Self-propulsion of slender micro-swimmers by curvature control: N-link swimmers

We discuss a reduced model to compute the motion of slender swimmers which propel themselves by propagating a bending wave along their body. Our approach is based on the use of resistive force theory for the evaluation of the viscous forces and torques exerted by the surrounding fluid, and on discretizing the kinematics of the swimmer by representing its body through an articulated chain of N rigid links capable of planar deformations.The resulting system of ODEs governing the motion of the swimmer is easy to assemble and to solve, making our reduced model a valuable tool in the design and optimization of bio-inspired engineered microdevices. We test the accuracy and robustness of our approach on three benchmark examples: Purcell's 3-link swimmer, Taylor's swimming sheet and some recent quantitative observations of circular motion of a sperm cell. An explicit formula for the displacement of Purcell's 3-link swimmer generated by a square stroke of small amplitude is also discussed.

• We study slender swimmers that move by controlling the curvature of their centerline.
• We present an algorithm to assemble and solve the equations of motion of the swimmer.
• We discuss connections between swimming and geometric control theory.
• We discuss connections between Purcell's square stroke and Lie brackets.
• We check the accuracy of our discrete model against several benchmark problems.