Session D7: Locomotion in Complex Fluids

Propulsion in viscoelastic fluids: waving, flapping

In this talk, we present recent results on low-Reynolds number locomotion in non-Newtonian fluids. We first consider waving motion, the prototypical biological situation arising e.g. in ciliary transport of mucus, or spermatozoa swimming in complex fluids. We use asymptotic methods to estimate the effect of viscoelastic stresses on the kinematics and energetics of locomotion and transport in complex fluids. In our second problem, we consider simple flapping motion. Because of Purcell's scallop theorem, reciprocal motion such as flapping is known to be ineffective in a Newtonian fluid. We show here instead that a fluid with normal stress differences - such as Oldroyd B - can be used to rectify flapping motion and generate non-zero average forces and flows.   

Theory of swimming filaments in viscoelastic media

Microorganisms often encounter and must move through complex media. What aspects of propulsion are altered when swimming in viscoelastic gels and fluids? Motivated by the swimming of sperm through the mucus of the female mammalian reproductive tract, we examine the swimming of filaments in nonlinearly viscoelastic fluids. We obtain the swimming velocity and hydrodynamic force exerted on an infinitely long cylinder with prescribed beating pattern. We apply these results to study the swimming of a simplified sliding-filament model for a sperm flagellum. Viscoelasticity tends to decrease swimming speed. The viscoelastic response of the fluid can change the shapes of beating patterns, and changes in the beating patterns can even lead to reversal of the swimming direction.   

Undulatory swimming in a viscoelastic fluid

Mammalian spermatozoa encounter complex, non-Newtonian fluid environments as they make their way through the female reproductive tract. The beat form realized by the flagellum varies tremendously along this journey. We will present recent progress on the development of computational models that couple the internal force generation of undulating flagella with the external dynamics of a complex fluid. An immersed boundary framework is used, with the complex fluid represented either by a continuum Oldroyd-B model, or a Newtonian fluid overlaid with discrete viscoelastic elements.   

Large and limbless: the locomotion of snakes

In efforts to understand snake locomotion, we consider one of their various gaits. By contracting and extending their bodies unidirectionally like a slinky, large snakes propel themselves in a straight line. In a combined experimental and theoretical investigation, we here report on the dynamics of a boa constrictor alongside the analysis of an n-linked extensible crawler model. Constraints on their locomotion are quantified and discussed, such as the elasticity, frictional anisotropy and abrasive wear of their skin. Also presented are certain snake behaviors that culminate in their tying themselves into knots.   

Biological and robotic movement through granular media

We discuss laboratory experiments and numerical simulations of locomotion of biological organisms and robots on and within a granular medium. Terrestrial locomotion on granular media (like desert and beach sand) is unlike locomotion on rigid ground because during a step the material begins as a solid, becomes a fluid and then re-solidifies. Subsurface locomotion within granular media is unlike swimming in water for similar reasons. The fluidization and solidification depend on the packing properties of the material and can affect limb penetration depth and propulsive force. Unlike aerial and aquatic locomotion in which the Navier-Stokes equations can be used to model environment interaction, models for limb interaction with granular media do not yet exist. To study how the fluidizing properties affect speed in rapidly running and swimming lizards and crabs, we use a trackway composed of a fluidized bed of of 250 $\mu m$ glass spheres. Pulses of air to the bed set the solid volume fraction $0.59<\phi<0.63$; a constant flow rate $Q$ below the onset of fluidization (at $Q=Q_f$) linearly reduces the material strength (resistance force per depth) at fixed $\phi$ for increasing $Q$. Systematic studies of four species of lizard and a species of crab (masses $\approx 20$ grams) reveal that as $Q$ increases, the average running speed of an animal decreases proportionally to $\sqrt{M/A-const}(1-Q/Q_f)$ where $M$ is the mass of the animal and $A$ is a characteristic foot area. While the crabs decrease speed by nearly $75 \%$ as the material weakens to a fluid, the zebra tailed lizard uses long toes and a plantigrade foot posture at foot impact to maintain high speed ($\approx 1.5$ m/sec). We compare our biological results to systematic studies of a physical model of an organism, a 2 kg hexapedal robot SandBot. We find that the robot speed sensitively depends on $\phi$ and the details of the limb trajectory. We simulate the robot locomotion by computing ground reaction forces on a numerical model of the robot using a soft-sphere Molecular Dynamics code.