= Kinematic reversibility of Stokes flow
For a fixed geometry, the steady <Stokes equation> maps $(\mathbf u,p',\mathbf f)$ to $(-\mathbf u,-p',-\mathbf f)$ when all imposed boundary <velocities> and disturbance <forces> are reversed; $p'$ is <pressure> relative to any unchanged <hydrostatic pressure>. <Linearity of Stokes flow> and <Uniqueness of Stokes flow> therefore reverse the entire <velocity field>. If the driving schedule and evolving geometry are retraced, fluid elements retrace their paths too. Reversing just one drive while leaving another applied <force> unchanged does not meet these conditions.
This reversal concerns the paths, while <viscous dissipation> remains positive: for a <sphere>, the required <power> $6\pi\mu aU^2$ is unchanged by $U\mapsto-U$. A reversed experiment must supply fresh <energy>. <Inertia>, <Brownian motion> and irreversible contact can prevent perfect retracing. Edward M. Purcell illustrated the mechanism with a <viscous fluid flow> demonstration in https://www.math.uwaterloo.ca/~mdunphy/purcell.pdf[Life at low Reynolds number, American Journal of Physics 45, 3–11 (1977)].
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