Locomotion of a wavy cylindrical tail through viscoplastic fluids

Beating of a long, thin cylindrical tail is a common means of locomotion at small scales. Here, the implications of plasticity in the ambient fluid are considered. Building on some previous work in which classical slender-body theory was generalised to describe motion in a viscoplastic fluid, solutions are presented to show how a yield stress affects the motion of a cylindical swimmer. Numerical solutions span the range of the two key parameters of the problem: the wave amplitude (relative to wavelenth) and a dimensionless measure of the yield stress; these are suplemented by discussion of some limits of the problem where analytical progress is possible. When the yield stress is large, 'burrowing' is possible at sufficiently high wave amplitudes, in which the swimmer moves along its centreline at the wave speed. Conversely, at low amplitudes the swimmer can be retarded by the ambient yield stress and, for some waveforms, even move backwards.