Piecewise isometries and mixing in granular tumblers Rob Sturman Department of Mathematics University of Leeds BIRS Workshop on Low Complexity Dynamics, 28 May 2008 Banff Joint work with Steve Meier, Julio Ottino, Northwestern Steve Wiggins, University of Bristol Rob Sturman Granular mixing
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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
Flow regimes
[from S. W. Meier et al., 2007]
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
[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
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?