Traveling waves are hydrodynamically optimal for long-wavelength flagella

Swimming eukaryotic microorganisms such as spermatozoa, algae, and ciliates self-propel in viscous fluids using traveling wavelike deformations of slender appendages called flagella. Waves are predominant because Purcell's scallop theorem precludes time-reversible kinematics for locomotion. Using the calculus of variations on a periodic long-wavelength model of flagellar swimming, we show that the planar flagellar kinematics maximizing the time-averaged propulsive force for a fixed amount of energy dissipated in the surrounding fluid correspond for all times to waves traveling with constant speed, potentially on a curved centerline, with propulsion always in the direction opposite to the wave.

Figure 1

An infinite flexible flagellum undergoing planar beating is described mathematically by a time-varying function $y(x,t)$ in Cartesian coordinates. The unit vectors $\mathbf{t}$ and $\mathbf{n}$ are the local tangent and normal to the instantaneous flagellar shape. The instantaneous velocity of the flagellum centerline, $\mathbf{u}$, can be decomposed into components along the directions tangent ($u_{\parallel }$) and normal to the flagellum ($u_{\perp }$), and similarly for the hydrodynamic force density acting on the flagellum, $\mathbf{f}$ (components $f_{\parallel }$ and $f_{\perp }$). The force and velocity components are related to one another through resistive-force theory. We use the calculus of variations to determine the class of functions $y(x,t)$ leading to the maximal average propulsive force for a given amount of average energy dissipated in the fluid.