Optimizing Low Reynolds Number Locomotion. Anette (Peko) Hosoi Hatsopoulos Microfluids Laboratory, MIT

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1 Optimizing Low Reynolds Number Locomotion Anette (Peko) Hosoi Hatsopoulos Microfluids Laboratory, MIT

2 What s in This Talk? 1. Optimal stroke patterns for 3-link swimmers 2. Building a better snail Swimming Crawling 2

3 optimal three-link Model of the swimmer.our modelswimmer. of the three-link swimmer inclu Modelthe of the swimmer.model ofofthethe threeof the surrounding flow, geometry andour kinematics swi Tiny Swimmers of the dynamics surrounding flow, the geometry kine equations governing of the system. Each linkand in the sw the dynamics of the system. rigid slender bodyequations of lengthgoverning 2l and radius b with characterisitic asp rigid slender body of lengthto2lbeand radius Figure 1). The Reynolds number is assumed small, Reb=with ρu l Figure 1).swimmer, The Reynolds number is assumed b characteristic speed of the ρ is the fluid density and µtothe characteristic speed of the effects, swimmer, is the fluid effects are thus neglected relative to viscous andρthe hydrody effects are thus neglected relative to viscous effect governed by Stokes equations governed by Stokes equations u = 0, p + µ 2 u = 0, subject to the boundary conditions u = 0, p + subject to the boundary conditions u = Us on S, u 0 at u = Us on S, u where u and p are the velocity and the pressure fields in the flu is the local velocity at the surface, of the swimmer. Thepressur hydro where u and p ares,the velocity and the is the local velocity at the surface, S, of the swi First two images from - Dennis Kunkel - Stephen Durr

4 Tiny Swimmers p + µ 2 u = 0 u = 0 The 3-link swimmer 4

5 Tiny Swimmers p + µ 2 u = 0 u = 0 The 3-link swimmer 4

6 3-link Swimmer Purcell (1977): proposed design Becker, Koehler and Stone (2003): optimized geometry (arm length/body length and stroke angle) 5

7 3-link Swimmer Purcell (1977): proposed design Becker, Koehler and Stone (2003): optimized geometry (arm length/body length and stroke angle) 5

8 3-link Swimmer Purcell (1977): proposed design Becker, Koehler and Stone (2003): optimized geometry (arm length/body length and stroke angle) Can we do better? 5

9 Optimizing Kinematics 6

10 Optimizing Kinematics Fixed geometry 6

11 Optimizing Kinematics Fixed geometry 6

12 Optimizing Kinematics Fixed geometry 6

13 2 Optimizing Kinematics FIG. 2: Stroke sequences of three-link swimmers in the (Ω1, Ω? square. Small swimmer diagrams correspond to successive config efficiency, ( ) optimal velocity and ( ) the optimal Purcell0s Fixed geometry -1 the stroke. The swimmer moves to the left when the trajectory to the right otherwise. 6-2

14 2 Optimizing Kinematics FIG. 2: Stroke sequences of three-link swimmers in the (Ω1, Ω? square. Small swimmer diagrams correspond to successive config efficiency, ( ) optimal velocity and ( ) the optimal Purcell0s Fixed geometry -1 the stroke. The swimmer moves to the left when the trajectory to the right otherwise. 6 Kanso and Marsden (2005) - 3-link fish Berman and Wang (2006) - insect flight -2

15 Model Swimmer F(s) U(s) Λ(s) F i = (F x i, F y i, τ i ) prescribe kinematics 2b s = 0 Ω 2 y R (s) x Ω 1 V i = (ẋ i, ẏ i, Θ i ) - Lowest order: resistive force theory - Next order: can incorporate effects of slenderness and interactions between links 2l i Λ i s = 2l R. Cox, Journal of fluid mechanics 44 (4), 791 (1970). force balance Solve linear system 3 F i = i=1 3 ( 3 j=1 i=1 A j i ) X j = 0. Constraint: links are attached Swimming velocities and efficiencies 7

16 Optimized Stroke Patterns Two cost functions - Efficiency - [useful work]/[energy dissipated] - Unique parametrization that optimizes efficiency for a given curve - Speed Symmetry axes 8

17 Optimized Stroke Patterns Two cost functions - Efficiency - [useful work]/[energy dissipated] - Unique parametrization that optimizes efficiency for a given curve - Speed Symmetry axes Distance Efficiency Arm Ratio Slenderness Purcell As slender Optimal as possible

18 Optimized Stroke Patterns Two cost functions - Efficiency - [useful work]/[energy dissipated] - Unique parametrization that optimizes efficiency for a given curve - Speed Symmetry axes Distance Efficiency Arm Ratio Slenderness Purcell As slender Optimal as possible

19 Optimized Stroke Patterns Two cost functions - Efficiency - [useful work]/[energy dissipated] - Unique parametrization that optimizes efficiency for a given curve - Speed Symmetry axes Distance Efficiency Arm Ratio Slenderness Purcell As slender Optimal as possible

