Modeling Running via Optimal Control for Shoe Design.

Shoe manufacturing technology is advancing faster than new shoe designs can viably be evaluated in human subject trials. To aid in the design process, this paper presents a model for estimating how new shoe properties will affect runner performance. This model assumes runners choose their gaits to optimize an intrinsic, unknown objective function. To learn this objective function, a simple two-dimensional mechanical model of runners was used to predict their gaits under different objectives, and the resulting gaits were compared to data from real running trials. The most realistic model gaits, i.e., the ones that best matched the data, were obtained when the model runners minimized the impulse they experience from the ground as well as the mechanical work done by their leg muscles. Using this objective function, the gait and thus performance of running under different shoe conditions can be predicted. The simple model is sufficiently sensitive to predict the difference in performance of shoes with disruptive designs but cannot distinguish between existing shoes whose properties are fairly similar. This model therefore is a viable tool for coarse-grain exploration of the design space and identifying promising behaviors of truly novel shoe materials and designs.

Confinement-induced stabilization of the Rayleigh-Taylor instability and transition to the unconfined limit.

The prevention of hydrodynamic instabilities can lead to important insights for understanding the instabilities' underlying dynamics. The Rayleigh-Taylor instability that arises when a dense fluid sinks into and displaces a lighter one is particularly difficult to arrest. By preparing a density inversion between two miscible fluids inside the thin gap separating two flat plates, we create a clean initial stationary interface. Under these conditions, we find that the instability is suppressed below a critical plate spacing. With increasing spacing, the system transitions from the limit of stability where mass diffusion dominates over buoyant forces, through a regime where the gap sets the wavelength of the instability, to the unconfined regime governed by the competition between buoyancy and momentum diffusion. Our study, including experiment, simulation, and linear stability analysis, characterizes all three regimes of confinement and opens new routes for controlling mixing processes.

Flagellar kinematics reveals the role of environment in shaping sperm motility.

Swimming spermatozoa from diverse organisms often have very similar morphologies, yet different motilities as a result of differences in the flagellar waveforms used for propulsion. The origin of these differences has remained largely unknown. Using high-speed video microscopy and mathematical analysis of flagellar shape dynamics, we quantitatively compare sperm flagellar waveforms from marine invertebrates to humans by means of a novel phylokinematic tree. This new approach revealed that genetically dissimilar sperm can exhibit strikingly similar flagellar waveforms and identifies two dominant flagellar waveforms among the deuterostomes studied here, corresponding to internal and external fertilizers. The phylokinematic tree shows marked discordance from the phylogenetic tree, indicating that physical properties of the fluid environment, more than genetic relatedness, act as an important selective pressure in shaping the evolution of sperm motility. More broadly, this work provides a physical axis to complement morphological and genetic studies to understand evolutionary relationships.

Passive phloem loading and long-distance transport in a synthetic tree-on-a-chip.

Vascular plants rely on differences in osmotic pressure to export sugars from regions of synthesis (mature leaves) to sugar sinks (roots, fruits). In this process, known as Münch pressure flow, the loading of sugars from photosynthetic cells to the export conduit (the phloem) is crucial, as it sets the pressure head necessary to power long-distance transport. Whereas most herbaceous plants use active mechanisms to increase phloem sugar concentration above that of the photosynthetic cells, in most tree species, for which transport distances are largest, loading seems, counterintuitively, to occur by means of passive symplastic diffusion from the mesophyll to the phloem. Here, we use a synthetic microfluidic model of a passive loader to explore the non-linear dynamics that arise during export and determine the ability of passive loading to drive long-distance transport. We first demonstrate that in our device, the phloem concentration is set by the balance between the resistances to diffusive loading from the source and convective export through the phloem. Convection-limited export corresponds to classical models of Münch transport, where the phloem concentration is close to that of the source; in contrast, diffusion-limited export leads to small phloem concentrations and weak scaling of flow rates with hydraulic resistance. We then show that the effective regime of convection-limited export is predominant in plants with large transport resistances and low xylem pressures. Moreover, hydrostatic pressures developed in our synthetic passive loader can reach botanically relevant values as high as 10 bars. We conclude that passive loading is sufficient to drive long-distance transport in large plants, and that trees are well suited to take full advantage of passive phloem loading strategies.

