scholarly journals Relativistic fluid dynamics: physics for many different scales

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Nils Andersson ◽  
Gregory L. Comer
Keyword(s):  
Fluid Model ◽  
Integrability Condition ◽  
Equations Of Motion ◽  
Model Systems ◽  
Theoretical Physics ◽  
Energy Tensor ◽  
Density Currents ◽  
Relativistic Fluids ◽  
Text Book ◽  
Relativistic Fluid

AbstractThe relativistic fluid is a highly successful model used to describe the dynamics of many-particle systems moving at high velocities and/or in strong gravity. It takes as input physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process—e.g., drawing on astrophysical observations—an understanding of relativistic features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as “small” as colliding heavy ions in laboratory experiments, and as large as the Universe itself, with “intermediate” sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theoretical physics underpinnings of the relativistic (multi-) fluid model. We focus on the variational principle approach championed by Brandon Carter and collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the “standard” text-book derivation of the equations of motion from the divergence of the stress-energy tensor in that one explicitly obtains the relativistic Euler equation as an “integrability” condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications of the general theory. The formalism provides a foundation for complex models, e.g., including electromagnetism, superfluidity and elasticity—all of which are relevant for state of the art neutron-star modelling.

scholarly journals On a viable first-order formulation of relativistic viscous fluids and its applications to cosmology

2017 ◽  
Vol 26 (13) ◽  
pp. 1750146 ◽  
Author(s):  
Marcelo M. Disconzi ◽  
Thomas W. Kephart ◽  
Robert J. Scherrer
Keyword(s):  
Equations Of Motion ◽  
Heavy Ion ◽  
Viscous Fluids ◽  
Energy Tensor ◽  
Stress Energy Tensor ◽  
Relativistic Fluids ◽  
First Order ◽  
Ion Collisions ◽  
Stress Energy ◽  
Barotropic Equation

We consider a first-order formulation of relativistic fluids with bulk viscosity based on a stress-energy tensor introduced by Lichnerowicz. Choosing a barotropic equation-of-state, we show that this theory satisfies basic physical requirements and, under the further assumption of vanishing vorticity, that the equations of motion are causal, both in the case of a fixed background and when the equations are coupled to Einstein's equations. Furthermore, Lichnerowicz's proposal does not fit into the general framework of first-order theories studied by Hiscock and Lindblom, and hence their instability results do not apply. These conclusions apply to the full-fledged nonlinear theory, without any equilibrium or near equilibrium assumptions. Similarities and differences between the approach explored here and other theories of relativistic viscosity, including the Mueller–Israel–Stewart formulation, are addressed. Cosmological models based on the Lichnerowicz stress-energy tensor are studied. As the topic of (relativistic) viscous fluids is also of interest outside the general relativity and cosmology communities, such as, for instance, in applications involving heavy-ion collisions, we make our presentation largely self-contained.

Relativistic dynamics of a spinning magnetic particle

1942 ◽  
Vol 38 (1) ◽  
pp. 40-60 ◽  
Author(s):  
Myron Mathisson
Keyword(s):  
Angular Momentum ◽  
Magnetic Particle ◽  
Magnetic Moments ◽  
Equations Of Motion ◽  
Energy Tensor ◽  
Centre Of Mass ◽  
Electric Moment ◽  
Electric And Magnetic Moments ◽  
Second Moments ◽  
External Forces

The author's general variational method is applied to the case of a particle for which second moments are important but third and higher moments are negligible. Equations of motion are obtained for the angular momentum and for the centre of mass, equations (12·35) and (12·41), with arbitrary external forces X.The angular forces are then calculated for a charged particle with electric and magnetic moments moving in a general electromagnetic field, on the assumption that the effect of a certain part of the energy tensor, Tiii of (15·17), is negligible. This leads to the equations of angular motion, (17·13), from which it is inferred that, in order that the magnitude of the angular momentum may be integrable, the angular momentum, electric and magnetic moments must all be parallel in a frame of reference in which the particle is instantaneously at rest.The linear forces are then calculated for the case of no electric moment, leading to the equations for linear motion (18·10). From these it is inferred that, in order that the mass may be integrable, the ratio of the magnetic moment to the angular momentum must be constant.

