A re-examination of the basic postulates of thermomechanics

1991 ◽  
Vol 432 (1885) ◽  
pp. 171-194 ◽  
Keyword(s):  
Heat Flow ◽  
Constitutive Equations ◽  
Equations Of Motion ◽  
Thermal Waves ◽  
Mechanical Theory ◽  
Usual Procedure ◽  
Special Cases ◽  
The Subject ◽  
Basic Equations ◽  
Classical Thermodynamics

This paper is mainly concerned with a re-examination of the basic postulates and the consequent procedure for the construction of the constitutive equations of material behaviour in thermomechanics. However, the implication of the basic postulates and the significance of the related procedure for the development of the constitutive equations is also illustrated in some detail in the context of flow of heat in a rigid solid with particular reference to the propagation of thermal waves at finite speed. More specifically, after briefly examining the nature of the basic equations of motion for a system of particles within the scope of the classical newtonian mechanics, the basic postulates of the purely mechanical theory for a continuum (including its specialization for a rigid body) is re-examined. This includes some differences from the usual procedure on the subject. Next, thermal variables are introduced and after observing a useful analogy between the thermal and mechanical variables, a discussion of a theory of heat (or a purely thermal theory) is provided which differs from the usual development in the classical thermodynamics. A detailed application of the latter development is then made to the problem of heat flow in a stationary rigid solid using several different and well-motivated constitutive equations. Special cases of these include linearized theories of the classical heat flow by conduction and of heat flow transmitted as thermal waves. The remainder of the paper is concerned with thermal mechanical theory of deformable media along with discussions of a number of related issues on the subject.

Heat Dissipation Through an Annular Disk or Fin of Uniform Thickness

1938 ◽  
Vol 5 (2) ◽  
pp. A78-A80
Author(s):  
William MacGregor Murray
Keyword(s):  
Heat Dissipation ◽  
Uniform Thickness ◽  
Annular Disk ◽  
Mathematical Expressions ◽  
General Proposition ◽  
Special Cases ◽  
Mathematical Investigation ◽  
The Subject ◽  
Basic Equations ◽  
Fin Design

Abstract The subject matter considered in this paper deals with the mathematical investigation of heat flow in an annular disk of uniform thickness. Originally, the investigation was carried on in connection with the design of fins for increasing the heat transfer in various kinds of heat exchangers and engines. The results of the study, however, might easily be applied to a number of other problems, since by altering the boundary conditions slightly one may use the same basic equations for calculating the temperature distribution and heat transmission in grinding wheels and disk clutches. The study of the problem in connection with fin design has brought forth other solutions for special cases of the general proposition considered here. The particular results obtained by previous investigators can be readily found from the general equations given in this paper. In order to assist in the numerical solution of the somewhat complicated equations a chart for evaluating the mathematical expressions has been included.

FLRW COSMOLOGY FROM YANG–MILLS GRAVITY WITH TRANSLATIONAL GAUGE SYMMETRY

2013 ◽  
Vol 28 (07) ◽  
pp. 1350018 ◽  
Author(s):  
DANIEL KATZ
Keyword(s):  
Gauge Invariance ◽  
Equations Of Motion ◽  
Flat Space ◽  
Metric Tensor ◽  
Yang Mills ◽  
Effective Metric ◽  
Starting Point ◽  
Special Cases ◽  
The Subject ◽  
Flrw Cosmology

We extend to basic cosmology the subject of Yang–Mills gravity — a theory of gravity based on local translational gauge invariance in flat space–time. It has been shown that this particular gauge invariance leads to tensor factors in the macroscopic limit of the equations of motion of particles which plays the same role as the metric tensor of general relativity (GR). The assumption that this "effective metric" tensor takes on the standard FLRW form is our starting point. Equations analogous to the Friedmann equations are derived and then solved in closed form for the three special cases of a universe dominated by (1) matter, (2) radiation and (3) dark energy. We find that the solutions for the scale factor are similar to, but distinct from, those found in the corresponding GR based treatment.

Model Theory: Geometrical and Set-Theoretic Aspects and Prospects

2003 ◽  
Vol 9 (2) ◽  
pp. 197-212 ◽  
Author(s):  
Angus Macintyre
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Theoretic Model ◽  
Topos Theory ◽  
Sheaf Theory ◽  
Special Cases ◽  
Algebraic Spaces ◽  
The Subject ◽  
Near Future ◽  
Modern Mathematics

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.

scholarly journals Relativistic equations for elementary particles

1948 ◽  
Vol 192 (1029) ◽  
pp. 195-218 ◽  
Keyword(s):  
Elementary Particle ◽  
Wave Equation ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Wave Form ◽  
Total Charge ◽  
Field Equations ◽  
Relativistic Approximation ◽  
Special Cases ◽  
Linear Differential

