Isentropic Equations Research Articles

Incompressible limit and stability of all-time solutions to 3-D full Navier-Stokes equations for perfect gases

This paper studies the incompressible limit and stability of global strong solutions to the three-dimensional full compressible Navier-Stokes equations, where the initial data satisfy the “well-prepared” conditions and the velocity field and temperature enjoy the slip boundary condition and convective boundary condition, respectively. The uniform estimates with respect to both the Mach number \(\varepsilon \in \left( {0,\overline \varepsilon } \right]'\) and time t ∈ [0,∞) are established by deriving a differential inequality with decay property, where \(\overline \varepsilon \in \left( {0,1} \right]\) is a constant. As the Mach number vanishes, the global solution to full compressible Navier-Stokes equations converges to the one of isentropic incompressible Navier-Stokes equations in t ∈ [0,+∞). Moreover, we prove the exponentially asymptotic stability for the global solutions of both the compressible system and its limiting incompressible system.

Hadron–quark crossover and hot neutron stars at birth

We construct a new isentropic equation of state (EOS) at finite temperature "CRover" on the basis of the hadron-quark crossover at high density. By using the new EOS, we study the structure of hot neutron stars at birth with the typical lepton fraction ($Y_l=0.3-0.4$) and the typical entropy per baryon ($S=1-2$). Due to the gradual appearance of quark degrees of freedom at high density, the temperature T and the baryon density at the center of the hot neutron stars with the hadron-quark crossover are found to be smaller than those without the crossover by a factor of 2 or more. Typical energy release due to the contraction of a hot neutron star to a cold neutron star with 1.4 solarmass is shown to be about 0.04 solarmass with the spin-up rate about 14%.

A lumped model of venting during thermal runaway in a cylindrical Lithium Cobalt Oxide lithium-ion cell

This paper presents a mathematical model built for analyzing the intricate thermal behavior of a 18650 LCO (Lithium Cobalt Oxide) battery cell during thermal runaway when venting of the electrolyte and contents of the jelly roll (ejecta) is considered. The model consists of different ODEs (Ordinary Differential Equations) describing reaction rates and electrochemical reactions, as well as the isentropic flow equations for describing electrolyte venting. The results are validated against experimental findings from Golubkov et al. [1] [Andrey W. Golubkov, David Fuchs, Julian Wagner, Helmar Wiltsche, Christoph Stangl, Gisela Fauler, Gernot Voitice Alexander Thaler and Viktor Hacker, RSC Advances, 4:3633–3642, 2014] for two cases - with flow and without flow. The results show that if the isentropic flow equations are not included in the model, the thermal runaway is triggered prematurely at the point where venting should occur. This shows that the heat dissipation due to ejection of electrolyte and jelly roll contents has a significant contribution. When the flow equations are included, the model shows good agreement with the experiment and therefore proving the importance of including venting.

Stability of viscous shock waves for the one-dimensional compressible navier-stokes equations with density-dependent viscosity

We study the large-time behavior toward viscous shock waves to the Cauchy problem of the one-dimensional compressible isentropic Navier-Stokes equations with density-dependent viscosity. The nonlinear stability of the viscous shock waves is shown for certain class of large initial perturbation with integral zero which can allow the initial density to have large oscillation. Our analysis relies upon the technique developed by Kanel' and the continuation argument.

Free boundary value problem for the cylindrically symmetric compressible navier-stokes equations with density-dependent viscosity

In this paper, we investigate the free boundary value problem (FBVP) for the cylindrically symmetric isentropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, we prove that there exists a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.

Linear isentropic Equation of State in formation of Black hole and Naked singularity

Linear isentropic Equation of State in formation of Black hole and Naked singularity

On the behavior of boundary layers of one-dimensional isentropic planar MHD equations with vanishing shear viscosity limit

In this paper, we consider an initial-boundary value problem for the one-dimensional equations of planar isentropic magnetohydrodynamics (MHD). The vanishing shear viscosity limit is justified. More important, both the thickness and the behavior of boundary layers are described. The proofs of these results are based on the full use of the material derivative, the “effective viscous flux”, and the mathematical structure of the equations.

On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

In this paper we present a sufficient condition for the global well-posedness of classical solutions to an initial value problem of compressible isentropic Navier-Stokes equations in the whole space $\mathbb{R}^3$. As an immediate result, the main theorem obtained implies that the Cauchy problem of compressible Navier-Stokes equations with vacuum has a global unique classical solution, provided the initial energy is sufficiently small, or the shear viscosity coefficient is sufficiently large, or the upper bound of the initial density is suitably small and the adiabatic exponent $\gamma\in (1,3/2)$. These results particularly extend the recent ones due to Huang-Li-Xin [7], where the global well-posedness of classical solutions with small initial energy was established.

