Abstract
Finding the fastest path to a desired destination is a vitally important task for microorganisms moving in a fluid flow. We study this problem by building an analytical formalism for overdamped microswimmers on curved manifolds and arbitrary flows. We show that the solution corresponds to the geodesics of a Randers metric, which is an asymmetric Finsler metric that reflects the irreversible character of the problem. Using the examples of spherical and toroidal surfaces, we demonstrate that the swimmer performance that follows this “Randers policy” always beats a more direct policy. Moreover, our results show that the relative gain grows significantly when specific structures related to either the geometry or the flow are exploited by the swimmer. A study of the shape of isochrones reveals features such as self-intersections, cusps, and abrupt nonlinear effects. Our work provides a link between microswimmer physics and geodesics in generalizations of general relativity.Received 15 October 2020Revised 2 February 2021Accepted 29 April 2021DOI:https://doi.org/10.1103/PhysRevResearch.3.023125Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasSwimmingPhysical SystemsActive matterMicroswimmersTechniquesGeometryBiological Physics
Highlights
Artificial micro- and nanoswimmers [1] with active external controls can increasingly be engineered to execute specialized tasks in complex environments
Adopting recent mathematical results from differential geometry [28,29], we show that this problem can be mapped onto geodesics of a Finsler-type geometry with a Randers metric [30]
We can analyze the optimal paths obtained by following the Finsler geometry-based approach, which we call the Randers policy (RP), in comparison with a benchmark, which we refer to as the direct policy (DP), in which the microswimmer always points in the direction of the target, regardless of the force field [21]
Summary
Artificial micro- and nanoswimmers [1] with active external controls (e.g., via chemical [2,3] and electromagnetic fields [4,5,6], feedback loops [7,8], and geometric features of boundaries [9]) can increasingly be engineered to execute specialized tasks in complex environments These have crucial technological and medical applications ranging from targeted delivery of drugs [10], genes [11], or other cargo [12], to prevention of dental biofilm [13].
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