Session BA: Biofluid Dynamics III: Swimming

Fin ray design and use in fish swimming

Fish fins have evolved over millions of years in a convergent fashion, leading to a highly-intricate fin-ray structure that is found in half of all fish species. This fin ray presumably arose for reasons of efficient hydrodynamic interaction. I will present a linear elasticity model of the fin ray, based on the physical picture of the ray which has emerged from past experiments. By comparing the model with recent experiments performed in the Lauder Lab in Harvard's Biology department, we find that the model helps to understand some of the fundamental properties of fin rays. I will also present some recent results from a fully-coupled fin-fluid model, which combines a model for flexible bodies in a fluid flow with the dynamics of vortex sheets.   

Microstructure of concentrated suspensions of swimming model micro-organisms

A swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, referred to as a squirmer. The centre of mass of the sphere may be displaced from the geometric centre, and the effects of inertia and Brownian motion are neglected. The well known Stokesian- dynamics method is modified in order to simulate squirmer motions in a concentrated suspension. The movement of up to 216 identical squirmers in a concentrated suspension without any imposed flow is simulated in a domain with periodic boundary conditions, and the microstructure of the suspension is investigated. The results show that; (a) an aggregation of cells appears by considering only the hydrodynamic interaction between cells, (b) the aggregated cells generate collective motions by the hydrodynamic interaction between themselves, and (c) the collective motions occur randomly in time and in space. These tendencies qualitatively agree well with earlier experiments.   

Momentum balance, hydrodynamic impulse and choreography in rapidly maneuvering live fish

It is well known that swimming fish can swim circles around underwater vehicles when it comes to maneuvering performance. A typical underwater vehicle sweeps a circular arc, about ten vehicle lengths in diameter, and this requires about 15 times the amount of time it takes for the vehicle to drive one body length. In contrast, a fish, such as the Danio aequipinnatus, can turn in a space that is approximately one third of its body length, and it requires about half the time it takes to swim one body length. High-speed Particle Imaging Velocimetry (PIV) is used to quantify the impulse imparted to the fluid during a maneuver which is compared to the change in momentum of the fish during the maneuver. In order to model the impulse of the fluid, we make the assumption that the wake may be modeled as an axisymmetric vortex ring. The evidence that this is an appropriate model comes from a recent study of rapidly maneuvering flapping foils. Our PIV results show that the fish in fact generates two such wakes, one generated by the tail and shed at the conclusion of stage one of the maneuver and the other generated by the mid-section of the body and shed at the conclusion of stage two of the maneuver. Both of these vertical impulses are required to balance the momentum change of the fish from its initial swimming trajectory to its final swimming trajectory in a classical C-shaped maneuver.   

A simple model for the diffusion of swimming model microorganisms.

At the 2004 DFD meeting, Pedley and Ishikawa presented computational simulations and results on the diffusion of swimming model micro-organisms (spherical squirmers) in a semi-dilute suspension. Now, a simple `gas' model is proposed to describe the diffusive behaviour. Organisms are treated as particles in a `constant speed gas', so that the motion of each organism is approximated by a straight line trajectory at constant speed punctuated by instantaneous collisions. While the model is crude, it accurately describes several scaling behaviours of the diffusion in the case of strongly squirming organisms, including: (i) the direction autocorrelation function decays exponentially with time, (ii) the steady state translational diffusivity and the time taken to approach that steady state are both inversely proportional to the concentration, (iii) the rotational diffusivity is directly proportional to the concentration but the time taken for the rotational diffusivity to approach its steady state value is independent of concentration. The gas model also allows one to estimate the numerical value of the translational diffusion coefficient on the basis of the hydrodynamics of two-body interactions.   

