Isentropic Equations Research Articles

A lattice Boltzmann model for two‐dimensional sound wave in the small perturbation compressible flows

AbstractA multi‐entropy‐level lattice Boltzmann model for two‐dimensional sound wave equation in the small perturbation flows is presented. In this model, we used higher‐order moment method, multi‐scale technique and the Chapman–Enskog expansion, and multi‐entropy‐level to obtain sound wave equation with isentropic equation. As numerical examples, the Doppler effects in the sound wave propagation, the sound scattering from circular cylinder are simulated. The numerical results show that this model can be used to simulate sound wave propagation. Copyright © 2010 John Wiley & Sons, Ltd.

A blow-up criterion for 3D non-resistive compressible magnetohydrodynamic equations with initial vacuum

In this paper, we prove a blow-up criterion of strong solutions to the 3D viscous and non-resistive isentropic compressible magnetohydrodynamic equations with initial vacuum. This blow-up criterion depends only on the gradient of velocity, which is analogous to the one for the compressible Navier–Stokes equations (cf. Huang et al. (2010) [40]).

Interface behavior of compressible navier-stokes equations with discontinuous boundary conditions and vacum

In this paper, we study a one-dimensional motion of viscous gas near vacuum. We are interested in the case that the gas is in contact with the vacuum at a finite interval. This is a free boundary problem for the one-dimensional isentropic Navier-Stokes equations, and the free boundaries are the interfaces separating the gas from vacuum, across which the density changes discontinuosly. Smoothness of the solutions and the uniqueness of the weak solutions are also discussed. The present paper extends results in Luo-Xin-Yang [12] to the jump boundary conditions case.

The limits of Riemann solutions to the isentropic magnetogasdynamics

The objective of this note is to prove that the Riemann solutions of the isentropic magnetogasdynamics equations converge to the corresponding Riemann solutions of the transport equations by letting both the pressure and the magnetic field vanish. The delta shock wave can be obtained as the limit of two shock waves and the vacuum state can be obtained as the limit of two rarefaction waves. Moreover the relation between the speed of formation of singular density and those of the vanishing pressure and the vanishing magnetic field is discussed in detail.

Nonlinear evolution systems and green's function

In this paper, we will introduce how to apply Green's function method to get the pointwise estimates for the solutions of Cauchy problem of nonlinear evolution equations with dissipative structure. First of all, we introduce the pointwise estimates of the time-asymptotic shape of the solutions of the isentropic Navier-Stokes equations and show to exhibit the generalized Huygen's principle. Then, for other nonlinear dissipative evolution equations, we will only introduce the result and give some brief explanations. Our approach is based on the detailed analysis of the Green's function of the linearized system and micro-local analysis, such as frequency decomposition and so on.

Local absorbent boundary condition for non-linear hyperbolic problems with unknown Riemann invariants

Generally, in problems where the Riemann invariants (RI) are known (e.g. the flow in a shallow rectangular channel, the isentropic gas flow equations), the imposition of non-reflective boundary conditions is straightforward. In problems where Riemann invariants are unknown (e.g. the flow in non-rectangular channels, the stratified 2D shallow water flows) it is possible to impose that kind of conditions analyzing the projection of the Jacobians of advective flux functions onto normal directions of fictitious surfaces or boundaries. In this paper a general methodology for developing absorbing boundary conditions for non-linear hyperbolic advective–diffusive equations with unknown Riemann invariants is presented. The advantage of the method is that it is very easy to implement in a finite element code and is based on computing the advective flux functions (and their Jacobian projections), and then, imposing non-linear constraints via Lagrange multipliers. The application of the dynamic absorbing boundary conditions to typical wave propagation problems with unknown Riemann invariants, like non-linear Saint-Venant system of conservation laws for non-rectangular and non-prismatic 1D channels and stratified 1D/2D shallow water equations, is presented. Also, the new absorbent/dynamic condition can handle automatically the change of Jacobians structure when the flow regime changes from subcritical to supercritical and viceversa, or when recirculating zones are present in regions near fictitious walls.

On the blow up criterion for the 2-D compressible Navier-Stokes equations

Motivated by [10], we prove that the upper bound of the density function j9 controls the finite time blow up of the classical solutions to the 2-D compressible isentropic Navier-Stokes equations. This result generalizes the corresponding result in [3] concerning the regularities to the weak solutions of the 2-D compressible Navier-Stokes equations in the periodic domain.

