A boundary-element analysis of flagellar propulsion

The swimming of a flagellar micro-organism by the propagation of helical waves along its flagellum is analysed by a boundary-element method. The method is not restricted to any particular geometry of the organism nor does it assume a specific wave motion for the flagellum. However, only results for an organism with a spherical or ellipsoidal cell body and a helically beating flagellum are presented here. With regard to the flagellum, it is concluded that the optimum helical wave (amplitude α and wavenumber k ) has α k ≈ 1 (pitch angle of 45°) and that for the optimum flagellar length L / A = 10 ( L being the flagellar length, A being the radius of the assumed spherical cell body) the optimum number of wavelengths N λ is about 1.5. Furthermore there appears to be no optimal value for the flagellar radius a , with the thinner flagella being favoured. These conclusions show excellent quantitative agreement with those of slender-body theory. For the case of an ellipsoidal cell body, the optimum aspect ratios B / A and C / A of the ellipsoid are about 0.7 and 0.3 respectively; A , B and C are the principal radii of the ellipsoid. These and all of the above conclusions show good qualitative agreement with experimental observations of efficiently swimming micro-organisms.