The effect of normal blowing on the flow near a rotating disk of infinite extent

1971 ◽  
Vol 47 (4) ◽  
pp. 789-798 ◽  
Author(s):  
H. K. Kuiken
Keyword(s):  
Asymptotic Analysis ◽  
Thin Layer ◽  
Rotating Disk ◽  
Complete Agreement ◽  
Viscous Effects ◽  
First Order ◽  
Numerical Integrations ◽  
Infinite Extent

The effect of blowing through a porous rotating disk on the flow induced by this disk is studied. For strong blowing the flow is almost wholly inviscid. First-order viscous effects are encountered only in a thin layer at some distance from the disk. The results of an asymptotic analysis are compared with numerical integrations of the full equations and complete agreement is found.

On the Vibration of a Rotating Disk

1972 ◽  
Vol 39 (4) ◽  
pp. 1143-1144 ◽  
Author(s):  
S. Barasch ◽  
Y. Chen
Keyword(s):  
Approximate Calculation ◽  
Initial Conditions ◽  
Rotating Disk ◽  
Equation Of Motion ◽  
Outer Radius ◽  
Order Equation ◽  
System Of Differential Equations ◽  
Fourth Order Equation ◽  
First Order ◽  
Inner Radius

The equation of motion of a rotating disk, clamped at the inner radius and free at the outer radius, is solved by reducing the fourth-order equation of motion to a set of four first-order equations subject to arbitrary initial conditions. A modified Adams’ method is used to numerically integrate the system of differential equations. Results show that Lamb-Southwell’s approximate calculation of the frequency is justified.

scholarly journals SMT-Based System Verification with DVF

10.29007/59rn ◽  
2018 ◽  
Author(s):  
Amit Goel ◽  
Sava Krstic ◽  
Rebekah Leslie ◽  
Mark Tuttle
Keyword(s):  
Thin Layer ◽  
Type System ◽  
Transition Systems ◽  
First Order ◽  
Verification Conditions ◽  
Logic Verification ◽  
System Verification ◽  
Selected Subset ◽  
Parametric Specification

We introduce the <i>Deductive Verificaton Framework</i> (DVF), a language and a tool for verifying properties of transition systems. The language is procedural and the system transitions are a selected subset of procedures. The type system and built-in operations are consistent with SMT-LIB, as are the multisorted first-order logical formulas that may occur in DVF programs as pre- and post-conditions, assumptions, assertions, and goals. A template mechanism allows parametric specification of complex types within the confines of this logic. Verification conditions are generated from specified goals and passed to SMT engine(s). A general assume-guarantee scheme supports a thin layer of interactive proving.

Boundary layers of an anchored magnetic field in a highly conductive flow

2010 ◽  
Vol 466 (2123) ◽  
pp. 3153-3166 ◽  
Author(s):  
Manuel Núñez ◽  
Alberto Lastra
Keyword(s):  
Magnetic Field ◽  
Boundary Layer ◽  
Asymptotic Analysis ◽  
Small Parameter ◽  
Order Approximation ◽  
First Order ◽  
Electrically Conducting ◽  
Electrically Conducting Fluid ◽  
First Order Approximation ◽  
Distance To The Boundary

The effects of the flow of an electrically conducting fluid upon a magnetic field anchored at the boundary of a domain are studied. By taking the resistivity as a small parameter, the first-order approximation of an asymptotic analysis yields a boundary layer for the magnetic potential. This layer is analysed both in general and in three particular cases, showing that while in general its effects decrease exponentially with the distance to the boundary, several additional effects are highly relevant.

INFLUENCE OF VIBRATION–ROTATION INTERACTION ON THE INTENSITIES OF PURE ROTATION LINES OF DIATOMIC MOLECULES

1956 ◽  
Vol 34 (11) ◽  
pp. 1119-1125 ◽  
Author(s):  
Robert Herman ◽  
Robert J. Rubin
Keyword(s):  
Line Intensity ◽  
Anharmonic Oscillator ◽  
Diatomic Molecules ◽  
Potential Energy Function ◽  
Complete Agreement ◽  
Contact Transformation ◽  
First Order ◽  
Pure Rotation ◽  
Rotation Line ◽  
Vibration Rotation

The magnitude of the effect of the vibration–rotation interaction on the intensities of pure rotation lines of diatomic molecules has been calculated for two different molecular models, the anharmonic oscillator and the rotating Morse or Pekeris oscillator. The intensity correction for the anharmonic oscillator has been obtained by adapting the contact transformation formalism for calculating second-order corrections to the energy to the calculation of first-order corrections to the matrix elements of the electric moment as suggested by H. H. Nielsen. The correction to the line intensity for vibrationless transitions of the anharmonic oscillator is found to be[Formula: see text]The results obtained here are also in complete agreement, to first order, with the results obtained earlier by Herman and Wallis for the 1–0 and 2–0 vibration–rotation line intensities. In the case of the Pekeris or rotating Morse oscillator the correction to the pure rotation line intensity is of the same form as above, namely,[Formula: see text]but exhibits minor differences which can be explained in terms of the difference in the vibrational potential energy function in the two cases.

