Fully nonlinear development of the most unstable goertler vortex in a three dimensional boundary layer

The nonlinear development is studied of the most unstable Gortler mode within a general 3-D boundary layer upon a suitably concave surface. The structure of this mode was first identified by Denier, Hall and Seddougui (1991) who demonstrated that the growth rate of this instability is O(G sup 3/5) where G is the Gortler number (taken to be large here), which is effectively a measure of the curvature of the surface. Previous researchers have described the fate of the most unstable mode within a 2-D boundary layer. Denier and Hall (1992) discussed the fully nonlinear development of the vortex in this case and showed that the nonlinearity causes a breakdown of the flow structure. The effect of crossflow and unsteadiness upon an infinitesimal unstable mode was elucidated by Bassom and Hall (1991). They demonstrated that crossflow tends to stabilize the most unstable Gortler mode, and for certain crossflow/frequency combinations the Gortler mode may be made neutrally stable. These vortex configurations naturally lend themselves to a weakly nonlinear stability analysis; work which is described in a previous article by the present author. Here we extend the ideas of Denier and Hall (1992) to the three-dimensional boundary layer problem. It is found that the numerical solution of the fully nonlinear equations is best conducted using a method which is essentially an adaption of that utilized by Denier and Hall (1992). The influence of crossflow and unsteadiness upon the breakdown of the flow is described

The effect of crossflow on Taylor vortices: A model problem

A number of practically relevant problems involving the impulsive motion or the rapid rotation of bodies immersed in fluid are susceptible to vortex-like instability modes. Depending upon the configuration of any particular problem the stability properties of any high-wavenumber vortices can take on one of two distinct forms. One of these is akin to the structure of Gortler vortices in boundary layer flows while the other is similar to the situation for classical Taylor vortices. Both the Gortler and Taylor problems have been extensively studied when crossflow effects are excluded from the underlying base flows. Recently, studies were made concerning the influence of crossflow on Gortler modes and a linearized stability analysis is used to examine crossflow properties for the Taylor mode. This work allows us to identify the most unstable vortex as the crossflow component increases and it is shown how, like the Gortler case, only a very small crossflow component is required in order to completely stabilize the flow. Our investigation forms the basis for an extension to the nonlinear problem and is of potential applicability to a range of pertinent flows

The transverse magnetic reflectivity minimum of metals

Copyright © 2008 Optical Society of America. This paper was published in Optics Express and is made available as an electronic reprint with the permission of OSA. The paper can be found at the following URL on the OSA website: http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7580 . Systematic or multiple reproduction or distribution to multiple locations via electronic or other means is prohibited and is subject to penalties under law.Metal surfaces, which are generally regarded as excellent reflectors of electromagnetic radiation, may, at high angles of incidence, become strong absorbers for transverse magnetic radiation. This effect, often referred to as the pseudo-Brewster angle, results in a reflectivity minimum, and is most strongly evident in the microwave domain, where metals are often treated as perfect conductors. A detailed analysis of this reflectivity minimum is presented here and it is shown why, in the limit of very long wavelengths, metals close to grazing incidence have a minimum in reflectance given by (√2−1)2

Neutral modes of a two-dimensional vortex and their link to persistent cat's eyes

Copyright © 2008 American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The following article appeared in Physics of Fluids 20 (2008) and may be found at http://link.aip.org/link/?PHFLE6/20/027101/1This paper considers the relaxation of a smooth two-dimensional vortex to axisymmetry after the application of an instantaneous, weak external strain field. In this limit the disturbance decays exponentially in time at a rate that is linked to a pole of the associated linear inviscid problem (known as a Landau pole). As a model of a typical vortex distribution that can give rise to cat's eyes, here distributions are examined that have a basic Gaussian shape but whose profiles have been artificially flattened about some radius rc. A numerical study of the Landau poles for this family of vortices shows that as rc is varied so the decay rate of the disturbance moves smoothly between poles as the decay rates of two Landau poles cross. Cat's eyes that occur in the nonlinear evolution of a vortex lead to an axisymmetric azimuthally averaged profile with an annulus of approximately uniform vorticity, rather like the artificially flattened profiles investigated. Based on the stability of such profiles it is found that finite thickness cat's eyes can persist (i.e., the mean profile has a neutral mode) at two distinct radii, and in the limit of a thin flattened region the result that vanishingly thin cat's eyes only persist at a single radius is recovered. The decay of nonaxisymmetric perturbations to these flattened profiles for larger times is investigated and a comparison made with the result for a Gaussian profile

