Locomotion of Helical Bodies in Viscoelastic Fluids: Enhanced Swimming at Large Helical Amplitudes

The motion of a rotating helical body in a viscoelastic fluid is considered. In the case of force-free swimming, the introduction of viscoelasticity can either enhance or retard the swimming speed and locomotive efficiency, depending on the body geometry, fluid properties, and the body rotation rate. Numerical solutions of the Oldroyd-B equations show how previous theoretical predictions break down with increasing helical radius or with decreasing filament thickness. Helices of large pitch angle show an increase in swimming speed to a local maximum at a Deborah number of order unity. The numerical results show how the small-amplitude theoretical calculations connect smoothly to the large-amplitude experimental measurements.

Figure 1

Experiments, theories, and simulations of swimming in viscoelastic fluids. With increasing fluid elasticity, (a) the helical bacterium Leptospira swims faster (see also Fig. 7 of Ref. 13, courtesy of J. H. Carr, CDC, 2103) [9, 14]; (b) the nematode C. elegans swims slower [15] (courtesy of X. N. Shen and P. E. Arratia, 2013); (c) a two-dimensional swimming sheet of small wave amplitude swims slower [19]; (d) a nearly cylindrical swimmer passing helical waves of small pitch angle swims slower with the same factor of reduction as for the two-dimensional sheet [20]; (e) a finite undulatory swimmer swims faster for a range of beating frequencies (instantaneous body velocity and mean-squared polymer distention field are shown [22], courtesy of J. Teran, L. Fauci, and M. Shelley, 2013); (f) and a thin helical body of arbitrary length can swim faster or slower, depending on the geometry and rotation rate [25]. (g) The axial component of fluid velocity generated by a rotating, force-free helical filament is shown; a helical fluid volume external to the coil is carried upward with the translating body, while a helical fluid volume internal to the coil is shuttled downward.

Figure 2

(a) Schematic of the dimensionless setup. The body rotates counterclockwise with unit angular speed when seen from above and swims in the axial direction with dimensionless speed $U$. (b) Helical-wave swimming speed (normalized by the Newtonian swimming speed) of filaments of varying thickness, $A=2^{n-2}$. Here $\psi =\pi /40$ and ${\eta }_s/\eta =0.5$. Deviations from the theoretical result of Fu et al. [20] (dotted line) are logarithmic in $A$. (c) Viscoelasticity leads to faster swimming for helices of a sufficiently large pitch angle. The filament thickness is fixed, $A=0.5$. Solid lines denote helical waves, dashed lines denote rigid body rotation, and symbols denote pitch angle, $\psi =\pi /40$, $\pi /10$, and $\pi /5$. (d) Larger swimming speeds are achieved by thinner filaments of the same pitch angles as in (c); here $A=0.2$. (e) The normalized swimming efficiency for helical waves and rigid body rotations. Symbols denote the same helices as in (c).

Figure 3

The mean-squared polymer distention fields, $\mathrm{tr}({\mathit{\tau }}_p)$, from the passage of helical waves are shown for three Deborah numbers, for filament thickness $A=0.2$ and pitch angles (a) $\psi =\pi /40$ and (b) $\psi =\pi /5$. The swimming speed decreases with the Deborah number in (a) and increases in (b). The path of the body during rotation is indicated by a white dashed line in the bottom left panel. The largest polymer distention appears along the second and fourth quadrants in case (a), and along the first and third quadrants in case (b), with the regions of maximal distention indicated by symbols.