Our experiments and molecular dynamics simulations on a projectile penetrating a two-dimensional granular medium reveal that the mean deceleration of the projectile is constant and proportional to the impact velocity. Thus, the time taken for a projectile to decelerate to a stop is independent of its impact velocity. The simulations show that the probability distribution function of forces on grains is time independent during a projectile's deceleration in the medium. At all times the force distribution function decreases exponentially for large forces.

Enhanced, passive transport is studied numerically in an oscillating vortex chain with stress-free boundary conditions. The long-range transport is found to be diffusive in the long-time limit with an effective diffusion coefficient D * that peaks dramatically in the vicinity of a few, well-defined resonant frequencies. Superdiffusive transients are also observed for frequencies near these resonant frequencies, with the duration of the transients diverging at the resonant frequencies. Standard analytical techniques based on the Melnikov approximation and on lobe dynamics fail to explain the behavior in the vicinity of these resonant peaks. An alternate explanation is provided, based on flights that have power-law scaling up to a maximum length that also diverges at the resonant frequencies. The long flights for frequencies near the resonant peaks occur because tracers in a lobe return (after an integer number of oscillation periods) to almost precisely the same location in the lobe of another vortex. These periodic orbits correspond to the formation -only at the resonant frequencies -of "tangle islands" within the chaotic region.

We have measured the trajectory and visualized the wake of a single sphere falling in a fluid confined between two closely spaced glass plates (a Hele-Shaw cell). The position of a sedimenting sphere was measured to better than 0.001d, where d is the sphere diameter, for Reynolds numbers (based on the terminal velocity) between 20 and 330, for gaps between the plates ranging from 1.014d to 1.4d. For gaps in the range 1.01d-1.05d, the behaviour of the sedimenting sphere is found to be qualitatively similar to that of an unconfined cylinder in a uniform flow, but our sedimenting sphere begins to oscillate and shed von Kármán vortices for Re > 200, which is far greater than the Re = 49 for the onset of vortex shedding behind cylinders in an open flow. When the gap is increased to 1.10d-1.40d, the vortices behind the sphere are different -they are qualitatively similar to those behind a sphere sedimenting in the absence of confining walls. Our precision measurements of the velocity of a sedimenting sphere and the amplitude and frequency of the oscillations provide a benchmark for numerical simulations of the sedimentation of particles in fluids. IntroductionThe motion of blunt bodies in a fluid has been studied since the early days of the development of fluid mechanics. Even now sedimenting bodies yield surprising discoveries and challenges for theoretical analyses, but there have been few experiments using modern imaging techniques to examine a sedimenting body.We have examined the sedimentation of a single sphere in a fluid contained between vertical plates separated by a distance only slightly greater than the sphere diameter. We focus on the flow dependence on the separation between the plates. We make measurements for Reynolds numbers ranging from about 30 to 300, but we do not examine in detail the values of the Reynolds numbers corresponding to successive bifurcations. We measure the sphere position in a co-moving reference frame and obtain high-precision results for the sphere velocity and the properties of the wake. Our results for the sphere terminal velocity should long serve as a benchmark for algorithms designed for the difficult general problem of numerical simulation of the Navier-Stokes equation in a system with moving boundaries.Our observations reveal some qualitative features that are common to bodies sedimenting in the absence of sidewalls, and to flow past a fixed sphere or a fixed cylinder in unconfined geometries. Hence we review these situations in the following section.