Equations For Perfect Fluid Research Articles

Hodge Decomposition of Conformal Vector Fields on a Riemannian Manifold and Its Applications

For a compact Riemannian m-manifold (Mm,g),m>1, endowed with a nontrivial conformal vector field ζ with a conformal factor σ, there is an associated skew-symmetric tensor φ called the associated tensor, and also, ζ admits the Hodge decomposition ζ=ζ¯+∇ρ, where ζ¯ satisfies divζ¯=0, which is called the Hodge vector, and ρ is the Hodge potential of ζ. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on Mm. The first result of this article states that a compact Riemannian m-manifold Mm is an m-sphere Sm(c) if and only if (1) for a nonzero constant c, the function −σ/c is a solution of the Poisson equation Δρ=mσ, and (2) the Ricci curvature satisfies Ricζ¯,ζ¯≥φ2. The second result states that if Mm has constant scalar curvature τ=m(m−1)c>0, then it is an Sm(c) if and only if the Ricci curvature satisfies Ricζ¯,ζ¯≥φ2 and the Hodge potential ρ satisfies a certain static perfect fluid equation. The third result provides another new characterization of Sm(c) using the affinity tensor of the Hodge vector ζ¯ of a conformal vector field ζ on a compact Riemannian manifold Mm with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold Mm, m>2, is a Euclidean m-space if and only if it admits a nontrivial conformal vector field ζ whose affinity tensor vanishes identically and ζ annihilates its associated tensor φ.

Lagrangian formulation for perfect fluid equations with the ℓ -conformal Galilei symmetry

Lagrangian formulation for perfect fluid equations which hold invariant under the ℓ-conformal Galilei group with half-integer ℓ is proposed. It is based on a Clebsch-type parametrization and reproduces Lagrangian description of the Euler fluid equations for ℓ=12. The transition from the Lagrangian formulation to the Hamiltonian one is analyzed in detail. Published by the American Physical Society 2024

Emergent cosmology in 4D Einstein Gauss Bonnet theory of gravity

In this paper, in an FLRW background and a perfect fluid equation of state, we explore the possibility of the realization of an emergent scenario in a 4D regularized extension of Einstein-Gauss-Bonnet gravity, with the field equations particularly expressed in terms of scalar-tensor degrees of freedom. By assuming non-zero spatial curvature (k = ± 1), the stability of the Einstein static universe (ESU) and its subsequent exit into the standard inflationary scenario is tested through different approaches. In terms of dynamical systems, a spatially closed universe rather than an open universe shows appealing behaviour to exhibit a graceful transition from the ESU to standard cosmological history. We found that under linear homogeneous perturbations, for some constraints imposed on the model parameters, the ESU is stable under those perturbations. Moreover, it is noted that for a successful graceful transition, the equation of state ω must satisfy the conditions −1 < ω < 0 and ω < − 1 for closed and open universes, respectively. Furthermore, the ESU is seen to be neutrally stable under matter perturbation in the Newtonian gauge.

Cosmological Einstein-λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-Perfect-Fluid Solutions with Asymptotic Dust or Asymptotic Radiation Equations of State

This article introduces the notions of asymptotic dust and asymptotic radiation equations of state. With these non-linear generalizations of the well known dust or (incoherent) radiation equations of state the perfect-fluid equations lose any conformal covariance or privilege. We analyse the conformal field equations induced with these equations of state. It is shown that the Einstein-λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}-perfect-fluid equations with an asymptotic radiation equation of state allow for large sets of Cauchy data that develop into solutions which admit smooth conformal boundaries in the future and smooth extensions beyond. In the case of asymptotic dust equations of state sharp results on the future asymptotic behaviour are not available yet.

Cosmologies with positive λ: hierarchies of future behaviour.

Smooth Cauchy data on [Formula: see text] for the Einstein-[Formula: see text]-vacuum field equations with cosmological constant [Formula: see text] that are sufficiently close to de Sitter data develop into a solution that admits a smooth conformal boundary [Formula: see text] in its future. The conformal Einstein equations determine a smooth conformal extension across [Formula: see text] that defines on 'the other side' again a [Formula: see text]-vacuum solution. In this article, we discuss to what extent these properties generalize to the future asymptotic behaviour of solutions to the Einstein-[Formula: see text] equations with matter. We study Friedmann-Lemaitre-Robertson-Walker (FLRW) solutions and the Einstein-[Formula: see text] equations coupled to conformally covariant matter transport equations, to conformally privileged matter equations, and to conformally non-covariant matter equations. We present recent results on the Einstein-[Formula: see text]-perfect-fluid equations with a nonlinear asymptotic dust or asymptotic radiation equation of state. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Ricci Vector Fields Revisited

