Passive hydrodynamic synchronization of two-dimensional swimming cells

Abstract

Spermatozoa flagella are known to synchronize when swimming in close proximity. We use a model consisting of two-dimensional sheets propagating transverse waves of displacement to demonstrate that fluid forces lead to such synchronization passively. Using two distinct asymptotic descriptions (small amplitude and long wavelength), we derive the synchronizing dynamics analytically for ar- bitrarily shaped waveforms in Newtonian fluids, and show that phase locking will always occur for sufficiently asymmetric shapes. We characterize the effect of the geometry of the waveforms and the separation between the swimmers on the synchronizing dynamics, the final stable conformations, and the energy dissipated by the cells. For two closely-swimming cells, synchronization always oc- curs at the in-phase or opposite-phase conformation, depending solely on the geometry of the cells. In contrast, the work done by the swimmers is always minimized at the in-phase conformation. As the swimmers get further apart, additional fixed points arise at intermediate values of the relative phase. In addition, computations for large-amplitude waves using the boundary integral method reveal that the two asymptotic limits capture all the relevant physics of the problem. Our results provide a theoretical framework to address other hydrodynamic interactions phenomena relevant to populations of self-propelled organisms.Comment: 29 pages, 12 figure

A note on the reciprocal theorem for the swimming of simple bodies

“…The vector of complex Fourier coefficients, c n , delineates wave shape (c 0 = 0), while the wave variable, z = k · r 0 − ωt, depends on the wave vector, k, and the actuation frequency, ω. The boundary conditions for an extensible sinusoidal sheet, c 1 = −i/2e y , and an inextensible sheet were first described by Taylor [3], and catalogued in detail elsewhere [28]. Non-dimensionalizing lengths by |k| and time by ω we obtain ∆r = n c n e inz = r 1 ,…”

Theory of Locomotion Through Complex Fluids

“…He found that energy dissipation in the fluid is minimized if the two sheets are oscillating with no phase difference [29]. However, in this symmetric sinusoidal setup, there can be no evolution of the phase from an arbitrary initial condition, due to the kinematic reversibility of the Stokes flow field equations [35,65]. If two such sheets are swimming in a viscoelastic fluid, such as those present along the path through the female reproductive system, then kinematic reversibility no longer constrains the dynamics.…”

“…[35] for general waveforms. Following Taylor, we take the approach that the amplitude of transverse oscillations of the sheet is small and expand all fields a regular perturbation series in ak = 1.…”

Gait synchronization in Caenorhabditis elegans

“…Although theoretical (19)(20)(21)(22)(23) studies of 2D waving sheets demonstrate that the sheets synchronize by hydrodynamic interactions, hydrodynamic forces in two dimensions are much greater than in three dimensions. To further verify that hydrodynamics did not play a significant role in synchronization, we carried out a controlled experiment in which we monitored the motions of an active (live) nematode and a paralyzed nematode suspended in a drop, far from the drop's boundaries.…”