Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2017 / Paper 2 / Section B / 23B / iii / Solution

For fluid density ${\rho }_f$, speed $U$ and characteristic length $d$, inertial kinetic energy scales as ${\rho }_f{d}^3{U}^2$ while viscous work over distance $d$ scales as $\eta U{d}^2$. Their ratio is the Reynolds number,Smaller particles at a fixed speed have smaller Re, so viscous stresses dominate fluid inertia when $\mathrm{Re}\ll 1$. Increasing speed increases Re and eventually makes inertia more important, rather than making viscous dominance stronger.

Particle inertia is a separate issue. Its particle velocity-relaxation time is ${\tau }_p={\rho }_p{d}^2/(18\eta )$; comparing it with a travel time $d/U$ gives a Stokes number ${\tau }_{p}U/d$. This uses particle density ${\rho }_p$, whereas the fluid Reynolds number uses fluid density ${\rho }_f$.