Symmetry breaking for drag minimization

For locomotion at high Reynolds numbers drag minimization favors fore-aft asymmetric slender shapes with blunt noses and sharp trailing edges. On the other hand, in an inertialess fluid the drag experienced by a body is independent of whether it travels forward or backward through the fluid, so there is no advantage to having a single preferred swimming direction. In fact numerically determined minimum drag shapes are known to exhibit almost no fore-aft asymmetry even at moderate \textit{Re}. We show that asymmetry persists, albeit extremely weakly, down to vanishingly small \textit{Re}, scaling asymptotically as \textit{Re}$^{3}$. The need to minimize drag to maximize speed for a given propulsive capacity gives one possible mechanism for the increasing asymmetry in the body plans seen in nature, as organisms increase in size and swimming speed from bacteria like E-Coli up to pursuit predator fish such as tuna. If it is the dominant mechanism, then this signature scaling will be observed in the shapes of motile micro-organisms.