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.

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.

Diblock copolymers confined to a two-dimensional surface may produce uniform features of macromolecular dimensions ($\sim 10–100\mathrm{nm}$). 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.

We have constructed two mechanical snails, Robosnail 1 and Robosnail 2, inspired by the propulsion mechanisms of live gastropods. Each uses a different mechanical strategy to move on a thin layer of viscous fluid. Robosnail 1 uses a flexible flapping sheet to generate lubrication pressures in a viscous Newtonian fluid which in turn propels the snail. Robosnail 2 uses a compressible sliding sheet on a layer of Laponite, a non-Newtonian, finite-yield stress fluid.

In this article, we consider the centralized optimal estimation problem in spatially distributed systems. We use the setting of spatially invariant systems as an idealization for which concrete and detailed results are given. Such estimators are known to have a degree of spatial localization in the sense that the estimator gains decay in space, with the spatial decay rates serving as a proxy for how far measurements need to be shared in an optimal distributed estimator. In particular, we examine the dependence of spatial decay rates on problem specifications such as system dynamics, measurement, and process noise variances, as well as their spatial autocorrelations. We propose nondimensional parameters that characterize the decay rates as a function of problem specifications. In particular, we find an interesting matching condition between the characteristic lengthscale of the dynamics and the measurement noise correlation lengthscale for which the optimal centralized estimator is completely decentralized. A new graphical technique—termed the branch point locus—is introduced to quantify spatial decay rates in terms of analyticity regions in the complex spatial frequency plane. Our results are illustrated through two case studies of systems with dynamics modeled by diffusion and the Swift–Hohenberg equation, respectively.