Existence of global solutions to isentropic gas dynamics equations with a source term

In this paper we prove existence of isentropic gas dynamic equations with a source term (1.2). To this end we construct a sequence of regular hyperbolic systems (1.1) to approximate the inhomogeneous system of isentropic gas dynamics (1.2). First, for each fixed approximation parameter δ and very general condition on P(ρ), we establish the existence of entropy solutions for the Cauchy problem (1.1) with bounded initial date (1.4). Second, letting e = o(δ), we obtain a complete proof of the H loc −1 compactness of weak entropy pairs of system (1.2) in the form η(ρ, u) = ρH(ρ, u) given in Chen-LeFloch (2003). Finally, for the conditions of P(ρ) given in Chen-LeFloch (2003), applied to the results in Theorems 1 and 2, we obtain the global existence of entropy solutions for the Cauchy problem (1.2) with bounded initial date (1.4).

Energy Transformation and Diabatic Processes in Developing and Nondeveloping African Easterly Waves Observed during the NAMMA Project of 2006

Abstract This paper provides an understanding of essential differences between developing and nondeveloping African easterly waves, which was a major goal of NAMMA, NASA’s field program in the eastern Atlantic, which functioned as an extension of the African Monsoon Multidisciplinary Analysis (AMMA) program during 2006. Three NAMMA waves are studied in detail using FNL analysis: NAMMA wave 2, which developed into Tropical Storm Debby; NAMMA wave 7, which developed into Hurricane Helene; and NAMMA wave 4, which did not develop within the NAMMA domain. Diagnostic calculations are performed on the analyzed fields using energy transformation equations and the isentropic potential vorticity equation. The results show that the two developing waves possess clear and robust positive barotropic energy conversion in conjunction with positive diabatic heating that includes a singular burst of heating at a particular time in the wave’s history. This positive barotropic energy conversion is facilitated in waves that have a northeast–southwest tilt to the trough axis and a wind maximum to the west of this axis. The nondeveloping wave is found to have the same singular burst of diabatic heating at one point in its history, but development of the wave does not occur due to negative barotropic energy conversion. Such conversion is facilitated by a northwest–southeast tilt to the trough axis and a wind maximum to the east of this axis. The conclusions about wave development and nondevelopment formulated in this research are viewed as important and significant, but they require additional testing with detailed observational- and numerical-based studies.

Uniform boundedness of the radially symmetric solutions of the Navier-Stokes equations for isentropic compressible fluids

We study the isentropic compressible Navier-Stokes equations with radially symmetric data and non-negative initial density in an annular domain. We prove the global existence of strong solutions for any $\gamma\geq 1$. Moreover, we obtain the uniform in time $L^{\infty}$-boundedness of the density and $H^{1}$-boundedness of the velocity, improving therefore the corresponding result in [2], where the condition $\gamma\geq 2$ is required to guarantee the existence.

Riemann-invariant solutions of the isentropic fluid flow equations

We use a new version of the conditional symmetry method to obtain rank-k solutions expressed in terms of Riemann invariants of the isentropic compressible ideal fluid flow in 3+1 dimensions. We describe the procedure for constructing bounded solutions in terms of the elliptic Weierstrass p-function in detail.

Structural conditions for full MHD equations

In this paper, we investigate the characteristic structure of the full equations of magnetohydrodynamics (MHD) and show that it satisfies the hypotheses of a general variable-multiplicity stability framework introduced by Métivier and Zumbrun, thereby extending to the general case various results obtained by Métivier and Zumbrun for the isentropic equations of MHD.

Global solutions to one-dimensional compressible Navier–Stokes–Poisson equations with density-dependent viscosity

In this paper, we prove the global existence of weak solutions to one-dimensional compressible isentropic Navier–Stokes–Poisson equations with density-dependent viscosity and free boundaries. The initial density ρ0∊W1,2n is bounded below by a positive constant, and the initial velocity u0∊L2n. In contrast to Jiang et al. [“Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity,” Methods Appl. Anal. 12, 239 (2005)], the Sobolev exponent n is less in this paper, and the viscosity coefficient μ=μ(ρ) is a general function of ρ including the cases μ(ρ)=c0ρθ (0<θ<1) and μ(ρ)=c0, where c0 is a positive constant.

Lattice QCD thermodynamic results with improved staggered fermions

We present results on the QCD equation of state, obtained with two different improved dynamical staggered fermion actions and almost physical quark masses. Lattice cut-off effects are discussed in detail as results for three different lattice spacings are available now, i.e. results have been obtained on lattices with temporal extent of N τ =4,6 and 8. Furthermore we discuss the Taylor expansion approach to non-zero baryon chemical potential and present the isentropic equation of state on lines of constant entropy per baryon number.

Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms

In this paper, we consider the local existence and a blow-up criterion for smooth solutions to the 2-D isentropic compressible Boussinesq equations, and obtain some new commutator estimates. In particular, we show that the time integral of the spatial maximum of gradient of velocity controls the breakdown of smooth solutions.

Low Mach Number Limit of Viscous Compressible Magnetohydrodynamic Flows

The relationship between the compressible magnetohydrodynamic flows with low Mach number and the incompressible magnetohydrodynamic flows is investigated. More precisely, the convergence of weak solutions of the compressible isentropic viscous magnetohydrodynamic equations to the weak solutions of the incompressible viscous magnetohydrodynamic equations is proved as the density becomes constant and the Mach number goes to zero; that is, the corresponding incompressible limits are justified when the spatial domain is a periodic domain, the whole space, or a bounded domain.

L1 Stability of Spatially Periodic Solutions in Relativistic Gas Dynamics

This paper proves the well posedness of spatially periodic solutions of the relativistic isentropic gas dynamics equations. The pressure is given by a γ-law with initial data of large amplitude, provided γ − 1 is sufficiently small. As a byproduct of our techniques, we obtain the same results for the classical case. At the limit c → + ∞, the solutions of the relativistic system converge to the solutions of the classical one, the convergence rate being 1/c2. We also construct the semigroup of solutions of the Cauchy problem for initial data with bounded total variation, which can be large, as long as γ − 1 is small.

Nonlinear Dynamical Stability of Newtonian Rotating and Non-rotating White Dwarfs and Rotating Supermassive Stars

We prove general nonlinear stability and existence theorems for rotating star solutions which are axi-symmetric steady-state solutions of the compressible isentropic Euler-Poisson equations in 3 spatial dimensions. We apply our results to rotating and non-rotating white dwarf, and rotating high density supermassive (extreme relativistic) stars, stars which are in convective equilibrium and have uniform chemical composition. This paper is a continuation of our earlier work ([28]).

Stability of polytropes

This paper is an investigation of the stability of some ideal stars. It is intended as a study in general relativity, with emphasis on the coupling to matter, aimed at a better understanding of strong gravitational fields and ``black holes.'' This contrasts with the usual attitude in astrophysics, where Einstein's equations are invoked as a refinement of classical thermodynamics and Newtonian gravity. Our work is based on action principles for systems of metric and matter fields, well-defined relativistic field models that we hope may represent plausible types of matter. The thermodynamic content must be extracted from the theory itself. When the flow of matter is irrotational, and described by a scalar density, we are led to differential equations that differ little from those of Tolman, but they admit a conserved current, and stronger boundary conditions that affect the matching of the interior solution to an external metric and imply a relation of mass and radius. We propose a complete revision of the treatment of boundary conditions. An ideal star in our terminology has spherical symmetry and an isentropic equation of state, $p=a{\ensuremath{\rho}}^{\ensuremath{\gamma}}$, $a$ and $\ensuremath{\gamma}$ piecewise constant. In our first work it was assumed that the density vanished beyond a finite distance from the origin and that the metric is to be matched at the boundary to an exterior Schwartzchild metric. But it is difficult to decide what the boundary conditions should be and we are consequently skeptical of the concept of a fixed boundary. We investigate the double polytrope, characterized by a polytropic index $n\ensuremath{\le}3$, in the bulk of the star and a value larger than five in an outer atmosphere that extends to infinity. It has no fixed boundary but a region of critical density where the polytropic index changes from a value that is appropriate for the bulk of the star to a value that provides a crude model for the atmosphere. The boundary conditions are now natural and unambiguous. The existence of a relation between mass and radius is confirmed, as well as an upper limit on the mass. The principal conclusion is that all the static configurations are stable. There is a solution that fits the Sun. The masses of white dwarfs respect the Chandrasekhar limit. The application to neutron stars has surprising aspects.

On the rotational compressible Taylor flow in injection-driven porous chambers

This work considers the compressible flow field established in a rectangular porous channel. Our treatment is based on a Rayleigh–Janzen perturbation applied to the inviscid steady two-dimensional isentropic flow equations. Closed-form expressions are then derived for the main properties of interest. Our analytical results are verified via numerical simulation, with laminar and turbulent models, and with available experimental data. They are also compared to existing one-dimensional theory and to a previous numerical pseudo-one-dimensional approach. Our analysis captures the steepening of the velocity profiles that has been reported in several studies using either computational or experimental approaches. Finally, explicit criteria are presented to quantify the effects of compressibility in two-dimensional injection-driven chambers such as those used to model slab rocket motors.