Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf ·...
Transcript of Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf ·...
![Page 1: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/1.jpg)
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
![Page 2: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/2.jpg)
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
![Page 3: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/3.jpg)
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
![Page 4: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/4.jpg)
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
![Page 5: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/5.jpg)
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
![Page 6: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/6.jpg)
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
![Page 7: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/7.jpg)
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
![Page 8: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/8.jpg)
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
![Page 9: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/9.jpg)
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.
Rob Sturman Granular mixing
![Page 10: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/10.jpg)
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.
Rob Sturman Granular mixing
![Page 11: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/11.jpg)
The draft lottery
0
50
100
150
200
250
300
350
0 50 100 150 200 250 300 350
Lotte
ry r
ank
Birthday
Rob Sturman Granular mixing
![Page 12: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/12.jpg)
Tumbler Mixers
Rob Sturman Granular mixing
![Page 13: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/13.jpg)
Different tumbler geometries
Rob Sturman Granular mixing
![Page 14: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/14.jpg)
Flow regimes
[from S. W. Meier et al., 2007]
Rob Sturman Granular mixing
![Page 15: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/15.jpg)
Tumblers
[from S. W. Meier et al., 2007]
Rob Sturman Granular mixing
![Page 16: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/16.jpg)
[Zuriguel, I., Gray, J.M.N.T., Peixinho, J. & T. Mullin (2006).Phys. Rev. E 73, 061302.]
Rob Sturman Granular mixing
![Page 17: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/17.jpg)
Tumblers
Modelling percolation and buoyancy
[from S. W. Meier et al., 2007]
Rob Sturman Granular mixing
![Page 18: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/18.jpg)
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
![Page 19: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/19.jpg)
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
![Page 20: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/20.jpg)
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
![Page 21: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/21.jpg)
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
![Page 22: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/22.jpg)
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
![Page 23: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/23.jpg)
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
![Page 24: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/24.jpg)
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
![Page 25: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/25.jpg)
Variable rotation rate
[Fiedor and Ottino, JFM 255 2005]
Rob Sturman Granular mixing
![Page 26: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/26.jpg)
Variable rotation rate
Key idea is that streamlines changes and cross
[Fiedor and Ottino, JFM 255 2005]
Rob Sturman Granular mixing
![Page 27: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/27.jpg)
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
![Page 28: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/28.jpg)
Poincaré sections
N = 2 N = 4
N = 6 N = 8
Rob Sturman Granular mixing
![Page 29: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/29.jpg)
Blinking experiments
Rob Sturman Granular mixing
![Page 30: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/30.jpg)
Linked Twist Maps on the plane
Domain is two intersect-ing annuli with two dis-tinct regions of intersec-tion
Rob Sturman Granular mixing
![Page 31: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/31.jpg)
Linked Twist Maps on the plane
The action of a twist mapis to take a line...
Rob Sturman Granular mixing
![Page 32: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/32.jpg)
Linked Twist Maps on the plane
... 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)
Rob Sturman Granular mixing
![Page 33: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/33.jpg)
Linked Twist Maps on the plane
... 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)
Rob Sturman Granular mixing
![Page 34: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/34.jpg)
Linked Twist Maps on the plane
... 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)
Rob Sturman Granular mixing
![Page 35: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/35.jpg)
The Blinking Vortex
Rob Sturman Granular mixing
![Page 36: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/36.jpg)
The Blinking Vortex
Rob Sturman Granular mixing
![Page 37: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/37.jpg)
The Blinking Vortex
Rob Sturman Granular mixing
![Page 38: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/38.jpg)
The Blinking Vortex
Rob Sturman Granular mixing
![Page 39: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/39.jpg)
Microfluidics — patterned walls
from [Stroock, A. D. et al., Science 295, 647–651 (2002)]
Rob Sturman Granular mixing
![Page 40: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/40.jpg)
Microfluidics — electroosmotic flow
from [Qian, S. & Bau, H. H., Anal. Chem., 74, 3616–3625 (2002)]
Rob Sturman Granular mixing
![Page 41: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/41.jpg)
Streamline crossing structure
Rob Sturman Granular mixing
![Page 42: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/42.jpg)
3-dimensional spherical tumbler
An obvious way to introduce some transversality...
Rob Sturman Granular mixing
![Page 43: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/43.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
Rob Sturman Granular mixing
![Page 44: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/44.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
Rob Sturman Granular mixing
![Page 45: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/45.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
Rob Sturman Granular mixing
![Page 46: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/46.jpg)
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
![Page 47: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/47.jpg)
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
![Page 48: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/48.jpg)
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
![Page 49: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/49.jpg)
Dynamical properties I
TheoremIf ω1/γ1 = ω2/γ2 then motion of a particle is constrained to asingle hemispherical surface.
Rob Sturman Granular mixing
![Page 50: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/50.jpg)
Dynamical properties II
TheoremPeriod 1 points for rotation about the x-axis form a bowl in theshape of a prolate spheroid.
Rob Sturman Granular mixing
![Page 51: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/51.jpg)
Intersection of bowl and spheroid
Rob Sturman Granular mixing
![Page 52: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/52.jpg)
Experiments
Experiments...are hard — how do you see what’s happening inside a spherefilled with a granular material?
Rob Sturman Granular mixing
![Page 53: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/53.jpg)
Experiments
Experiments...are hard — how do you see what’s happening inside a spherefilled with a granular material?
Rob Sturman Granular mixing
![Page 54: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/54.jpg)
Rob Sturman Granular mixing
![Page 55: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/55.jpg)
Rob Sturman Granular mixing
![Page 56: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/56.jpg)
Piecewise isometries
[Goetz, 2003]
Rob Sturman Granular mixing
![Page 57: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/57.jpg)
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
Rob Sturman Granular mixing
![Page 58: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/58.jpg)
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
Rob Sturman Granular mixing
![Page 59: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/59.jpg)
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
Rob Sturman Granular mixing
![Page 60: Granular mixing in 2 and 3 dimensions - University of Leedsrsturman/talks/surrey_granular.pdf · Granular mixing in 2 and 3 dimensions Rob Sturman ... is important — Science 125th](https://reader033.fdocuments.net/reader033/viewer/2022042708/5aa0cf097f8b9a89178e9335/html5/thumbnails/60.jpg)
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
Rob Sturman Granular mixing