Barotropic Equation Research Articles

A new convex integration approach for the compressible Euler equations and failure of the local maximal dissipation criterion

Abstract In this paper we establish a new convex integration approach for the barotropic compressible Euler equations in two space dimensions. In contrast to existing literature, our new method generates not only the momentum for given density, but also the energy and the energy flux. This allows for a simple way to construct admissible solutions, i.e. solutions which satisfy the energy inequality. Moreover using the convex integration method developed in this paper, we show that the local maximal dissipation criterion fails in the following sense: There exist wild solutions which beat the self-similar solution of the one-dimensional Riemann problem extended to two dimensions. Hence the local maximal dissipation criterion rules out the self-similar solution. The convex integration machinery itself is carried out in a very general way. Hence this paper provides a universal framework for convex integration which not even specifies the form of the partial differential equations under consideration. Therefore this general framework is applicable in many different situations.

Numerical study on cavitation generation induced by the high-speed jet impact on the water surface

The complex interaction between shock waves and two-phase interfaces can generate cavitation. In this study, the cavitation induced by the high-speed jet impact on the water surface was investigated. The mixture fluid is modeled using the barotropic equation of state under the framework of the two-phase flow model, which can describe the mixture of air, water, and vapor with any proportion. Through constructing a 1D Riemann problem for the impact-induced cavitation phase transition, it indicates that the coupling effect of multiple rarefaction waves emitted from the two-phase interface is responsible for the cavitation phase transition inside the liquid. Then, a 3D (three-dimensional) simulation regarding the impact of a high-speed jet on the water surface was conducted and validated against previous experiments that captured the cavitation phase transition phenomenon in the central region after the jet impact. The 3D simulation results revealed the spatial structure and development process of shock waves in detail. The coupling effects of shock waves and two-phase interfaces generate a ring-shaped rarefaction wave, which develops radially inward and superimposes, resulting in the formation of acorn-shaped cavitation bubble nuclei inside the water. The 3D simulation can provide spatial shock/rarefaction wave structures and internal flow details that have never been obtained in experiments, such as shock generation and propagation, rarefaction wave generation and center convergence, and the internal structure of acorn-shaped cavitation nucleation. Furthermore, the influence of the jet velocity on the cavitation intensity was analyzed, and a quantitative relationship was provided.

Compact anisotropic stellar distribution with linear barotropic equation of state in energy–momentum trace coupling modification of general relativity

There are no known exact isotropic or anisotropic stellar models with an equation of state in energy–momentum trace-coupling (EMTC) modifications of general relativity — the simplest linear case of f(R,T) gravity. The difficulty lies in the intricate entanglement of the density and pressure functions that generate intractable governing equations when an equation of state is introduced. For example there is no interior Schwarzschild incompressible star analogue in the well studied f(R,T) theory. In Einstein’s theory it is straightforward to find anisotropic stellar models with a linear equation of state since the system is under-determined and there remains one more choice to nominate any of the variables. This is also true in EMTC theories however, the master field equation is more formidable. If interpreted as a linear second order equation, no viable solutions emerge by prescribing one of the gravitational potentials or by suppressing some terms in the spirit of Tolman (1939). Rewriting as a nonlinear first order equation and speculating on a power-law form of the temporally directed gravitational potential results in success in finding a physically viable compact star distribution. Specifically the model has monotonic decrease of both pressure and density and a surface of vanishing pressure exists. Several stability tests are imposed and the model performs according to expectations. The singularity at the stellar center may be cured by insertion of a regular core enveloped by our fluid model in a multi-layered star.

Cosmological solution through gravitational decoupling in [formula omitted] gravity

This paper aims to formulate anisotropic cosmological solution of a non-static spherical structure with the help of gravitational decoupling scheme through minimal geometric deformation in f(G,T) gravity. This technique transforms only the radial metric function while the temporal component remains unchanged. Consequently, the field equations are separated into two independent arrays: one is related to the seed source and the other characterizes the extra sector. In order to derive the solution corresponding to the isotropic sector, we use the Friedmann–Lemaitre–Robertson–Walker cosmic model and employ the barotropic equation of state as well as power-law model. Finally, we study the impact of decoupling parameter to describe different eras of the universe through graphical analysis. It is found that physically viable and stable trends of the resulting solution are achieved for both radiation-dominated as well as matter-dominated epochs in this modified theory.

