On the Poincaré problem for a compressible medium

1974 ◽  
Vol 62 (4) ◽  
pp. 657-675 ◽  
Author(s):  
Roger F. Gans
Keyword(s):  
Rotation Frequency ◽  
Compressible Medium ◽  
Zeroth Order ◽  
Frequency Shifts ◽  
First Order ◽  
Specific Heats ◽  
Confluent Hypergeometric Functions ◽  
Negative Case ◽  
Poincaré Problem ◽  
Order Set

By the ‘Poincaré problem’ is meant the determination of the free oscillations of a contained rotating fluid, its velocity being linearized around a state of solid rotation. Compressibility requires one to introduce a basic thermodynamic profile as well as a basic velocity distribution. Here the temperature gradient has been supposed proportional to the adiabatic gradient, by introduction of a proportionality constant α (α = 0 in the isothermal case; α = 1 in the adiabatic case). In this formulation the system is reducible to a single second-order ordinary differential equation and its boundary condition.It is proved that if α = 1 the oscillation frequencies in the rotating system cannot equal plus or minus twice the rotation frequency. The negative case is pathological in the sense that there are solutions arbitrarily near the forbidden solution, and a solution curve of frequency as a function of rotation rate crosses the forbidden frequency.The basic system is expanded in terms of a power series in γ − 1, where γ is the ratio of specific heats. The zeroth-order set of equations is solved in terms of confluent hypergeometric functions, and a solvability condition on the first-order set gives frequency shifts as functions of α. Several zeroth-order frequencies have been calculated, together with four first-order frequency shifts.

Categoricity and the sets

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Set Theory ◽  
Second Order ◽  
Order Logic ◽  
First Order ◽  
The Status ◽  
Second Order Logic ◽  
Philosophy Of Set Theory ◽  
Order Set

In this chapter, the focus shifts from numbers to sets. Again, no first-order set theory can hope to get anywhere near categoricity, but Zermelo famously proved the quasi-categoricity of second-order set theory. As in the previous chapter, we must ask who is entitled to invoke full second-order logic. That question is as subtle as before, and raises the same problem for moderate modelists. However, the quasi-categorical nature of Zermelo's Theorem gives rise to some specific questions concerning the aims of axiomatic set theories. Given the status of Zermelo's Theorem in the philosophy of set theory, we include a stand-alone proof of this theorem. We also prove a similar quasi-categoricity for Scott-Potter set theory, a theory which axiomatises the idea of an arbitrary stage of the iterative hierarchy.

scholarly journals Perturbation Solution for One-Dimensional Flow to a Constant-Pressure Boundary in a Stress-Sensitive Reservoir

2021 ◽  
Author(s):  
Amarjot Singh Bhullar ◽  
Gospel Ezekiel Stewart ◽  
Robert W. Zimmerman
Keyword(s):  
Approximate Solution ◽  
Pore Pressure ◽  
Constant Pressure ◽  
Initial Permeability ◽  
Order Perturbation ◽  
Zeroth Order ◽  
One Dimensional ◽  
First Order ◽  
Outflow Boundary ◽  
Perturbation Solutions

Abstract Most analyses of fluid flow in porous media are conducted under the assumption that the permeability is constant. In some “stress-sensitive” rock formations, however, the variation of permeability with pore fluid pressure is sufficiently large that it needs to be accounted for in the analysis. Accounting for the variation of permeability with pore pressure renders the pressure diffusion equation nonlinear and not amenable to exact analytical solutions. In this paper, the regular perturbation approach is used to develop an approximate solution to the problem of flow to a linear constant-pressure boundary, in a formation whose permeability varies exponentially with pore pressure. The perturbation parameter αD is defined to be the natural logarithm of the ratio of the initial permeability to the permeability at the outflow boundary. The zeroth-order and first-order perturbation solutions are computed, from which the flux at the outflow boundary is found. An effective permeability is then determined such that, when inserted into the analytical solution for the mathematically linear problem, it yields a flux that is exact to at least first order in αD. When compared to numerical solutions of the problem, the result has 5% accuracy out to values of αD of about 2—a much larger range of accuracy than is usually achieved in similar problems. Finally, an explanation is given of why the change of variables proposed by Kikani and Pedrosa, which leads to highly accurate zeroth-order perturbation solutions in radial flow problems, does not yield an accurate result for one-dimensional flow. Article Highlights Approximate solution for flow to a constant-pressure boundary in a porous medium whose permeability varies exponentially with pressure. The predicted flowrate is accurate to within 5% for a wide range of permeability variations. If permeability at boundary is 30% less than initial permeability, flowrate will be 10% less than predicted by constant-permeability model.

scholarly journals ANISOTROPIC NULL STRING COSMOLOGIES

1998 ◽  
Vol 13 (39) ◽  
pp. 3169-3177 ◽  
Author(s):  
IOANNIS GIANNAKIS ◽  
K. KLEIDIS ◽  
A. KUIROUKIDIS ◽  
D. PAPADOPOULOS
Keyword(s):  
Equations Of Motion ◽  
String Tension ◽  
Perturbative Expansion ◽  
Speed Of Light ◽  
Zeroth Order ◽  
First Order ◽  
Cosmological Background ◽  
String Propagation ◽  
Analytical Expressions ◽  
Null String

We study string propagation in an anisotropic, cosmological background. We solve the equations of motion and the constraints by performing a perturbative expansion of the string coordinates in powers if c2 — the worldsheet speed of light. To zeroth order the string is approximated by a tensionless string (since c is proportional to the string tension T). We obtain exact, analytical expressions for the zeroth- and first-order solutions and we discuss some cosmological implications.

