On Elastocapillary Coalescence

Dynamics of a Viscous Liquid within an Elastic Shell with Application to Soft Robotics

Shai B. Elbaz and Amir D. Gat

Technion - Israel Institute of Technology

Faculty of Mechanical Engineering

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12/01/2013

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Background (1/2) Soft Robotics

Emerging field of experimental soft robotics. (Stokes et al. ,2013, Shepherd et al. 2013, others)

Embeded fluidic networks

Constant spatial pressure - inflation/deflation

Essentially: creating a solid deformations field by a fluidic stress field.

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Wed like to offer our version of a visco-elastic soft-robot

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Background (2/2) - Biological Flows

Interaction between fluid and solid dynamics involving viscous flow through elastic cylinders extensively studied.

Heil & Pedley 1996,1997 studied the stability of cylindrical shells conveying viscous flow and stokes flow in collapsible tubes.

Paidoussis (1998) extensively studied fluid-structure Interactions for the case of axial flow in slender structures.

Canic & Mikelic 2003 studied viscous incomp. flow through a long elastic tube in the context of arterial blood flow.

Many others.

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Our Goal

Apply models and methods used in biological flows to study time varying deformation patterns in soft-robotics.

Add a new level of control to soft-robotics

Introduce visco-elastic motion to traditional mechanical eng. applications.

(Math. Overview)

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Problem Definition

Fluid-structure interaction between:

Viscous, Newtonian, incompressible flow.

Slender, linearly elastic cylindrical shell closed at one end.

Assume negligible inertia in liquid and solid.

External stress and pressure

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Were going to give a broad mathematical overview due to time limitations

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Elastic Medium (1/2) Governing Eq.

Conservation of Momentum,

Strain Displacement Relations,

Hooks Law,

,

,

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Elastic Medium (2/2) Final Formulation

Follow elastic axi-symmetric shell theory.

Boundary conditions imposed on stress field at fluidic and external interface.

We relate the deformations to fluidic pressure and stress.

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Looking to relate the displacements to fluidic pressure, an equivalent law of laplace i.o. to enter fluidic domain

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Fluidic medium (1/2) Governing Eq.

Conservation of Momentum,

Conservation of Mass,

Velocity boundary conditions - no-slip and no-penetration imposed at solid-liquid interface.

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Fluidic Medium (2/2) Final Formulation

Axial velocity profile,

Non-zero velocity boundary conditions yield char. time scale,

Integrating continuity Eq.,

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Will attempt to show how small E or large mu materials have the tendency to retain energy over time

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Coupled Fluidic-Elastic System

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Subs. elastic relations into integrated continuity relation,

are known rational functions of .

For incompressible materials, , effect of .

We may formulate an IBVP on the pressure field,

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Results Overview

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Solid-liquid material properties,

Slenderness ratio

Wall thickness ratio

Liquid - Silicone oil at

Shell material - rubber

Examine response to,

Constant Pressure inflation step response

Oscillating pressure at inlet of the form - ,

Response to an external sudden force

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Results (1/2)

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Constant pressure inflation of a slender elastic cylinder with internal viscous flow

Results (2/2)

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Quasi-steady diffusion of a slender elastic cylinder with internal viscous flow

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Oscillatory Time Response

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Oscillatory Frequency Response

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Response to External Obstacle (1/2)

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How will the fluid inside the shell react to an obstacle.

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Response to External Obstacle (2/2)

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Concluding Remarks

Closed analytic solution for pressure, velocity and deformation fields.

Characteristic time scale of the visco-elatic interaction.

Analysis of governed inlet pressure and external domain on the deformation field of the shell.

Elastic material compressibility.

Inducing the flow off the base frequency.

Phase reversal.

Boundary pressure feedback movement detection.

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Compressibility two effects. Tendency of compressible material to retain pressure.

Effect of induced boundary is canceled out for incompressible materials due flow volume conservation.

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Future Research

The current study lays the foundation for treatment of an external fluidic domain.

Control based on plant-model and boundary feedback to navigate/propel the vehicle.

Multi-channel networks complex deformations.

A new breed of visco-elastic robots?

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Pressure Field P(Z,T)

Axial Cylinder Coordinate - Z

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|P(Z,T)|

Amplitude of P(Z,T)

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Axial Cylinder Coordinate - Z

Arg(P(Z,T)) [deg]

Phase of P(Z,T)

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Transient Solution to an External Pressure Shock Applied to a Slender Cylindrical shell with Internal Viscous Flow

Axial Cylinder Coordinate - Z

Pressure Field P(Z,T)