scholarly journals Constitutive Equations for Magnetic Active Liquids

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1910
Author(s):  
Simona Fialová ◽  
František Pochylý
Keyword(s):  
Stress Tensor ◽  
Constitutive Equations ◽  
Magnetization Vector ◽  
Irreversible Thermodynamic ◽  
Strain Rate Tensor ◽  
3D Space ◽  
Velocity Tensor ◽  
Magnetic Liquids ◽  
Curie Principle ◽  
Definition Of

This article is focused on the derivation of constitutive equations for magnetic liquids. The results can be used for both ferromagnetic and magnetorheological fluids after the introduced simplifications. The formulation of constitutive equations is based on two approaches. The intuitive approach is based on experimental experience of non-Newtonian fluids, which exhibit a generally non-linear dependence of mechanical stress on shear rate; this is consistent with experimental experience with magnetic liquids. In these general equations, it is necessary to determine the viscosity of a liquid as a function of magnetic induction; however, these equations only apply to the symmetric stress tensor and can only be used for an incompressible fluid. As a result of this limitation, in the next part of the work, this approach is extended by the asymmetry of the stress tensor, depending on the angular velocity tensor. All constitutive equations are formulated in Cartesian coordinates in 3D space. The second approach to determining constitutive equations is more general: it takes the basis of non-equilibrium thermodynamics and is based on the physical approach, using the definition of density of the entropy production. The production of entropy is expressed by irreversible thermodynamic flows, which are caused by the effect of generalized thermodynamic forces after disturbance of the thermodynamic equilibrium. The dependence between fluxes and forces determines the constitutive equations between stress tensors, depending on the strain rate tensor and the magnetization vector, which depends on the intensity of the magnetic field. Their interdependencies are described in this article on the basis of the Curie principle and on the Onsager conditions of symmetry.

LES Investigation of Boussinesq Constitutive Relation Validity in a Corner Separation Flow

2018 ◽  
Author(s):  
Jean-François Monier ◽  
Nicolas Poujol ◽  
Mathieu Laurent ◽  
Feng Gao ◽  
Jérôme Boudet ◽  
...  
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Constitutive Relation ◽  
Strain Rate Tensor ◽  
Separation Flow ◽  
Compressor Cascade ◽  
Corner Separation ◽  
Reynolds Stress Tensor ◽  
Mean Strain

The present study aims at analysing the Boussinesq constitutive relation validity in a corner separation flow of a compressor cascade. The Boussinesq constitutive relation is commonly used in Reynolds-averaged Navier-Stokes (RANS) simulations for turbomachinery design. It assumes an alignment between the Reynolds stress tensor and the zero-trace mean strain-rate tensor. An indicator that measures the alignment between these tensors is used to test the validity of this assumption in a high fidelity large-eddy simulation. Eddy-viscosities are also computed using the LES database and compared. A large-eddy simulation (LES) of a LMFA-NACA65 compressor cascade, in which a corner separation is present, is considered as reference. With LES, both the Reynolds stress tensor and the mean strain-rate tensor are known, which allows the construction of the indicator and the eddy-viscosities. Two constitutive relations are evaluated. The first one is the Boussinesq constitutive relation, while the second one is the quadratic constitutive relation (QCR), expected to render more anisotropy, thus to present a better alignment between the tensors. The Boussinesq constitutive relation is rarely valid, but the QCR tends to improve the alignment. The improvement is mainly present at the inlet, upstream of the corner separation. At the outlet, the correction is milder. The eddy-viscosity built with the LES results are of the same order of magnitude as those built as the ratio of the turbulent kinetic energy k and the turbulence specific dissipation rate ω. They also show that the main impact of the QCR is to rotate the mean strain-rate tensor in order to realign it with the Reynolds stress tensor, without dilating it.

scholarly journals Damage Tensors and Tensors of Continuity in Stress Analysis

1983 ◽  
Vol 38 (12) ◽  
pp. 1383-1390 ◽  
Author(s):  
J. Betten
Keyword(s):  
Linear Operator ◽  
Coordinate System ◽  
Stress Tensor ◽  
Constitutive Equations ◽  
Symmetric Tensor ◽  
Diagonal Form ◽  
Symmetric Part ◽  
Third Order ◽  
Symmetric Property ◽  
Damage Tensor