The comparison of the Riemann solutions in gas dynamics

In this paper, we compare two Riemann solution constructions of isentropic and non-isentropic gas dynamic equations, with the same initial values. We obtain that when rarefaction waves occur in the non-isentropic solution, the same family rarefaction waves occur in the isentropic solution, however, when advancing or reflected shock waves occur in the isentropic case, the same kind shock waves occur in the non-isentropic case. Moreover, by numerical calculation, it can be found that the wave intensities of the two solutions are quite close. Our conclusion reflects that almost non-isentropic gas dynamic process can be simulated by isentropic gas dynamic process in practice.

Magnetogenesis through convection in barotropic fluids

It is shown that an unmagnetized plasma with non-uniform bulk velocity can generate magnetic fields through consideration of the non-relativistic isentropic two-fluid equations, even when the initial conditions contain with no fields or currents, uniform densities and pressures, and a divergence-free bulk velocity. This effect does not depend on the baroclinicity of the plasma and is therefore relevant even in barotropic flows, where the Biermann battery is absent. It also does not rely on kinetic effects or shear discontinuities. Instead, our magnetogenesis effect arises from convection terms proportional to the electron mass in the generalized Ohm's law. The resulting magnetic fields are typically weak but may still serve as seed fields for dynamo mechanisms.

Global classical solutions of 3D isentropic compressible MHD with general initial data

In this paper, we consider the Cauchy problem of three-dimensional isentropic compressible magnetohydrodynamic equations with general initial data which could be either vacuum or non-vacuum under the assumption that the viscosity coefficient μ is large enough.

Incompressible limit of full compressible magnetohydrodynamic equations with well-prepared data in 3-D bounded domains

This paper studies the singular limits of the non-isentropic compressible magnetohydrodynamic equations for viscous and heat-conductive ideal polytropic flows with magnetic diffusions in a three-dimensional bounded domain as the Mach number goes to zero. Provided that the initial data are well-prepared, we establish the uniform estimates with respect to the Mach number, which gives the convergence from the full compressible magnetohydrodynamic equations to isentropic incompressible magnetohydrodynamic equations.

Variability of stratospheric mean age of air and of the local effects of residual circulation and eddy mixing

AbstractWe analyze the variability of mean age of air (AoA) and of the local effects of the stratospheric residual circulation and eddy mixing on AoA within the framework of the isentropic zonal mean continuity equation. AoA for the period 1988–2013 has been simulated with the Lagrangian chemistry transport model CLaMS driven by ERA‐Interim winds and diabatic heating rates. Model simulated AoA in the lower stratosphere shows good agreement with both in situ observations and satellite observations from Michelson Interferometer for Passive Atmospheric Sounding, even regarding interannual variability and changes during the last decade. The interannual variability throughout the lower stratosphere is largely affected by the quasi‐biennial‐oscillation‐induced circulation and mixing anomalies, with year‐to‐year AoA changes of about 0.5 years. The decadal 2002–2012 change shows decreasing AoA in the lowest stratosphere, below about 450 K. Above, AoA increases in the Northern Hemisphere and decreases in the Southern Hemisphere. Mixing appears to be crucial for understanding AoA variability, with local AoA changes resulting from a close balance between residual circulation and mixing effects. Locally, mixing increases AoA at low latitudes (40°S–40°N) and decreases AoA at higher latitudes. Strongest mixing occurs below about 500 K, consistent with the separation between shallow and deep circulation branches. The effect of mixing integrated along the air parcel path, however, significantly increases AoA globally, except in the polar lower stratosphere. Changes of local effects of residual circulation and mixing during the last decade are supportive of a strengthening shallow circulation branch in the lowest stratosphere and a southward shifting circulation pattern above.

On regular solutions of the $3$D compressible isentropic Euler-Boltzmann equations with vacuum

In this paper, we discuss the Cauchy problem for the compressible isentropic Euler-Boltzmann equations with vacuum in radiation hydrodynamics. We establish the existence of a unique local regular solution with vacuum by the theory of quasi-linear symmetric hyperbolic systems and some techniques dealing with the complexity caused by the coupling between fluid and radiation field under some physical assumptions for the radiation quantities. Moreover, it is interesting to show the non-global existence of regular solutions caused by the effect of vacuum for polytropic gases with adiabatic exponent $1<\gamma\leq 3$ via some observations on the propagation of the radiation field. Compared with [11][15][20], some new initial conditions that will lead to the finite time blow-up for classical solutions have been introduced. These blow-up results tell us that the radiation effect on the fluid is not strong enough to prevent the formation of singularities caused by the appearance of vacuum.

