Low-Reynolds-number swimmer utilizing surface traveling waves: Analytical and experimental study

Abstract

Microscale slender swimmers are frequently encountered in nature and are now used in microrobotic applications. The swimming mechanism examined in this paper is based on small transverse axisymmetric traveling wave deformations of a cylindrical long shell. The thin-shelled device is assumed to be inextensible at the middle surface and extensible at the surface wetted by the fluid. Assuming low-Reynolds-number hydrodynamics, an analytical solution is derived for waves of small amplitudes compared with the cylinder diameter. We show that swimming velocity increases with β(1) (the ratio of cylinder radius to wavelength) and that swimming velocity is linearly dependent on wave propagation velocity, increasing to leading order with the square of the ratio of wave amplitude to wavelength β(2) and decreasing with the wall thickness. A fourth-order correction in β(2) was also calculated and was found to have a negative effect on the swimming velocity. The results for a shell of negligible-wall thickness were compared with Taylor's solution for an inextensible two-dimensional flat membrane undergoing a waving motion and Felderhof's results [Phys. Fluids 22, 113604 (2010)] for an unbounded flow field and negligible-wall thickness. We show that Taylor's analytic solution is a particular limiting case of the present solution, assuming zero wall thickness and infinite values of β(1). The present mechanism was also compared with Taylor's well known solutions of waving planar and helical circular tails. We show that the present approach yields higher velocities as β(1) increases, whereas, the opposite occurs for waving tails. Indeed, in the region where β(1)>15, the present approach yields velocities nearly as fast as Taylor's helical waving tail while consuming less power and with a design that is considerably more compact. In this regime, the axisymmetric swimmer is twice as fast as Taylor's planar-tail swimmer for an additional investment of only one-third of the power. Experiments were conducted using a macroscale autonomous model immersed in highly viscous silicone fluid. We outlined how the proposed mechanism was realized to propel an elongated, yet finite, swimmer. Measured data demonstrate the effects of wave velocity and wavelength on swimming speed, showing good agreement with analytical results.