20 Optimized Stroke Patterns Two cost functions - Efficiency - [useful work]/[energy dissipated] - Unique parametrization that optimizes efficiency for a given curve - Speed Symmetry axes Distance Efficiency Arm Ratio Slenderness Purcell As slender Optimal as possible

Optimized Stroke Patterns Two cost functions - Efficiency - [useful work]/[energy dissipated] - Unique parametrization that optimizes efficiency for a given curve

21 Optimized Stroke Patterns Two cost functions - Efficiency - [useful work]/[energy dissipated] - Unique parametrization that optimizes efficiency for a given curve - Speed Symmetry axes Distance Efficiency Arm Ratio Slenderness Purcell As slender Optimal as possible

22 3-Link Race D = distance W = Total work (viscous dissipation) Thin line = path of center of mass Optimize geometry Optimize geometry AND kinematics Distance Efficiency Arm Ratio Slenderness Purcell As slender Optimal as possible 9

23 Multiple Links Large N snake Analytic solution by Lighthill (in Mathematical Biofluiddynamics) 41 degree angle

24 Multiple Links Large N snake Analytic solution by Lighthill (in Mathematical Biofluiddynamics) 41 degree angle small drag coefficient large drag coefficient

25 Multiple Links Large N snake Analytic solution by Lighthill (in Mathematical Biofluiddynamics) 41 degree angle small drag coefficient large drag coefficient

26 Effect of Slenderness Efficiency E from Becker et al Fully optimized stroke Purcell stroke Slenderness log 1/κ 11

27 Effect of Slenderness Efficiency E Biological systems from Becker et al Slenderness log 1/κ Fully optimized stroke Purcell stroke 11

28 Gastropod Locomotion Locomotion is directly coupled to stresses in the thin fluid film 12

29 Gastropod Locomotion Locomotion is directly coupled to stresses in the thin fluid film 12

30 Gastropod Locomotion Retrograde vs direct waves F. Vles, C. R. Acad. Sci., Paris 145, 276 (1907) Locomotion is directly coupled to stresses in the thin fluid film 12

31 What is Required for Locomotion? 13

32 What is Required for Locomotion? 13

33 What is Required for Locomotion? H F 13

34 What is Required for Locomotion? H F Fshear stress 13

35 What is Required for Locomotion? H F = τ 1 A = µ(τ 1 )V 1 A/H ( V cm = [V 2 (N 1) + V 1 ] /N ( ) 1 1 F Fshear stress Couette flow in small gap Each pad carries equal mass 13

36 What is Required for Locomotion? H F = τ 1 A = µ(τ 1 )V 1 A/H ( V cm = [V 2 (N 1) + V 1 ] /N ( ) V cm = F H ( AN µ(τ 1 ) 1 ) µ(τ 2 ) F Fshear stress Couette flow in small gap Each pad carries equal mass 13

37 What is Required for Locomotion? H F = τ 1 A = µ(τ 1 )V 1 A/H ( V cm = [V 2 (N 1) + V 1 ] /N ( ) V cm = F H ( AN µ(τ 1 ) 1 ) µ(τ 2 ) F Fshear stress Couette flow in small gap Each pad carries equal mass Nonlinear characteristics first measured by: M. Denny, J. Exp. Biol. 91, 195 (1981) M. Denny, Nature 285, 160 (1980) 13

38 RoboSnail II Laponite Carbopol 14

39 Tune Material Properties stress? strain rate 15

40 Tune Material Properties stress? strain rate stress Shear Thickening Newtonian Shear Thinning strain rate 15

41 Tune Material Properties stress? Perturb rheology of Newtonian fluid (snail will crawl on any non-newtonian fluid) stress strain rate Shear Thickening Newtonian Shear Thinning strain rate shear thickening shear thinning Which material properties are favorable? 15

42 Tune Material Properties stress stress? strain rate Shear Thickening Newtonian Shear Thinning strain rate Perturb rheology of Newtonian fluid (snail will crawl on any non-newtonian fluid) γ = σ µ ( 1 ɛ σ σ ɛ > 0, shear thickening ɛ < 0, shear thinning Which material properties are favorable? ) 15

43 Rheology Cost Function ( ) Mechanical work done in crawling ( = rate of viscous dissipation) E = λ h 0 0 σ γ dy dx. E = hλ 4µ σ2 ɛ 17 hλ σ 2 σ µσ ( Chemical cost associated with mucus production ( ~ flux in frame moving with snail) E Q = h 0 u s (x)dy Q s = ɛ 79 h 2 ( 185 σ 3 σ σ σ σ σ 1 + ɛ 432 µσ 8532σ σ σ )

