Small-amplitude swimmers can self-propel faster in viscoelastic fluids.
Many small organisms self-propel in viscous fluids using travelling wave-like deformations of their bodies or appendages. Examples include small nematodes moving through soil using whole-body undulations or spermatozoa swimming through mucus using flagellar waves. When self-propulsion occurs in a non-Newtonian fluid, one fundamental question is whether locomotion will occur faster or slower than in a Newtonian environment. Here we consider the general problem of swimming using small-amplitude periodic waves in a viscoelastic fluid described by the classical Oldroyd-B constitutive relationship. Using Taylor's swimming sheet model, we show that if all travelling waves move in the same direction, the locomotion speed of the organism is systematically decreased. However, if we allow waves to travel in two opposite directions, we show that this can lead to enhancement of the swimming speed, which is physically interpreted as due to asymmetric viscoelastic damping of waves with different frequencies. A change of the swimming direction is also possible. By analysing in detail the cases of swimming using two or three travelling waves, we demonstrate that swimming can be enhanced in a viscoelastic fluid for all Deborah numbers below a critical value or, for three waves or more, only for a finite, non-zero range of Deborah numbers, in which case a finite amount of elasticity in the fluid is required to increase the swimming speed.