Past exam of the fluids course of the University of Oxford / 2022 / 4 / a / Solution

A small Reynolds number, $\mathrm{Re}=\rho UL/\eta \ll 1$, means that inertia is small compared with viscous forces. It can arise from small speed, small length or large dynamic viscosity. Examples are the locomotion of a microswimmer and the slow sedimentation of a small particle in a viscous liquid.

To neglect local acceleration as well, an externally imposed time scale $T$ must satisfy $\rho L^2/(\eta T)=L^2/(\nu T)\ll 1$. This holds for a steady flow or a sufficiently slowly varying quasi-steady Stokes approximation. Small Reynolds number alone is insufficient for that extra step. For an incompressible flow with constant material coefficients and uniform gravity, the Navier-Stokes equation then reduces to the Stokes equation:Taking the divergence and using commuting spatial derivatives givesTaking the curl, which annihilates the pressure gradient and uniform gravitational force, givesThese are the Harmonic pressure and vorticity in Stokes flow results.

A concrete counterexample to inferring both solely from small Reynolds number is Stokes' second problem: an oscillating plate with arbitrarily small velocity amplitude has negligible inertia, but ${\partial }_t\mathbf{\omega }=\nu {\nabla }^2\mathbf{\omega }$ is nonzero. Its vorticity is not a harmonic function. The quasi-steady qualification is therefore necessary.