Razor clam to RoboClam: burrowing drag reduction mechanisms and their robotic adaptation.

Estimates based on the strength, size, and shape of the Atlantic razor clam (Ensis directus) indicate that the animal's burrow depth should be physically limited to a few centimeters; yet razor clams can dig as deep as 70 cm. By measuring soil deformations around burrowing E. directus, we have found the animal reduces drag by contracting its valves to initially fail, and then fluidize, the surrounding substrate. The characteristic contraction time to achieve fluidization can be calculated directly from soil properties. The geometry of the fluidized zone is dictated by two commonly-measured geotechnical parameters: coefficient of lateral earth pressure and friction angle. Calculations using full ranges for both parameters indicate that the fluidized zone is a local effect, occurring between 1-5 body radii away from the animal. The energy associated with motion through fluidized substrate-characterized by a depth-independent density and viscosity-scales linearly with depth. In contrast, moving through static soil requires energy that scales with depth squared. For E. directus, this translates to a 10X reduction in the energy required to reach observed burrow depths. For engineers, localized fluidization offers a mechanically simple and purely kinematic method to dramatically reduce energy costs associated with digging. This concept is demonstrated with RoboClam, an E. directus-inspired robot. Using a genetic algorithm to find optimal digging kinematics, RoboClam has achieved localized fluidization burrowing performance comparable to that of the animal, with a linear energy-depth relationship, in both idealized granular glass beads and E. directus' native cohesive mudflat habitat.

Flagellar waveform dynamics of freely swimming algal cells.

We present quantitative measurements of time-dependent flagellar waveforms for freely swimming biflagellated algal cells, for both synchronous and asynchronous beating. We use the waveforms in conjunction with resistive force theory as well as a singularity method to predict a cell's time-dependent velocity for comparison with experiments. While net propulsion is thought to arise from asymmetry between the power and recovery strokes, we show that hydrodynamic interactions between the flagella and cell body on the return stroke make an important contribution to enhance net forward motion.

Capillary breakup of discontinuously rate thickening suspensions.

Using discontinuously rate thickening suspensions (DRTS) as a model system, we show that beads-on-a-string morphologies can arise as a result of external viscous drag acting during capillary-driven breakup of a non-Newtonian fluid. To minimize the perturbative effect of gravity, we developed a new experimental test platform in which the filament is supported in a horizontal position at the surface of an immiscible oil bath. We show that the evolution of thin DRTS filaments during the capillary thinning process is well described by a set of one-dimensional slender filament equations. The strongly rate-dependent rheology of the test fluid and the aspect ratio of the filament couple to control the thinning dynamics and lead to a simple criterion describing the localized arrest of the capillary thinning process and the subsequent formation of complex, high aspect ratio beads-on-a-string structures.

Localized fluidization burrowing mechanics of Ensis directus.

Muscle measurements of Ensis directus, the Atlantic razor clam, indicate that the organism only has sufficient strength to burrow a few centimeters into the soil, yet razor clams burrow to over 70 cm. In this paper, we show that the animal uses the motions of its valves to locally fluidize the surrounding soil and reduce burrowing drag. Substrate deformations were measured using particle image velocimetry (PIV) in a novel visualization system that enabled us to see through the soil and watch E. directus burrow in situ. PIV data, supported by soil and fluid mechanics theory, show that contraction of the valves of E. directus locally fluidizes the surrounding soil. Particle and fluid mixtures can be modeled as a Newtonian fluid with an effective viscosity based on the local void fraction. Using these models, we demonstrate that E. directus is strong enough to reach full burrow depth in fluidized soil, but not in static soil. Furthermore, we show that the method of localized fluidization reduces the amount of energy required to reach burrow depth by an order of magnitude compared with penetrating static soil, and leads to a burrowing energy that scales linearly with depth rather than with depth squared.

Identification and evaluation of the Atlantic razor clam (Ensis directus) for biologically inspired subsea burrowing systems.

