On the Lagrangian description of unsteady boundary layer separation. Part 1: General theory

Although unsteady, high-Reynolds number, laminar boundary layers have conventionally been studied in terms of Eulerian coordinates, a Lagrangian approach may have significant analytical and computational advantages. In Lagrangian coordinates the classical boundary layer equations decouple into a momentum equation for the motion parallel to the boundary, and a hyperbolic continuity equation (essentially a conserved Jacobian) for the motion normal to the boundary. The momentum equations, plus the energy equation if the flow is compressible, can be solved independently of the continuity equation. Unsteady separation occurs when the continuity equation becomes singular as a result of touching characteristics, the condition for which can be expressed in terms of the solution of the momentum equations. The solutions to the momentum and energy equations remain regular. Asymptotic structures for a number of unsteady 3-D separating flows follow and depend on the symmetry properties of the flow. In the absence of any symmetry, the singularity structure just prior to separation is found to be quasi 2-D with a displacement thickness in the form of a crescent shaped ridge. Physically the singularities can be understood in terms of the behavior of a fluid element inside the boundary layer which contracts in a direction parallel to the boundary and expands normal to it, thus forcing the fluid above it to be ejected from the boundary layer

On the Jacobi-Metric Stability Criterion

We investigate the exact relation existing between the stability equation for the solutions of a mechanical system and the geodesic deviation equation of the associated geodesic problem in the Jacobi metric constructed via the Maupertuis-Jacobi Principle. We conclude that the dynamical and geometrical approaches to the stability/instability problem are not equivalent.Comment: 14 pages, no figure

Stellar models with Schwarzschild and non-Schwarzschild vacuum exteriors

A striking characteristic of non-Schwarzschild vacuum exteriors is that they contain not only the total gravitational mass of the source, but also an {\it arbitrary} constant. In this work, we show that the constants appearing in the "temporal Schwarzschild", "spatial Schwarzschild" and "Reissner-Nordstr{\"o}m-like" exteriors are not arbitrary but are completely determined by star's parameters, like the equation of state and the gravitational potential. Consequently, in the braneworld scenario the gravitational field outside of a star is no longer determined by the total mass alone, but also depends on the details of the internal structure of the source. We show that the general relativistic upper bound on the gravitational potential $M/R < 4/9$, for perfect fluid stars, is significantly increased in these exteriors. Namely, $M/R < 1/2$, $M/R < 2/3$ and $M/R < 1$ for the temporal Schwarzschild, spatial Schwarzschild and Reissner-Nordstr{\"o}m-like exteriors, respectively. Regarding the surface gravitational redshift, we find that the general relativistic Schwarzschild exterior as well as the braneworld spatial Schwarzschild exterior lead to the same upper bound, viz., $Z < 2$. However, when the external spacetime is the temporal Schwarzschild metric or the Reissner-Nordstr{\"o}m-like exterior there is no such constraint: $Z < \infty$. This infinite difference in the limiting value of $Z$ is because for these exteriors the effective pressure at the surface is negative. The results of our work are potentially observable and can be used to test the theory.Comment: 19 pages, 3 figures and caption

Brane world solutions of perfect fluid in the background of a bulk containing dust or cosmological constant

The paper presents some solutions to the five dimensional Einstein equations due to a perfect fluid on the brane with pure dust filling the entire bulk in one case and a cosmological constant (or vacuum) in the bulk for the second case. In the first case, there is a linear relationship between isotropic pressure, energy density and the brane tension, while in the second case, the perfect fluid is assumed to be in the form of chaplygin gas. Cosmological solutions are found both for brane and bulk scenarios and some interesting features are obtained for the chaplygin gas on the brane which are distinctly different from the standard cosmology in four dimensions.Comment: 10 Latex pages, 5 figure

Wave-like Solutions for Bianchi type-I cosmologies in 5D

We derive exact solutions to the vacuum Einstein field equations in 5D, under the assumption that (i) the line element in 5D possesses self-similar symmetry, in the classical understanding of Sedov, Taub and Zeldovich, and that (ii) the metric tensor is diagonal and independent of the coordinates for ordinary 3D space. These assumptions lead to three different types of self-similarity in 5D: homothetic, conformal and "wave-like". In this work we present the most general wave-like solutions to the 5D field equations. Using the standard technique based on Campbell's theorem, they generate a large number of anisotropic cosmological models of Bianchi type-I, which can be applied to our universe after the big-bang, when anisotropies could have played an important role. We present a complete review of all possible cases of self-similar anisotropic cosmologies in 5D. Our analysis extends a number of previous studies on wave-like solutions in 5D with spatial spherical symmetry

Equivalence Between Space-Time-Matter and Brane-World Theories

We study the relationship between space-time-matter (STM) and brane theories. These two theories look very different at first sight, and have different motivation for the introduction of a large extra dimension. However, we show that they are equivalent to each other. First we demonstrate that STM predicts local and non-local high-energy corrections to general relativity in 4D, which are identical to those predicted by brane-world models. Secondly, we notice that in brane models the usual matter in 4D is a consequence of the dependence of five-dimensional metrics on the extra coordinate. If the 5D bulk metric is independent of the extra dimension, then the brane is void of matter. Thus, in brane theory matter and geometry are unified, which is exactly the paradigm proposed in STM. Consequently, these two 5D theories share the same concepts and predict the same physics. This is important not only from a theoretical point of view, but also in practice. We propose to use a combination of both methods to alleviate the difficult task of finding solutions on the brane. We show an explicit example that illustrate the feasibility of our proposal.Comment: Typos corrected, three references added. To appear in Mod. Phys. Let

