Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf ·...

Granular mixing in 2 and 3 dimensions

Rob Sturman

Department of MathematicsUniversity of Leeds

Dynamical Systems Seminar, 19 October 2007Surrey

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




Mixing




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


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


The draft lottery

In 1969, the Selective Service Systemof the USA held a lottery to determinethe order of draft into the U.S. Army forthe Vietnam War.Days of the year numbered 1 to 366drawn once to assign lottery number,and again to determine draft number.


The draft lottery

In 1969, the Selective Service Systemof the USA held a lottery to determinethe order of draft into the U.S. Army forthe Vietnam War.Days of the year numbered 1 to 366drawn once to assign lottery number,and again to determine draft number.


The draft lottery

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350

Lotte

ry r

ank

Birthday


Tumbler Mixers


Different tumbler geometries


Flow regimes

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


Tumblers



[Zuriguel, I., Gray, J.M.N.T., Peixinho, J. & T. Mullin (2006).Phys. Rev. E 73, 061302.]


Tumblers

Modelling percolation and buoyancy



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))




ρ = 0φ = 2π/T (ρ)


P(ρ, φ) = (ρ, φ+ 2πτ/T (ρ))




ρ = 0φ = 2π/T (ρ)


P(ρ, φ) = (ρ, φ+ 2πτ/T (ρ))




ρ = 0φ = 2π/T (ρ)


P(ρ, φ) = (ρ, φ+ 2πτ/T (ρ))




ρ = 0φ = 2π/T (ρ)


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.


Poincaré sections

N = 2 N = 4

N = 6 N = 8


Blinking experiments


Linked Twist Maps on the plane

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



The action of a twist mapis to take a line...



... and twist it around theannulus.Then do the same withpoints in the other annu-lus.

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










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


3-dimensional spherical tumbler

An obvious way to introduce some transversality...


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




x = ωyy = −ωxz = 0


x = γ1(δ1(x , z) + y)

y = ω1xy/δ1(x , z)

z = 0


δ1(x , z) = δ0

√1 − x2/L2

=√ω1/γ1

√R2 − x2 − z2




x = ωyy = −ωxz = 0


x = γ1(δ1(x , z) + y)

y = ω1xy/δ1(x , z)

z = 0


δ1(x , z) = δ0

√1 − x2/L2

=√ω1/γ1

√R2 − x2 − z2


Rotation about the x-axis


x = 0y = −ωzz = ωy


x = 0y = ω2zy/δ2(x , z)

z = γ2(δ2(x , z) + y)


δ2(x , z) = δ0

√1 − z2/L2

=√ω2/γ2

√R2 − x2 − z2




x = 0y = −ωzz = ωy


x = 0y = ω2zy/δ2(x , z)

z = γ2(δ2(x , z) + y)


δ2(x , z) = δ0

√1 − z2/L2

=√ω2/γ2

√R2 − x2 − z2




x = 0y = −ωzz = ωy


x = 0y = ω2zy/δ2(x , z)

z = γ2(δ2(x , z) + y)


δ2(x , z) = δ0

√1 − z2/L2

=√ω2/γ2

√R2 − x2 − z2


Dynamical properties I

TheoremIf ω1/γ1 = ω2/γ2 then motion of a particle is constrained to asingle hemispherical surface.


Dynamical properties II

TheoremPeriod 1 points for rotation about the x-axis form a bowl in theshape of a prolate spheroid.


Intersection of bowl and spheroid


Experiments

Experiments...are hard — how do you see what’s happening inside a spherefilled with a granular material?


Experiments

Experiments...are hard — how do you see what’s happening inside a spherefilled with a granular material?




Piecewise isometries

[Goetz, 2003]


Outstanding problems

3 dimensional transport through the hemispherethe role of segregationthe role of the dynamics of piecewise isometrieshow to measure, or even see, all the above in anexperiment











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