scholarly journals Relation between dynamic and bulk viscosities

2020 ◽  
Author(s):  
В.А. Павловский ◽  
Д.В. Никущенко
Keyword(s):  
Strain Rate ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Strain Rate Tensor ◽  
Fluid Viscosity ◽  
Material Constants ◽  
Symmetry Properties ◽  
Volume Viscosity ◽  
The Relationship ◽  
Viscous Stresses

Во многих технических устройствах реализуются течения, при описании которых фигурируют различные виды вязкостей. В работе рассмотрены вопросы возникновения этих вязкостей и связь между ними. Показано, как тензор материальных констант 4C - тензор 4-го ранга, имеющий 81 компоненту, переводит тензор скоростей деформаций в тензор вязких напряжений. При рассмотрении связи между тензорами скоростей деформаций и вязких напряжений после перехода из трехмерного пространства в шестимерное, и использования свойств симметрии среды, можно заметить, что число независимых материальных констант для изотропных жидкостей сводится к двум константам Ламе. Рассмотрение скорости объемного расширения жидкости приводит к появлению объёмной вязкости, которая выражается через эти константы. В случае равенства нулю объёмной вязкости имеет место модель ньютоновской стоксовой жидкости, в которой постоянные Ламе становятся пропорциональными друг другу. Далее записаны уравнения Навье-Стокса при разных выражениях для вязкости жидкости. When describing many real flows in technical systems, different kinds of viscosities are usually used. In the present article, we discuss how these kinds of viscosities occur and what are relations between them. We show how material constants tensor 4C (4th rank tensor containing 81 terms) transforms strain rate tensor into viscous stresses tensor. Considering the relationship between the strain rate tensors and viscous stresses after the transition from three-dimensional to six-dimensional space and using the symmetry properties of the medium, it can be obtained that the number of independent material constants for isotropic fluids decreases to two Lamé constants. Taking into account volume expansion rate leads to volume viscosity, which is expressed with that constants. When bulk viscosity is equal to zero there is a model of Newtonian liquid, where Lamé constants become proportional to each other. Finally Navier-Stocks equations are written for different expressions of fluid viscosity.

A Research on the Morphology and Composition of Flocs (Part 1: Floc Breaage Disregarded)

2013 ◽  
Vol 726-731 ◽  
pp. 1566-1572 ◽  
Author(s):  
Shi Qiang Ding ◽  
Qing Na Li ◽  
Xin Rong Pang ◽  
Ji Run Xu
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
New Classification ◽  
Suspension Viscosity ◽  
Primary Particles ◽  
Long Time ◽  
Relationship Of ◽  
The Relationship ◽  
Three Dimensional Space ◽  
Floc Breakage

The characteristics of flocs aggregated in flocculation have been paid more and more attention for a long time. In this paper, a new classification and analyses method dealing with the flocs is developed. The flocs formed after flocculation is divided into four kinds, including the left primary particles, linear flocs with all component particles in a line, planar flocs with all component particles on a plane and volumetric flocs with all component particles in a three-dimensional space. By analyzing the formation approaches of different kind of flocs regardless of the floc breakage, the number of every kind of floc is analyzed to be related with the suspension concentration mathematically. After comparing the different items in the models describing the relationship of floc number and concentration, a series of simplified expressions are presented. Lastly, a mathematical equation relating the measurable suspension viscosity with the numbers of different flocs is obtained.

scholarly journals The relationship between diversity and productivity from a three-dimensional space view in a natural mesotrophic lake

2021 ◽  
Vol 121 ◽  
pp. 107069
Author(s):  
Ai-Ping Wu ◽  
Shi-Yun Ye ◽  
Jin-Rui Yuan ◽  
Liang-Yu Qi ◽  
Zheng-Wu Cai ◽  
...  
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Mesotrophic Lake ◽  
The Relationship ◽  
Three Dimensional Space

The molecular orbital theory of chemical valency. XII. The excited states of diatomic molecules

1953 ◽  
Vol 216 (1124) ◽  
pp. 1-10 ◽  
Keyword(s):  
Wave Equation ◽  
Excited States ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Wave Functions ◽  
Triplet States ◽  
Orbital Theory ◽  
Lennard Jones ◽  
Singlet States ◽  
Symmetry Properties

A method is developed for the solution of the wave equation for two electrons in the presence of two centres. The work of Lennard-Jones & Pople (1951) on the ground state of such a system is generalized so as to apply to all the excited states. Full advantage is taken of the symmetry properties of the wave functions, both in three-dimensional and six-dimensional space, to reduce the wave equation to a number of component parts, each of a particular symmetry type. This leads to sets of equations with characteristic symmetry properties appropriate to singlet states and triplet states, whether even or odd, positive or negative in the standard notation ( 1 ∑ - g ).