Game (wildlife) meat safety: understanding the game meat supply chain and the roles of role-players in a multifaceted control environment

Moderator: Lyle Renecker.Presented at the 8th international congress for wildlife and livelihoods on private and communal lands: livestock, tourism, and spirit, that was held on September 7-12, 2014 in Estes Park, Colorado.As in other parts of the world, the South African game industry is expanding and game meat is finding its way in the local and international food supply chain. In order to do effective game meat control, one must have an understanding of the game meat supply chain and the stakeholders therein. The term 'food supply chain' refers to the total supply process from agricultural production, harvest / slaughter, through processing and handling of a food and its ingredients, storage and distribution up to consumption. In the case of game meat, the supply chain will include the relevant food and meat control authorities, primary producers (farmers), suppliers of feed and veterinary drugs to farmers, hunters, abattoirs, small and large scale processors, wholesale and retail, suppliers of processing materials, import and export agencies, transport and the consumer. From this it is clear that several role-players are involved, which leads to a multifaceted and fragmented food control environment. It was necessary to determine the control points essential for meat safety and the responsibilities of the role-players for the execution of these control points along the game meat supply chain. Information regarding the study was obtained through a desk top study and analysis of questionnaire responses from game farmers, hunters and national, provincial and municipal control authorities. Game meat is often entering the supply chain in an uncontrolled manner while there is a lack of a single formal framework for its operation amongst the stakeholders. To address this issue, the research looked for insight into the game meat supply chain; the differences between the game meat supply chain and the conventional meat supply chain; the respective role-players in the supply chain; and the essential meat safety control points along the supply chain. The crucial meat safety link between the supply chain and the roles of stakeholders in ensuring that safe meat is supplied to the consumer is often vague. The introduction of a mechanism to reduce the indistinctness is essential for this growing industry. The mechanism suggested relates to the establishment of a game meat safety forum whereby all role-players are involved

Self-similar cosmologies in 5D: spatially flat anisotropic models

In the context of theories of Kaluza-Klein type, with a large extra dimension, we study self-similar cosmological models in 5D that are homogeneous, anisotropic and spatially flat. The "ladder" to go between the physics in 5D and 4D is provided by Campbell-Maagard's embedding theorems. We show that the 5-dimensional field equations $R_{AB} = 0$ determine the form of the similarity variable. There are three different possibilities: homothetic, conformal and "wave-like" solutions in 5D. We derive the most general homothetic and conformal solutions to the 5D field equations. They require the extra dimension to be spacelike, and are given in terms of one arbitrary function of the similarity variable and three parameters. The Riemann tensor in 5D is not zero, except in the isotropic limit, which corresponds to the case where the parameters are equal to each other. The solutions can be used as 5D embeddings for a great variety of 4D homogeneous cosmological models, with and without matter, including the Kasner universe. Since the extra dimension is spacelike, the 5D solutions are invariant under the exchange of spatial coordinates. Therefore they also embed a family of spatially {\it inhomogeneous} models in 4D. We show that these models can be interpreted as vacuum solutions in braneworld theory. Our work (I) generalizes the 5D embeddings used for the FLRW models; (II) shows that anisotropic cosmologies are, in general, curved in 5D, in contrast with FLRW models which can always be embedded in a 5D Riemann-flat (Minkowski) manifold; (III) reveals that anisotropic cosmologies can be curved and devoid of matter, both in 5D and 4D, even when the metric in 5D explicitly depends on the extra coordinate, which is quite different from the isotropic case.Comment: Typos corrected. Minor editorial changes and additions in the Introduction and Summary section

Ordering in a frustrated pyrochlore antiferromagnet proximate to a spin liquid

We perform a general study of spin ordering on the pyrochlore lattice with a 3:1 proportionality of two spin polarizations. Equivalently, this describes valence bond solid conformations of a quantum dimer model on the diamond lattice. We determine the set of likely low temperature ordered phases, on the assumption that the ordering is weak, i.e the system is close to a ``U(1)'' quantum spin liquid in which the 3:1 proportionality is maintained but the spins are strongly fluctuating. The nature of the 9 ordered states we find is determined by a ``projective symmetry'' analysis. All the phases exhibit translational and rotational symmetry breaking, with an enlarged unit cell containing 4 to 64 primitive cells of the underlying pyrochlore. The simplest of the 9 phases is the same ``R'' state found earlier in a theoretical study of the ordering on the magnetization plateau in the $S=3/2$ materials \cdaf and \hgaf. We suggest that the spin/dimer model proposed therein undergoes a direct transition from the spin liquid to the R state, and describe a field theory for the universal properties of this critical point, at zero and non-zero temperatures