scholarly journals A dynamical system analysis of f(R,T) gravity

2016 ◽  
Vol 13 (09) ◽  
pp. 1650108 ◽  
Author(s):  
Behrouz Mirza ◽  
Fatemeh Oboudiat
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Equation Of State ◽  
System Analysis ◽  
Equations Of Motion ◽  
Energy Tensor ◽  
Stress Energy Tensor ◽  
Future Singularity ◽  
First Case ◽  
Stress Energy

We investigate equations of motion and future singularities of [Formula: see text] gravity where [Formula: see text] is the Ricci scalar and [Formula: see text] is the trace of stress-energy tensor. Future singularities for two kinds of equation of state (barotropic perfect fluid and generalized form of equation of state) are studied. While no future singularity is found for the first case, some kind of singularity is found to be possible for the second. We also investigate [Formula: see text] gravity by the method of dynamical systems and obtain some fixed points. Finally, the effect of the Noether symmetry on [Formula: see text] is studied and the consistent form of [Formula: see text] function is found using the symmetry and the conserved charge.

Semiclassical relativistic fluid theory for electrostatic envelope modes in dense electron–positron–ion plasmas: Modulational instability and rogue waves

2013 ◽  
Vol 79 (6) ◽  
pp. 1089-1094 ◽  
Author(s):  
IOANNIS KOURAKIS ◽  
MICHAEL MC KERR ◽  
ATA UR-RAHMAN
Keyword(s):  
Fluid Model ◽  
Modulational Instability ◽  
Rogue Waves ◽  
Relativistic Electrons ◽  
Relativistic Fluid ◽  
Ion Plasma ◽  
Electron Positron ◽  
Electrons And Positrons ◽  
Perturbative Method ◽  
Fluid Theory

AbstractA fluid model is used to describe the propagation of envelope structures in an ion plasma under the influence of the action of weakly relativistic electrons and positrons. A multiscale perturbative method is used to derive a nonlinear Schrödinger equation for the envelope amplitude. Criteria for modulational instability, which occurs for small values of the carrier wavenumber (long carrier wavelengths), are derived. The occurrence of rogue waves is briefly discussed.

scholarly journals Field-Dependent Symmetries of a Non-relativistic Fluid Model

2000 ◽  
Vol 282 (2) ◽  
pp. 218-246 ◽  
Author(s):  
M. Hassaíne ◽  
P.A. Horváthy
Keyword(s):  
Fluid Model ◽  
Relativistic Fluid ◽  
Field Dependent

Nonlinear Vibrations of Plates in Axial Pulsating Flow

2014 ◽  
Author(s):  
E. Tubaldi ◽  
M. Amabili ◽  
F. Alijani
Keyword(s):  
Flow Velocity ◽  
Nonlinear Vibrations ◽  
Plate Theory ◽  
Fluid Model ◽  
Equations Of Motion ◽  
Transmural Pressure ◽  
Pulsating Flow ◽  
Pulsation Frequency ◽  
Rectangular Plates ◽  
Flow Perturbation

A theoretical approach is presented to study nonlinear vibrations of thin infinitely long rectangular plates subjected to pulsatile axial inviscid flow. The case of plates in axial uniform flow under the action of constant transmural pressure is also addressed for different flow velocities. The equations of motion are obtained based on the von Karman nonlinear plate theory retaining in-plane inertia via Lagrangian approach. The fluid model is based on potential flow theory and the Galerkin method is applied to determine the expression of the flow perturbation potential. The effect of different system parameters such as flow velocity, pulsation amplitude, pulsation frequency, and channel pressurization on the stability of the plate and its geometrically nonlinear response to pulsating flow are fully discussed. In case of zero uniform transmural pressure numerical results show hardening type behavior for the entire flow velocity range when the pulsation frequency is spanned in the neighbourhood of the plate’s fundamental frequency. Conversely, a softening type behavior is presented when a uniform transmural pressure is introduced.

