= Solution
For a microscopic swimmer, the <Reynolds number> $\mathrm{Re}=\rho UL/\eta$ is usually small. Taking $L=2\ \mu\mathrm m$, $U=20\ \mu\mathrm{m\,s^{-1}}$ and water-like properties gives $\mathrm{Re}\simeq4\times10^{-5}$. The <Stokes equations> then balance viscous <stress> and <pressure> almost instantaneously, so a swimmer cannot coast appreciably when its propulsive stroke stops.
The <scallop theorem> rules out a reciprocal opening-and-closing stroke in an otherwise unchanged Newtonian environment: reversing the sequence of shapes reverses the motion, and changing the stroke speed does not prevent cancellation. Organisms therefore use nonreciprocal motion. A sperm-like bending wave travels along a flagellum; a bacterium can rotate a helical flagellar bundle. The unequal parallel and perpendicular drag of a slender filament converts those motions into a net thrust. A freely suspended swimmer is <force-free> and <torque-free> overall, so body drag balances thrust and the body counter-rotates against the rotating bundle. Tracking a body and its flagellum together can reveal these opposite rotations. The <helical microswimmer with a spherical head> makes that force and torque balance quantitative.
Back to article page
