Swimming and pumping by helical waves in viscous and viscoelastic fluids

Abstract

We study helical bodies of arbitrary cross-sectional profile as they swim or transport fluid by the passage of helical waves. Many cases are explored: the external flow problem of swimming in a cylindrical tube or an infinite domain, the internal fluid pumping problem, and confined/unconfined swimming and internal pumping in a viscoelastic (Oldroyd-B) fluid. A helical coordinate system allows for the analytical calculation of swimming and pumping speeds and fluid velocities in the asymptotic regime of nearly cylindrical bodies. In a Newtonian flow, a matched asymptotic analysis results in corrections to the swimming speed accurate to fourth-order in the small wave amplitude, and the results compare favorably with full numerical simulations. We find that the torque-balancing rigid body rotation generally opposes the direction of wave passage, but not always. Confinement can result in local maxima and minima of the swimming speed in the helical pitch, and the effects of confinement decrease exponentially fast with the diameter of the tube. In a viscoelastic fluid, we find that the effects of fluid elasticity on swimming and internal pumping modify the Newtonian results through the mode-dependent complex viscosity, even in a confined domain.