Parabolic Region Research Articles

Oxidation kinetics of the pack siliconized TiAl-based alloy and microstructure evolution of the coating

The cyclic oxidation resistance of a two-phase TiAl-based alloy was remarkably improved with the formation of composite coating by siliconization with mixed powder of 15 wt%Si + 85 wt%Al 2O 3. The composite coating consists of a Ti 5Si 3-based inner layer and an Al 2O 3-based outer layer. The cyclic oxidation test at 900 °C showed that increasing the siliconization temperature benefits the oxidation resistance. The higher the siliconization temperature, the stabler the Ti 5Si 3-based layer and the more Al 2O 3 concentrated in the outer layer. Usually, the oxidation curves consist of three regions, the parabolic, the linear and the quadratic. For the specimen that was siliconized at 1250 °C for 2 h, however, only the parabolic region appeared during the whole cyclic oxidation test at 900 °C up to 1000 h. The weight gain is less than 0.3 mg cm −2 after cyclic oxidation at 900 °C for 1000 h, corresponding to a parabolic oxidation rate constant, K p ≈ 6.03 × 10 −5 mg 2/(cm 4 h). Such a low oxidation rate is attributed to the composite layer of the specimen with a stable Ti 5Si 3-based layer and a dense Al 2O 3-based layer.

Robust H/sub /spl infin// filter design with variance constraints and parabolic pole assignment

In this letter, we consider a multiobjective filtering problem for uncertain linear continuous time-invariant systems subject to error variance constraints. A linear filter is used to estimate a linear combination of the system states. The problem addressed is the design of a filter such that, for all admissible parameter uncertainties, the following three objectives are simultaneously achieved: 1) the filtering process is P-stable, i.e., the poles of the filtering matrix are located inside a parabolic region; 2) the steady-state variance of the estimation error of each state is not more than the individual prespecified value; and 3) the transfer function from exogenous noise inputs to error state outputs meets the prespecified H/sub /spl infin// norm upper-bound constraint. An effective algebraic matrix inequality approach is developed to derive both the existence conditions and the explicit expression of the desired filters. An illustrative example is used to demonstrate the usefulness of the proposed design approach.

Si1-xGex층의 건식산화 동안 Ge 재 분포와 상호 확산의 영향

We have studied the Ge redistribution after dry oxidation and the oxide growth rate of <TEX>$Si_{1-x}Ge_x$</TEX> epitaxial layer. Oxidation were performed at 700, 800, 900, and <TEX>$1,000\;^{\circ}C$</TEX>. After the oxidation, the results of RBS (Rutherford Back Scattering) & AES(Auger Electron Spectroscopy) showed that Ge was completely rejected out of the oxide and pile up at <TEX>$Si_{1-x}Ge_x$</TEX> interface. It is shown that the presence of Ge at the <TEX>$Si_{1-x}Ge_x$</TEX> interface changes the dry oxidation rate. The dry oxidation rate was equal to that of pure Si regardless of Ge mole fraction at 700 and 800<TEX>$^{\circ}C$</TEX>, while it was decreased at both 900 and <TEX>$1,000^{\circ}C$</TEX> as the Ge mole fraction was increased. The dry of idation rates were reduced for heavy Ge concentration, and large oxiidation time. In the parabolic growth region of <TEX>$Si_{1-x}Ge_x$</TEX> oxidation, the parabolic rate constant are decreased due to the presence of Ge-rich layer. After the longer oxidation at the <TEX>$1,000^{\circ}C$</TEX>, AES showed that Ge peak distribution at the <TEX>$Si_{1-x}Ge_x$</TEX> interface reduced by interdiffusion of silicon and germanium.

Spectral multipliers for sub-Laplacians with drift on Lie groups

We study spectral multipliers of right invariant sub-Laplacians with drift Open image in new window on a connected Lie group G. The operators we consider are self-adjoint with respect to a positive measure Open image in new window , whose density with respect to the left Haar measure λ G is a nontrivial positive character Open image in new window of G. We show that if p≠2 and G is amenable, then every Open image in new window spectral multiplier of Open image in new window extends to a bounded holomorphic function on a parabolic region in the complex plane, which depends on p and on the drift. When G is of polynomial growth we show that this necessary condition is nearly sufficient, by proving that bounded holomorphic functions on the appropriate parabolic region which satisfy mild regularity conditions on its boundary are Open image in new window spectral multipliers of Open image in new window .

