Session Q14: Low Reynolds Number Locomotion: Non-Newtonian Fluids

Swimming of microorganisms in complex fluids; Part 1: Viscoelastic effects

Most motile microorganisms dwell in complex fluids. We aim to understand the dynamics of microbial swimming using a macroscopic force-free, torque-free robot that emulates the swimming strategy of real bacteria. In particular, to address the effects of viscoelasticity, we use Boger fluids (constant viscosity but elastic). Additionally, we quantified the flow field around the swimmers using standard Particle image velocimetry techniques (PIV). The measurements are conducted for a range of helical wavelength, λ, Deborah number, De, and two head geometries. We observe a significant swimming speed enhancement over the varied range of parameters. To understand this behavior, we decompose the thrust and drag forces into viscous and elastic contributions. Using resistive force theory, we can predict the enhancement and impedance in swimming speeds. The appearance of negative wakes in elastic fluids can be associated with improved swimmer performance. This provides insights into propulsion mechanisms of micro-swimmers in complex fluid environments and the kinematics of swimming gaits, extending it to hydrodynamic interactions with their surroundings and other swimmers (these issues are discussed in Part 2 of this presentation).   

Swimming of microorganisms in complex fluids; Part 2: Interactions

Bacteria exhibit various collective motions, many of which have important implications for the properties of the suspension they dwell in. The flows created influence their mutual interactions and modify the rheology of their suspensions. Although the flow field around real bacteria has been studied, a few have studied the effect of changing geometrical parameters and fluid rheology. We carry out experiments using the techniques described in Part 1 to study the interactions of swimmers in both purely viscous and viscoelastic fluids for a range of helical wavelength, λ, Deborah number, De, and two head geometries. We found strikingly different behaviors when the fluid has elastic properties; for instance, instead of attraction, we observe repulsion. Also, the orbiting behavior observed in a Newtonian fluid is no longer seen in the Boger fluid. We rationalize the changes in the interactions by analyzing the velocity fields around swimmers. The dominant Stokeslet-like velocity field observed in Newtonian fluids is significantly modified by elastic effects. Many types of interactions will be shown and discussed, addressing the effects of elasticity. The work provides insights into understanding predator-prey interactions and complex bio-fluid systems of flagellar microorganisms.   

Frustrated run and tumble of swimming E-coli bacteria in nematic liquid crystals

In many situations bacteria move in complex environments associated with non-Newtonian rheology. In liquid crystals the kinetics of bacterial motion is constrained by the orientational molecular order of the fluid and new dynamics arise from this orientational constraint. In this work, we study the swimming reorientation of a single bacterium, E. coli, constrained to move along the director field of a lyotropic chromonic liquid crystal (LCLC) that is confined to a planar cell. In such an environment, the spontaneous run and tumble motion of the bacterium gets frustrated: the elasticity of the liquid crystal prevents flagella from unbundling. Interestingly, in order to change direction, bacteria execute a reversal motion along the director field, driven by the relocation of a single flagellum to the other side of the bacterial body, coined as a frustrated tumble. We present a detailed experimental characterization of this phenomenon, exploiting exceptional spatial and temporal resolution of bacteria and flagella dynamics during swimming, obtained using a two color Lagrangian tracking technique. We suggest a possible mechanism behind the frustrated run and tumble motion, accounting for these observations.   

Anisotropic swimming and reorientation of an undulatory microswimmer in liquid-crystalline polymers

Microorganisms can efficiently navigate in anisotropic complex fluids, but the precise swimming mechanisms remain largely unexplored. Their dynamics are determined by the interplay between multiple effects, including the fluid's orientation order, swimmer's undulatory gait, and the finite length. Here we perform the numerical study of the two-dimensional undulatory motions of a flexible swimmer in lyotropic liquid-crystalline polymers (LCPs). The swimmer is modeled as a nearly inextensible yet flexible fiber with imposed traveling-wave-like actuation, and initially may have an arbitrary swimming direction with respect to the nematic director. We investigate the orientation-dependent swimming behaviors in nematic LCPs for an infinite long sheet (i.e., Taylor's swimming sheet model) and finite-length swimmers. We demonstrate that the swimmer must be sufficiently stiff to produce undulatory deformations to gain net motions. Moreover, a motile finite-length swimmer can reorient itself to swim parallel with the nematic director, due to a net body torque arising from the asymmetric distribution of the polymer force along the body.   

Rheotaxis of E. coli in Complex Fluids

The positive rheotaxis of microorganisms in Newtonian fluids encompasses the spontaneous orientation of individual swimmers against a unidirectional flow. This mechanism is now understood as being governed mainly by the positioning of the swimmer at an angle in the high shear flow region close to solid boundaries. Further studies underline a similar behavior for artificial self-propelled swimmers and demonstrates the hydrodynamic interplays that prescribes the swimmer’s gait.

