The efficiency of propulsion by a rotating flagellum

Abstract

At very low Reynolds number, the regime in which fluid dynamics is governed by Stokes equations, a helix that translates along its axis under an external force but without an external torque will necessarily rotate. By the linearity of the Stokes equations, the same helix that is caused to rotate due to an external torque will necessarily translate. This is the physics that underlies the mechanism of flagellar propulsion employed by many microorganisms. Here, I examine the linear relationships between forces and torques and translational and angular velocities of helical objects to understand the nature of flagellar propulsion.

Figure 1

An isolated propeller, subjected to an external force F and an external torque N. It rotates at angular velocity ω and translates at velocity v.

Figure 2

Two propellers with propulsion matrices P₁ and P₂ (Upper) connected by a thin axial wire (Lower). The propulsion matrix of the composite propeller is P₁ + P₂.

Figure 3

A special propeller with a symmetrical propulsion matrix (B = C). See the text.

Figure 4

A propeller in the shape of a right-handed helix connected to a spherical cell, both moving to the right at velocity v. The propeller and cell rotate in opposite directions, at angular velocities ω and Ω, respectively. The external force F acting on the propeller is directed to the left.

Figure 5

Two propellers made of steel wire allowed to sink under their own weight, with axes vertical. b is a composite propeller consisting of propeller a joined rigidly to its mirror image.