Simulation of swimming strings immersed in a viscous fluid flow

In nature, many phenomena involve interactions between flexible bodies and their surrounding viscous fluid, such as a swimming fish or a flapping flag. The intrinsic dynamics is complicate and not well understood. A flexible string can be regarded as a one-dimensional flag model. Many similarities can be found between the flapping string and swimming fish, although different wake speed results in a drag force for the flapping string and a propulsion force for the swimming fish. In the present study, we propose a mathematical formulation for swimming strings immersed in a viscous fluid flow. Fluid motion is governed by the Navier-Stokes equations and a momentum forcing is added in order to bring the fluid to move at the same velocity with the immersed surface. A flexible inextensible string model is described by another set of equations with an additional momentum forcing which is a result of the fluid viscosity and the pressure difference across the string. The momentum forcing is calculated by a feedback loop. Simulations of several numerical examples are carried out, including a hanging string which starts moving under gravity without ambient fluid, a swinging string immersed in a quiescent viscous fluid, a string swimming within a uniform surrounding flow, and flow over two side-by-side strings. The numerical results agree well with the theoretical analysis and previous experimental observations. Further simulation of a swimming fish is under consideration.   

3D Gyrotactic Bioconvection Simulations

We introduce a new code for solving the continuum equations for gyrotactic bioconvection [1] in \emph{three spatial dimensions} to study pattern formation in suspensions of swimming micro-organisms such as the single-celled alga, \emph{Clamydomonas nivalis}. The governing equations consist of Navier--Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation. These equations are solved efficiently using a semi-implicit second-order accurate conservative finite-difference method. The structure and stability of a three-dimensional plume in a deep chamber with stress-free sidewalls are investigated. The solutions are compared with previous studies on two-dimensional [2] and axisymmetric bioconvection [3]. The evolution of the plumes is rich and varied, with quasi-steady states giving way to varicose and then meandering modes. 1. Pedley, T.J., Hill, N.A. \& Kessler, J.O., 1988. The growth of bioconvection patterns in a uniform suspension of gyrotactic microorganisms. J.~Fluid Mech.~\textbf{195}, 223---238. 2. Ghorai, S. \& Hill, N.A., 2000. Wavelengths of gyrotactic plumes in bioconvection. Bull.~Math.~Biol.~\textbf{62}, 429---450. 3. Ghorai, S. \& Hill, N.A., 2002. Axisymmetric bioconvection in a cylinder. J.~Theor.~Biol.~\textbf{219}, 137---152.   

Rotational dynamics of a towed superhelix in a Stokes fluid

Most bacteria use trailing and rotating flagella to locomote through a fluid. The spirochete bacterium instead locomotes itself by rotating its entire super-helically shaped body, driven by a flagellum that is threaded along its body. A natural question is how the effectiveness of locomotion depends on the detailed helix and super-helix arrangement (e.g. different handedness, different pitches). With this motivation, we study the rotational dynamics of super-helical bodies being towed through highly viscous fluids. A typical body is short-pitched helix whose axis is itself shaped as a helix of larger pitch and opposite handedness. We find that the direction and rate of the rotation of the body is a result of competition between these two super-posed helices; For small axial helix amplitude, the body dynamics is controlled by the short-pitched helix, while there is a cross-over at larger amplitude to control by the axial helix. We also investigate this system by numerically constructing solutions to the Stokes equations using the method of regularized Stokeslets and find excellent agreement between our numerical and experimental results.   

Oscillations of Eukaryotic Cilia and Flagella

The undulating beat of eukaryotic flagella and cilia produces forces that move cells and cause locomotion. The timing mechanisms that generate these periodic undulations are still mysterious and the question of how these oscillations arise is still a subject of much research - both experimental and theoretical. Recent experimental results on paralyzed and reconstituted flagella offer new insight into the dynamical mechanisms that could result in sustained waveform generation. Motivated by these recent experimental results we propose a model that mimics the flagellar structure as motor driven elastic, inextensible filaments. We hypothesize that the oscillations arise due to motor (dynein) driven, constrained, relative sliding of parts of the flagella. The dynamical equations describing the evolution of the populations of attached and detached motors is actively coupled to the local configuration as well as local sliding velocities via strain and configuration dependent kinetic reaction rates. At the same time, the filament configuration is actively coupled to the motor densities via the dependence of the active internal torque densities on the motor populations as well as their internal state. Appropriate ensemble averaged force-velocity relationships for the motors completes the set of equations. Numerical solutions reveal onset of dynamical instabilities via Hopf-bifurcations with oscillatory waveforms emerging from a trivial base state corresponding to a straight, non-moving flagellum.