Oxidation Behavior of Al-25Ti-8Mn Intermetallic Compound Fabricated by Mechanical Alloying and Spark Plasma Sintering

The oxidation behavior and the thermal stability of nanocrystalline Al-25Ti-8Mn intermetallic compound were investigated. intermetallic compound, which has a potential for high temperature structural material, was fabricated by mechanical alloying(MA) with Mn to enhance the thermal stability and ductility. And Al-25Ti-8Mn intermetallic compound was sintered by spark plasma sintering(SPS) at . After sintering process, cubic structure was maintained without phase transformation and the grain size was about 50nm. To investigate the oxidation behavior of the specimens, thermal gravimetric analysis(TGA) was performed at 700, 800, 900, and for 24 h in . As the temperature increased from the weight gain of specimens increased. However at , unlike the oxidation behavior of , the weight gain of specimen decreased drastically and the transition from linear rate region to parabolic rate region occurred rapidly due to the dense .

Fundamental solutions of the Tricomi operator, III

In this paper we complete the results of our papers [2], [3] and show how to generate from the hypergeometric function F⁡1/6,1/6;1;ζ fundamental solutions for the classical Tricomi operator relative to any point in the elliptic, parabolic, or hyperbolic region of the operator.

Field-induced Bose–Einstein condensation of a strongly correlated spin liquid in BaCuSi 2O 6

Low-temperature specific heat and magnetization experiments were performed in single-crystal samples of BaCuSi 2O 6 at high magnetic fields. These properties show changes upon entering a roughly parabolic region of the ( H, T) phase diagram defined by critical magnetic fields H c 1 = 23.5 T and H c 2 = 49 T , and a critical temperature T c = 3.78 K . The specific heat measured at 37 T shows a λ -shaped anomaly, and the magnetic susceptibility drops sharply on cooling. We used analytical methods and a directed-loop Monte Carlo algorithm to reproduce the experimental results, which are consistent with Bose–Einstein condensation of strongly correlated spin degrees of freedom.

Starlikeness associated with parabolic regions

A parabolic starlike function f of order ρ in the unit disk is characterized by the fact that the quantity zf′(z)/f(z) lies in a given parabolic region in the right half‐plane. Denote the class of such functions by PS∗(ρ). This class is contained in the larger class of starlike functions of order ρ. Subordination results for PS∗(ρ) are established, which yield sharp growth, covering, and distortion theorems. Sharp bounds for the first four coefficients are also obtained. There exist different extremal functions for these coefficient problems. Additionally, we obtain a sharp estimate for the Fekete‐Szegö coefficient functional and investigate convolution properties for PS∗(ρ).

Invariant manifolds and their zero-viscosity limits for Navier-Stokes equations

First we prove a general spectral theorem for the linear Navier- Stokes (NS) operator in both 2D and 3D. The spectral theorem says that the spectrum consists of only eigenvalues which lie in a parabolic region, and the eigenfunctions (and higher order eigenfunctions) form a complete basis in H ' (' = 0,1,2,···). Then we prove the existence of invariant manifolds. We are also interested in a more challenging problem, i.e. studying the zero-viscosity limits ( ! 0 + ) of the invariant manifolds. Under an assumption, we can show that the sizes of the unstable manifold and the center-stable manifold of a steady state are O( p ), while the sizes of the stable manifold, the center manifold, and the center-unstable manifold are O( ), as ! 0+. Finally, we study three examples. The first example is defined on a rectangular periodic domain, and has only one unstable eigenvalue which is real. A complete esti- mate on this eigenvalue is obtained. Existence of an 1D unstable manifold and a codim 1 stable manifold is proved without any assumption. For the other two examples, partial estimates on the eigenvalues are obtained.

Numerical algorithms for one parabolic-elliptic problem

In this paper we solve numerically a parabolic-elliptic problem. Two finite difference schemes are proposed. The first scheme is a modification of the backward Euler algorithm and it requires to solve an elliptic problem at each time step. The spectral estimates of the obtained matrix are presented. The second scheme is a modification of the stability-correction scheme. This scheme is used as a classical splitting scheme in the parabolic region of the problem definition and as a new iterative algorithm in the elliptic part of the problem. We prove the convergence of the proposed scheme.

