Swimming of flagellated microorganisms

The swimming motion of a microorganism with a single flagellum is investigated for both helical and planar flagellar motion. First the force and torque exerted on the organism by the surrounding fluid are calculated in terms of the specified flagellar motion and the unknown linear and angular velocity of the whole organism. Then these unknown velocities are determined by the condition that the net force and torque on the organism are zero. Using these velocities, the trajectory of the organism is found. In the case of helical flagellar motion, the path of the entire organism is found to be a helix of small radius. The axis of the flagellum is not parallel to the axis of the helical path, but makes a small angle with it and precesses around it. If the flagellar motion is planar and sinusoidal, then the trajectory of the organism is found to be a straight line with small oscillations about it. Each point of the flagellum also oscillates longitudinally with double the frequency of the transverse oscillation, producing a figure eight motion. However if the flagellar motion is planar and asymmetric, then the trajectory is found to be a circle with small superposed oscillations. These conclusions account for the observed helical and circular trajectories of sperm, and for the figure eight motion of the tip of the flagellum in the planar case