44 Rheology Cost Function ( ) Mechanical work done in crawling ( = rate of viscous dissipation) E = λ h 0 0 σ γ dy dx. > 0 E = hλ 4µ σ2 ɛ 17 hλ σ 2 σ µσ ( Chemical cost associated with mucus production ( ~ flux in frame moving with snail) E Q = h 0 u s (x)dy Q s = ɛ 79 h 2 ( 185 σ 3 σ σ σ σ σ 1 + ɛ 432 µσ 8532σ σ σ )

45 Rheology Cost Function ( ) Mechanical work done in crawling ( = rate of viscous dissipation) E = λ h 0 0 σ γ dy dx. ɛ > 0, shear thickening Chemical cost associated with mucus production ( ~ flux in frame moving with snail) E Q = h 0 u s (x)dy > 0 E = hλ 4µ σ2 ɛ 17 hλ σ 2 σ µσ ( Q s = ɛ 79 h 2 ( 185 σ 3 σ σ σ σ σ 1 + ɛ 432 µσ 8532σ σ σ )

46 Rheology Cost Function ( ) Mechanical work done in crawling ( = rate of viscous dissipation) E = λ h 0 0 σ γ dy dx. ɛ > 0, shear thickening Chemical cost associated with mucus production ( ~ flux in frame moving with snail) E Q = h 0 u s (x)dy > 0 E = hλ 4µ σ2 ɛ 17 hλ σ 2 σ µσ ( Q s = ɛ 79 h 2 ( 185 σ 3 σ σ σ σ σ 1 + ɛ 432 µσ 8532σ σ σ )

47 Rheology Cost Function ( ) Mechanical work done in crawling ( = rate of viscous dissipation) E = λ h 0 0 σ γ dy dx. ɛ > 0, shear thickening Chemical cost associated with mucus production ( ~ flux in frame moving with snail) E Q = h 0 u s (x)dy > 0 E = hλ 4µ σ2 ɛ 17 hλ σ 2 σ µσ ( > 0 Q s = ɛ 79 h 2 ( 185 σ 3 σ σ σ σ σ 1 + ɛ 432 µσ 8532σ σ σ )

48 Rheology Cost Function ( ) Mechanical work done in crawling ( = rate of viscous dissipation) E = λ h 0 0 σ γ dy dx. ɛ > 0, shear thickening Chemical cost associated with mucus production ( ~ flux in frame moving with snail) E Q = h 0 u s (x)dy > 0 E = hλ 4µ σ2 ɛ 17 hλ σ 2 σ µσ ( > 0 Q s = ɛ 79 h 2 ( 185 σ 3 σ σ σ σ σ 1 + ɛ 432 µσ 8532σ σ σ ) +... ɛ < 0, shear thinning 16

49 Cost of Locomotion The high cost is primarily due to the cost of mucus production, which alone is greater than the total cost of movement for a mammal or reptile of similar weight,... shear thinning Mark Denny, Science, 208, No (1980) 17

50 Cost of Locomotion The high cost is primarily due to the cost of mucus production, which alone is greater than the total cost of movement for a mammal or reptile of similar weight,... shear thinning Mark Denny, Science, 208, No (1980) Pedal mucus from common garden snail, Helix aspera is strongly shearthinning. 8mm plate with sandpaper 100 micron gap, T=22ºC 17

Rely on the nonlinear response of pedal mucus to crawl We can tune viscous material

51 Final Comments 3-link (and n-link) swimmer (low Reynolds number) Optimizing kinematics Snails Trade-off between efficiency and robustness in biological systems? Rely on the nonlinear response of pedal mucus to crawl We can tune viscous material properties to find which weakly nonlinear response is energetically favorable shear thinning Mechanical wall-climber 18

52 Acknowledgments Collaborators: Daniel Tam (GS, MIT) Brian Chan (GS, MIT) Eric Lauga (MIT, Dept of Math) - Optimizing 3-link swimmer - Robosnails + mechanical swimmer - Optimizing crawling Funding: Schlumberger-Doll Research (SDR) NSF 19

mechanical swimmer - Optimizing Tuesday crawling 10:45-11:10 Slip, Swim, Mix,

53 Acknowledgments Collaborators: Daniel Tam (GS, MIT) Brian Chan (GS, MIT) Eric Lauga (MIT, Dept of Math) Funding: - Optimizing 3-link swimmer - Robosnails + mechanical swimmer - Optimizing Tuesday crawling 10:45-11:10 Slip, Swim, Mix, Pack: Fluid Mechanics at the Micron Scale Schlumberger-Doll Research (SDR) NSF 19

Optimizing Low Reynolds Number Locomotion: Crawling. Anette (Peko) Hosoi Hatsopoulos Microfluids Laboratory, MIT

Optimizing Low Reynolds Number Locomotion: Crawling Anette (Peko) Hosoi Hatsopoulos Microfluids Laboratory, MIT What s in This Talk? Last time... Swimming Today... Crawling Crawlers of Krogerup Why Snails?