The hydrodynamics of flagellar propulsion: helical waves

Abstract

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 α, wavenumber k). A wide range of parameter values is considered to determine the optimal swimming motion. The optimal helical wave has ak ≈ 1, corresponding to a pitch angle of 45°. The optimum number of waves on the flagellum increases as the flagellar length L/A increases, such that the optimum wavelength decreases as L/A increases. The efficiency is relatively insensitive to the flagellar radius a/A. The optimum flagellar length is L/A ≈ 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 (L/A > 15) by 10%. The Lighthill coefficients underestimate the power consumption up to 50% for L/A < 10, and overestimate up to 25% for L/A > 10. Overall, the Lighthill coefficients are superior to the Gray-Hancock coefficients in modelling swimming by helical waves.

Modelling bacterial behaviour close to a no-slip plane boundary: the influence of bacterial geometry

“…Phan-Thien et al (1987) used a boundary-element method (BEM) to analyse the motion of an ellipsoidal cell body propelled by a rigidly rotating helical flagellum. Good agreement was found when compared to the SBT results of Higdon (1979b) for swimming in unbounded fluids. More recently, Fujita & Kawai (2001) have drawn the same conclusions with quantitatively very similar results in their own BEM study.…”

Increased filtration efficiency of attached compared to free-swimming flagellates

“…The intense complexity of models of this type made progress impossible without the aid of computers. Higdon (1979a,b,c) used the approach to set up systems of singular integral equations to model the swimming motion of whole organisms with smooth flagella moving in a helical (Higdon 1979c) and a planar wave (Higdon 1979a), and the fluid motions generated by planar waves if the organism is sessile (Higdon 1979b). There is no analytical solution to the problems; a complex analytical-numerical approach is required.…”

A 3D Motile Rod-Shaped Monotrichous Bacterial Model

“…Higdon [60,61,59] studied the hydrodynamics of microorganism swimming using stokeslets, dipoles and rotlets to represent the cell body and applied slender body theory to represent the flagellum with a set of stokeslets and dipoles. A similar approach was used by Ramia, et al [86] to study microorganism motility.…”