The forward undulatory locomotion of Ceanorhabditis elegans in viscoelastic fluids

Caenorhabditis elegans is a soil dwelling roundworm that has served as model organisms for studying a multitude of biological and engineering phenomena. We study the undulatory locomotion of nematode in viscoelastic fluids with zero-shear viscosity varying from 0.03--75~Pa$\cdot$s and relaxation times ranging from 0--350~s. We observe that the averaged normalized wavelength of swimming worm is essentially the same as that in Newtonian fluids. The undulatory frequency f shows the same reduction rate with respect to zero-shear viscosity in viscoelastic fluids as that found in the Newtonian fluids, meaning that the undulatory frequency is mainly controlled by the fluid viscosity. However, the moving speed $V_m$ of the worm shows more distinct dependence on the elasticity of the fluid and exhibits a 4\% drop with each 10-fold increase of the Deborah number $\mathrm{De}$, a dimensionless number characterizing the elasticity of a fluid. To estimate the swimming efficiency coefficient and the ratio $K=C_N/C_L$ of resistive coefficients of the worm in various viscoelastic fluids, we show that whereas it would take the worm around 7 periods to move a body length in a Newtonian fluid, it would take 27 periods to move a body length in a highly viscoelastic fluid.