We continue studying the σ-Ricci vector field u on a Riemannian manifold (Nm,g), which is not necessarily closed. A Riemannian manifold with Ricci operator T, a Coddazi-type tensor, is called a T-manifold. In the first result of this paper, we show that a complete and simply connected T-manifold(Nm,g), m&gt;1, of positive scalar curvature τ, admits a closed σ-Ricci vector field u such that the vector u−∇σ is an eigenvector of T with eigenvalue τm−1, if and only if it is isometric to the m-sphere Sαm. In the second result, we show that if a compact and connected T-manifold(Nm,g), m&gt;2, admits a σ-Ricci vector field u with σ≠0 and is an eigenvector of a rough Laplace operator with the integral of the Ricci curvature Ricu,u that has a suitable lower bound, then (Nm,g) is isometric to the m-sphere Sαm, and the converse also holds. Finally, we show that a compact and connected Riemannian manifold (Nm,g) admits a σ-Ricci vector field u with σ as a nontrivial solution of the static perfect fluid equation, and the integral of the Ricci curvature Ricu,u has a lower bound depending on a positive constant α, if and only if (Nm,g) is isometric to the m-sphere Sαm.

New Characterizations of Hyperspheres and Spherical Hypercylinders in Euclidean Space

Let x be an isometric immersion of a Riemannian n‐manifold M into a Euclidean (n + 1)‐space which does not pass through the origin of . Then, the tangential part of the position vector field x of x is called the canonical vector field, and the normal part gives rise to a scalar function called the support function. Using the canonical vector field, support function, and mean curvature, we establish three new characterizations of hyperspheres. In addition, we prove that if the energy function of M satisfies the static perfect fluid equation, then M has at most two distinct principal curvatures. As an application, we prove that a complete noncompact hypersurface M is a spherical hypercylinder if the energy function of M satisfies the static perfect fluid equation, and it has exactly two distinct nonsimple principal curvatures.

Euclidean hypersurfaces isometric to spheres

&lt;p&gt;Given an immersed hypersurface $ M^{n} $ in the Euclidean space $ E^{n+1} $, the tangential component $\boldsymbol{\omega }$ of the position vector field of the hypersurface is called the basic vector field, and the smooth function of the normal component of the position vector field gives a function $ \sigma $ on the hypersurface called the support function of the hypersurface. In the first result, we show that on a complete and simply connected hypersurface $ M^{n} $ in $ E^{n+1} $ of positive Ricci curvature with shape operator $ T $ invariant under $\boldsymbol{\omega }$ and the support function $ \sigma $ satisfies the static perfect fluid equation if and only if the hypersurface is isometric to a sphere. In the second result, we show that a compact hypersurface $ M^{n} $ in $ E^{n+1} $ with the gradient of support function $ \sigma $, an eigenvector of the shape operator $ T $ with eigenvalue function the mean curvature $ H $, and the integral of the squared length of the gradient $ \nabla \sigma $ has a certain lower bound, giving a characterization of a sphere. In the third result, we show that a compact and simply connected hypersurface $ M^{n} $ of positive Ricci curvature in $ E^{n+1} $ has an incompressible basic vector field $\boldsymbol{\omega }$, if and only if $ M^{n} $ is isometric to a sphere.&lt;/p&gt;

Hamiltonian formulation for perfect fluid equations with the ℓ–conformal Galilei symmetry

The Hamiltonian formulation for perfect fluid equations with the ℓ-conformal Galilei symmetry is proposed. For an arbitrary half-integer value of the parameter ℓ, the Hamilton and non-canonical Poisson brackets are found, in terms of which the original higher derivative equations of motion take the conventional Hamiltonian form. The full set of conserved charges is found and their algebra is established.

Group-theoretic approach to perfect fluid equations with conformal symmetry

The method of nonlinear realizations is a convenient tool for building dynamical realizations of a Lie group, which relies solely upon structure relations of the corresponding Lie algebra. The goal of this work is to discuss advantages and limitations of the method, which is here applied to construct perfect fluid equations with conformal symmetry. Four cases are studied in detail, which include the Schrodinger group, the l-conformal Galilei group, the Lifshitz group, and the relativistic conformal group.