Impact of cold dark matter and variable equations of state on the stability of thin-shell wormholes

The goal of this research is to better understand how two identical black hole copies, bounded by a cold dark matter halo, produce and stabilize thin-shell wormholes. The occurrence of a cold dark matter halo is discovered to increase the black hole’s horizon radius. Investigating the stable geometry of these wormholes by linearized radial perturbation analysis is the main objective of the work. It is noteworthy that the development of thin-shell wormholes with reduced energy limitations is associated with the presence of a cold dark matter halo. We study how the stability of the wormholes is affected by variable equations of state, including phantom-like, variable Chaplygin, and barotropic equations of state. The question emphasizes how crucial the existence of a cold dark matter halo is to preserving stable thin-shell wormhole configurations. The findings shed light on the interactions between a cold dark matter halo and wormholes, which improved our knowledge of both phenomena and their possible consequences for extraterrestrial travel.

A linear assessment of barotropic Rossby wave propagation in different background flow configurations

Abstract. The horizontal propagation of Rossby waves in the upper troposphere has been a long-standing topic in dynamical meteorology. The concept of “waveguidability’’, i.e., the capability of the background flow to act as a zonal waveguide for Rossby waves, may prove useful to address this problem, but developing a systematic definition and quantification of such a property remains challenging. With an eye to such issues, the current paper suggests a novel and efficient algorithm to solve the linearized barotropic vorticity equation on a sphere in a forced-dissipative configuration. The algorithm allows one to obtain linear wave solutions resulting from arbitrary combinations of the forcing and the background zonal wind. These solutions can be used to systematically study single- and double-jet configurations and are employed here to show that the latitude of the jet stream does not appear to affect waveguidability. The onset of barotropic instability might hinder the applicability of the linear framework, but it is shown that the nonlinear flow evolution can still be retrieved qualitatively from the linearized solution, both for the stationary component of the wave field and for the temporal evolution of transient waves.

Group foliations of the β-plane barotropic vorticity equation

Despite the large number of publications on symmetry analysis of the barotropic vorticity equation on the β-plane, its group foliations have not been considered previously. The present publication aims to address this shortcoming. Group foliations are constructed for the equation, and based on them, invariant solutions are derived, some of which generalize previously known exact analytical solutions. There is also a discussion of the pros and cons of the group foliation approach including consideration of some numerical issues.

Probing the existence of wormhole solutions in [formula omitted] gravity with conformal symmetry

This study presents novel and physically plausible wormhole solutions within the framework of f(Q,T) gravity theory, incorporating conformal symmetries. The investigation explores the feasibility of traversable wormholes under diverse scenarios, considering traceless, anisotropic, and barotropic equations of state. Additionally, the influence of model parameters on the existence and characteristics of these wormhole structures is thoroughly examined. Notably, the derived shape function satisfies all the necessary criteria. Furthermore, in one of the cases, the presence of non-exotic fluid is confirmed, while in others, exotic matter is only required near the wormhole throat.

Exploring wormhole solutions with global monopole charge in the context of f(Q) gravity

This study explores the potential existence of traversable wormholes influenced by a global monopole charge within the f(Q) gravity framework. To elucidate the characteristics of these wormholes, we conducted a comprehensive analysis of wormhole solutions employing three different forms of redshift function under a linear f(Q) model. Wormhole shape functions were derived for barotropic, anisotropic, and isotropic Equations of State (EoS) cases. However, in the isotropic EoS case, the calculated shape function failed to satisfy the asymptotic flatness condition. Additionally, we observed that our obtained shape functions adhered to the flaring-out conditions under an asymptotic background for the remaining EoS cases. Furthermore, we examined the energy conditions at the wormhole throat with a radius r0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0$$\\end{document}. We noted the influences of the global monopole’s parameter η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document}, the EoS parameter ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document}, and n in violating energy conditions, particularly the null energy conditions. Finally, we conducted a stability analysis utilizing the Tolman–Oppenheimer–Volkov (TOV) equation and found that our obtained wormhole solution is stable.

