Computational Microswimmers

1. Computational Microswimmers Susan Haynes Eastern Michigan University Computer Science

2. 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

3. 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

4. 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

5. 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.

6. 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.

7. 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)

8. 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.

9. 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.

10. 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.

11. 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!

12. 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.

13. 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.

14. 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.

15. Where else? 3. EMU’s Physics department has a new focus on computational physics -- possible collaboration with respected colleagues there. • Pretty pictures:

16. 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).

17. 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