Abstract Starting from a third order skew-symmetric tensor of continuity to represent area vectors (bivectors) of Cauchy's tetrahedron in a damaged state, a second order damage tensor is found which has the diagonal form with respect to the considered coordinate system. The second part of the paper is concerned with the stresses in a damaged continuum. Introducing a linear operator of rank four a net-stress tensor is formulated. This tensor can be decomposed into a symmetric part and into an antisymmetric one, where only the symmetric part is equal to the net-stress tensor introduced by Rabotnov [7].In view of the formulation of constitutive equations the non-symmetric property of the actual net-stress tensor is a disadvantage. Therefore, a pseudo-net-stress tensor is introduced, which is symmetric.

scholarly journals Asymmetry of the atomic-level stress tensor in homogeneous and inhomogeneous materials

2018 ◽  
Vol 474 (2217) ◽  
pp. 20180155 ◽  
Author(s):  
Ji Rigelesaiyin ◽  
Adrian Diaz ◽  
Weixuan Li ◽  
Liming Xiong ◽  
Youping Chen
Keyword(s):  
Stress Tensor ◽  
Unit Area ◽  
Symmetric Tensor ◽  
Shear Loading ◽  
Homogeneous Material ◽  
Inhomogeneous Materials ◽  
Virial Stress ◽  
Definition Of ◽  
Dynamics Simulations ◽  
Symmetric Property

The stress tensor is described as a symmetric tensor in all classical continuum mechanics theories and in most existing statistical mechanics formulations. In this work, we examine the theoretical origins of the symmetry of the stress tensor and identify the assumptions and misinterpretations that lead to its symmetric property. We then make a direct measurement of the stress tensor in molecular dynamics simulations of four different material systems using the physical definition of stress as force per unit area acting on surface elements. Simulation results demonstrate that the stress tensor is asymmetric near dislocation cores, phase boundaries, holes and even in homogeneous material under a shear loading. In addition, the atomic virial stress and Hardy stress formulae are shown to significantly underestimate the stress tensor in regions of stress concentration.

Dynamic Effects in Stress Analysis for Discrete Elements Modeling: Application to Masonry

2005 ◽  
Author(s):  
Claude Bohatier ◽  
Brahim Chetouane ◽  
Marc Vinches
Keyword(s):  
Stress Tensor ◽  
Relative Weight ◽  
Load Level ◽  
Discrete Element Modelling ◽  
Dynamic Effects ◽  
Discrete Elements ◽  
A Cell ◽  
Definition Of ◽  
Dynamic Excitations ◽  
Final Expression

The discrete element modelling allows the study of the mechanical behaviour of a collection of solids, submitted to dynamic excitations. The proposed definition of a stress tensor concerns a cell constituted of its kernel, and its neighbouring solids. We demonstrate that taking into account the inertial effects allows the symmetry of the stress tensor. The relative weight of the centrifugal effects remaining in the final expression of the stress tensor is evaluated in order to determine whether or not it has to be taken into account, depending on the application. The proposed definition allows the definition of the load level on different parts of the discontinuous structure. Applications to real masonry cases are presented.

On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet

1994 ◽  
Vol 275 ◽  
pp. 83-119 ◽  
Author(s):  
Shewen Liu ◽  
Charles Meneveau ◽  
Joseph Katz
Keyword(s):  
Stress Tensor ◽  
Eddy Viscosity ◽  
Weighting Function ◽  
A Priori ◽  
Scale Model ◽  
Solid Boundary ◽  
Strain Rate Tensor ◽  
Subgrid Scale ◽  
The Real ◽  
Subgrid Scale Model

