Gwynn J. Elfring

Hydrodynamic Phase Locking of Swimming Microorganisms

Some microorganisms, such as spermatozoa, synchronize their flagella when swimming in close proximity. Using a simplified model (two infinite, parallel, two-dimensional waving sheets), we show that phase locking arises from hydrodynamics forces alone, and has its origin in the front-back asymmetry of the geometry of their flagellar waveform. The time evolution of the phase difference between coswimming cells depends only on the nature of this geometrical asymmetry, and microorganisms can phase lock into conformations which minimize or maximize energy dissipation.

Two-dimensional flagellar synchronization in viscoelastic fluids

Experimental studies have demonstrated that spermatozoa synchronize their flagella when swimming in close proximity. In a Newtonian fluid, it was shown theoretically that such synchronization arises passively due to hydrodynamic forces between the two swimmers if their waveforms exhibit a front-back geometrical asymmetry. Motivated by the fact that most biological fluids possess a polymeric microstructure, here we address synchronization in a viscoelastic fluid analytically. Using a two-dimensional infinite sheet model, we show that the presence of polymeric stresses removes the geometrical asymmetry constraint and therefore even symmetric swimmers synchronize. Such synchronization occurs on asymptotically faster time scales than in a Newtonian fluid, and the swimmers are seen to be driven into a stable in-phase conformation minimizing the energy dissipated in the surrounding fluid.

Squirming through shear-thinning fluids

Many micro-organisms find themselves immersed in fluids displaying non-Newtonian rheological properties such as viscoelasticity and shear-thinning viscosity. The effects of viscoelasticity on swimming at low Reynolds numbers have already received considerable attention, but much less is known about swimming in shear-thinning fluids. A general understanding of the fundamental question of how shear-thinning rheology influences swimming still remains elusive. To probe this question further, we study a spherical squirmer in a shear-thinning fluid using a combination of asymptotic analysis and numerical simulations. Shear-thinning rheology is found to affect a squirming swimmer in non-trivial and surprising ways; we predict and show instances of both faster and slower swimming depending on the surface actuation of the squirmer. We also illustrate that while a drag and thrust decomposition can provide insights into swimming in Newtonian fluids, extending this intuition to problems in complex media can prove problematic.

Passive hydrodynamic synchronization of two-dimensional swimming cells

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 arbitrarily 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 occurs 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 fix...

Taylor's swimming sheet: Analysis and improvement of the perturbation series

Abstract In G.I. Taylor’s historic paper on swimming microorganisms, a two-dimensional sheet was proposed as a model for flagellated cells passing traveling waves as a means of locomotion. Using a perturbation series, Taylor computed swimming speeds up to fourth order in amplitude. Here we systematize the expansion so that it can be carried out formally to arbitrarily high order. The resultant series diverges for an order one value of the wave amplitude, but may be transformed into a series with much improved convergence properties and which yields results comparing favorably to those obtained numerically via a boundary integral method for moderate and large values of the wave amplitudes.

Synchronization of flexible sheets

When swimming in close proximity, some microorganisms such as spermatozoa synchronize their flagella. Previous work on swimming sheets showed that such synchronization requires a geometrical asymmetry in the flagellar waveforms. Here we inquire about a physical mechanism responsible for such symmetry breaking in nature. Using a two-dimensional model, we demonstrate that flexible sheets with symmetric internal forcing deform when interacting with each other via a thin fluid layer in such a way as to systematically break the overall waveform symmetry, thereby always evolving to an in-phase conformation where energy dissipation is minimized. This dynamics is shown to be mathematically equivalent to that obtained for prescribed waveforms in viscoelastic fluids, emphasizing the crucial role of elasticity in symmetry breaking and synchronization.

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

The use of the reciprocal theorem has been shown to be a powerful tool to obtain the swimming velocity of bodies at low Reynolds number. The use of this method for lower-dimensional swimmers, such as cylinders and sheets, is more problematic because of the undefined or ill-posed resistance problems that arise in the rigid-body translation of these shapes. Here, we show that this issue can be simply circumvented and give concise formulas obtained via the reciprocal theorem for the self-propelled motion of deforming two-dimensional bodies. We also discuss the connection between these formulae and Faxen’s laws.

The effect of gait on swimming in viscoelastic fluids

Abstract In this paper, we give formulas for the swimming of simplified two-dimensional bodies in complex fluids using the reciprocal theorem. By way of these formulas we calculate the swimming velocity due to small-amplitude deformations on the simplest of these bodies, a two-dimensional sheet, to explore general conditions on the swimming gait under which the sheet may move faster, or slower, in a viscoelastic fluid compared to a Newtonian fluid. We show that in general, for small amplitude deformations, a speed increase can only be realized by multiple deformation modes in contrast to slip flows. Additionally, we show that a change in swimming speed is directly due to a change in thrust generated by the swimmer.

External losses in high-bypass turbo fan air engines

The external irreversible losses of air engines, due to equilibration of the hot and fast exhaust with the environment, are discussed based on the second law of thermodynamics. The effect of the bypass ratio on thermomechanical exergy destruction in the exhaust stream is demonstrated. The analysis gives a strong motivation for the use of high bypass turbo fan engines in modern aircraft.

Theory of Locomotion Through Complex Fluids

Microorganisms such as bacteria often swim in fluid environments that cannot be classified as Newtonian. Many biological fluids contain polymers or other heterogeneities which may yield complex rheology. For a given set of boundary conditions on a moving organism, flows can be substantially different in complex fluids, while non-Newtonian stresses can alter the gait of the microorganisms themselves. Heterogeneities in the fluid may also be characterized by length scales on the order of the organism itself leading to additional dynamic complexity. In this chapter we present a theoretical overview of small-scale locomotion in complex fluids with a focus on recent efforts quantifying the impact of non-Newtonian rheology on swimming microorganisms.