Skip to main content

Biological Fluid Dynamics, Non-linear Partial Differential Equations

  • Reference work entry
Encyclopedia of Complexity and Systems Science

Definition of the Subject

Swimming, i. e., being able to advance in a fluid in the absence of external propulsive forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers (Re). This is the regime of interest for micro‐organisms and micro‐robots or nano‐robots, where hydrodynamics is governed by Stokes equations. Thus, besides the rich mathematics it generates, low Re propulsion is of great interest in biology (How do microorganism swim? Are their strokes optimal and, if so, in which sense? Have theseoptimalswimming strategies been selected by evolutionary pressure?) and biomedicine (can small-scale self-propelled devices be engineered for drug delivery, diagnostic, or therapeutic purposes?).

For a microscopic swimmer, moving and changing shape at realistically low speeds, the effects of inertia are negligible. This is true for both the inertia of the fluid and the inertia of the swimmer. As pointed out by Taylor [10], this implies that the swimming...

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Abbreviations

Swimming :

The ability to advance in a fluid in the absence of external propulsive forces by performing cyclic shape changes.

Navier–Stokes equations:

A system of partial differential equations describing the motion of a simple viscous incompressible fluid (a Newtonian fluid)

$$\rho\left(\frac{\partial v}{\partial t} + (v\cdot\nabla)v\right) = -\nabla p + \eta \Delta v$$
$$\text{div}\ v = 0$$

where v and p are the velocity and the pressure in the fluid, ρ is the fluid density, and η its viscosity. For simplicity external forces, such as gravity, have been dropped from the right hand side of the first equation, which expresses the balance between forces and rate of change of linear momentum. The second equation constrains the flow to be volume preserving, in view of incompressibility.

Reynolds number:

A dimensionless number arising naturally when writing Navier–Stokes equations in non‐dimensional form. This is done by rescaling position and velocity with \( { x^\ast = x/L } \) and \( { v^\ast = v/V } \), where L and V are characteristic length scale and velocity associated with the flow. Reynolds number (Re) is defined by

$${\text{Re}} = \frac{VL\rho}{\eta} = \frac{VL}{\nu}$$

where \( { \nu = \eta/\rho } \) is the kinematic viscosity of the fluid, and it quantifies the relative importance of inertial versus viscous effects in the flow.

Steady Stokes equations:

A system of partial differential equations arising as a formal limit of Navier–Stokes equations when \( { \text{Re} \to 0 } \) and the rate of change of the data driving the flow (in the case of interest here, the velocity of the points on the outer surface of a swimmer) is slow

$$\begin{aligned}-\eta\Delta v + \nabla p &= 0\\\text{div}\ v &= 0\;.\end{aligned}$$

Flows governed by Stokes equations are also called creeping flows.

Microscopic swimmers:

Swimmers of size \( { L = 1\,\ifx\letex\relax\text{\textmu m}\else\upmu\text{m}\fi } \) moving in water (\( { \nu \sim 1\,\text{mm}^{2}/\text{s} } \) at room temperature) at one body length per second give rise to \( { \text{Re} \sim 10^{-6} } \). By contrast, a 1 m swimmer moving in water at \( { V = 1\,\text{m/s} } \) gives rise to a Re of the order 106.

Biological swimmers:

Bacteria or unicellular organisms are microscopic swimmers; hence their swimming strategies cannot rely on inertia. The devices used for swimming include rotating helical flagella, flexible tails traversed by flexural waves, and flexible cilia covering the outer surface of large cells, executing oar-like rowing motion, and beating in coordination. Self propulsion is achieved by cyclic shape changes described by time periodic functions (swimming strokes). A notable exception is given by the rotating flagella of bacteria, which rely on a submicron‐size rotary motor capable of turning the axis of an helix without alternating between clockwise and anticlockwise directions.

Swimming microrobots:

Prototypes of artificial microswimmers have already been realized, and it is hoped that they can evolve into working tools in biomedicine. They should consist of minimally invasive, small-scale self-propelled devices engineered for drug delivery, diagnostic, or therapeutic purposes.

Bibliography

Primary Literature

  1. Alouges F, DeSimone A, Lefebvre A (2008) Optimal strokes for low Reynolds number swimmers: an example. J Nonlinear Sci 18:277–302

    MathSciNet  ADS  MATH  Google Scholar 

  2. Alouges F, DeSimone A, Lefebvre A (2008) Optimal strokes for lowReynolds number axisymmetric swimmers. Preprint SISSA 61/2008/M

    Google Scholar 

  3. Avron JE, Kenneth O, Oakmin DH (2005) Pushmepullyou: an efficientmicro‐swimmer. New J Phys 7:234-1–8

    Google Scholar 

  4. Becker LE, Koehler SA, Stone HA (2003) On self-propulsion ofmicro‐machines at low Reynolds numbers: Purcell's three-link swimmer. J FluidMechanics 490:15–35

    MathSciNet  ADS  MATH  Google Scholar 

  5. Berg HC, Anderson R (1973) Bacteria swim by rotating theirflagellar filaments. Nature 245:380–382

    ADS  Google Scholar 

  6. Lighthill MJ (1952) On the Squirming Motion of Nearly SphericalDeformable Bodies through Liquids at Very Small Reynolds Numbers. Comm PureAppl Math 5:109–118

    MathSciNet  MATH  Google Scholar 

  7. Najafi A, Golestanian R (2004) Simple swimmer at low Reynoldsnumbers: Three linked spheres. Phys Rev E 69:062901-1–4

    ADS  Google Scholar 

  8. Purcell EM (1977) Life at low Reynolds numbers. Am J Phys 45:3–11

    ADS  Google Scholar 

  9. Tan D, Hosoi AE (2007) Optimal stroke patterns for Purcell'sthree-link swimmer. Phys Rev Lett 98:068105-1–4

    ADS  Google Scholar 

  10. Taylor GI (1951) Analysis of the swimming of microscopicorganisms. Proc Roy Soc Lond A 209:447–461

    ADS  MATH  Google Scholar 

Books and Reviews

  1. Agrachev A, Sachkov Y (2004) Control Theory from the GeometricViewpoint. In: Encyclopaedia of Mathematical Sciences, vol 87, Control Theory andOptimization. Springer, Berlin

    Google Scholar 

  2. Childress S (1981) Mechanics of swimming and flying. CambridgeUniversity Press, Cambridge

    MATH  Google Scholar 

  3. Happel J, Brenner H (1983) Low Reynolds numberhydrodynamics. Nijhoff, The Hague

    Google Scholar 

  4. Kanso E, Marsden JE, Rowley CW, Melli-Huber JB (2005) Locomotionof Articulated Bodies in a Perfect Fluid. J Nonlinear Sci 15:255–289

    MathSciNet  ADS  MATH  Google Scholar 

  5. Koiller J, Ehlers K, Montgomery R (1996) Problems and Progress inMicroswimming. J Nonlinear Sci 6:507–541

    MathSciNet  ADS  MATH  Google Scholar 

  6. Montgomery R (2002) A Tour of Subriemannian Geometries, TheirGeodesics and Applications. AMS Mathematical Surveys and Monographs, vol 91. American Mathematical Society, Providence

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag

About this entry

Cite this entry

DeSimone, A., Alouges, F., Lefebvre, A. (2009). Biological Fluid Dynamics, Non-linear Partial Differential Equations. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_36

Download citation

Publish with us

Policies and ethics