Robosnail 1 SnailInspired Fluid Locomotion Brian Chan M

Robosnail 1 Snail-Inspired Fluid Locomotion Brian Chan, M. S Theresa Guo, undergraduate researcher Advisors: Anette Hosoi, Julio Guerrero (SLB) Hatsopolous Microfluids Laboratory Department of Mechanical Engineering Massachusetts Institute of Technology Schlumberger - Doll Research, SDR

Contents: n n n Snail locomotion Type 1 Robosnails ¨ Theory ¨ Simulations Robosnail 1 A ¨ Design ¨ Experiment Robosnail 1 B ¨ Design ¨ Experiment Robosnail 1 C ¨ In progress Conclusions Robosnail 1 A Robosnail 1 B

Motivation To evaluate the feasibility of using snail-like locomotion, and to optimize the performance of mechanical Robosnails. Advantages of snail locomotion: n can be configured as a versatile flexible robot n A sealed Robosnail mechanism can be robust in muddy conditions n Effective locomotion for environments with little or no traction

Snail Locomotion basics - All snails are separated from the substrate by a fluid layer (mucus) Locomotive forces must be transferred through this layer to the substrate Snails, equipped with a single flexible foot must find a way to generate fluid forces parallel to the substrate.

Classifying Snail locomotion: Direct waves: waves of compression (used by most land snails) Direct waves/Retrograde waves (Denny 1989) Limax maximus (moving with direct waves) Retrograde waves: waves of expansion (used by most aquatic snails) – also possible flapping motion …

Robosnail 1: Design using Retrograde Waves (Joint SLB/MIT U. S. Patent Pending) Driving a flexible waving membrane with a motor

Governing physics: Analysis of thin fluid layers: Lubrication Theory n Assumptions: Height scale much smaller than length scale ¨ pressure varies only in the x direction ¨ Inertia effects negligible ¨ n The lubrication equation: (slight modifications for non. Newtonian fluids) Conservation of momentum

Robosnail 1: Theory - Physical mechanism Using lubrication pressures for propulsion Pressure under a sinusoidal flapping membrane (1 wavelength): Immediately before the wave trough, fluid is being compressed (high pressure), Behind the wave trough, fluid is being pulled apart (low pressure) Pressure acting on sloped surface creates a propulsive force.

Robosnail 1: 2 D Theory x – velocity profile Volume flux Q: Pressure p: From conservation of momentum and mass we find velocity

Robosnail 1: 2 D Theory Steady- state horizontal force balance: horizontal component of pressure force and shear stress at membrane balances tractoring force

Robosnail 1: 2 D Theory We derive a simple linear tractoring force – velocity function, which resembles a motor torque-speed curve. where

Robosnail 1: Full 3 D Theory For real Robosnails, we always experience side leakage, hence losses. By analyzing a differential control volume, we can derive a 3 D lubrication equation. Where p is pressure, h is height, and η is viscosity.

Robosnail 1: Full 3 D Theory Deriving a force-speed relationship [A]: stalled robosnail [B] pure shearing force In 3 D, the force-velocity relationship is still linear

Robosnail 1: 3 D Simulations Numerical solutions for the pressure show the losses due to leakage; we can integrate pressure to solve for the tractoring force (maximum, stalled)

Robosnail 1: 3 D Simulations Sinusoidal foot profile: Comparing the tractoring force of 3 D Robosnails to the ideal 2 D case: As we expect, the wider the foot, the closer it behaves like the 2 D snail.

Robosnail 1 A: Experiment

Robosnail 1 A: Results: Free velocity Fluid: glycerol b/l = 0. 6

Robosnail 1 A: Results: Stall Force Strain gauge Fluid: Silicone oil b/l = 0. 6

Robosnail 1 A: Results: Force-velocity Fluid: Silicone oil b/l = 0. 6 By varying the payload m and the waving velocity vw, we measure different values of vs

Robosnail 1 B: Apparatus - - Periodic foot design to eliminate entrance/exit anomalies. Replaceable tracks define the height profile.

Robosnail 1 B: Apparatus Core mechanism: Various replaceable tracks:

Robosnail 1 B: Experiment

Robosnail 1 B: Results: Free Velocity (Sinusoidal foot) RS-1 B performs better than the 3 d solution (due to partial sealing effect of the tank walls) but understandably still not as well as the 2 D solution.

Robosnail 1 C: Faster-than-wave locomotion Snail speed is a function of wave shape. For some non-sinusoidal wave shapes we can predict Vs/Vw >1. That is, a Robosnail that moves faster than it ‘steps’! We replace the sinusoidal foot with a foot composed of two parabolas, varying the size ratios:

Conclusions - Lubrication theory predicts a linear force and velocity relationship for both 2 D and 3 D Robosnails. - Analytic solutions exist for the 2 D case for any given wave height function. - Numerical simulations give a similar linear force-velocity relation for 3 D snails, but with losses dependent on the ratio of snail width to length. - We have experimental data for sinusoidal wave Robosnails that confirms the numerical results. - In theory, certain wave shapes exhibit regimes where the snail speed is faster than the wave speed; future experiments will test this theory.

Appendix: 2 D theory (detail) To more easily analyze the fluid flow in the lubrication layer we switch to a reference frame following the waves. In the new reference frame, Q = constant.

Appendix: Dimensionless variables Experimental constants: L μ h 0 vw wavelength viscosity average fluid thickness waving velocity Dimensionless variables x = x*/L b = b*/L p= Dimensional variables: x* b* p* h* a* v s* Fx* x-position half-width of foot pressure height foot amplitude snail velocity tractoring force h = h*/h 0 a = a*/h 0 vs= vs*/ vw Fx=

Acknowledgements n National Science Foundation n Schlumberger Limited

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