Coins falling in water

When a coin falls in water, its trajectory is one of four types determined by its dimensionless moment of inertia $I^\ast$ and Reynolds number Re: (A) steady; (B) fluttering; (C) chaotic; or (D) tumbling. The dynamics induced by the interaction of the water with the surface of the coin, however, makes the exact landing site difficult to predict a priori. Here, we describe a carefully designed experiment in which a coin is dropped repeatedly in water, so that we can determine the probability density functions (pdf) associated with the landing positions for each of the four trajectory types, all of which are radially symmetric about the center-drop line. In the case of the steady mode, the pdf is approximately Gaussian distributed, with variances that are small, indicating that the coin is most likely to land at the center, right below the point it is dropped from. For the other falling modes, the center is one of the least likely landing sites. Indeed, the pdf's of the fluttering, chaotic and tumbling modes are characterized by a "dip" around the center. For the tumbling mode, the pdf is a ring configuration about the center-line, with a ring width that depends on the dimensionless parameters $I^\ast$ and Re and height from which the coin is dropped. For the chaotic mode, the pdf is generally a broadband distribution spread out radially symmetrically about the center-line. For the steady and fluttering modes, the coin never flips, so the coin lands with the same side up as was dropped. For the chaotic mode, the probability of heads or tails is close to 0.5. In the case of the tumbling mode, the probability of heads or tails based on the height of the drop which determines whether the coin flips an even or odd number of times during descent

The N-vortex problem on a rotating sphere: IV. Ring configurations coupled to a background field

We study the evolution of N-point vortices in ring formation embedded in a background flowfield that initially corresponds to solid-body rotation on a sphere. The evolution of the point vortices are tracked numerically as an embedded dynamical system along with the M contours which separate strips of constant vorticity. The full system is a discretization of the Euler equations for incompressible flow on a rotating spherical shell, hence a ‘barotropic’ model of the one-layer atmosphere. We describe how the coupling creates a mechanism by which energy is exchanged between the ring and the background, which ultimately serves to break-up the structure. When the center-of-vorticity vector associated with the ring is initially misaligned with the axis of rotation of the background field, it sets up the propagation of Rossby-Haurwitz waves around the sphere which move retrograde to the solid-body rotation. These waves pass energy to the ring (in the case when the solid-body field and the ring initially co-rotate), or extract energy from the ring (when the solid-body field and the ring initially counter-rotate), hence the Hamiltonian and the center-of-vorticity vector associated with the N-point vortices are no longer conserved as they are for the one-way coupled model described in Newton & Shokraneh (2006a). In the first case, energy is transferred to the ring, the length of the center-of-vorticity vector increases, while its tip spirals in a clockwise manner towards the North Pole. The ring stays relatively intact for short times but ultimately breaks-up on a longer timescale. In the later case, energy is extracted from the ring, the length of the center-of-vorticity vector decreases while its tip spirals towards the North Pole and the ring loses its coherence more quickly than in the co-rotating case. The special case where the ring is initially oriented so that its center-of-vorticity vector is perpendicular to the axis of rotation is also examined as it shows how the coupling to the background field breaks this symmetry. In this case, both the length of the center-of-vorticity vector and the Hamiltonian energy of the ring achieve a local maximum at roughly the same time

A Check-list and identification key for succulent plants in general cultivation in Nairobi

Volume: 8

Vortex crystals

Vortex crystals is one name in use for the subject of vortex patterns that move without change of shape or size. Most of what is known pertains to the case of arrays of parallel line vortices moving so as to produce an essentially two-dimensional flow. The possible patterns of points indicating the intersections of these vortices with a plane perpendicular to them have been studied for almost 150 years. Analog experiments have been devised, and experiments with vortices in a variety of fluids have been performed. Some of the states observed are understood analytically. Others have been found computationally to high precision. Our degree of understanding of these patterns varies considerably. Surprising connections to the zeros of 'special functions' arising in classical mathematical physics have been revealed. Vortex motion on two-dimensional manifolds, such as the sphere, the cylinder (periodic strip) and torus (periodic parallelogram) has also been studied, because of the potential applications, and some results are available regarding the problem of vortex crystals in such geometries. Although a large amount of material is available for review, some results are reported here for the first time. The subject seems pregnant with possibilities for further development.published or submitted for publicationis peer reviewe

A preliminary study on amphizoic amoebae with special reference to their preference for bacterial food

The present study was planned to screen the growth pattern of six different species of pathogenic and non pathogenic amphizoic amoebae viz. Naegleria fowleri, N. gruberi, Acanthamoeba culbertsoni, A. rhysodes, A. polyphaga and A. glebae using six different bacterial species like Escherichia coli (Strain E1 and E2 and E. coli lactose +ve), Proteus, Klebsiella and Pseudomonas as food in their in vitro growth on non-nutrient agar medium. It was observed that out of six amoebae used; the pathogenic N. fowleri and A. culbertsoni were feeding on E. coli (all the strains). Feeding these two species of bacteria, the growth of these two amoebae was luxuriant, but not so good while feeding other strains of bacteria though they fed, survived and formed cysts. The remaining four amoebae were found to feed and survive only on E. coli (all the strain) and formed cysts but showed very poor growth while feeding on other four bacterial strains. It was inferred that E. coli is the most suitable bacterial species for in vitro growth of amphizoic amoebae for various purposes. This also reiterates that there exists a complex inter-relationship between amoebae and bacteria in different habitats

Viscous evolution of point vortex equilibria: The collinear state

When point vortex equilibria of the 2D Euler equations are used as initial conditions for the corre- sponding Navier-Stokes equations (viscous), typically an interesting dynamical process unfolds at short and intermediate time scales, before the long time single peaked, self-similar Oseen vortex state dom- inates. In this paper, we describe the viscous evolution of a collinear three vortex structure that cor- responds to an inviscid point vortex fixed equilibrium. Using a multi-Gaussian 'core-growth' type of model, we show that the system immediately begins to rotate unsteadily, a mechanism we attribute to a 'viscously induced' instability. We then examine in detail the qualitative and quantitative evolution of the system as it evolves toward the long-time asymptotic Lamb-Oseen state, showing the sequence of topological bifurcations that occur both in a fixed reference frame, and in an appropriately chosen rotating reference frame. The evolution of passive particles in this viscously evolving flow is shown and interpreted in relation to these evolving streamline patterns.Comment: 17 pages, 15 figure