From the general principles of quantum mechanics it is deduced that the wave equation of a particle can always be written as a linear differential equation of the first order with matrix coefficients. The principle of relativity and the elementary nature of the particle then impose certain restrictions on these coefficient matrices. A general theory for an elementary particle is set up under certain assumptions regarding these matrices. Besides, two physical assumptions concerning the particle are made, namely, (i) that it satisfies the usual second-order wave equation with a fixed value of the rest mass, and (ii) either the total charge or the total energy for the particle-field is positive definite. It is shown that in consequence of (ii) the theory can be quantized in the interaction free case. On introducing electromagnetic interaction it is found that the particle exhibits a pure magnetic moment in the non-relativistic approximation. The well-known equations for the electron and the meson are included as special cases in the present scheme. As a further illustration of the theory the coefficient matrices corresponding to a new elementary particle are constructed. This particle is shown to have states of spin both 3/2 and 1/2. In a certain sense it exhibits an inner structure in addition to the spin. In the non-relativistic approximation the behaviour of this particle in an electromagnetic field is the same as that of the Dirac electron. Finally, the transition from the particle to the wave form of the equations of motion is effected and the field equations are given in terms of tensors and spinors.

Unification of Software Reliability Models by Self-Exciting Point Processes

1997 ◽  
Vol 29 (2) ◽  
pp. 337-352 ◽  
Author(s):  
Yiping Chen ◽  
Nozer D. Singpurwalla
Keyword(s):  
Software Reliability ◽  
Point Processes ◽  
Computer Software ◽  
Probability Models ◽  
Reliability Models ◽  
Common Structure ◽  
Software Reliability Models ◽  
Special Cases ◽  
Active Debate ◽  
The Subject

Assessing the reliability of computer software has been an active area of research in computer science for the past twenty years. To date, well over a hundred probability models for software reliability have been proposed. These models have been motivated by seemingly unrelated arguments and have been the subject of active debate and discussion. In the meantime, the search for an ideal model continues to be pursued. The purpose of this paper is to point out that practically all the proposed models for software reliability are special cases of self-exciting point processes. This perspective unifies the very diverse approaches to modeling reliability growth and provides a common structure under which problems of software reliability can be discussed.

Thermal Stresses Due to Disturbance of Uniform Heat Flow by an Insulated Ovaloid Hole

1960 ◽  
Vol 27 (4) ◽  
pp. 635-639 ◽  
Author(s):  
A. L. Florence ◽  
J. N. Goodier
Keyword(s):  
Heat Flow ◽  
Closed Form ◽  
Thermal Stresses ◽  
Thermoelastic Problem ◽  
Uniform Heat ◽  
Special Cases

The linear thermoelastic problem is solved for a uniform heat flow disturbed by a hole of ovaloid form, which includes the ellipse and circle as special cases. Results for stress and displacement are found in closed form, by reducing the problem to one of boundary loading solvable by a method of Muskhelishvili.

scholarly journals Thermal Disturbances in Twinned Orthotropic Thermoelastic Material

2018 ◽  
Vol 23 (4) ◽  
pp. 897-910 ◽  
Author(s):  
L. Rani ◽  
V. Singh
Keyword(s):  
Fourier Transforms ◽  
Crystallographic Axis ◽  
Physical Domain ◽  
Matrix Differential Equation ◽  
Special Cases ◽  
Eigen Value ◽  
Basic Equations ◽  
Matrix Differential ◽  
Thermally Conducting ◽  
Vector Matrix

Abstract This paper deals with deformation in homogeneous, thermally conducting, single-crystal orthotropic twins, bounded symmetrically along a plane containing only one common crystallographic axis. The Fourier transforms technique is applied to basic equations to form a vector matrix differential equation, which is then solved by the eigen value approach. The solution obtained is applied to specific problems of an orthotropic twin crystal subjected to triangular loading. The components of displacement, stresses and temperature distribution so obtained in the physical domain are computed numerically. A numerical inversion technique has been used to obtain the components in the physical domain. Particular cases as quasi-static thermo-elastic and static thermoelastic as well as special cases are also discussed in the context of the problem.

scholarly journals Symmetry analysis of rotating fluid

2005 ◽  
Vol 47 (1) ◽  
pp. 65-74 ◽  
Author(s):  
K. Fakhar ◽  
Zu-Chi Chen ◽  
Xiaoda Ji
Keyword(s):  
Steady State ◽  
Infinite Number ◽  
Special Function ◽  
Equations Of Motion ◽  
Time Dependent ◽  
Lie Theory ◽  
Rotating Fluid ◽  
Type Solution ◽  
Function Type ◽  
Basic Equations

AbstractThe machinery of Lie theory (groups and algebras) is applied to the unsteady equations of motion of rotating fluid. A special-function type solution for the steady state is derived. It is then shown how the solution generates an infinite number of time-dependent solutions via three arbitrary functions of time. This algebraic structure also provides the mechanism to search for other solutions since its character is inferred from the basic equations.

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