scholarly journals Developable or Not Related to Information Loss

2018 ◽  
Vol 10 (5) ◽  
pp. 28
Author(s):  
William Chen
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Information Loss ◽  
Surface Model ◽  
Euclidean Spaces ◽  
Developable Surfaces ◽  
The Relationship ◽  
Theoretical Results ◽  
Three Dimensional Space

In this paper we present a lemma and two theorems. These theoretical results will be used to test whether or not a given surface model can be developed. We then choose some examples to demonstrate how to perform these tests. All of these theories and examples are for general purposes, and are not restricted to any particular field. Although all examples are in three-dimensional space, it can be expanded to finite n-dimensional Euclidean spaces. The objective of this paper is to link the relationship between developable surfaces and information loss.

scholarly journals Constructive Geometric Models with Imaginary Objects

2020 ◽  
pp. short27-1-short27-9
Author(s):  
Denis Voloshinov ◽  
Alexandra Solovjeva
Keyword(s):  
Projective Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Order ◽  
Second Order ◽  
The Other ◽  
Significant Obstacle ◽  
The Relationship ◽  
Planar Drawing ◽  
Three Dimensional Space

The article is devoted to the consideration of a number of theoretical questions of projective geometry related to specifying and displaying imaginary objects, especially, conics. The lack of development of appropriate constructive schemes is a significant obstacle to the study of quadratic images in three-dimensional space and spaces of higher order. The relationship between the two circles, established by the inversion operation with respect to the other two circles, in particular, one of which is imaginary, allows obtain a simple and effective method for indirect setting of imaginary circles in a planar drawing. The application of the collinear transformation to circles with an imaginary radius also makes it possible to obtain unified algorithms for specifying and controlling imaginary conics along with usual real second-order curves. As a result, it allows eliminate exceptional situations that arise while solving problems with quadratic images in spaces of second and higher order.

scholarly journals Body Metamorphosis and Animality: Volatile Bodies and Boulder Artworks from Lepenski Vir

2005 ◽  
Vol 15 (1) ◽  
pp. 35-69 ◽  
Author(s):  
Dušan Borić
Keyword(s):  
Life Cycle ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Migratory Fish ◽  
The Social ◽  
Social Efficacy ◽  
The Relationship ◽  
Three Dimensional Space ◽  
The Way

This article discusses the notion of body metamorphosis as a theory of phenomenal change by examining carved representational and ‘aniconic’ boulders from Lepenski Vir and other Meso-Neolithic sites in the Danube Gorges. The voluminous size of the boulders at Lepenski Vir, the way in which they occupy the three-dimensional space within buildings and around hearths, and the carvings over their surfaces suggest that they were understood as volatile bodies, undergoing continuous metamorphoses. The relationship between the seasonal recurrence of the Danube's migratory fish and these boulders is explored through the notion of animality. These boulders indicate prescribed stages of life-cycle metamorphosis that affected inextricably-linked realms of human and animal worlds. Prescribed stages of social embodiment at Lepenski Vir are discerned by looking at the archaeological context of representational boulders that sometimes directly commemorate particular deceased individuals. The possibility that boulder artworks acted as sacred heirlooms of particular buildings is connected to the social efficacy they might have acquired.

Invariants of the reduced velocity gradient tensor in turbulent flows

2013 ◽  
Vol 716 ◽  
pp. 597-615 ◽  
Author(s):  
J. I. Cardesa ◽  
D. Mistry ◽  
L. Gan ◽  
J. R. Dawson
Keyword(s):  
Strain Rate ◽  
Turbulent Flows ◽  
Velocity Gradient ◽  
Three Dimensional ◽  
Joint Probability ◽  
Strain Rate Tensor ◽  
Gradient Tensor ◽  
Local Isotropy ◽  
Velocity Gradient Tensor ◽  
Asymmetric Shape