scholarly journals Achieving Seventh-Order Amplitude Accuracy in Leapfrog Integrations

2013 ◽  
Vol 141 (9) ◽  
pp. 3037-3051 ◽  
Author(s):  
Paul D. Williams
Keyword(s):  
Nonlinear System ◽  
Third Order ◽  
First Order ◽  
Time Stepping ◽  
The Third ◽  
Numerical Integrations ◽  
Seventh Order ◽  
Ocean Models ◽  
Stability And Accuracy ◽  
Fifth Order

Abstract The leapfrog time-stepping scheme makes no amplitude errors when integrating linear oscillations. Unfortunately, the Robert–Asselin filter, which is used to damp the computational mode, introduces first-order amplitude errors. The Robert–Asselin–Williams (RAW) filter, which was recently proposed as an improvement, eliminates the first-order amplitude errors and yields third-order amplitude accuracy. However, it has not previously been shown how to further improve the accuracy by eliminating the third- and higher-order amplitude errors. Here, it is shown that leapfrogging over a suitably weighted blend of the filtered and unfiltered tendencies eliminates the third-order amplitude errors and yields fifth-order amplitude accuracy. It is further shown that the use of a more discriminating (1, −4, 6, −4, 1) filter instead of a (1, −2, 1) filter eliminates the fifth-order amplitude errors and yields seventh-order amplitude accuracy. Other related schemes are obtained by varying the values of the filter parameters, and it is found that several combinations offer an appealing compromise of stability and accuracy. The proposed new schemes are tested in numerical integrations of a simple nonlinear system. They appear to be attractive alternatives to the filtered leapfrog schemes currently used in many atmosphere and ocean models.

Rotating Disk Shape: Is the Equation Nonlinear?

1989 ◽  
Vol 111 (4) ◽  
pp. 456-458
Author(s):  
R. R. Jettappa
Keyword(s):  
Differential Equation ◽  
Linear Equation ◽  
Rotating Disk ◽  
Variable Coefficients ◽  
Thin Disk ◽  
Second Order ◽  
First Order ◽  
Governing Differential Equation ◽  
Centrifugal Loading

The determination of the shape of a rotating disk under centrifugal loading is considered. It is shown that the governing differential equation for the shape of a rotating thin disk is reducible to a linear equation of second order with variable coefficients. However, the form of this equation is such that it can be treated as an equation of first order thereby facilitating the integration by quadratures. All this is possible without any additional mathematical assumptions so that the results are exact within the limitations of the thin disk theory.

MODELING AND ANALYSIS OF SALT TRANSPORT IN THE SINGLE CYLINDRICAL ROOT WITH FINITE LENGTH

2013 ◽  
Vol 06 (06) ◽  
pp. 1350040
Author(s):  
ARUN KUMAR ◽  
BEENA SHARMA ◽  
ASHU RANI
Keyword(s):  
Mathematical Model ◽  
Soil Moisture ◽  
Asymptotic Analysis ◽  
Salt Solution ◽  
Salt Transport ◽  
Moisture Concentration ◽  
Modeling And Analysis ◽  
Infinite Extent ◽  
Asymptotic Matching ◽  
Solution Uptake

A mathematical model for salt transport by a cylindrical root in an infinite extent of soil is derived and solved analytically by asymptotic matching of the inner and outer solutions. By asymptotic analysis it is shown that the salt solution uptake by a single cylindrical root in the absence of competition does not influence the overall salt concentration in the soil even when the soil moisture concentration is less than full saturation.

scholarly journals Generalized impedance boundary conditions and shape derivatives for 3D Helmholtz problems

2016 ◽  
Vol 26 (10) ◽  
pp. 1995-2033 ◽  
Author(s):  
Djalil Kateb ◽  
Frédérique Le Louër
Keyword(s):  
Thin Layer ◽  
Three Dimensional ◽  
Transmission Problem ◽  
Impedance Boundary Condition ◽  
Constant Thickness ◽  
Laplacian Operator ◽  
Shape Sensitivity ◽  
Impedance Boundary ◽  
First Order ◽  
Exterior Boundary

This paper is concerned with the shape sensitivity analysis of the solution to the Helmholtz transmission problem for three-dimensional sound-soft or sound-hard obstacles coated by a thin layer. This problem can be asymptotically approached by exterior problems with an improved condition on the exterior boundary of the coated obstacle, called generalized impedance boundary condition (GIBC). Using a series expansion of the Laplacian operator in the neighborhood of the exterior boundary, we retrieve the first-order GIBCs characterizing the presence of an interior thin layer with a constant thickness. The first shape derivative of the solution to the original Helmholtz transmission problem solves a new thin layer transmission problem with non-vanishing jumps across the exterior and the interior boundary of the thin layer. We show that we can interchange the first-order differentiation with respect to the shape of the exterior boundary and the asymptotic approximation of the solution. Numerical experiments are presented to highlight the various theoretical results.

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