An explicit well-balanced scheme on staggered grids for barotropic Euler equations

In this paper, we introduce a specific modification of the numerical fluxes in order to insure the well-balanced property of schemes on staggered grids for the Euler equations. This property is crucial for the numerical representation of equilibrium solutions of balance laws with source terms, like when describing flows subjected to gravity and a complex topography. We propose first and second order versions of the well-balanced scheme. The performances of the method are evaluated through a series of 1D and 2D benchmarks.

Cosmological implications of the Weyl geometric gravity theory

We consider the cosmological implications of the Weyl geometric gravity theory. The basic action of the model is obtained from the simplest conformally invariant gravitational action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The total action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To maintain the conformal invariance of the action the trace condition is imposed on the matter energy–momentum tensor, thus making the matter sector of the action conformally invariant. The field equations are derived by varying the action with respect to the metric tensor, the Weyl vector field, and the scalar field, respectively. We investigate the cosmological implications of the theory, and we obtain first the cosmological evolution equations for a flat, homogeneous and isotropic geometry, described by Friedmann–Lemaitre–Robertson–Walker metric, which generalize the Friedmann equations of standard general relativity. In this context we consider two cosmological models, corresponding to the vacuum state, and to the presence of matter described by a linear barotropic equation of state. In both cases we perform a detailed comparison of the predictions of the theory with the cosmological observational data, and with the standard 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} CDM model. By assuming that the presence of the Weyl geometric effects induce small perturbations in the homogeneous and isotropic cosmological background, and that the anisotropy parameter is small, the equations of the cosmological perturbations due to the presence of the Weyl geometric effects are derived. The time evolution of the metric and matter perturbations are explicitly obtained. Therefore, if Weyl geometric effects are present, the Universe would acquire some anisotropic characteristics, and its geometry will deviate from the standard FLRW one.

Approaching the structure of rotating bodies from dimension reduction

We show that the two-dimensional structure of a rigidly rotating self-gravitating body is accessible with relatively good precision by assuming a purely spheroidal stratification. With this hypothesis, the two-dimensional problem becomes one-dimensional, and consists in solving two coupled fixed-point equations in terms of equatorial mass density and eccentricity of isopycnics. We propose a simple algorithm of resolution based on the self-consistent field method. Compared to the full unconstrained-surface two-dimensional problem, the precision in the normalized enthalpy field is better than 10−3 in absolute, and the computing time is drastically reduced. In addition, this one-dimensional approach is fully appropriate to fast rotators, works for any density profile (including any barotropic equation of state), and can account for mass density jumps in the system, including the existence of an ambient pressure. Several tests are given.

Stability analysis of anisotropic stellar structures in Rastall theory of gravity utilizing cracking technique

In this manuscript, we employ the cracking technique in the Rastall gravity framework to assess how local density perturbations affect the stability of anisotropic stellar structures. To achieve this, we developed field equations for spherically symmetric spacetime with the charge inclusion in the context of Rastall gravity and then employed the krori Barua approach and barotropic equation of state while considering Reissner–Nordstrom metric as exterior spacetime, to develop the solution of the field equations. Further, we formulated the Tolman–Oppenheimer–Volkoff (TOV) equation by considering the non-conservation of energy–momentum tensor in the context of Rastall gravity. The stability of configuration is then disrupted by incorporating local density perturbations. Subsequently, our proposed approach is employed on the following compact stars: Her X-1, SAX J1808.4-3658, 4U 1820-30, PSR J1614-2230, Vela X-1, Cen X-3, and RXJ1856-37 to demonstrate its viability. The findings demonstrate that all stars under consideration experience cracking. This work particularly emphasizes the crucial role of cracking method in offering insightful information on the stability analysis of anisotropic stellar structures.