AbstractIn this paper we examine the invariants $p$ and $q$ of the reduced $2\times 2$ velocity gradient tensor (VGT) formed from a two-dimensional (2D) slice of an incompressible three-dimensional (3D) flow. Using data from both 2D particle image velocimetry (PIV) measurements and 3D direct numerical simulations of various turbulent flows, we show that the joint probability density functions (p.d.f.s) of $p$ and $q$ exhibit a common characteristic asymmetric shape consistent with $\langle pq\rangle \lt 0$. An explanation for this inequality is proposed. Assuming local homogeneity we derive $\langle p\rangle = 0$ and $\langle q\rangle = 0$. With the addition of local isotropy the sign of $\langle pq\rangle $ is proved to be the same as that of the skewness of $\partial {u}_{1} / \partial {x}_{1} $, hence negative. This suggests that the observed asymmetry in the joint p.d.f.s of $p{{\ndash}}q$ stems from the universal predominance of vortex stretching at the smallest scales. Some advantages of this joint p.d.f. compared with that of $Q{{\ndash}}R$ obtained from the full $3\times 3$ VGT are discussed. Analysing the eigenvalues of the reduced strain-rate matrix associated with the reduced VGT, we prove that in some cases the 2D data can unambiguously discriminate between the bi-axial (sheet-forming) and axial (tube-forming) strain-rate configurations of the full $3\times 3$ strain-rate tensor.

scholarly journals Simultaneous Invariants of Strain and Rotation Rate Tensors and Their Admitted Region

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Igor Vigdorovich ◽  
Holger Foysi
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Turbulent Flows ◽  
Reynolds Stress ◽  
Building Block ◽  
Rotation Rate ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Strain Rate Tensor ◽  
Reynolds Stress Tensor

The purpose of this paper is to establish the admitted region for five simultaneous, functionally independent invariants of the strain rate tensorSand rotation rate tensorΩand calculate some simultaneous invariants of these tensors which are encountered in the theory of constitutive relations for turbulent flows. Such a problem, as far as we know, has not yet been considered, though it is obviously an integral part of any problem in which scalar functions of the tensorsSandΩare studied. The theory provided inside this paper is the building block for a derivation of new algebraic constitutive relations for three-dimensional turbulent flows in the form of expansions of the Reynolds-stress tensor in a tensorial basis formed by the tensorsSandΩ, in which the scalar coefficients depend on simultaneous invariants of these tensors.

Three-Dimensional Computed Tomographic Reconstructions of Intracranial Meningiomas

Neurosurgery ◽  
1988 ◽  
Vol 23 (5) ◽  
pp. 570-575 ◽  
Author(s):  
David J. Chalif ◽  
Craig R. Dufresne ◽  
Joseph Ransohoff ◽  
Joseph A. McCarthy
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Computer Software ◽  
Computed Tomographic ◽  
Intracranial Meningiomas ◽  
Three Dimensional Images ◽  
Ct Data ◽  
Surgical Adjunct ◽  
Relationship Of ◽  
The Relationship

Abstract In a series of four patients with intracranial meningiomas, three-dimensional computed tomographic (CT) reconstructions were a useful diagnostic and surgical adjunct. Three-dimensional images are created from standard CT data by a boundary-detecting computer software program. Three-dimensional images of tumor invading or adjacent to the bony calvarium are projected about the x, y, and z axes. Axial and sagittal sections delineate the lesions. The images created allow a surgical view of the meningiomas in three-dimensional space and demonstrate the relationship of these masses to the skull. The use of three-dimensional reconstructions in craniofacial surgery and in neurosurgery is reviewed.

Graphical representation of stereochemical configuration (IUPAC Recommendations 2006)

2006 ◽  
Vol 78 (10) ◽  
pp. 1897-1970 ◽  
Author(s):  
Jonathan Brecher
Keyword(s):  
Graphical Representation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Wide Audience ◽  
Stereochemical Configuration ◽  
Stereogenic Centers ◽  
Relationship Of ◽  
Stereogenic Center ◽  
The Relationship

Stereochemical configuration is determined by the relationship of atoms in three-dimensional space, yet remains most commonly represented in two-dimensional media such as printed publications or computer screens. Recommendations are provided for the display of three-dimensional stereochemical information in two-dimensional diagrams in ways that avoid ambiguity and are likely to be understood correctly by all viewers. Examples are provided for all types of stereochemical configuration, with explanation of which styles are preferred and which should be avoided. Principal recommendations include:Know your audience: Diagrams that have a wide audience should be drawn as simply as possible.Avoid ambiguous drawing styles.Avoid the use of perspective diagrams and class-specific drawing styles (Fischer projections, Haworth projections, etc.) when structures are to be interpreted by computers.Use solid wedges to indicate bonds that project above the plane of the paper and hashed wedges to indicate bonds that project below the plane of the paper; in both cases, the bonds must be oriented with the narrow end at the stereogenic center.Avoid connecting stereogenic centers with a stereobond.

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