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 (v s ), viscosity of fluid (  ), density of fluid (  ).

R = v S L /  = ms -1 m / m 2 s -1 ---> dimensionless

R, generally speaking

     R increases with increasing velocity (v S ), 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 cm 2 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: 10 8   Cessna flying: 10 6 Human swimming: 10 5 - 10 6   Flying duck: 10 5 Tiny guppy swimming: 10 2 (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?

1.

2.

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.

4.

Pretty pictures:

What are pedagogical advantages of this for parallel programming?

     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 Dover, pp 175-188 2002. Introduction to Mathematical Biology, 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