The properties of turbulence subgrid-scale stresses are studied using experimental data in the far field of a round jet, at a Reynolds number of Rλ ≈ 310. Measurements are performed using two-dimensional particle displacement velocimetry. Three elements of the subgrid-scale stress tensor are calculated using planar filtering of the data. Using a priori testing, eddy-viscosity closures are shown to display very little correlation with the real stresses, in accord with earlier findings based on direct numerical simulations at lower Reynolds numbers. Detailed analysis of subgrid energy fluxes and of the velocity field decomposed into logarithmic bands leads to a new similarity subgrid-scale model. It is based on the ‘resolved stress’ tensor Lij, which is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale. The correlation coefficient of this model with the real stress is shown to be substantially higher than that of the eddy-viscosity closures. It is shown that mixed models display similar levels of correlation. During the a priori test, care is taken to only employ resolved data in a fashion that is consistent with the information that would be available during large-eddy simulation. The influence of the filter shape on the correlation is documented in detail, and the model is compared to the original similarity model of Bardina et al. (1980). A relationship between Lij and a nonlinear subgrid-scale model is established. In order to control the amount of kinetic energy backscatter, which could potentially lead to numerical instability, an ad hoc weighting function that depends on the alignment between Lij and the strain-rate tensor, is introduced. A ‘dynamic’ version of the model is shown, based on the data, to allow a self-consistent determination of the coefficient. In addition, all tensor elements of the model are shown to display the correct scaling with normal distance near a solid boundary.

scholarly journals An Improved Hydrodynamics Formulation for Multiphase Flow Lattice-Boltzmann Models

1998 ◽  
Vol 09 (08) ◽  
pp. 1393-1404 ◽  
Author(s):  
D. J. Holdych ◽  
D. Rovas ◽  
J. G. Georgiadis ◽  
R. O. Buckius
Keyword(s):  
Multiphase Flow ◽  
Stress Tensor ◽  
Lattice Boltzmann ◽  
Effective Field ◽  
Navier Stokes ◽  
Benchmark Problems ◽  
Physical Parameters ◽  
Two Phase ◽  
Navier Stokes Equation ◽  
Definition Of

Lattice-Boltzmann (LB) models provide a systematic formulation of effective-field computational approaches to the calculation of multiphase flow by replacing the mathematical surface of separation between the vapor and liquid with a thin transition region, across which all magnitudes change continuously. Many existing multiphase models of this sort do not satisfy the rigorous hydrodynamic constitutive laws. Here, we extend the two-dimensional, seven-speed Swift et al. LB model1 to rectangular grids (nine speeds) by using symbolic manipulation (MathematicaTM) and compare the LB model predictions with benchmark problems, in order to evaluate its merits. Particular emphasis is placed on the stress tensor formulation. Comparison with the two-phase analogue of the Couette flow and with a flow involving shear and advection of a droplet surrounded by its vapor reveals that additional terms have to be introduced in the definition of the stress tensor in order to satisfy the Navier–Stokes equation in regions of high density gradients. The use of Mathematica obviates many of the difficulties with the calculations "by-hand," allowing at the same time more flexibility to the computational analyst to experiment with geometrical and physical parameters of the formulation.

Methods of Classical Mechanics Applied to Turbulence Stresses in a Tip Leakage Vortex

1995 ◽  
Author(s):  
Joan G. Moore ◽  
Scott A. Schorn ◽  
John Moore
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Classical Mechanics ◽  
Surface Model ◽  
Strain Rate Tensor ◽  
Reynolds Stresses ◽  
Tip Leakage Vortex ◽  
Tip Leakage ◽  
Stress Tensors ◽  
Principal Directions

Moore et al. measured the six Reynolds stresses in a tip leakage vortex in a linear turbine cascade. Stress tensor analysis, as used in classical mechanics, has been applied to the measured turbulence stress tensors. Principal directions and principal normal stresses are found. A solid surface model, or 3-d glyph, for the Reynolds stress tensor is proposed and used to view the stresses throughout the tip leakage vortex. Modelled Reynolds stresses using the Boussinesq approximation are obtained from the measured mean velocity strain rate tensor. The comparison of the principal directions and the 3-d graphical representations of the strain and Reynolds stress tensors aids in the understanding of the turbulence and what is required to model it.

scholarly journals The Jahn–Teller and Pseudo-Jahn–Teller Effects: A Unique and Only Source of Spontaneous Symmetry Breaking in Atomic Matter

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1577
Author(s):  
Isaac B. Bersuker
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
External Perturbation ◽  
Broken Symmetry ◽  
High Symmetry ◽  
Atomic Systems ◽  
Jahn Teller ◽  
Curie Principle ◽  
Adiabatic Potential Energy Surface ◽  
Definition Of

