The impact of fluid elasticity on the motility of swimming microorganisms
Keywords: Chemical Engineering.

In recent years, there has been a great interest in understanding the physics of motility at the microscopic scale. Most of this work, however, has focused on motion in simple Newtonian fluids, like water. Yet, most biological fluids relevant to swimming microorganisms are in fact rheologically complex, since they are laden with large biopolymers that give them a rich microstructure. In particular, these fluids are often viscoelastic, meaning they display both a viscous and elastic response to stress. Understanding motility in these environments is not only important from a scientific point of view, but may also aid researchers in a number of engineering applications, e.g. preventing the spread of disease by disrupting biofilm formation. Thus, the goal of this work is to determine how fluid elasticity affects the motility of swimming microorganisms, through the use of a combination of numerical simulations, microhydrodynamic theory, and experiments. We begin by considering organisms that swim via an undulatory, or wavelike, gait. In particular, we consider the nematode C. elegans, a model biological organism that has received a great deal of interest in recent years for its use in experimental studies of motility in complex fluids. We model the motion of C. elegans computationally through the Immersed Finite Element Method (IFEM), a computational fluid dynamics method that explicitly models the motion of an immersed body (such as a swimming organism) as a Lagrangian mesh moving in a background Eulerian mesh representing the surrounding fluid. We find that in all cases C. elegans swims slower in an elastic fluid than it otherwise would in a corresponding Newtonian fluid of the same viscosity, in agreement with what is observed in experiments. Furthermore, we find that the maximum speed reduction in elastic fluids is determined by the polymer concentration in the surrounding fluid. An analysis of the stress in the surrounding flow field reveals that this reduction in speed comes as a result of regions of predominantly extensional flow concentrated at the ends (i.e. the head and the tail) of the swimming worm. In these regions, polymer molecules are readily stretched, leading to a buildup of extra polymer stress that acts to impede the swimmer's forward motion. Finally, to compare the effect fluid elasticity has on different swimming gaits, we also consider the motion of a swimming amoeba. Again using a modified version of the IFEM method, we find that swimming amoebae, like C. elegans, experience a reduction in their swimming speed when traversing a viscoelastic fluid. In contrast to C. elegans, we observe a much greater speed for the case of a swimming amoeba. We believe this is a consequence of the fact that the amoeba generates regions of relatively high polymer stress around its entire body --- regions that it is unable to move out of at any point in time in its swim cycle. Oftentimes, fluid elasticity is seen to cause a reduction in motility, as measured by the organism's swimming speed. Recent experiments with the bacteria E. coli, however, have shown that it experiences an enhancement of its speed when swimming in a viscoelastic fluid. To explain this phenomenon, we adopt the theoretical squirmer model to describe the microorganism and its gait. In the squirmer model, a swimmer's gait is captured by a prescribed slip velocity at the surface of its body, with different terms in the general expression for the slip velocity representing different modes of swimming. In our work, we show that the inclusion of a higher-order mode acting in the azimuthal direction (seen as a rotlet dipole in the far-field) is critical for properly describing swimming dynamics in an elastic fluid. Specifically, we show that systematically increasing this mode, which amounts to increasing the relative extent of "swirling" or rotational flow present in the organism's gait, causes the swimmer to transition from experiencing a speed reduction to a speed enhancement in a viscoelastic fluid. Thus, it is seen that elasticity increases the speed of swimmers with swirl present in their gait (like E. coli, which propels itself via a rotating flagellar bundle), while reducing the speed of those without swirl. As described above, swimmers with swirl present in their gait swim faster in elastic fluids than they do in viscous Newtonian fluids. Now, we ask a more fundamental question: can the combination of elasticity and swirling flow enable new propulsion strategies that otherwise would not work in a Newtonian fluid under the constraints of Stokes flow? To answer this, we consider a model swimmer consisting of two counter-rotating bodies of revolution aligned along their axis of revolution; in particular, we examine a swimmer consisting of two counter-rotating spheres of unequal sizes. Using a combination of numerical simulations and asymptotic theory valid for weakly elastic fluids (as measured by the Deborah number of the fluid), we show that such a swimmer exhibits zero net motion in a viscous Newtonian fluid, but propels itself in the direction of the larger of the two spheres in a viscoelastic fluid. In particular, we find that the swimming speed is nearly linear in the Deborah number, the dimensionless group quantifying the relative magnitude of elastic effects in the fluid, and in the polymer concentration in the fluid. An analysis of the surrounding flow field reveals that thrust in the viscoelastic case originates from an imbalance of pressure acting across the body, as a result of a pressure wake found directly behind the swimmer. Finally, we conclude our work by creating a robotic swimmer to test our hypotheses with experiments. Indeed, we find that the swimming robot is unable to propel itself in a viscous Newtonian fluid (corn syrup) but exhibits a steady swimming motion in a viscoelastic fluid containing polyacrylamide. Furthermore, we find good agreement between experimental measurements of the swimming speed and numerical simulations designed to match the geometry of the swimming robot and the rheology of the viscoelastic fluid used in experiments. In summary, we use a combination of numerical simulations, microhydrodynamic theory, and experiments to ask fundamental questions about how fluid elasticity impacts the motility of swimming microorganisms. In this, we focus both on quantifying the effect of viscoelasticity and in understanding the underlying mechanisms that drive these phenomena. Thus, this work serves as a key step in understanding the physics of motility in complex biological fluids