In this article, we identify and analyze a subsea organism to serve as a model for biologically inspired burrowing technology to be used in applications such as anchoring, installation of cables, and recovery of oil. After inspecting myriad forms of life that live on or within ocean substrates, the Atlantic razor clam, Ensis directis, stood out as an attractive basis for new burrowing technology because of its low-energy requirements associated with digging (0.21 J/cm), its speed and depth of burrrowing (∼1 cm/s and 70 cm, respectively), and its size and simplicity relative to man-made machines. As anchoring is a prime application for the technology resulting from this work, the performance of an Ensis directus-based anchoring system was compared to existing technologies. In anchoring force per embedment energy, the E. directus-based anchor beats existing technology by at least an order of magnitude. In anchoring force per weight of device, the biologically inspired system weighs less than half that of current anchors. The article concludes with a review of E. directus's digging strategy, which involves motions of its valves to locally fluidize the substrate to reduce burrowing drag and energy, and the successful adaptation of E. directus's burrowing mechanisms into an engineering system: the RoboClam burrowing robot, which, like the animal, uses localized fluidization to achieve digging energy that scales linearly with depth, rather than depth squared, for moving through static soil.

Optimal kinematics and morphologies for spermatozoa.

We investigate the role of hydrodynamics in the evolution of the morphology and the selection of kinematics in simple uniflagellated microorganisms. We find that the most efficient swimming strategies are characterized by symmetrical, nonsinusoidal bending waves propagating from the base of the head to the tip of the tail. In addition, we show that the ideal tail-to-head length ratio for such a swimmer is ≈12 and that this predicted ratio is consistent with data collected from over 400 species of mammalian sperm.

Optimal feeding and swimming gaits of biflagellated organisms.

Locomotion is widely observed in life at micrometric scales and is exhibited by many eukaryotic unicellular organisms. Motility of such organisms can be achieved through periodic deformations of a tail-like projection called the eukaryotic flagellum. Although the mechanism allowing the flagellum to deform is largely understood, questions related to the functional significance of the observed beating patterns remain unresolved. Here, we focus our attention on the stroke patterns of biflagellated phytoplanktons resembling the green alga Chlamydomonas. Such organisms have been widely observed to beat their flagella in two different ways--a breaststroke and an undulatory stroke--both of which are prototypical of general beating patterns observed in eukaryotes. We develop a general optimization procedure to determine the existence of optimal swimming gaits and investigate their functional significance with respect to locomotion and nutrient uptake. Both the undulatory and the breaststroke represent local optima for efficient swimming. With respect to the generation of feeding currents, we found the breaststroke to be optimal and to enhance nutrient uptake significantly, particularly when the organism is immersed in a gradient of nutrients.

Vertical hydrodynamic focusing in glass microchannels.

Vertical hydrodynamic focusing in microfluidic devices is investigated through simulation and through direct experimental verification using a confocal microscope and a novel form of stroboscopic imaging. Optimization for microfluidic cytometry of biological cells is examined. By combining multiple crossing junctions, it is possible to confine cells to a single analytic layer of interest. Subtractive flows are investigated as a means to move the analysis layer vertically in the channel and to correct the flatness of this layer. The simulation software (ADINA and Coventor) is shown to accurately capture the complex dependencies of the layer interfaces, which vary strongly with channel geometry and relative flow rates.

Soft swimming: exploiting deformable interfaces for low reynolds number locomotion.

Reciprocal movement cannot be used for locomotion at low Reynolds number in an infinite fluid or near a rigid surface. Here we show that this limitation is relaxed for a body performing reciprocal motions near a deformable interface. Using physical arguments and scaling relationships, we show that the nonlinearities arising from reciprocal flow-induced interfacial deformation rectify the periodic motion of the swimmer, leading to locomotion. Such a strategy can be used to move toward, away from, and parallel to any deformable interface as long as the length scales involved are smaller than intrinsic scales, which we identify. A macroscale experiment of flapping motion near a free surface illustrates this new result.

Optimal stroke patterns for Purcell's three-link swimmer.