A Slender-Ship Theory of Wave Resistance

1983 ◽  
Vol 27 (01) ◽  
pp. 13-33
Author(s):  
Francis Noblesse
Keyword(s):  
Froude Number ◽  
Wave Resistance ◽  
Successive Approximations ◽  
Zeroth Order ◽  
Remarkable Effect ◽  
Ship Wave ◽  
First Order ◽  
Wigley Hull ◽  
Low Froude Number ◽  
The Waves

A new slender-ship theory of wave resistance is presented. Specifically, a sequence of explicit slender-ship wave-resistance approximations is obtained. These approximations are associated with successive approximations in a slender-ship iterative procedure for solving a new (nonlinear integro-differential) equation for the velocity potential of the flow caused by the ship. The zeroth, first, and second-order slender-ship approximations are given explicitly and examined in some detail. The zeroth-order slender-ship wave-resistance approximation, r(0) is obtained by simply taking the (disturbance) potential, ϕ, as the trivial zeroth-order slender-ship approximation ϕ(0) = 0 in the expression for the Kochin free-wave amplitude function; the classical wave-resistance formulas of Michell [1]2 and Hogner [2] correspond to particular cases of this simple approximation. The low-speed wave-resistance formulas proposed by Guevel [3], Baba [4], Maruo [5], and Kayo [6] are essentially equivalent (for most practical purposes) to the first-order slender-ship low-Froude-number approximation, rlF(1), which is a particular case of the first-order slender-ship approximation r(1): specifically, the first-order slender-ship wave-resistance approximation r(1) is obtained by approximating the potential ϕ in the expression for the Kochin function by the first-order slender-ship potential ϕ1 whereas the low-Froude-number approximation rlF(1) is associated with the zero-Froude-number limit ϕ0(1) of the potentialϕ(1). A major difference between the first-order slender-ship potential ϕ(1) and its zero-Froude-number limit ϕ0(1) resides in the waves that are included in the potential ϕ(1) but are ignored in the zero-Froude-number potential ϕ0(1). Results of calculations by C. Y. Chen for the Wigley hull show that the waves in the potential ϕ(1) have a remarkable effect upon the wave resistance, in particular causing a large phase shift of the wave-resistance curve toward higher values of the Froude number. As a result, the first-order slender-ship wave-resistance approximation in significantly better agreement with experimental data than the low-Froude-number approximation rlF(1) and the approximations r(0) and rM.

Synthesis, Properties and Application in Optical Memory of a Photochromic Diarylethene Bearing Pyrimidine and Pyridine Ring

2012 ◽  
Vol 490-495 ◽  
pp. 3733-3737
Author(s):  
Shu Hong Jing ◽  
Shou Zhi Pu ◽  
Shi Qiang Cui
Keyword(s):  
Uv Light ◽  
Optical Memory ◽  
Order Reaction ◽  
Pmma Film ◽  
Zeroth Order ◽  
First Order ◽  
Synthesis Properties ◽  
Fluorescence Switching ◽  
Photochromic Reaction ◽  
Hexane Solution

A new photochromic diarylethene compound, 1-(2,4-dimethoxy-5-pyrimidine)-2-[2-methyl-5-(3-pyridine)-3-thienyl]perfluorocyclopentene(1a), was synthesized, and its photochromic reactivity, fluorescent and electrochemical property were also investigated. Diarylethene 1a changed the color from colorless to pink upon irradiation with UV light, in which absorption maxima were observed at 520 and 519 nm in hexane and PMMA film, respectively. The the photochromic reaction kinetics indicated that the cyclization processes of 1 belong to the zeroth order reaction and the cycloreversion process belong to the first order reaction. This new photochromic system also exhibited remarkable fluorescence switching in hexane solution and this new photochromic system also exhibited remarkable optical storage character.

scholarly journals The Effects of Rapid Rotation on Pulsation

2002 ◽  
Vol 185 ◽  
pp. 177-185
Author(s):  
Hideyuki Saio
Keyword(s):  
Low Frequency ◽  
Rotation Frequency ◽  
Rotating Stars ◽  
Frequency Shifts ◽  
Rapid Rotation ◽  
The Stability

AbstractWe discuss the properties of pulsations in rotating stars, which include frequency shifts, the effects on the stability of p- and g-modes, and the properties of low-frequency modes whose frequencies are comparable to or less than the rotation frequency.

scholarly journals Study on the Effect of Fractional Derivative on the Hyperspectral Data of Soil Organic Matter Content in Arid Region

