Dispersion relations for active undulators in overdamped environments

Area of Science:

• Biomechanics
• Fluid Dynamics
• Locomotion

Background:

• Undulatory swimmers like nematodes and fish adapt gait parameters (frequency ω, wave number k) to maintain performance across substrates.
• Environmental rheology significantly impacts locomotion efficiency.

Purpose of the Study:

• To identify a unifying relationship between gait parameters (ω, k) for overdamped undulatory swimmers.
• To model organism locomotion as actively driven viscoelastic beams in fluid.

Main Methods:

• Experimental observation of undulatory swimmers (nematodes, spermatozoa, fish) in varied rheologies.
• Development of a viscoelastic beam model to simulate locomotion dynamics.
• Analysis of scaling relationships between frequency and wave number.

Main Results:

• A universal active dispersion relation ω∝k^{±2} was identified for undulatory swimmers.
• The model successfully reproduced experimentally observed scaling.
• The observed scaling (k^{-2} or k^{2}) depends on the relative strength of body versus environmental dissipation.

Conclusions:

• Mechanics constrains gait parameter relationships, allowing continuous variation along the dispersion curve.
• The findings unify the understanding of locomotion across different species and environments.
• Body length independence from boundary conditions is observed when body dissipation dominates.