Session PV: Micro Propulsion II

Simulation of the flow field and tumbling dynamics of multiply flagellated bacterium

To study the hydrodynamics of swimming of multi-flagellated bacteria, such as Escherichia coli, we develop a simulation method using a bead-spring model to account for the hydrodynamic and the mechanical interactions between multiple flagella and the cell body, the reversal of the rotation of a flagellum in a tumble and associated polymorphic transformations of the flagellum. This simulation reproduces the experimentally observed behaviors of E. coli, namely, a three-dimensional random-walk trajectory in run-and-tumble motion and steady clockwise swimming near a wall. Here we show using a modeled cell that the polymorphic transformation of flagellum in a tumble facilitates the reorientation of the cell, and that the time-averaged flow field near a cell in a run has double-layered helical streamlines. Moreover, the instantaneous flow field, which is strongly time-dependent, is more than 10-fold larger in magnitude than the time-averaged flow, large enough to affect the migration behavior of surrounding chemoattractants, with the Peclet number for these molecules being larger than one near a swimming cell.   

A PIV study of the flow field around a rigid rotating helix

Micro-organisms such as bacteria and spermatozoa propel themselves through viscous fluid using flagella which exhibit helical waves. Their typical swimming Reynolds number is 10$^{-3}$ or less. This study presents an attempt of examining the flow field around a rotating flagellum using a scaled-up model. The helical models are made of 2mm diameter metal wires. They have a helical diameter of 17mm and an axial length of 220mm. Three helices with a different pitch angle of 30$^{\circ}$, 45$^{\circ}$ and 60$^{\circ}$ were tested. The experiment was performed in a rectangular silicone oil bath with a dynamic viscosity of 4.875 Pa.s. The helix was rotated at a frequency of 0.25, 0.5 and 1Hz respectively. The Reynolds number based on the tangential velocity of the rotating helix ranged from 0.02 to 0.09. PIV measurements were undertaken on the central plane parallel to the helical axis. The thrust produced by the rotating helices were also measured using a force sensor.   

Of cilium and flagellum kinematics

The kinematics of propulsion of small animals such as paramecium and spermatozoa is considered. Larger scale models of the cilium and flagellum have been built and a four-motor apparatus has been constructed to reproduce their known periodic motions. The cilium model has transverse deformational ability in one plane only, while the flagellum model has such ability in two planes. When the flagellum model is given a push-pull in one diametral plane, instead of transverse deflection in one plane, it forms a coil. Berg {\&} Anderson's postulation (\textit{Nature} \textbf{245} 1973) that a flagellum rotates, is recalled. The kinematics of cilia of paramecium, of the whipping motion of the spermatozoa flagella, and of the flapping motion (rolling and pitching) of the pectoral fins of much larger animals such penguins, have been reproduced in the same basic paramecium apparatus. The results suggest that each of the tiny individual paramecium propulsors have the intrinsic dormant kinematic and structural building blocks to optimize into higher Reynolds number propulsors. A synthetic hypothesis on how small might have become large is animated.   

External vortex pumping by oscillating plate arrays of mayfly nymphs

Mayfly nymphs are aquatic insects, many of which can generate ventilation currents by beating two linear arrays of external plate-like gills. The oscillation Reynolds number associated with the gill motion changes with animal size, varying from $Re \sim$ 2 to 50 depending on age and species. Thus mayflies provide a novel system model for studying ontogenetic changes in pumping mechanisms associated with transitions from a more viscous- to inertia-dominated flow. Observation of the 3-D kinematics of the gill motion of the species \textit{C. triangulifer} reveal that the mayfly makes a transition in stroke motion when $Re>$5, with a corresponding shift in mean flow from the ventral to the dorsal direction. Time-resolved PIV measurements within the inter-gill space reveal the basic elements of the flow consist of vortex rings generated by the strokes of the individual gills. For the larger $Re$ case, the phasing of the plate motion generates a complex array of small vortices that interact to produce an intermittent dorsally directed jet. For $Re<$5, distinct vortices are still observed, but increased diffusion creates vortices that simultaneously envelope several gills, forcing a new flow pattern to emerge and preventing the effective use of the high $Re$ stroke kinematics. Thus we argue the transition in the kinematics is a reflection of a single mechanism adapted over the traversed $Re$ range, rather than a shift to a completely new mechanism. This work is supported by the NSF under grant CBET-0730907.   

