Charles Bouton and the Navier-Stokes Global Regularity Conjecture

The present article examines the Lie group invariants of the Navier-Stokes equation (NSE) for incompressible fluids. This is accomplished by applying the invariant theory of Charles Bouton which shows that the self-similar solutions of the NSE are relative invariants of the scaling group. The scaling transformation admitted by the NSE has been recently revisited and a general form of the transformation has been discovered from which it follows that Leray's self-similar solutions are an isolated case. The general form of such solutions is derived by the application of Bouton's first theorem and shows that the standard NSE system is not always supercritical, but can be critical or subcritical. Criticality criteria are derived. Using the criterion of Beale-Kato-Majda, we rule out blow-up for a subset of Bouton's self-similar solutions. For another subset, we show that the system exhibits a conserved quantity, the cavitation number of the fluid. It is coercive, scale- and rotationally invariant. By extending the analysis of Bouton to higher-dimensioned manifolds and by virtue of Bouton's theorems, additional conserved quantities are found, which could further elucidate the physics of fluid turbulence.Comment: The main results are unchanged; new developments: The general scaling transform of the NSE; Bouton's self-similar solutions; Leray's self-similar solutions are an isolated case; Ruled out blow-up for Bouton's self-similar solutions via Beale-Kato-Majda argument and showed existence. Other minor correction

Porosity and density of spark-processed silicon

Abstract Spark-processed Si (sp-Si) is a porous solid-state material. Due to the nature of its structure and morphology, the traditional methods for porosity measurements cannot be utilized. Using the measure theory and the expected value theorems of stereology, we have calculated the porosity of sp-Si to be 43%. Stereological analysis was applied to sp-Si specimen, prepared within a fixed set of growth parameters. Over 60 cross-sectional scanning electron micrographs of the specimen were utilized in this work. The sp-Si sample has a characteristic cylindrical symmetry due to the uniform surface resistance of the Si substrate and to the random nature of spark processing. However, sp-Si is not isotropic, uniform and random (IUR), exhibiting radial and axial anisotropy of porosity. To avoid bias in the calculation, we chose random areas of the cross-sectional surface of sp-Si and calculated their porosities. The calculated values entered into a weighted statistical distribution, in which the statistical weights were determined from the symmetry properties of the sample. The statistical approach and the fact that volume is an additive quantity, allowed us to use a two-dimensional population of points in the calculation of the three-dimensional pore volume fraction and to satisfy the requirement for IUR sample. In the course of the present work we examined fourteen sp-Si samples, prepared under different processing conditions. Ten of these samples were studied qualitatively by measuring the area of the pores relative to the total area in a cross-sectional cut of the sample. Four samples were studied quantitatively using the stereological method outlined above and exhibited porosities in the neighborhood of 43%. One of these studies is described in detail in the present paper and provides a consistent value for the porosity of sp-Si materials, processed in air. Small-spot X-ray photoelectron spectroscopy studies of sp-Si were used in the calculation of its density. In the case of inhomogeneous materials, the density is a weighted (with respect to volume) average of the densities of all participating phases. Taking into account the already calculated porosity, we have estimated the density of sp-Si to be 1.36 g/cm 3 . The main contribution to this value comes from amorphous SiO 2 , which occupies most of the volume of sp-Si

Charles Bouton and the Navier-Stokes Global Regularity Conjecture

The present article examines the Lie group invariants of the Navier-Stokes equation for incompressible fluids. This is accomplished by applying the invariant theory of Charles Bouton. His analyis shows that since the solutions of the NSE are relative invariants of the scaling group, they must be isobaric polynomials of x,y,z,t and thus infinitely differentiable. Then, bounded energy follows from conservation law. The total angular momentum per unit mass is a scale-invariant vector; it is analyzed and conclusions are drawn about its role in turbulence

The Vortex Tube Effect - Short Introduction

The Vortex Tube Effect - Short Introductio