Past exam of the quantum condensed matter course of the University of Cambridge / 2007 / Section A / A2 / b

At a micrometre scale, viscous forces normally dominate inertia. As an illustrative Newtonian estimate, $L=1\mu \mathrm{m}$, $U=10\mu \mathrm{m}{\mathrm{s}}^{-1}$, $\rho =10^3\mathrm{kg}{\mathrm{m}}^{-3}$ and $\eta =10^{-3}\mathrm{Pa}\mathrm{s}$ give a Reynolds number $\rho UL/\eta =10^{-5}$. The surrounding flow is approximately a Stokes flow, satisfying $-\nabla p+\eta {\nabla }^2\mathbf{u}=0$ and $\nabla \cdot \mathbf{u}=0$.

A body of radius $0.5\mu \mathrm{m}$ translated at that speed in an unbounded Newtonian liquid would need Stokes drag $6\pi \eta aU\simeq 9.4\times 10^{-14}\mathrm{N}$. It stops almost immediately when propulsion stops. The scallop theorem rules out propulsion by a single reciprocal stroke in this ideal regime; a rotating helix or a nonreciprocal cycle of shapes can propel it. Brownian motion also randomizes position and orientation: Stokes–Einstein relation scales as $a^{-1}$ and rotational diffusion as $a^{-3}$, so retaining heading becomes harder as size decreases.

The Newtonian calculation is a scale estimate, not a complete model of blood. Vessel walls alter drag, the imposed flow advects the body, and nearby red blood cells make the medium heterogeneous at this scale. Non-Newtonian fluid effects and deformable obstacles can matter; steering must be assessed against those effects rather than inertial coasting.