Article Outline
Glossary
Definition of the Subject
Introduction
The Mathematics of Swimming
The Scallop Theorem Proved
Optimal Swimming
The Three-Sphere Swimmer
Future Directions
Bibliography
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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
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Books and Reviews
Agrachev A, Sachkov Y (2004) Control Theory from the Geometric Viewpoint. In: Encyclopaedia of Mathematical Sciences, vol 87, Control Theory and Optimization. Springer, Berlin
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DeSimone, A., Alouges, F., Lefebvre, A. (2012). Biological Fluid Dynamics, Non-linear Partial Differential Equations. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_3
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