Theory of Swimming Filaments in Viscoelastic Media

Abstract

Motivated by the swimming of sperm in the non-Newtonian fluids of the female mammalian reproductive tract, we examine the swimming of filaments in the nonlinear viscoelastic Upper Convected Maxwell model. We obtain the swimming velocity and hydrodynamic force exerted on an infinitely long cylinder with prescribed beating pattern. We use these results to examine the swimming of a simplified sliding-filament model for a sperm flagellum. Viscoelasticity tends to decrease swimming speed, and changes in the beating patterns due to viscoelasticity can reverse swimming direction.The physical environment of the cell places severe constraints on mechanisms for motility. For example, viscous effects dominate inertial effects in water at the scale of a few microns. Therefore, swimming cells use viscous resistance to move, since mechanisms that rely on imparting momentum to the surrounding fluid, such as waving a rigid oar, do not work [1,2]. The fundamental principles of swimming in the low-Reynolds number regime of small-scale, slow flows have been established for many years [2,3,4,5], yet continue to be an area of active research. However, when a sperm cell moves through the viscoelastic mucus of the female mammalian reproductive tract, the theory of swimming in a purely viscous fluid is inapplicable. Observations of sperm show that they are strongly affected by differences between viscoelastic and viscous fluids. In particular, the shape of the flagellar beating pattern as well as swimming trajectories and velocities depend on the properties of the medium [6,7,8].The interplay of medium properties and flagellar motility or transport also arises in other situations, such as spirochetes swimming in a gel [9], and cilia beating in mucus to clear foreign particles in the human airway [10]. Motivated by these phenomena, we develop a theory for swimming filaments in a viscoelastic medium. We begin by analyzing the swimming of an infinite filament with a prescribed beating pattern in a fluid described by the Upper Convected Maxwell (UCM) model [11]. We deduce the hydrodynamic force per unit length acting on the filament and the swimming velocity to leading order in the deformation of the filament. Our results extend the findings of Lauga [12], who considered a variety of fading memory models for the case of a prescribed beat pattern on a planar sheet. We further apply our results to a simple model flagellum with active internal forces, and find that changes in flagellum shapes play a crucial role in distinguishing the effects of viscoelastic media.Newtonian fluids are characterized by a simple constitutive relation, in which stress is proportional to strain rate. Non-newtonian fluids cannot be characterized by a simple universal constitutive relation, and exhibit a range of phenomena such as elasticity, shear thinning, and yield stress behavior. We choose to focus our attention on fluids with fading memory, in which the stress relaxes over time to the viscous stress. We consider small amplitude deflections of an infinite filam...

Force-free swimming of a model helical flagellum in viscoelastic fluids

“…The importance of elastic effects are quantified by the Deborah number De, defined as the ratio of the relaxation time τ of the fluid to the characteristic time scale of the swimmer. For an infinite sheet or filament deformed by low-amplitude traveling waves and suspended in a dilute polymer solution described by the Oldroyd-B model, the swimming speed is always less than the Newtonian speed and decreases with increasing Deborah number (7)(8)(9). These theoretical predictions agree roughly with the measurements of swimming C. elegans (6).…”

“…There are several natural directions for further study. The discrepancy between the experiment and the predictions for the swimming speed of a filament supporting small-amplitude helical traveling waves (8) suggests that unlike the Newtonian case, rigidly rotating helices and filaments with propagating helical waves behave differently in a viscoelastic medium. Because the discrepancy emerges at small Deborah number, it would be useful to make an expansion about the Newtonian case for small De, as has been done for spheres falling near a wall in a weakly viscoelastic fluid (28).…”

The heterogeneous motility of the Lyme disease spirochete in gelatin mimics dissemination through tissue

“…From the parameterized shape, we computed the center of mass and the average orientation of the bacteria and then calculated swimming speed along the direction of the mean orientation. We found that translocating bacteria in the 3-5% gelatin matrices swam without slipping (i.e., the swimming speed was equal to the wave speed), as has been observed for Leptonema illini in 1% methylcellulose solution (29) but is not predicted by any current theories for swimming in nonNewtonian fluids (9,12,13,30). In the 2% matrices, translocating bacteria sometimes slip with respect to the matrix but still typically swim without slipping.…”

“…Changes in the wave shape and frequency of the beating flagellum may enhance the ability of mammalian sperm to move through viscoelastic fluids, such as cervical mucus (8). Recent theoretical work has tried to explain how viscoelastic or gel-like media affect the swimming of microorganisms (9)(10)(11)(12)(13)(14), but, to date, there have been very few empirical studies to test the theoretical predictions (7).…”

Swarming in viscous fluids: Three-dimensional patterns in swimmer- and force-induced flows

“…One exception is the literature on swimmers wherein models have been developed for a single organism or a few organisms that propel themselves in viscous [48][49][50][51][52][53][54][55][56][57][58] and non-Newtonian fluids [57,[59][60][61][62][63][64][65][66][67]. In particular, swarming hydrodynamic theories have been derived wherein swimmer densities with or without fluid flows are described as continuous fields [4,19,68,69].…”