Generalized Chaplygin gas and accelerating universe in f(Q,T) gravity

The generalized Chaplygin gas (GCG), which has an unusual perfect fluid equation of state, is another promising candidate for dark energy. We investigate the GCG scenario coupled with a baryonic matter in a newly suggested $f(Q,T)$ gravity, an arbitrary function of non-metricity $Q$ and the trace of energy-momentum tensor $T$. We consider the functional form of $f(Q, T)$ as a linear combination of $Q$ and an arbitrary function of $T$, denoted by $h(T)$. Furthermore, we obtain two different functional forms of the $f(Q,T)$ model under high pressure and high-density scenarios of GCG. We also test each model with the recent Pantheon supernovae data set of 1048 data points, Hubble data set of 31 points, and baryon acoustic oscillations. The deceleration parameter is constructed using $OHD+SNeIa+BAO$, predicting a transition from decelerated to accelerated phases of the universe expansion. Also, the equation of state parameter acquires a negative behavior depicting acceleration. Finally, we analyze the statefinder diagnostic to discriminate between the GCG and other dark energy models.

Inherent nonlinearity of fluid motion and acoustic gravitational wave memory

We consider the propagation of nonlinear sound waves in a perfect fluid at rest. By employing the Riemann wave equation of nonlinear acoustics in one spatial dimension, it is shown that waves carrying a constant density perturbation at their tails produce an acoustic analogue of gravitational wave memory. For the acoustic memory, which is in general nonlinear, the nonlinearity of the effective spacetime dynamics is not due to the Einstein equations, but due to the nonlinearity of the perfect fluid equations. For concreteness, we employ a box-trapped Bose-Einstein condensate, and suggest an experimental protocol to observe acoustic gravitational wave memory.

On Minimal Hypersurfaces of a Unit Sphere

Minimal compact hypersurface in the unit sphere Sn+1 having squared length of shape operator A2&lt;n are totally geodesic and with A2=n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in Sn+1. One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of Sn+1. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in Sn+1, (n&gt;2), provided the scalar curvature τ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in Sn+1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field Aw.

Trans-Sasakian static spaces

Static spaces (with perfect fluids) appeared in a natural way both in physics (cf. Hawking and Ellis, 1975) and mathematics (cf. A. Fischer and J. Marsden, Duke Math. J. 42 (3) (1975), 519–547). These arise as solutions of the perfect static fluid equation and play a key role in general relativity, providing patterns for some celestial objects. In this paper, we investigate compact and simply connected trans-Sasakian spaces having type (α,β), whose defining functions α and β satisfy the static perfect fluid equation. In particular, we derive some conditions that ensure these spaces are isometric to a 3-sphere. First result of this work shows that the function α satisfying static perfect fluid equation and the scalar curvature τ satisfying certain inequality are not only necessary, but also sufficient conditions for a 3-dimensional compact and simply connected trans-Sasakian manifold to be isometric to a 3-sphere. In the second result of this paper, we prove that the function β satisfying the static perfect fluid equation, scalar curvature τ satisfying certain inequality and the Ricci operator satisfying a Coddazi-type equation are also requirements ensuring that a trans-Sasakian space (again compact and also simply connected) is isometric with a 3-sphere.

Noncommutative-Geometry Wormholes with Isotropic Pressure

The strategy adopted in the original Morris-Thorne wormhole was to retain complete control over the geometry at the expense of certain engineering considerations. The purpose of this paper is to obtain several complete wormhole solutions by assuming a noncommutative-geometry background with a concomitant isotropic pressure condition. This condition allows us to consider a cosmological setting with a perfect-fluid equation of state. An extended form of the equation generalizes the first solution and subsequently leads to the generalized Chaplygin gas model and hence to a third solution. The solutions obtained extend several previous results. This paper also reiterates the need for a noncommutative-geometry background to account for the enormous radial tension that is a characteristic of Morris-Thorne wormholes.

Longitudinal spin polarization in a thermal model

We use a thermal model with single freeze-out to determine longitudinal polarization of $\Lambda$ hyperons emitted from a hot and rotating hadronic medium. We consider the top RHIC energies and use the model parameters determined in the previous analyses of particle spectra and elliptic flow. Using a direct connection between the spin polarization tensor and thermal vorticity, we reproduce earlier results which indicate a quadrupole structure of the longitudinal component of the polarization three-vector with an opposite sign compared to that found in the experiment. We further use only the spatial components of the thermal vorticity in the laboratory system to define polarization and show that this leads to the correct sign and magnitude of the quadrupole structure. This procedure resembles a non-relativistic connection between the polarization three-vector and vorticity employed in other works. In general, our results bring further evidence that the spin polarization dynamics in heavy-ion collisions may be not directly related to the thermal vorticity. The additional material explains the construction of the hydrodynamicaly consistent gradients of fluid velocity and temperature in thermal models with the help of the perfect-fluid equations of motion.