Introduction

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Mirror Symmetry ◽  
Complex Geometry ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Hamiltonian Formalism ◽  
Theoretical Physics ◽  
Lagrange Equations ◽  
Low Dimensional Topology ◽  
The Subject ◽  
Low Dimensional

Symplectic topology has a long history. It has its roots in classical mechanics and geometric optics and in its modern guise has many connections to other fields of mathematics and theoretical physics ranging from dynamical systems, low-dimensional topology, algebraic and complex geometry, representation theory, and homological algebra, to classical and quantum mechanics, string theory, and mirror symmetry. One of the origins of the subject is the study of the equations of motion arising from the Euler–Lagrange equations of a one-dimensional variational problem. The Hamiltonian formalism arising from a Legendre transformation leads to the notion of a ...

BOUNDARY CONDITIONS IN THE MECHANICS OF FIELDS

1952 ◽  
Vol 30 (6) ◽  
pp. 684-698
Author(s):  
S. M. Neamtan ◽  
E. Vogt
Keyword(s):  
Variational Principle ◽  
Boundary Conditions ◽  
Vector Fields ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Second Order ◽  
Dirac Field ◽  
Energy Tensor ◽  
Complex Scalar ◽  
Set Up

A variational principle has been set up for the description of relativistic fields with the aid of Lagrangians involving second order derivatives of the field functions. This constitutes a generalization of the usual formulation in that, besides the boundary conditions usually imposed, it admits also linear homogeneous boundary conditions. The formulation has been developed for the complex scalar and complex vector fields. The variational principle then yields not only the wave equations but also the allowed boundary conditions. A Hamiltonian and equations of motion in canonical form can be set up. A symmetric stress–energy tensor and a charge–current vector are defined, yielding the usual conservation equations. For the vector field, π4 is not identically zero; also the Lorentz condition arises out of the variational principle and does not have to be separately imposed. For the Dirac field an extension to Lagrangians with second order derivatives is not possible, but for this field also the variational principle yields the allowed boundary conditions.

scholarly journals A systematic treatment of moving axes in hydrodynamics

1949 ◽  
Vol 196 (1045) ◽  
pp. 285-300 ◽  
Keyword(s):  
Heat Transfer ◽  
Tropical Cyclones ◽  
Equations Of Motion ◽  
Rate Of Change ◽  
Energy Tensor ◽  
Mathematical Form ◽  
Tensor Calculus ◽  
Systematic Treatment ◽  
The Earth ◽  
Special Cases

A method is developed for transforming the equations of hydrodynamics to a system of curvilinear co-ordinates in motion relative to fixed axes using the tensor-calculus and without employing Coriolis’s theorem .The basic entity is the kinetic metric, a four-dimensional quadratic form defined through the kinetic energy of unit mass of fluid.The mechanics of special relativity are used to obtain, by approximation in terms of 1/ c 2 , where c is the velocity of light, the classical formulae.The equations of motion in terms of the energy-tensor, the four-dimensional vorticity-tensor and the˙ velocity-components are successively obtained and the equation of continuity is shown to be independent (in mathematical form) of the motion of the co-ordinate-system . This property holds also for the equation of heat-transfer in a non-viscous fluid. Applications are made to the case of local Cartesian and local cylindrical polar co-ordinates on the Earth ’s surface. Formulae for the rate of change of vorticity due to Helmholtz and to Petterssen, respectively, are obtained as special cases and Sawyer’s theory of tropical cyclones is also discussed.

scholarly journals The Boltzmann Equations in General Relativity

1936 ◽  
Vol 4 (4) ◽  
pp. 238-253 ◽  
Author(s):  
A. G. Walker
Keyword(s):  
General Relativity ◽  
Continuous Medium ◽  
Equations Of Motion ◽  
Free Particle ◽  
Energy Tensor ◽  
Principle Of Equivalence ◽  
Boltzmann Equations ◽  
Conservation Equations ◽  
Proper Mass ◽  
Conservation Of Momentum

In a recent paper, J. L. Synge gives an interesting derivation of the conservation equations Tij,j = 0 satisfied by the energy tensor Tij of a continuous medium. Previous to the appearance of this paper, these equations were generally obtained by assuming the classical equations of motion and continuity, after which it was necessary to appeal to the Principle of Equivalence. It then follows that the path of a free particle is a geodesic. Synge however starts with the hypothesis that the path of a particle between collisions is a geodesic and that the proper mass is constant. The conservation equations are then deduced exactly from the law of conservation of momentum for collisions.

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