swimming at low reynolds number
TRANSCRIPT
![Page 1: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/1.jpg)
Swimming at low Reynolds number
Gareth Alexander Chris Pooley Vic Putz
![Page 2: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/2.jpg)
http://mcb.harvard.edu/Faculty/Berg.htmlsynechococcus
![Page 3: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/3.jpg)
Low Reynolds number swimming
Re ~ inertial terms / viscous terms~ length x velocity / viscosity ~10-6
Navier-Stokes equations → Stokes equations which are time reversibleNeed a non-reciprocal cycle for swimming (Purcell, Shapere, Wilczek)
( ) ( )t nu nu u
1p u u u3
γ α γ α
α β α β β α γ γ αβ
∂ + ∂ =
⎛ ⎞− ∂ + ν∂ ∂ + ∂ + ∂ δ⎜ ⎟⎝ ⎠
inertial
viscous
![Page 4: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/4.jpg)
Micron scale swimmers
Dreyfus et al, Nature 2005
![Page 5: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/5.jpg)
Low Reynolds number swimming
three sphere swimmerNajafi and Golestanian, 2004
Snake swimmer
![Page 6: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/6.jpg)
Radial and tangential motion
forward motion
turning motion
A generalised three sphere swimmer
![Page 7: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/7.jpg)
1. Low Re swimming
2. The three-sphere swimmer and its flow fieldnot a generic swimmer
3. Swimmer-swimmer interactions the importance of relative phase
4. Swimmer-swimmer scatteringconsequence of time reversal invariance
5. Dumbbell swimmers
6. Many swimmers
Swimming with friends at low Reynolds number
![Page 8: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/8.jpg)
Oseen tensor hydrodynamics
FLUID
jF
iv
ijr
( )
1
6
1ˆ ˆ
8
,
,
ii
ij ij i j
i ij j
H i jR
H rr i jr
v H F
πη
δπη
= =
= + ≠
=
![Page 9: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/9.jpg)
i i ii i
= × =∑ ∑F 0 F r 0
Shape change :
Enforce: no external forces or torques
Oseen tensor (Stokes flow)
1
N
i ij jj =
= ∑v H F
2 1
1 3
− =− =
v v Wv v 0
2 1 3
![Page 10: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/10.jpg)
271 2
RDε⎛ ⎞
⎜ ⎟⎝ ⎠
Displacement in a swimming stroke
D
R sphere radius
length of swimming strokeε
![Page 11: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/11.jpg)
r1
( )21r
( )31r
1. Time-averaged flow field – large r
dipole + quadrupole
![Page 12: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/12.jpg)
r1
( )21r
( )31r
Time-averaged flow field – large r
dipole + quadrupole
![Page 13: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/13.jpg)
T-duality
![Page 14: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/14.jpg)
self T-dual
not self T-dual
front arm longerextensile
back arm longer contractile
equal arm lengths
![Page 15: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/15.jpg)
r1
( )21r ( )31
r
1. Time-averaged flow field – large r
3-sphere, extensionsdifferent
3-sphere, extensions same (self T-dual)
![Page 16: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/16.jpg)
2. Two Golestanian swimmers in phase
![Page 17: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/17.jpg)
л out of phase л/2 out of phase
Two Golestanian swimmers
![Page 18: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/18.jpg)
л out of phase л/2 out of phase
Relative phase matters – cannot treat a swimmer as a passive scalar
Two Golestanian swimmers
![Page 19: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/19.jpg)
Non self T-dual swimmers
![Page 20: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/20.jpg)
1. Low Re swimming
2. The three-sphere swimmer and its flow fieldnot a generic swimmer
3. Swimmer-swimmer interactions the importance of relative phase
4. Swimmer-swimmer scatteringconsequence of time reversal invariance
5. Dumbbell swimmers
6. Many swimmers
Swimming with friends at low Reynolds number
![Page 21: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/21.jpg)
Swimmer-swimmer scattering
![Page 22: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/22.jpg)
Oseen tensor method breaks down EXCHANGE trajectories
Both swimmers TURN in same direction
Hydrodynamic scattering
![Page 23: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/23.jpg)
turn exchange
![Page 24: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/24.jpg)
angle betweenincoming trajectories
impact parameter
![Page 25: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/25.jpg)
turn exchange
![Page 26: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/26.jpg)
![Page 27: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/27.jpg)
![Page 28: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/28.jpg)
![Page 29: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/29.jpg)
![Page 30: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/30.jpg)
![Page 31: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/31.jpg)
![Page 32: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/32.jpg)
OK if
ie swimmers are duals under time reversal
=
![Page 33: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/33.jpg)
If A,B are mutually T-dual swimmers the final state after scattering is the same as theinitial state before scattering
![Page 34: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/34.jpg)
Swimmers that are not T-dual
![Page 35: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/35.jpg)
![Page 36: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/36.jpg)
phase locking: two collinear swimmers
phase differencebetween the swimmers
time
extensile
![Page 37: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/37.jpg)
Phase locking: two dimensions
![Page 38: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/38.jpg)
phase locking: three co-linear swimmers
relative phase
time
![Page 39: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/39.jpg)
Phase locking: three co-linear pumps
![Page 40: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/40.jpg)
Dumbbell swimmers
![Page 41: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/41.jpg)
![Page 42: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/42.jpg)
0 5 10 15 20 1000
2
4
6
8
0
2
4
6
8
20
number of dipoles
displacement after a long time
![Page 43: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/43.jpg)
![Page 44: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/44.jpg)
Prof Ray GoldsteinProf Ray GoldsteinRay Goldsteinhttp://www.dampt.cam.ac.uk/user/gold/movies.html
![Page 45: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/45.jpg)
Many dumbbell swimmers
![Page 46: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/46.jpg)
4. Many Golestanian swimmers
![Page 47: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/47.jpg)
Comparing apolar and polar swimmers
dumbbells polar swimmers
![Page 48: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/48.jpg)
Comparing apolar and polar swimmers
![Page 49: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/49.jpg)
Comparing apolar and polar swimmers
![Page 50: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/50.jpg)
Many snake swimmers
![Page 51: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/51.jpg)
What next?
1.Matching simulations and the continuum theories
2.What is the continuum theory for self T-dual swimmers?
3. Noise
4. When does the zero Re approximation fail?
5. Real swimmers: which results are generic?
6. Shear flow, obstacles, swimmer rheology
![Page 52: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/52.jpg)
Summary
1.Two swimmer hydrodynamic interactions beautiful and complex
2. Relative swimmer phase is important
3. Scattering parameters are preserved ifthe swimmers are mutually T-dual
With many thanks to Gareth Alexander, Chris Pooley,Vic Putz
![Page 53: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/53.jpg)
![Page 54: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/54.jpg)
![Page 55: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/55.jpg)
Non self T-dual swimmers
![Page 56: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/56.jpg)
Scattering: swimmers that are not T-dual
![Page 57: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/57.jpg)
In Phase
Random Phase
![Page 58: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/58.jpg)
In Phase
Random Phase
![Page 59: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/59.jpg)
![Page 60: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/60.jpg)
Prof Ray GoldsteinProf Ray GoldsteinRay Goldsteinhttp://www.dampt.cam.ac.uk/user/gold/movies.html
![Page 61: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/61.jpg)
Random Phases
![Page 62: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/62.jpg)
In Phase
![Page 63: Swimming at low Reynolds number](https://reader031.vdocuments.mx/reader031/viewer/2022013005/61cc3252563f2d2df425aa63/html5/thumbnails/63.jpg)