Approximation formulas and inequalities for the Euler-Mascheroni constant

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
Chao-Ping Chen
Keyword(s):  
Approximation Formulas

The determination of the pressure—viscosity coefficient of a lubricant through an accurate film thickness formula and accurate film thickness measurements

2009 ◽  
Vol 223 (8) ◽  
pp. 1143-1163 ◽  
Author(s):  
H van Leeuwen
Keyword(s):  
Film Thickness ◽  
Viscosity Coefficient ◽  
Test Fluid ◽  
Limited Data ◽  
Ehd Lubrication ◽  
Spacer Layer ◽  
Wide Range ◽  
Approximation Formulas ◽  
Thickness Measurements

The pressure—viscosity coefficient is an indispensable property in the elastohydrodynamic (EHD) lubrication of hard contacts, but often not known. A guess will easily lead to enormous errors in the film thickness. This article describes a method to deduct this coefficient by adapting the value of the pressure—viscosity coefficient until the differences between accurate film thickness approxi-mation values and accurate film thickness measurements over a wide range of values are at a minimum. Eleven film thickness approximation formulas are compared in describing the film thickness of a test fluid with known value of the pressure—viscosity coefficient. The measurement method is based on spacer layer interferometry. It is concluded that for circular contacts the newer more versatile expressions are not better than some older approximations, which are limited to a smaller region of conditions, and that the older fits are as least as appropriate to find the pressure—viscosity coefficient of fluids, in spite of the limited data where they have been based on.

scholarly journals Study on divergence approximation formula for pressure calculation in particle method

2018 ◽  
Vol 10 (4) ◽  
pp. 159-169
Author(s):  
Zhu Yue ◽  
Jiang Shengyao ◽  
Yang Xingtuan ◽  
Duan Riqiang
Keyword(s):  
Pressure Oscillation ◽  
Particle Method ◽  
Free Surface Flows ◽  
Implicit Method ◽  
Approximation Formula ◽  
Pressure Poisson Equation ◽  
Pressure Calculation ◽  
Approximation Formulas ◽  
Moving Particle ◽  
Dam Breaking

The moving particle semi-implicit method is a meshless particle method for incompressible fluid and has proven useful in a wide variety of engineering applications of free-surface flows. Despite its wide applicability, the moving particle semi-implicit method has the defects of spurious unphysical pressure oscillation. Three various divergence approximation formulas, including basic divergence approximation formula, difference divergence approximation formula, and symmetric divergence approximation formula are proposed in this paper. The proposed three divergence approximation formulas are then applied for discretization of source term in pressure Poisson equation. Two numerical tests, including hydrostatic pressure problem and dam-breaking problem, are carried out to assess the performance of different formulas in enhancing and stabilizing the pressure calculation. The results demonstrate that the pressure calculated by basic divergence approximation formula and difference divergence approximation formula fluctuates severely. However, application of symmetric divergence approximation formula can result in a more accurate and stabilized pressure.

scholarly journals Smoothest Approximation Formulas

1949 ◽  
Vol 20 (4) ◽  
pp. 612-615 ◽  
Author(s):  
Arthur Sard
Keyword(s):  
Approximation Formulas

On the approximation of convex functions using linear positive operators

2017 ◽  
Vol 26 (2) ◽  
pp. 137-143
Author(s):  
DAN BARBOSU
Keyword(s):  
Convex Functions ◽  
Remainder Term ◽  
Mean Value ◽  
Positive Operators ◽  
Linear Functionals ◽  
Linear Positive Operators ◽  
Famous Theorem ◽  
Approximation Formulas ◽  
Phd Thesis ◽  
Higher Order Convexity

The goal of the paper is to present some results concerning the approximation of convex functions by linear positive operators. First, one recalls some results concerning the univariate real valued convex functions. Next, one presents the notion of higher order convexity introduced by Popoviciu [Popoviciu, T., Sur quelques propri´et´ees des fonctions d’une ou deux variable r´eelles, PhD Thesis, La Faculte des Sciences de Paris, 1933 (June)] . The Popoviciu’s famous theorem for the representation of linear functionals associated to convex functions of m−th order (with the proof of author) is also presented. Finally, applications of the convexity to study the monotonicity of sequences of some linear positive operators and also mean value theorems for the remainder term of some approximation formulas based on linear positive operators are presented.

Small-angle scattering from collections of interacting hard ellipsoids of revolution studied by Monte Carlo simulations and other methods of statistical thermodynamics

1999 ◽  
Vol 32 (5) ◽  
pp. 917-923 ◽  
Author(s):  
Bo Sjöberg
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Simulations ◽  
Small Angle ◽  
Chemical Potential ◽  
Compressibility Factor ◽  
Particle Method ◽  
Spherical Particles ◽  
Angle Scattering ◽  
Approximation Formulas ◽  
Experimental Values

Computer simulations using Monte Carlo methods are used to investigate the effects of interparticle correlations on small-angle X-ray and neutron scattering from moderate or highly concentrated systems of ellipsoids of revolution. Both oblate and prolate ellipsoids, of varying eccentricities and concentrations, are considered. The advantage with Monte Carlo simulation is that completely general models, both regarding particle shapes and interaction potentials, can be considered. Equations are also given that relate the nonideal part of the chemical potential, βμni, with the scattering at zero angle,I(0), and the compressibility factor,z. The quantity βμnican be obtained during the Monte Carlo simulations by using Widom's test-particle method. For spherical particles, the simulations are compared with approximation formulas based on the Percus–Yevick equation. A method is also suggested for the calculation of both βμniandzfrom experimental values ofI(0) recorded as a function of concentration.

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