In a mostly review paper, we show that the important problem of symmetry, broken symmetry, and spontaneous broken symmetry of polyatomic systems is directly related to the Jahn–Teller (JT) and pseudo-Jahn–Teller (PJT) effects, including the hidden-JT and hidden-PJT effects, and these JT effects (JTEs) are the only source of spontaneous symmetry breaking in matter. They are directly related to the violation of the adiabatic approximation by the vibronic and other nonadiabatic couplings (jointly termed nonadiabaticity) in the interaction between the electrons and nuclei, which becomes significant in the presence of two or more degenerate or pseudodegenerate electronic states. In a generalization of this understanding of symmetry, we suggest an improved (quantum) definition of stereo-chemical polyatomic space configuration, in which, starting with their high-symmetry configuration, we separate all atomic systems into three distinguishable groups: (1) weak nonadiabaticity, stable high-symmetry configurations; (2) moderate-to-strong nonadiabaticity, unstable high-symmetry configurations, JTEs, spontaneous symmetry breaking (SSB); (3) very strong nonadiabaticity, stable distorted configurations. The JTEs, inherent to the second group of systems, produce a rich variety of novel properties, based on their multiminimum adiabatic potential energy surface (APES), leading to a short lifetime in the distorted configuration. We show the role of the Curie principle in the possibilities to observe the SSB in atomic matter, and mention briefly the revealed recently gamma of novel properties of matter in its interaction with external perturbation that occur due to the SSB, including ferroelectricity and orientational polarization, leading to enhanced permittivity and flexoelectricity.

scholarly journals MATHEMATICAL KNOWLEDGE OF THE RENAISSANCE ERA AND ESTABLISHMENT OF DRAWING AS AN INDEPENDENT FORM OF ART

2019 ◽  
pp. 109-115
Author(s):  
Yuliya Maystrenko-Vakulenko
Keyword(s):  
Mathematical Knowledge ◽  
Western Europe ◽  
Modern Science ◽  
Educational Models ◽  
Energy Field ◽  
3D Space ◽  
Art Form ◽  
Independent Form ◽  
Definition Of ◽  
Art Academies

This research makes it possible to better understand the reasons why the artistic mindset for world perception and reproduction has drastically changed during a relatively short transition from the Medieval times to the Renaissance, as well as the factors that accelerated the birth of drawing as an independent form of art and prompted the artists of those times to search for ways to picture the tridimensionality of space.Drawing originated as a tool to research the relations between an object and space, as it’s the definition of space that became perceived and imagined on a basis completely different from that of the Medieval times, thanks to scientific discoveries. The crucial change in the art of the Renaissance era: reproduction of the illusion of 3D space and depth was performed through using new mathematical knowledge. Thanks to the research of M. Kuzansky, M. Kopernik, as well as L. B. Alberti, F. Brunelleschi and other scientists, the philosophic thought of the XV century has redeemed the definition of movement, bringing it to the level of eternity: while earlier, in the Medieval tradition, movement was imagined as a sign of earthly imperfection, transience and impermanence. A person’s own view of the world gained its value. It’s exactly this energy of potential movement of the Universe that we can see all the way through drawings by artists of the High Renaissance. The Renaissance space, that has now gained tridimensionality, is imagined as cosmic and immense, with the objects in it being clusters of an energy field. This Renaissance mindset has kickstarted the birth of art academies in Western Europe, where academic drawing was essential in art education. But today requires an update to the principles of teaching. Recognition of the immediate connection between scientific discoveries and direction of development for the art of drawing (both as an independent art form and an academic subject) will make it possible to define ways for its further progression, and avoid stagnation and obsolescence of educational models. Today, drawing is still a tool of world awareness that also allows exploration of its new properties discovered by A. Einstein at the beginning of the XX century. It is also a modern science with more complex, n-dimensional capabilities.

scholarly journals ANALYSIS OF INFLAMMABLE STATE OF PURCHASE AT THE WALKING

2019 ◽  
pp. 80-85
Author(s):  
Irina Bubnovska
Keyword(s):  
Stress State ◽  
Strain Rate ◽  
Stress Tensor ◽  
Plane Strain ◽  
Equilibrium Equation ◽  
State Parameter ◽  
Strain Rate Tensor ◽  
Workpiece Material

On the basis of the equilibrium equation, the plasticity conditions in the zones of plane strain, the equation of the connection of the components of the stress tensor and the strain rate tensor, an expression was obtained for determining the stress state parameter, which makes it possible to estimate the deformity of the workpiece material during rolling.

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