Computational Microswimmers

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# Computational Microswimmers - PowerPoint PPT Presentation

Computational Microswimmers Susan Haynes Eastern Michigan University Computer Science The small world is different Macro swimmer: Inertial effects are significant: Can coast Turbulence effects, drag In water (I.e., with water’s viscosity), water is like, well, water.

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### Computational Microswimmers

Susan Haynes

Eastern Michigan University

Computer Science

The small world is different
• Macro swimmer: Inertial effects are significant:
• Can coast
• Turbulence effects, drag
• In water (I.e., with water’s viscosity), water is like, well, water.
• Micro swimmer: inertial effects are zero
• No coast -- swimmer stops movement almost immediately after propulsive force stops
• No turbulence
• In water, at micro-scale, viscosity is like viscosity of cold molasses at macro-scale
• ==> Intuition frequently fails
Reynolds number, R, describes a body moving in a fluid.
• A fluid means gas or liquid.
• It is the ratio of inertial forces to viscous forces (dimensionless)
• Variables: ‘size’ of body (L), velocity (vs), viscosity of fluid (), density of fluid ().
• R = vS L /  = ms-1 m / m2s-1 ---> dimensionless
R, generally speaking
• R increases with increasing velocity (vS), fluid density (), size of object (L)
• R decreases with increasing fluid viscosity ()
• Crudely put: large things have higher R than small things.
• Fast things have higher R than slow things
• Things moving in air have higher R than things in water ( dominates )
• For water,  = 10-2 cm2 s-1
• For life on earth, air or water:
• Macro-scale R > 1
• Micro-scale R < 1
Example Reynolds numbers
• Large ( > 1) (inertial effects dominate)
• Blue whale: 108
• Cessna flying: 106
• Human swimming: 105 - 106
• Flying duck: 105
• Tiny guppy swimming: 102 (viscosity starts to matter)
• Small ( < 1) (viscous effects dominate)
• Spermatozoa swimming: 10 -2
• E. coli approx 10-6
• Earth’s mantle <<< 1 (maybe 10-15?)
• We have no intuition for what happens when R << 1.
Fantastic Voyage, Oscar-winning film with early babe scientist Raquel Welch, 1966, is completely wrong.

You should imagine instead, being immersed in a vat of molasses (that’s what the viscosity of water feels like to micro-swimmers), no part of your body can move at greater than 1 cm/min. If, in two weeks, you’re able to move 10 meters -- you are a very successful low Reynolds number swimmer.

Navier-Stokes equations
• The Navier-Stokes equations are a set of non-linear partial differential equations that describe fluid flow.
• They are the starting point for simulating fluid flow.
• Possible to solve only in very limited cases.
• Generally, one has to do numerical simulations -- but there are many evil effects when used in CFD simulations (nonconvergence, truncation errors, instability, etc)
The good news
• Fortunately! In the low Reynolds number world, the inertial terms can be removed from the Navier-Stokes equations and this linearizes the equations! Numerical simulations will be better behaved.
• Throw away the inertial terms. Throw away “other forces” (f), because they relate to gravity and centrifugal forces (that don’t apply to neutral buoyancy, slow swimmer).
• You’re left with linear PDEs:

