A hydrodynamic analysis of flagellar propulsion

The swimming of a micro-organism by flagellar propulsion is examined. The organism consists of a spherical cell body (radius A) propelled by waves travelling down a long slender flagellum (radius a, length L). Slender-body theory for Stokes flow is used to replace the flagellum with distributions of Stokeslets and dipoles along its centre-line. The cell body is represented by a Stokeslet, dipole andrrotlet for translation and rotation, and by an image system to cancel the velocity induced by the singularities along the flagellum. The error in the slender-body theory is O(a/L), while the images cancel the velocity on the surface of the sphere exactly. With these approximations, the boundary-value problem for the Stokes equations is transformed into a system of singular integral equations. The unknowns are the velocity and angular velocity of the organism and the force distribution along the flagellum. An iteration procedure is used to solve the equations numerically. 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.