Numerical techniques for flow problems with singularities

AbstractThis paper deals with grid approximations to Prandtl's boundary value problem for boundary layer equations on a flat plate in a region including the boundary layer, but outside a neighbourhood of its leading edge. The perturbation parameter ε= Re‐1 takes any values from the half‐interval (0,1] ; here Re is the Reynolds number. To demonstrate our numerical techniques we consider the case of the self‐similar solution. By using piecewise uniform meshes, which are refined in a neighbourhood of the parabolic boundary layer, we construct a finite difference scheme that converges ε‐uniformly. We present the technique of experimental substantiation of ε‐uniform convergence for both the numerical solution and its normalized (scaled) difference derivatives, outside a neighbourhood of the leading edge of the plate. By numerical experiments we demonstrate the efficiency of numerical techniques based on the fitted mesh method. We discuss also the applicability of fitted operator methods for the numerical approximation of the Prandtl problem. It is shown that the use of meshes refined in the parabolic boundary layer region is necessary for achieving ε‐uniform convergence. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Linear stability analysis in fluid–structure interaction with transpiration. Part I: Formulation and mathematical analysis

The aim of this work is to provide a new Linearization Principle approach particularly suited for problems in fluid–structure stability. The complexity here, and the main difference with respect to the classical approach, comes from the fact that the full non-linear fluid equations are written in a moving (i.e. time dependent) domain. The underlying idea of our approach uses transpiration techniques [J. Fluid Mech. 4 (1958) 383; G. Mortchéléwicz, Application of linearized Euler equations to flutter, in: 85th AGARD SMP Meeting, Aalborg, Denmark, 1997; P. Raj, B. Harris, Using surface transpiration with an Euler method for cost-effective aerodynamic analysis, in: AIAA 24th Applied Aerodynamics Conference, number 93-3506, Monterey, Canada, 1993; AIAA 27(6) (1989) 777], with the formalization and linearization recently developed in [Rév. Européenne Élém. Finis, 9(6–7) (2000) 681, A. Dervieux (Ed.), Fluid–Structure Interaction, Kogan Page Science, London, 2003 (Chapter 3)]. This allows us to obtain a new grid independent coupled spectral problem involving the linearized Navier–Stokes equations and those of a reduced linear structure. The coupling is realized through specific transpiration conditions acting on a fixed interface, while keeping a fixed fluid domain. We provide a rigorous mathematical treatment of this eigenproblem. We prove that the corresponding eigenmodes, characterizing the free evolution of the system, can be obtained from the characteristic values of a compact operator acting on a Hilbert space. Moreover, we localize the eigenfrequencies of the system in a parabolic region of the complex plan centered along the positive real axis.

Micromachined variable capacitors with wide tuning range

Abstract In this paper, ultra-thin silicon wafers, SU-8 bonding and deep reactive ion etching (DRIE) technology have been combined for the fabrication of folded spring, dual electrostatic drive, vertical plate variable capacitor devices with displacement limiting bumpers. The SU-8 bonding replaces the use of expensive SOI wafers and enables a more flexible design. The thick SU-8 also decreases the parasitic substrate capacitance. Due to the presence of the bumpers, our variable capacitor with parallel plate drive electrodes has two tuning voltage regimes: first a parabolic region that achieves roughly a 290% tuning range, then a linear region that achieves an additional 310%, making the total tuning range about 600%. Our variable capacitor with comb drive electrodes has a parabolic region that achieves roughly a 205% tuning range, then a linear region that achieves an additional 37%, making its total tuning range about 242%. The variable capacitors have Q factors around 100 at 1 MHz owing to the use of silicon electrodes other than lower resistivity metal.

The region division and NC machining of compound surfaces

Compound surfaces are divided into three kinds of regions by the theory of differential geometry. The correlative mathematical models for defining the cutter parameters, the step-forward length, and the path intervals in the NC machining of the surfaces are described based on the feature of the regions. In the NC machining of the hyperbolic and parabolic region, the authors propose that the step-forward moves along the asymptotic direction and should turn a small angle towards the convex region so as to avoid interference. The prolonged step-forward length would result in improvement in machining efficiency.

Sufficient conditions for starlikeness associated with parabolic region

An analytic function f(z) = z + a n+1 z n+1 + ⋯, defined on the unit disk △ = {z : |z| &lt; 1}, is in the class S p if z f′(z)/f(z) is in the parabolic region Rew &gt; |w − 1|. This class is closely related to the class of uniformly convex functions. Sufficient conditions for function to be in S p are obtained. In particular, we find condition on λ such that the function f(z), satisfying (1 − α)(f(z)/z) μ + αf′(z)(f(z)/z) μ−1≺1 + λz, is in S p.

