Numerical results are presented for planar sinusoidal waves (amplitude α, wave-number k). The average swimming speed and power consumption are computed for a wide range of the parameters. The optimal sine wave for minimizing power consumption is found to be a single wave with amplitude αk ≈ 1. The power consumption is found to be relatively insensitive to changes in the flagellar radius. The optimal flagellar length is found to be in the range L/A = 20–40. The instantaneous force distribution and flow field for a typical organism are presented. The trajectory of the organism through one cycle shows that a wave of constant amplitude may have the appearance of increasing amplitude owing to the yawing motion of the organism.The results are compared with those obtained using resistance coefficients. For organisms with small cell bodies (A/L = 0.05), the average swimming speed predicted by Gray-Hancock coefficients is accurate to within 10%. For large cell bodies (A/L = 0.2), the error in swimming speed is approximately 20%. The relative error in the predicted power consumption is 25–50%. For the coefficients suggested by Lighthill, the power is consistently underestimated. The Gray-Hancock coefficients underestimate the power for small cell bodies and overestimate it for large cell bodies.

A method is described for solving the integral equations governing Stokes flow in arbitrary two-dimensional domains. It is demonstrated that the boundary-integral method provides an accurate, efficient and easy-to-implement strategy for the solution of Stokes-flow problems. Calculations are presented for simple shear flow in a variety of geometries including cylindrical and rectangular, ridges and cavities. A full description of the flow field is presented including streamline patterns, velocity profiles and shear-stress distributions along the solid surfaces. The results are discussed with special relevance to convective transport processes in low-Reynolds-number flows.

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.