stemsfrom intersectingstreamlinesConstant rotationrate gives analogywith blinking flowsThis can bemathematicallyformalised usinglinked twist maps
Rob Sturman Granular mixing
Linked Twist Maps on the plane
../FIGURES/annuli.jpg
Domain is two intersect-ing annuli with two dis-tinct regions of intersec-tion.
Rob Sturman Granular mixing
Linked Twist Maps on the plane
../FIGURES/big_blobs1_small.pdf
A twist map takes pointsin an annulus...
Rob Sturman Granular mixing
Linked Twist Maps on the plane
../FIGURES/big_blobs2_small.pdf
... and performs a shear,wrapping this initial setaround the annulus
F (r , θ) = (r , θ + f (r))
(centred at the centre ofleft annulus)
Rob Sturman Granular mixing
Linked Twist Maps on the plane
../FIGURES/big_blobs3_small.pdf
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)
Rob Sturman Granular mixing
Linked Twist Maps on the plane
../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
Rob Sturman Granular mixing
Microfluidics — patterned walls
from [Stroock, A. D. et al., Science 295, 647–651 (2002)]
Rob Sturman Granular mixing
Microfluidics — electroosmotic flow
from [Qian, S. & Bau, H. H., Anal. Chem., 74, 3616–3625 (2002)]
Rob Sturman Granular mixing
Streamline crossing structure
Dynamicalbehaviour stemsfrom intersectingstreamlinesConstant rotationrate gives analogywith blinking flowsThis can bemathematicallyformalised usinglinked twist maps
Rob Sturman Granular mixing
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.
Rob Sturman Granular mixing
Three dimensional blinking system
../FIGURES/3d_a.jpg ../FIGURES/3d_c.jpg
../FIGURES/3d_d.jpg
../FIGURES/3d_e.jpg
Rob Sturman Granular mixing
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
Rob Sturman Granular mixing
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
Rob Sturman Granular mixing
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)
Rob Sturman Granular mixing
Dynamics of piecewise isometries
Piecewise isometries can possess efficient mixing behaviour inthe absence of any stretching and folding.
Rob Sturman Granular mixing
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
Rob Sturman Granular mixing
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?
Rob Sturman Granular mixing
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