The hydrodynamics of flagellar propulsion: helical waves

The swimming of a micro-organism by the propagation of helical waves on a long slender flagellum is analysed. The model developed by Higdon (1979) is used to study the motion of an organism with a spherical cell body (radius A ) propelled by a cylindrical flagellum (radius a, length L). The average swimming speed and power consumption are calculated for helical waves (amplitude a, wavenumber k). A wide range of parameter values is considered to determine the optimal swimming motion. The optimal helical wave has ak w 1, corresponding to a pitch angle of 45". The optimum number of waves on the flagellum increases as the flagellar length LIA increases, such that the optimum wavelength decreases as LIA increases. The efficiency is relatively insensitive to the flagellar radius a/A. The optimum flagellar length is LIA M 10. The results are compared to calculations using two different forms of resistance coefficients. Gray-Hancock coefficients overestimate the swimming speed by approximately 20% and underestimate the power consumption by 50%. The coefficients suggested by Lighthill (1976) overestimate the swimming speed for large cell bodies (L/A < 15) by 20% and underestimate for small cell bodies (LIA > 15) by 10%. The Lighthill coefficients underestimate the power consumption up to 50% for LIA < 10, and overestimate up to 25% for LIA > 10. Overall, the Lighthill coefficients are superior to the Gray-Hancock coefficients in modelling swimming by helical waves.