Stroke patterns for Purcell's three-link swimmer are optimized. We model the swimmer as a jointed chain of three slender rods moving in an inertialess flow. The swimmer is optimized for efficiency and speed. We were able to attain swimmer designs significantly more efficient than those previously suggested by authors who only consider geometric design rather than kinematic criteria. The influence of slenderness on optimality is considered as well.

Rheological fingerprinting of gastropod pedal mucus and synthetic complex fluids for biomimicking adhesive locomotion.

Nonlinear rheological properties are often relevant in understanding the response of a material to its intended environment. For example, many gastropods crawl on a thin layer of pedal mucus using a technique called adhesive locomotion, in which the gel structure is periodically ruptured and reformed. We present a mechanical model that captures the key features of this process and suggests that the most important properties for optimal inclined locomotion are a large, reversible yield stress, followed by a small shear viscosity and a short thixotropic restructuring time. We present detailed rheological measurements of native pedal mucus in both the linear and nonlinear viscoelastic regimes and compare this "rheological fingerprint" with corresponding observations of two bioinspired slime simulants, a polymer gel and a clay-based colloidal gel, that are selected on the basis of their macroscopic rheological similarities to gastropod mucin gels. Adhesive locomotion (of snails or mechanical crawlers) imposes a large-amplitude pulsatile simple shear flow onto the supporting complex fluid, motivating the characterization of nonlinear rheological properties with large amplitude oscillatory shear (LAOS). We represent our results in the form of Lissajous curves of oscillatory stress against time-varying strain. The native pedal mucus gel is found to exhibit a pronounced strain-stiffening response, which is not imitated by either simulant.

Two-dimensional self-assembly in diblock copolymers.

Diblock copolymers confined to a two-dimensional surface may produce uniform features of macromolecular dimensions (approximately 10-100 nm). We present a mathematical model for nanoscale pattern formation in such polymers that captures the dynamic evolution of a solution of poly(styrene)-b-poly(ethylene oxide), PS-b-PEO, in solvent at an air-water interface. The model has no fitting parameters and incorporates the effects of surface tension gradients, entanglement or vitrification, and diffusion. The resultant morphologies are quantitatively compared with experimental data.

Theory for shock dynamics in particle-laden thin films.

We present a theory to explain the emergence of a particle-rich ridge observed experimentally in a thin film particle-laden flow on an incline. We derive a lubrication theory for this system which is qualitatively compared to preliminary experimental data. The ridge formation arises from the creation of two shocks due to the differential transport rates of fluid and particles. This parallels recent findings of double shocks in thermal-gravity-driven flow [A. L. Bertozzi, Phys. Rev. Lett. 81, 5169 (1998); J. Sur, Phys. Rev. Lett. 90, 126105 (2003); A. Munch, Phys. Rev. Lett. 91, 016105 (2003)]. However, here the emergence of the shocks arises from a new mechanism involving the settling rates of the species.

Peeling, healing, and bursting in a lubricated elastic sheet.

We consider the dynamics of an elastic sheet lubricated by the flow of a thin layer of fluid that separates it from a rigid wall. By considering long wavelength deformations of the sheet, we derive an evolution equation for its motion, accounting for the effects of elastic bending, viscous lubrication, and body forces. We then analyze various steady and unsteady problems for the sheet, such as peeling, healing, levitating, and bursting, using a combination of numerical simulation and dimensional analysis. On the macroscale, we corroborate our theory with a simple experiment, and, on the microscale, we analyze an oscillatory valve that can transform a continuous stream of fluid into a series of discrete pulses.

Periodic knolls and valleys: coexistence of solid and liquid states in granular suspensions.

We report the spontaneous emergence of a doubly periodic train of sedimented knolls in a dense suspension. These solidified knolls rise out of, and coexist alongside, a sea of freely flowing liquid in a slowly rotating horizontal bottle. We apply a variable viscosity model that permits simultaneous analysis of fluidlike and solidlike behaviors that are ubiquitous in a variety of sedimenting flows. The model generates qualitative agreement with experiments, and produces new insights into mechanisms by which sedimented structures form.