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Cheng-Biao Fu ◽  
Hei-Gang Xiong ◽  
An-Hong Tian
Keyword(s):  
Organic Matter ◽  
Soil Organic Matter ◽  
Fractional Derivative ◽  
Arid Region ◽  
Arid Zone ◽  
Organic Matter Content ◽  
Matter Content ◽  
Second Order ◽  
Zeroth Order ◽  
First Order

Discussion on the application of fractional derivative algorithm in monitoring organic matter content in field soil is scarce. This study is aimed at improving the accuracy of soil organic matter (SOM) content estimation in arid region, and the undesirable model precision caused by the missing information associated with the larger discrepancy between conventional integer-order, i.e., first order and second order, derivative, and raw spectral data. We utilized fractional derivative (of zeroth order to second order in 0.2-order interval) processing on the field spectral reflectance (R) of the salinized soil sample from Fukang, Xinjiang, and its square root-transformed (R), log-transformed (lgR), inverse-transformed (1/R), and inverse log-transformed (1/lgR) values. The correlation coefficient of each fractional derivative of transformed value with SOM content was calculated. The simulation showed the derivative reflectance value approximates zero. When increasing from zeroth order to first order, the derivative curve gradually aligns to the first-order curve, and the destination alignment was also seen while increasing from first order to second order. The significance test of 0.05 showed initial increase and later decay of bands in the five spectral transformations as the order increases. For specific bands, the derivative algorithm clearly justifies the correlation between soil spectra and organic matter content, and all of the absolute highest correlation coefficient values were obtained at fractional orders. When compared with integer-order derivative, fractional derivative is significantly better in improving correlation, showing overall superiority. The result supports the application of fractional derivative in the hyperspectral remote monitor of SOM in arid zone, which may in turn realize the timely and accurate SOM monitor in arid zone, and provides the basis for ecological restoration.

scholarly journals An Orientation-Sensitive Vassiliev Invariant for Virtual Knots

2003 ◽  
Vol 12 (06) ◽  
pp. 767-779 ◽  
Author(s):  
Jörg Sawollek
Keyword(s):  
Virtual Link ◽  
Zeroth Order ◽  
Opposite Orientation ◽  
Vassiliev Invariants ◽  
Virtual Knot ◽  
First Order ◽  
Virtual Knots ◽  
Conway Polynomial ◽  
Vassiliev Invariant ◽  
Open Question

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.

A Continuum Model for Diffusion in Laminated Composite Media

1976 ◽  
Vol 98 (1) ◽  
pp. 133-138 ◽  
Author(s):  
A. Maewal ◽  
T. C. Bache ◽  
G. A. Hegemier
Keyword(s):  
Effective Conductivity ◽  
Continuum Theory ◽  
Laminated Composite ◽  
Order Theory ◽  
Attractive Alternative ◽  
Zeroth Order ◽  
Diffusion Type ◽  
First Order ◽  
First Order Theory ◽  
Higher Order Theory

Using a method developed for studying wave propagation problems, a continuum theory is developed for diffusion-type processes in a laminated composite with periodic micro-structure. Construction is based upon an asymptotic scheme in which a typical macrodimension is assumed large compared to a microdimension. The order of truncation of the asymptotic sequence so obtained defines a hierarchy of models. Solutions are given for the lowest-order models and compared with the results from a finite difference code. For most cases the zeroth-order “effective conductivity” theory yields good results. For exceptional problems requiring a higher-order theory, a modified version of the first-order theory is shown to suffice. For many applications these elementary equations may offer an attractive alternative to other means for obtaining solutions.

Convergent-Tapered Annular Seals: Analysis and Testing for Rotordynamic Coefficients

1985 ◽  
Vol 107 (3) ◽  
pp. 307-316 ◽  
Author(s):  
D. W. Childs ◽  
J. B. Dressman
Keyword(s):  
Dynamic Pressure ◽  
Momentum Equation ◽  
Computational Method ◽  
Centrifugal Pumps ◽  
Zeroth Order ◽  
Taper Angle ◽  
First Order ◽  
Dynamic Coefficients ◽  
Direct Stiffness ◽  
Annular Seals

A combined analytical-computational method is developed to calculate the pressure field and dynamic coefficients for tapered high-pressure annular seals typical of neck-ring and interstage seals employed in multistage centrifugal pumps. Completely developed turbulent flow is assumed in both the circumferential and axial directions and is modeled by Hirs’ bulk-flow turbulent-lubrication equations. Linear zeroth- and first-order perturbation equations are developed for the momentum equations and continuity equations. The development of the circumferential velocity field is defined from the zeroth-order circumferential-momentum equation, and a leakage relationship is defined from the zeroth-order axial-momentum equation. A short-bearing approximation is used to derive an analytical expression for the first-order (dynamic) pressure gradient. This expression is integrated numerically to define dynamic coefficients for the seal. Numerical results are presented and compared to previous results for straight and tapered seals. The direct stiffness and leakage increase with increasing taper angle, while the remaining dynamic coefficients decrease. An optimal taper angle is shown to exist with respect to (a) the direct stiffness, and (b) the ratio of direct stiffness to leakage. Stiffness increases on the order of 40-50 percent are predicted. Experimental results are presented for seals with three taper angles which show generally good agreement between theory and prediction.

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