On Classical Solutions of the Compressible Magnetohydrodynamic Equations with Vacuum

In this paper, we consider the three-dimensional compressible isentropic magnetohydrodynamic equations with infinite electric conductivity. The existence of unique local classical solutions is established when the initial data are arbitrarily large, contain vacuum, and satisfy some initial layer compatibility condition. The initial mass density need not be bounded away from zero and may vanish in some open set or decay at infinity. Moreover, we prove that the $L^\infty$ norm of the deformation tensor of velocity gradients controls the possible blow-up (see [O. Rozanova, “Blow Up of Smooth Solutions to the Barotropic Compressible Magnetohydrodynamic Equations with Finite Mass and Energy,” in Hyperbolic Problems: Theory, Numerics and Applications, AMS, Providence, RI, 2009, pp. 911--917], [Z. Xin, Comm. Pure Appl. Math., 51 (1998), pp. 229--240]) for classical (or strong) solutions, which means that if a solution of the compressible magnetohydrodynamic equations is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of the deformation tensor as the critical time approaches. Our criterion (see (1.12)) is the same as Ponce's criterion for three-dimensional incompressible Euler equations [G. Ponce, Comm. Math. Phys., 98 (1985), pp. 349--353] and Huang, Li, and Xin's criterion for the three-dimensional compressible Navier--Stokes equations [X. Huang, J. Li, and Z. Xin, Comm. Math. Phys., 301 (2011), pp. 23--35].

Structural Stability of Supersonic Contact Discontinuities in Three-Dimensional Compressible Steady Flows

In this paper, we study the structurally nonlinear stability of supersonic contact discontinuities in three-dimensional compressible isentropic steady flows. Based on the weakly linear stability result and the $L^2$-estimates obtained in [Y.-G. Wang and F. Yu, J. Differential Equations, 255 (2013), pp. 1278--1356] for the linearized problems of three-dimensional compressible isentropic steady equations at a supersonic contact discontinuity satisfying certain stability conditions, we first derive tame estimates of solutions to the linearized problem in higher order norms by exploring the behavior of vorticities. Since the supersonic contact discontinuities are only weakly linearly stable, so the tame estimates of solutions to the linearized problems have loss of regularity with respect to both background states and initial data. Thus, to use the tame estimates to study the nonlinear problem we adapt the Nash--Moser--Hörmander iteration scheme to conclude that supersonic contact discontinuities in three-dimensional compressible steady flows satisfying the stability conditions of Wang and Yu are structurally nonlinearly stable at least locally in space.

Vanishing viscosity of isentropic Navier-Stokes equations for interacting shocks

We study the vanishing viscosity of the Navier-Stokes equations for interacting shocks. Given an entropy solution to p -system which consists of two different families of shocks interacting at some positive time, we show that such entropy solution is the vanishing viscosity limit of a family of global smooth solutions to the isentropic Navier-Stokes equations. The key point of the proofs is to derive the estimates separately before and after the interaction time and connect the incoming and outgoing viscous shock profiles.

On local structural stability of one-dimensional shocks in radiation hydrodynamics

In this paper, we are concerned with the local structural stability of one-dimensional shock waves in radiation hydrodynamics described by the isentropic Euler-Boltzmann equations. Even though in this radiation hydrodynamics model, the radiative effects can be understood as source terms to the isentropic Euler equations of hydrodynamics, in general the radiation field has singularities propagated in an angular domain issuing from the initial point across which the density is discontinuous. This is the major difficulty in the stability analysis of shocks. Under certain assumptions on the radiation parameters, we show there exists a local weak solution to the initial value problem of the one dimensional Euler-Boltzmann equations, in which the radiation intensity is continuous, while the density and velocity are piecewise Lipschitz continuous with a strong discontinuity representing the shock-front. The existence of such a solution indicates that shock waves are structurally stable, at least local in time, in radiation hydrodynamics.

Separation of acoustic waves in isentropic flow perturbations

The present contribution investigates the mechanisms of sound generation and propagation in the case of highly-unsteady flows. It is based on the linearisation of the isentropic Navier-Stokes equation around a new pathline-averaged base flow. As a consequence of this unsteady and non-radiating base flow, the perturbation equations satisfy a conservation law. It is demonstrated that this flow perturbations can be split into acoustic and vorticity modes, with the acoustic modes being independent of the vorticity modes. Moreover, we conclude that the present acoustic perturbation is propagated by the convective wave equation and fulfills Lighthill's acoustic analogy. Therefore, we can define the deviations from the convective wave equation as the “true” sound sources. In contrast to other authors, no assumptions on a slowly varying or irrotational flow are necessary.

Blow-up of smooth solutions to the compressible magnetohydrodynamic flows

In this paper, we prove the blow-up phenomena of smooth solutions to the Cauchy problem for the full compressible magnetohydrodynamic equations and isentropic compressible magnetohydrodynamic equations with constant and degenerate viscosities under some restrictions on the initial data. In particular, our results do not require that the initial data have compact support or contain vacuum in any finite region.