Master-equation approach to the study of phase-change processes in data storage media

We study the dynamics of crystallization in phase-change materials using a master-equation approach in which the state of the crystallizing material is described by a cluster size distribution function. A model is developed using the thermodynamics of the processes involved and representing the clusters of size two and greater as a continuum but clusters of size one (monomers) as a separate equation. We present some partial analytical results for the isothermal case and for large cluster sizes, but principally we use numerical simulations to investigate the model. We obtain results that are in good agreement with experimental data and the model appears to be useful for the fast simulation of reading and writing processes in phase-change optical and electrical memories

Determinant Structure of the Rational Solutions for the Painlev\'e IV Equation

Rational solutions for the Painlev\'e IV equation are investigated by Hirota bilinear formalism. It is shown that the solutions in one hierarchy are expressed by 3-reduced Schur functions, and those in another two hierarchies by Casorati determinant of the Hermite polynomials, or by special case of the Schur polynomials.Comment: 19 pages, Latex, using theorem.st

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Backlund transformations for the equations considered. We are also able to derive Backlund transformations onto other ODEs in the Painleve classification.Comment: To appear in Nonlinearity (22 pages

Cyclotron resonance of the quasi-two-dimensional electron gas at Hg1-xCdxTe grain boundaries

The magnetotransmission of a p-type Hg0.766Cd0.234Te bicrystal containing a single grain boundary with an inversion layer has been investigated in the submillimetre wavelength range. For the first time the cyclotron resonance lines belonging to the various electric subbands of a quasi-two-dimensional carrier system at a grain boundary could be detected. The measured cyclotron masses and the subband densities determined from Shubnikov-de Haas experiments are compared with theoretical predictions and it is found that the data can be explained very well within the framework of a triangular well approximation model which allows for non-parabolic effects

B\"acklund transformations for the second Painlev\'e hierarchy: a modified truncation approach

The second Painlev\'e hierarchy is defined as the hierarchy of ordinary differential equations obtained by similarity reduction from the modified Korteweg-de Vries hierarchy. Its first member is the well-known second Painlev\'e equation, P2. In this paper we use this hierarchy in order to illustrate our application of the truncation procedure in Painlev\'e analysis to ordinary differential equations. We extend these techniques in order to derive auto-B\"acklund transformations for the second Painlev\'e hierarchy. We also derive a number of other B\"acklund transformations, including a B\"acklund transformation onto a hierarchy of P34 equations, and a little known B\"acklund transformation for P2 itself. We then use our results on B\"acklund transformations to obtain, for each member of the P2 hierarchy, a sequence of special integrals.Comment: 12 pages in LaTeX 2.09 (uses ioplppt.sty), to appear in Inverse Problem

Dependence of magnetic field generation by thermal convection on the rotation rate: a case study

Dependence of magnetic field generation on the rotation rate is explored by direct numerical simulation of magnetohydrodynamic convective attractors in a plane layer of conducting fluid with square periodicity cells for the Taylor number varied from zero to 2000, for which the convective fluid motion halts (other parameters of the system are fixed). We observe 5 types of hydrodynamic (amagnetic) attractors: two families of two-dimensional (i.e. depending on two spatial variables) rolls parallel to sides of periodicity boxes of different widths and parallel to the diagonal, travelling waves and three-dimensional "wavy" rolls. All types of attractors, except for one family of rolls, are capable of kinematic magnetic field generation. We have found 21 distinct nonlinear convective MHD attractors (13 steady states and 8 periodic regimes) and identified bifurcations in which they emerge. In addition, we have observed a family of periodic, two-frequency quasiperiodic and chaotic regimes, as well as an incomplete Feigenbaum period doubling sequence of bifurcations of a torus followed by a chaotic regime and subsequently by a torus with 1/3 of the cascade frequency. The system is highly symmetric. We have found two novel global bifurcations reminiscent of the SNIC bifurcation, which are only possible in the presence of symmetries. The universally accepted paradigm, whereby an increase of the rotation rate below a certain level is beneficial for magnetic field generation, while a further increase inhibits it (and halts the motion of fluid on continuing the increase) remains unaltered, but we demonstrate that this "large-scale" picture lacks many significant details.Comment: 39 pp., 22 figures (some are low quality), 5 tables. Accepted in Physica