“Purcell’s Scallop Theorem” with Non-viscous Friction

Many mechanical engineers learn about the “Purcell’s Scallop Theorem”: that a single Degree-of-Freedom moving back and forth cannot generate net motion in a viscous medium. This talk will define and analyze Purcell’s classical result, and tie it to the motility map used in geometric mechanics. Using this tooling, we will define and analyze: swimming in biologically relevant non-newtonian fluids (1) Ostwald–de Waele fluids; (2) Carreau-Yasuda fluids; crawling, sliding, or slipping on a surface with (3) dynamic Coulomb friction; (4) asymmetric friction represented by Finsler metrics. For each of these cases we provide some mathematical analysis and some new insights. In each case, the motility map and its geometry lead to approximations of the body velocity integral, and from there to the net motion of the body through space. This talk is suitable for graduate students in mechanical engineering, applied mathematics, and physics. Early parts of this work were funded by the Dan D. and Betty Kahn Foundation Autonomous systems Mega-Project, ARO MURI W911NF-17-1-0306, and NSF CMMI 1825918.