Emulating dark energy models with known equation of state via the created cold dark matter scenario

In this work we establish only at background dynamics level, the equivalence between the Created Cold Dark Matter (CCDM) model and diverse dark energy (DE) models. We find that for a barotropic or linear equation of state (EoS) for the DE, p=wρ, and standard matter sector the corresponding CCDM model is described by the same functional structure of the ratio of particle production, Γ. For a different EoS the functional form of the Γ term is not longer maintained, however, in the case of a polytropic EoS given by the Chaplygin gas the resulting Γ term can be written as the one obtained in the barotropic case under certain considerations.

Dynamics of the Isotropic Star Differential System from the Mathematical and Physical Point of Views

The following differential quadratic polynomial differential system dxdt=y−x, dydt=2y−yγ−12−γy−5γ−4γ−1x, when the parameter γ∈(1,2] models the structure equations of an isotropic star having a linear barotropic equation of state, being x=m(r)/r where m(r)≥0 is the mass inside the sphere of radius r of the star, y=4πr2ρ where ρ is the density of the star, and t=ln(r/R) where R is the radius of the star. First, we classify all the topologically non-equivalent phase portraits in the Poincaré disc of these quadratic polynomial differential systems for all values of the parameter γ∈R∖{1}. Second, using the information of the different phase portraits obtained we classify the possible limit values of m(r)/r and 4πr2ρ of an isotropic star when r decreases.

Stability of a class of solutions of the barotropic vorticity equation on a sphereequation on a sphere

Stability of a class of solutions of the barotropic vorticity equation on a sphereequation on a sphere

Boundary feedback control for hyperbolic systems

We are interested in the feedback stabilization of general linear multi-dimensional first order hyperbolic systems in ℝd. Using a Lyapunov function with a suited weight function depending on the system under consideration we show stabilization in L2 for the studied system using a suitable feedback control. Therefore the controllability of the studied system is related to the feasibility of an associated linear matrix inequality.We show the applicability discussing the barotropic Euler equations.

Thin-shell wormholes with AdS black holes surrounded by Chaplygin dark fluid

In this paper, we inspect the stable geometry of thin-shell wormholes in the background of static spherically-symmetric AdS black holes solution bounded by dark fluid and Chaplygin-like equation of state [Formula: see text], where [Formula: see text] and [Formula: see text] denote the pressure and energy density that are related through positive constant [Formula: see text]. For this purpose, we consider the two equivalent copies of black holes and connected them at the hypersurface through cut and paste approach. Then, we employ the linearized radial perturbation to discuss the stability of the developed wormhole geometry by assuming variable equations of state. We obtain the maximum stable configuration for massive black holes for both barotropic and Chaplygin variable equations of state while for higher values of [Formula: see text], stable regions decrease. It is found that for every selection of physical parameters of black hole geometry with low intensity of Chaplygin dark fluid, we get the stable configuration of thin-shell wormholes.

Compressibility-induced destabilisation of falling liquid films: an integral approach

We revisit the classical 2D problem of a gravity-driven liquid layer down an inclined plate (Kapitza, 1948), relaxing the usual assumption of homogeneous fluid. We set out to answer three major issues. When the fluid density is allowed to vary, (i) how does this feature structurally affect the formulation of a low-dimensional depth-averaged model? (ii) To what extent and (iii) by virtue of which physical mechanism does compressibility participate in the long-wave interfacial instability? To provide the relevant answers, (i) we first make use of a second-order asymptotic expansion in the shallowness parameter to develop a weakly-compressible boundary-layer system: starting from a two-equation momentum-integrated model, an additional barotropic equation of state is required for closure purposes. In this respect, (ii) a temporal linear stability analysis is performed: it is revealed that compressibility plays a destabilising role whose magnitude is enhanced at intermediately tilted configurations, and the more the Reynolds number approaches the critical threshold in the incompressible limit. (iii) We finally interpret the ensuing dispersion relation under the convenient framework of two-wave hierarchy, initiated by Whitham (1974): the primary instability gets promoted by the flow compressibility as it contributes to deceleration of dynamic waves most significantly in the low-inertia regime. Indeed, compressibility locally acts as a further boost to the inertia-based mechanism of Kapitza instability by amplifying flow-rate variations within the liquid film.

A convergent finite volume scheme for the stochastic barotropic compressible Euler equations

In this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure-valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system.