Size segregation of granular matter in silo discharges

We present an experimental study of segregation of granular matter in a quasi-two-dimensional silo emptying out of an orifice. Size separation is observed when multisized particles are used with the larger particles found in the center of the silo in the region of fastest flow. We use imaging to study the flow inside the silo and quantitatively measure the concentration profiles of bidisperse beads as a function of position and time. The angle of the surface is given by the angle of repose of the particles, and the flow occurs in a few layers only near the top of this inclined surface. The flowing region becomes deeper near the center of the silo and is confined to a parabolic region centered at the orifice which is approximately described by the kinematic model. The experimental evidence suggests that the segregation occurs on the surface and not in the flow deep inside the silo where velocity gradients also are present. We report the time development of the concentrations of the bidisperse particles as a function of size ratios, flow rate, and the ratio of initial mixture. The qualitative aspects of the observed phenomena may be explained by a void filling model of segregation.

Interpretation of equatorial electrojet irregularities observed with a broad beam HF zenithal radar

Observations of the equatorial ionosphere were performed in Africa with the radar of the Laboratoire de Détection et de Géophysique near the magnetic equator during the International Equatorial Electrojet Year (1993–1994). This high‐resolution zenithal HF radar uses a broad beam antenna, in which complex echo structures are observed. A diversity of irregularities appears in the 100–500 km radar range up to the F region. Some irregularities appear in range‐time‐intensity diagrams as a parabolic region between 140 and 180 km with a minimum at noon. This region is only observed for radar frequencies between 4 and 5 MHz. At these frequencies, type II irregularities arising from vertical and oblique directions in the 100–140 km range dominate the spectra, and type I irregularities are difficult to clearly identify. Ray tracing and simulations show that type I irregularities are a possible source of the observed parabolic region. Irregularities from the equatorial “150 km region” as observed with the Jicamarca radar appear at slightly higher radar ranges.

A P1 - P1 Finite Element Method for a Phase Relaxation Model II: Adaptively Refined Meshes

We examine the effect of adaptively generated refined meshes on the P1 - P1 finite element method with semi-explicit time stepping of part I, which applies to a phase relaxation model with small parameter $\ep>0$. A typical mesh is highly graded in the so-called refined region, which exhibits a local meshsize proportional to the time step $\tau$, and is coarse in the remaining parabolic region where the meshsize is of order $\sqrt\tau$. Three admissibility tests guarantee mesh quality and, upon failure, lead to remeshing and so to incompatible consecutive meshes. The most severe test checks whether the transition region, where phase changes take place, belongs to the refined region. The other two tests monitor equidistribution of pointwise interpolation errors. The resulting adaptive scheme is shown to be stable in various Sobolev norms and to converge with a rate of order $\Or(\tau/\sqrt\ep)$ in the natural energy spaces. Several numerical experiments illustrate the scheme's efficiency and enhanced performance as compared with those of part I.

Stress across an elastoplastic boundary of a mode I crack: parabolic to hyperbolic plasticity transition

In previous work, the stresses of a mode I elastic–plastic fracture mechanics problem were analytically continued across a prescribed elastoplastic boundary for plane stress loading conditions involving a linear elastic/perfectly plastic material obeying the Tresca yield condition. Immediately across the elastic-plastic boundary, a nonlinear parabolic partial differential equation governs the plastic stress field. The present solution deals with stresses extending beyond the parabolic region into the hyperbolic region of the plastic zone. This analytical solution is obtained through a tranformation of the original system of nonlinear partial differential equations into a linear system with constant coefficients. The solution, so obtained, is expressible in terms of elementary transcendental functions. It also exhibits a limiting line which passes through the crack tip. This feature of the solution suggests the formation of a plastic hinge in the material.

Evanescent field Raman scattering by phonons with parabolic dispersion

Evanescent field Raman scattering (EFRS) of light by phonons in perfect crystals is proposed as a method to investigate spectra of vibrations from the entire Brillouin zone. A boundary between the crystal, and the waveguide providing the evanescent field into the crystal depth ensures violation of the momentum conservation perpendicular to the boundary. As a consequence, Raman scattering by phonons from the entire Brillouin zone can be observed. A parabolic region of the phonon dispersion curve demonstrates a specific asymmetric EFRS line as a function of the energy transfer. Location of the parabolic regions in the momentum space, and the mode dispersion can be identified.