= Solution
{c}
A small <Reynolds number>, $\mathrm{Re}=\rho UL/\eta\ll1$, 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 $\rho L^2/(\eta T)=L^2/(\nu T)\ll1$. 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>:
$$
0=-\nabla p-\rho g\hat{\mathbf z}+\eta\nabla^2\mathbf u,\qquad \nabla\cdot\mathbf u=0.
$$
Taking the <divergence> and using commuting spatial <derivatives> gives
$$
0=-\nabla^2p+\eta\nabla^2(\nabla\cdot\mathbf u),\qquad
\boxed{\nabla^2p\simeq0.}
$$
Taking the <curl>, which annihilates the <pressure gradient> and uniform gravitational force, gives
$$
0=\eta\nabla^2(\nabla\times\mathbf u),\qquad
\boxed{\nabla^2\boldsymbol\omega\simeq0.}
$$
These 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 $\partial_t\boldsymbol\omega=\nu\nabla^2\boldsymbol\omega$ is nonzero. Its <vorticity> is not a <harmonic function>. The quasi-steady qualification is therefore necessary.
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