Regularized Slender Body Theory

Various slender body theories allow for the representation of filaments in Stokes' flow by a distribution of fundamental solutions along the filament center line. The idea is revisited here in the more general setting of regularized forces in a small neighborhood along the center line. The regularity in the forces produces a smooth final expression that helps eliminate the computational instabilities of the unregularized formulas. The derivations of the regular slender body theories corresponding with the standard theories of Lighthill and of Keller and Rubinow are outlined. Consistency with these theories is verified in the limit as the smoothing parameter vanishes. Numerical issues of the resulting theories are addressed in the context of test problems.   

Simulation of flagellar motions near a rigid surface

Simulations of the hydrodynamic interaction of rotating flagella with a neighboring solid surface are presented using the method of Regularized Stokeslets. We include in the method the use of regularized rotlets and a system of images that exactly cancels the fluid velocity at the wall. The results show features such as an attraction towards the surface and rotations that generate a drag force that allows the flagellum to ``roll'' along the surface. Other compute flows resemble observed features of the flow when organisms are near the bottom of the plate in an experimental setting.   

The Optimal Elastic Flagellum

We address the question of optimality for slender swimming bodies or flagella in viscous fluid environments. Our novel approach is to define an energy which includes not only the work performed against the surrounding fluid, but also the energy stored elastically in the bending of the body, the energy stored elastically in internal shearing (such as the relative sliding of microtubules internal to a flagellum), and viscous dissipation due to the presence of an internal fluid. The shape of the optimal periodic planar wave is determined numerically and in some cases analytically which maximizes a related efficiency measure. We find that bending or internal dissipation costs regularize the optimal shape, but elastic shearing costs do not. For bodies of finite length, we show that the number of wavelengths expressed by the body is determined by a competition between bending costs and the work done on the fluid associated with body rotations. The hydrodynamic efficiency is shown to be less sensitive to the morphology than the bending costs, which may help us to better understand the locomotory forms observed in nature.   

A mechanism of low-Reynolds-number propulsion enhancement

Is has been known for some time that some microorganisms can swim faster in high-viscosity gel-forming polymer solutions. The qualitative explanation of this phenomena first offered by H. Berg and L. Turner (Nature {\bf 278}, 349, 1979) suggested that propulsion enhancement is a result of flagellum pushing on quasi-rigid loose polymer network. Following this hypothesis we consider inertia-less propulsion in a model heterogeneous environment composed of sparse array of stationary obstacles embedded into a viscous Newtonian solvent. It is demonstrated that for some propulsion techniques, including transverse surface waves and rotating helical filament, the propulsion speed (for the prescribed swimming gait) is enhanced when compared to the locomotion through viscous solvent. It is shown that locomotion is not only advantageous speed-wise, but is also more hydrodynamically efficient. The results of the rigorous numerical simulations of the rotating ``shish- kebab'' filament propelled through a random sparse matrix of stationary spherical obstructions are in a very close agreement with the predictions of proposed modified resistive force theory based on effective media approximation.   

Unsteady low-Re swimming

In this talk, we focus on unsteady effects relevant to the fluid-based locomotion of micro-organisms. First, we consider transient effects in locomotion arising from the inertia of both the swimmer and the surrounding fluid. We discuss and derive the relevant time scales governing transient effects in low Reynolds number swimming, and illustrate them using the prototypical problem of a 2D swimmer starting from rest. Second, we address geometrical unsteadiness resulting from the finite-size of the swimmer. We solve numerically for the swimming kinematics of active (internally-forced) filaments, as models for eukaryotic flagella, and discuss the resulting unsteadiness of the cell body.