 2 u -  p = 0

• Linear PDEs are much better behaved in simulation.
• Linear PDEs are easily to implement in a CFD simulation.
• Linear PDEs are way easier to solve.
• PLUS, a few of the artificial micro-swimmers have had their equations solved analytically, so it is possible to compare numerical results with actual solutions.
Simplest Morphology -- and the starting place to think about swimming nanobots
• E.M. Purcell:
• Reciprocal motion will not work for low R animals. Reciprocal motion means to change body shape, then return to original state through the sequence in reverse.
• The ‘Scallop Theorem’: A scallop moves by opening its shell slowly, then closing it fast (‘jet propulsion’!) -- This strategy won’t work for low R animals. An animal with a single degree of freedom (like a scallop with its single hinge) is forced to do “reciprocal motion”. Movement in one direction is completely undone by the reciprocal motion in the reverse direction.
Purcell swimmer
• This strategy is proposed for low R (artificial) animal.
• The Purcell swimmer has been solved (in 2003), and built (at macro-scale though run in high viscous liquid) http://web.mit.edu/chosetec/www/robo/3link/
• (At least) two degrees of freedom are necessary to effect displacement.
Najafi and Golestanian had a better idea (building on Purcell) - simpler to model and to solve
• Three linked spheres. Center sphere has two ‘motors’ on opposite sides that each connect to an retractable rod.
• Non-reciprocal motion.
• Center sphere’s action to move itself to the right.
• Pull in left
• Pull in right
• Push out left
• Push out right
• Modelled and solved!
Many other proposed morphologies and propulsive strategies (all non-reciprocating)
• Lay an enzymatic site on one side of a sphere. The enzyme promotes reaction in its area. The reaction creates chemical particles that are denser near the enzymatic site. The particles propel the sphere by osmotic force.
• An elongated swimmer that treadmills on the surface.
• Three spheres, linked like spokes of a wheel.
• Squirmers: spherical and toroidal.
• And let’s not forget the real-world: cilia and flagella (whip-like) abound.
What’s the point of artifical low Reynold’s swimmers?

Aside from just being cool, think nanobots for

drug (or other therapy) delivery, sensors,

localized control.

Where am I going with this?
• Test novel structures for nanobots through computational fluid dynamics simulations (FEATFLOW is open source http://www.featflow.de).
• Engage students in our parallel programming class in more interesting problems than parallelizing the trapezoid rule, odd-even sort, cellular automata, and simple heat diffusion or wave propagation problems.
Where else?

3. EMU’s Physics department has a new focus on computational physics -- possible collaboration with respected colleagues there.

• Pretty pictures:
• Numerical problems are easily parallelizable - we’re still using MPI and it lends itself well to numerical problems.
• Standard implementation techniques: mesh, finite element, finite volume, …
• You can generate very pretty pictures.
• High niftiness factor.
• Once you discretize the PDEs, the algorithms are simply iterative updating -- very simple to conceptualize (unlike, e.g., dynamic programming which has simple, even trivial, operations, but is very hard to conceptualize).
REFERENCES: THE Wonderful, the Good and the Not So Good.
• E.M. Purcell, ‘Life at Low Reynolds Number’, Am J of Physics vol 45, pp 3-11, 1977.
• S.I. Rubinow, ‘The swimming of microorganisms’ in Introduction to Mathematical Biology, Dover, pp 175-188 2002.
• Najafi, Golestanian, ‘Simple swimmer at low Reynolds number: Three linked spheres’, Physical Review E, 69, 062901, 2004.
• Becker, Koehler, Stone, ‘On self-propulsion of micro-machines at low Reynolds number: Purcell’s three link swimmer, J. Fluid Mech (2003), vol 490, pp 15-35.
• Golestanian, Liverpool, Ajdari, ‘Propulsion of a molecular machine by asymmetric distribution of reaction products’, Physical Review Letters, 94, 220801 (2005).
• Dreyfus, Baudry, Stone, ‘Purcell’s “rotator”: mechanical rotation at low Reynolds number’, European Physical Journal B, vol 47, pp 161-164, 2005.
• Lighthill, Mathematical Biofluiddynamics , SIAM, vol 17, 1975.
• Childress, Mechanics of Swimming and Flying, C.U. Press, 1981.
• Kuzmin, Introduction to Computational Fluid Dynamics, web tutorial, http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/cfd.html
• wikipedia.com
• CFD-Wiki: http://www.cfd-online.com/Wiki/Main_Page
• http://www.prism.gatech.edu/~gtg635r/Lift-Drag%20Ratio%20Optimization%20of%20Cessna%20172.html