Shock waves dynamics and evolution of vortex formation during the interaction of spherical air pulse with aqueous foam layer

The numerical study of the powerful air spherical shock-wave pulse interaction with the protective aqueous foam barrier with the initial liquid volume fraction of 0.2 is carried out. The foam layer thickness is selected to satisfy the condition of the non-reflection of compression wave from the foam external boundary at the considered time intervals. In studying the wave flows dynamics, we used the assumption of the foam structure destruction into the microdrops suspension behind the strong shock wave front. The two-phase medium is described on the basis of the gas-droplet mixture model, which includes the laws of conservation of mass, momentum and energy for each phase in accordance with the single-pressure, two-speed, two-temperature approximations in a two-dimensional axisymmetric formulation. The Schiller–Naumann model is used for taking into account the interfacial drag forces. The contact heat transfer influence at the interface between the phases is taken into account by the Ranz–Marshall model. To describe the properties of air and water, the Peng–Robinson and perfect fluid equations of state are used. The numerical implementation of the model is carried out using the OpenFOAM open-source software with the two-step PIMPLE algorithm. The numerical study results are presented as spatial distributions of pressure fields, velocities and streamlines. The significant attenuation of the spherical shock wave intensity during its interaction with the aqueous foam layer has been established. The causes and dynamics of the toroidal vortices series formation in the gas region behind the shock front are investigated. The results reliability is confirmed by comparison with the solutions of the similar problem, found by another numerical method.

Thermodynamics of Various Entropies in Specific Modified Gravity with Particle Creation

We consider the particle creation scenario in the dynamical Chern-Simons modified gravity in the presence of perfect fluid equation of state p=(γ-1)ρ. By assuming various modified entropies (Bekenstein entropy logarithmic entropy, power law correction, and Renyi entropy), we investigate the first law of thermodynamics and generalized second law of thermodynamics on the apparent horizon. In the presence of particle creation rate, we discuss the generalized second law of thermodynamics and thermal equilibrium condition. It is found that thermodynamics laws and equilibrium condition remain valid under certain conditions of parameters.

Perfect fluids

We present a systematic treatment of perfect fluids with translation and rotation symmetry, which is also applicable in the absence of any type of boost symmetry. It involves introducing a fluid variable, the kinetic mass density, which is needed to define the most general energy-momentum tensor for perfect fluids. Our analysis leads to corrections to the Euler equations for perfect fluids that might be observable in hydrodynamic fluid experiments. We also derive new expressions for the speed of sound in perfect fluids that reduce to the known perfect fluid models when boost symmetry is present. Our framework can also be adapted to (non-relativistic) scale invariant fluids with critical exponent zz. We show that perfect fluids cannot have Schrödinger symmetry unless z=2z=2. For generic values of zz there can be fluids with Lifshitz symmetry, and as a concrete example, we work out in detail the thermodynamics and fluid description of an ideal gas of Lifshitz particles and compute the speed of sound for the classical and quantum Lifshitz gases.

Brane world models with bulk perfect fluid and broken 4D Poincaré invariance

We consider 5D brane world models with broken global 4D Poincar\a'{e} invariance (4D part of the spacetime metric is not conformal to the Minkowski spacetime). The bulk is filled with the negative cosmological constant and may contain a perfect fluid. In the case of empty bulk (the perfect fluid is absent), it is shown that one brane solution always has either a physical or a coordinate singularity in the bulk. We cut off these singularities in the case of compact two brane model and obtain regular exact solutions for both 4D Poincar\a'{e} broken and restored invariance. When the perfect fluid is present in the bulk, we get the master equation for the metric coefficients in the case of arbitrary bulk perfect fluid equation of state (EoS) parameters. In two particular cases of EoS, we obtain the analytic solutions for thin and thick branes. First one generalizes the well known Randall-Sundrum model with one brane to the case of the bulk anisotropic perfect fluid. In the second solution, the 4D Poincar\a'{e} invariance is restored. Here, the spacetime goes asymptotically to the anti-de Sitter one far from the thick brane.