= Solution
\b[TODO: user-deferred biology. The bacterial-motion application is deferred. The following covers the Reynolds number and its general fluid-mechanical implications.]
The <Reynolds number>
$$
\boxed{\mathrm{Re}=\frac{\rho UL}{\eta}=\frac{UL}{\nu}}
$$
compares advective <inertia> with viscous <force> on a specified <length> scale $L$ and <velocity> scale $U$. Here $\rho$ is fluid <mass density>, $\eta$ is <dynamic viscosity> and $\nu=\eta/\rho$ is <kinematic viscosity>. In the <Navier-Stokes equation>, advective acceleration scales as $U^2/L$, while the viscous term per unit <mass> scales as $\nu U/L^2$; their ratio is <Reynolds number>.
When $\mathrm{Re}\ll1$ and unsteady fluid <inertia> is also negligible, the <incompressible flow> equations reduce to <Stokes flow>:
$$
-\nabla p+\eta\nabla^2\mathbf u+\mathbf f=\mathbf0,
\qquad \nabla\cdot\mathbf u=0.
$$
Thus <pressure>, applied body <forces> and viscous stresses balance at each instant. For a translating rigid <sphere>, <drag> is proportional to <velocity>, $F_D=6\pi\eta aU$, rather than the inertial scale $\rho a^2U^2$. Given the same instantaneous geometry, reversing the imposed boundary velocities reverses the <Stokes flow>; there is no fluid-inertial coasting in this approximation. This statement concerns the fluid limit and does not discard a separately significant particle <inertia>.
A low steady <Reynolds number> alone does not justify dropping rapid unsteady acceleration. For forcing frequency $\omega$, the additional dimensionless ratio is $\rho\omega L^2/\eta$. No organism-specific sizes, velocities or propulsion mechanism are assumed here.
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