Past exam of the flows, fluctuations and complexity course of the University of Oxford / 2016 / 2 / a

For an incompressible Newtonian fluid at low Reynolds number, the quasi-steady Stokes equations areThe first equation expresses local conservation of volume for constant density; the second balances pressure, viscous stresses and body force with negligible inertia. Convective inertia is negligible when $\mathrm{Re}=U\ell /\nu \ll 1$ at the relevant flow scale. Quasi-steady flow also requires the viscous diffusion time ${\ell }^2/\nu$ to be short compared with the evolution time. Low convective Reynolds number alone does not justify neglecting a rapidly oscillating unsteady acceleration.

For a thin layer, slenderness further changes the comparison of longitudinal and transverse derivatives; these scalings, as well as low inertia, are needed for lubrication theory.