 

Microorganisms have evolved to thrive in fluids that exhibit non-Newtonian behavior; these complex fluids are ubiquitous in nature and particularly in the human body. Our study focuses on the description of E. coli’s rheotaxis in different dilute polymer solutions. Experiments show that the flux upstream is increased by an order of magnitude in viscoelastic shear-thinning fluids as opposed to strictly Newtonian viscous fluids. The analysis of E. coli orientation distribution allows us to quantify the relative contribution of the elastic and viscous stresses on the phenomena.

 

    

Squirming in viscoplastic fluids

An axisymmetric squirmer in a Bingham viscoplastic fluid is studied numerically to determine the effect of a yield stress environment on locomotion. The effects of stroke modes, both pure and combined, are investigated and it is found that for the treadmill or ``neutral'' mode, the swimmer in a yield stress fluid has a lower swimming velocity and uses more power. However, the efficiency of swimming reaches its maximum at a finite yield limit. In addition, for higher yield limits, higher stroke modes can increase the swimming velocity and hydrodynamic efficiency of the treadmill swimmer. The higher-order odd-numbered squirming modes, particularly the third stroke mode, can generate propulsion by themselves that increases in strength as the viscoplastic nonlinearity increases at moderate yield limits. These results show that swimmers in viscoplastic environments, both biological and artificial, could potentially employ other non-standard swimming strategies to optimize their locomotion.

    

Slender body theory for a flagellated bacterium in a two-fluid model of a polymeric solution

We analyse the motion of a flagellated bacterium in a polymer solution modelled as a two-fluid Newtonian medium using slender body theory. The radius of the flagellar bundle is comparable or smaller than the microstructural length scale of the polymer so that the polymer and solvent satisfy different boundary conditions on the flagella surface. This is true for bacterial motion in biological fluids such as mucus where the polymer forms a mesh with typical pore sizes larger than the flagellar radius but smaller than the size of the cell. If the polymer relaxes rapidly, the scenario can be effectively modelled as a medium composed of two interpenetrating Newtonian fluids, the solvent and polymer, with different viscosities. The interphase drag yields a screening length L_B, within which the relative tangential velocity between the two phases is screened. From our calculations, we observe either a monotonic or non-monotonic variation of swimming speed with the viscosity ratio, depending on L_B. The results are sensitive to the way the polymer interacts with the flagella and we discuss scenarios corresponding to different types of interaction and compare our predictions with experimental observations.   

Forces on a Flexible Helix in Dilute Suspensions

To understand the effect of geometry and elasticity on the swimming dynamics of small-scale organisms, we study the behavior of deformable helices of varying pitch lengths in fluid suspensions of varying particle concentration. The helices are submerged in a tank that is rotated at a constant rotational speed. We measure thrust force and torque directly using the transducer on a TA ARES G2 rheometer to determine the normal and tangential force coefficient ratio. The tails are also placed on a swimmer with a cylindrical head that is actuated by a rotating magnetic field. The speed of the swimmers is compared for different pitch angles and fluids. We use neutrally buoyant fluid suspensions. Results for the flexible helices are compared with results for rigid helices. For rigid helices, the drag coefficient ratio—and consequently, the swimming speed—increases with particle concentration for most pitch angles. For flexible helices, the increase of drag coefficient ratio disappears. Possible explanations for these findings are discussed.

    

The role of viscoelasticity in mucociliary clearance - a continuum approach

We present numerical simulations and analytical predictions of mucociliary clearance based on a continuum description for a viscoelastic mucus film, where momentum transfer from the beating cilia is represented via a moving-envelope boundary condition introduced by Bottier et al. (PLoS Comp. Biol., 2017). Mucus viscoelasticity is represented via the Oldroyd-B model, where we have fitted the relaxation time and the total viscosity to the dynamic moduli of real mucus, ranging from `healthy'  to `diseased' conditions. We solve the nonlinear governing equations using the code Basilisk and also obtain an analytical solution in the weakly-viscoelastic limit. Both approaches predict a drop in the mucus flow rate q versus its Newtonian limit q_N as the cilia beat frequency, f, is increased. In the case of diseased mucus, q/q_N drops by 40% in the physiological range of f. Further, we find that this drop increases with decreasing cilia beat amplitude and film thickness. For healthy mucus, q/q_N remains close to unity, even at large f. This contrasts with the drop in swimming speed observed for microorganisms (Lauga, CUP, 2021) and suggests that, while elasticity of mucus hinders the propulsion of pathogens, it does not deteriorate mucociliary clearance.