Compressible Fluid Equations Research Articles

Unique solvability of the initial boundary value problems for compressible viscous fluids

We study the Navier–Stokes equations for compressible barotropic fluids in a domain Ω⊂ R 3 . We first prove the local existence of the unique strong solution, provided the initial data satisfy a natural compatibility condition. The initial density needs not be bounded away from zero; it may vanish in an open subset ( vacuum) of Ω or decay at infinity when Ω is unbounded. We also prove a blow-up criterion for the local strong solution, which is new even for the case of positive initial densities. Finally, we prove that if the initial vacuum is not so irregular, then the compatibility condition of the initial data is necessary and sufficient to guarantee the existence of a unique strong solution.

Concentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids

Numerical simulations [2-D Riemann problem in gas dynamics and formation of spiral, in: Nonlinear Problems in Engineering and Science—Numerical and Analytical Approach (Beijing, 1991), Science Press, Beijing, 1992, pp. 167–179; Discrete Contin. Dyn. Syst. 1 (1995) 555–584; 6 (2000) 419–430] for the Euler equations for gas dynamics in the regime of small pressure showed that, for one case, the particles seem to be more sticky and tend to concentrate near some shock locations, and for the other case, in the region of rarefaction waves, the particles seem to be far apart and tend to form cavitation in the region. In this paper we identify and analyze the phenomena of concentration and cavitation by studying the vanishing pressure limit of solutions of the full Euler equations for nonisentropic compressible fluids with a scaled pressure. It is rigorously shown that any Riemann solution containing two shocks and possibly one-contact-discontinuity to the Euler equations for nonisentropic fluids tends to a δ-shock solution to the corresponding transport equations, and the intermediate densities between the two shocks tend to a weighted δ-measure that, along with the two shocks and possibly contact-discontinuity, forms the δ-shock as the pressure vanishes. By contrast, it is also shown that any Riemann solution containing two rarefaction waves and possibly one-contact-discontinuity to the Euler equations for nonisentropic fluids tends to a two-contact-discontinuity solution to the transport equations, and the nonvacuum intermediate states between the two rarefaction waves tend to a vacuum state as the pressure vanishes. Some numerical results exhibiting the processes of concentration and cavitation are presented as the pressure decreases.

Axisymmetric solutions of the 3D Navier–Stokes equations for compressible isentropic fluids

We prove the existence of global weak solutions to the Navier–Stokes equations for compressible isentropic fluids for any γ>1 when the Cauchy data are axisymmetric, where γ is the specific heat ratio. Moreover, we obtain a new integrability estimate of the density in any neighborhood of the symmetric axis (the singularity axis).

On the Cauchy Problem for the Equations of Ideal Compressible MHD Fluids with Radiation

We consider a system of balance laws describing the motion of an ionized compressible fluid interacting with magnetic fields and radiation effects. The local-in-time existence of a unique smooth solution for the Cauchy problem is proven. The proof follows from the method of successive approximations.

One-Dimensional Traffic Flow by Compressible Flow Model Including Several Types of Optimal Velocity Functions

In this paper, we considered a model which describes the one-dimensional traffic flow observed in a highway. Optimal velocity model which has been introduced in the microscopic model was applied to the macroscopic compressible fluid equations. Numerical simulations were performed by applying the finite difference method to this model. Effect of sensitivity which depend on the stability of asymptotic numerical solutions is discussed. Furthermore, we adopted several types of optimal velocity functions, the well-known optimal velocity function proposed by Bando et al., the linear one and the non-smooth one. Different non-trivial asymptotic structure of clusters of congestion are observed between the cases using two types of optimal velocity functions except the linear one.

Exponential stability and universal attractors for the Navier–Stokes equations of compressible fluids between two horizontal parallel plates in [formula omitted

In this paper we establish the exponential stability of solutions and existence of universal attractors for the Navier–Stokes equations of a compressible viscous heat-conductive fluid between two horizontal parallel plates in R 3 .

Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity

The Navier-Stokes equations for compressible barotropic fluid in 1D with the mass force under zero velocity boundary conditions are studied. We prove the uniform upper and lower bounds for the density ρ as well as the uniform in time L 2(Ω)-estimates for ρ x and u x (u is the velocity). Moreover, a collection of the decay rate estimates for ρ-ρ∞ (with ρ∞ being the stationary density) and u in 2(Ω)-norm and H 1(Ω)-norm as time t → ∞ are established. The results are given for general state function p(ρ) (but mainly monotone) and viscosity coefficient µ(ρ) of arbitrarily fast (or slow) growth as well as for the large data.

Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations

AbstractWe establish the existence and stability of multidimensional transonic shocks (hyperbolic‐elliptic shocks) for the Euler equations for steady compressible potential fluids in infinite cylinders. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for velocity, can be written as a second order nonlinear equation of mixed elliptic‐hyperbolic type for the velocity potential. The transonic shock problem in an infinite cylinder can be formulated into the following free boundary problem: The free boundary is the location of the multidimensional transonic shock which divides two regions of C1,α flow in the infinite cylinder, and the equation is hyperbolic in the upstream region where the C1,α perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem in unbounded domains. Our results indicate that there exists a solution of the free boundary problem such that the equation is always elliptic in the unbounded downstream region, the uniform velocity state at infinity in the downstream direction is uniquely determined by the given hyperbolic phase, and the free boundary is C1,α, provided that the hyperbolic phase is close in C1,α to a uniform flow. We further prove that, if the steady perturbation of the hyperbolic phase is C2,α, the free boundary is C2,α and stable under the steady perturbation. © 2003 Wiley Periodicals Inc.

Viscous potential flow

Potential flows vanishes, but the viscous contribution to the stress in an incompressible fluid (Stokes 1850) does not vanish in general. Here, we show how the viscosity of a viscous fluid in potential flow away from the boundary layers enters Prandtl's boundary layer equations. Potential flow equations for viscous compressible fluids are derived for sound waves which perturb the Navier–Stokes equations linearized on a state of rest. These linearized equations support a potential flow with the novel features that the Bernoulli equation and the potential as well as the stress depend on the viscosity. The effect of viscosity is to produce decay in time of spatially periodic waves or decay and growth in space of time-periodic waves.In all cases in which potential flows satisfy the Navier–Stokes equations, which includes all potential flows of incompressible fluids as well as potential flows in the acoustic approximation derived here, it is neither necessary nor useful to put the viscosity to zero.

Strong solutions of the Navier–Stokes equations for isentropic compressible fluids

We study strong solutions of the isentropic compressible Navier–Stokes equations in a domain Ω⊂ R 3 . We first prove the local existence of unique strong solutions provided that the initial data ρ 0 and u 0 satisfy a natural compatibility condition. The important point in this paper is that we allow the initial vacuum: the initial density may vanish in an open subset of Ω. We then prove a new uniqueness result and stability result. Our results are valid for unbounded domains as well as bounded ones.

DYNAMIC PROCESSES IN THE OIL PIPELINES. PART II. THE LEAK IDENTIFICATION BY NUMERICAL METHODS

The analysis of identification method of leaks occurring due to damages of the linear part of the main oil pipelines is carried out. The momentum and continuity equations of viscous compressible fluid in a pressure pipeline are presented. Differential equations of fluid movement in the oil pipeline are solved by the method of characteristics. Variation diagrams of pressure and velocity of fluid of a leaking oil pipeline are presented.

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.

Global Entropy Solutions in <i>L</i><sup>infinity</sup> to the Euler Equations and Euler-Poisson Equations for Isothermal Fluids with Spherical Symmetry

We prove the existence of global entropy solutions in Linfinity to the multidimensional Euler equations and Euler-Poisson equations for compressible isothermal fluids with spherically symmetric initial data that allows vacuum and unbounded velocity outside a solid ball. The multidimensional existence problem can be reduced to the existence problem for the one-dimensional Euler equations and Euler-Poisson equations with geometrical source terms. Due to the presence of the geometrical source terms, new variables- weighted density and momentum-are first introduced to transform the nonlinear system into a new nonlinear hyperbolic system to reduce the geometric source effect. We then develop a shock capturing scheme of Lax-Friedrichs type to construct approximate solutions for the weighted density and momentum. Since the velocity may be unbounded, the Courant-Friedrichs-Lewy stability condition may fail for the standard fractional-step Lax-Friedrichs scheme; hence we introduce a cut-off technique to modify the approximate density functions and adjust the ratio of the space and time mesh sizes to construct our approximate solutions. Finally we establish the convergence and consistency of the approximate solutions using the method of compensated compactness and obtain global entropy solutions in Linfinity. The solutions we obtain allow unbounded velocity near vacuum, one of the essential difficulties here, which is different from the isentropic case.

On Large-Time Behavior of Solutions to the Compressible Navier-Stokes Equations in the Half Space in R 3

The Navier-Stokes equation for compressible viscous fluid is considered on the half space in R3 under the zero-Dirichlet boundary condition for the momentum with initial data near an arbitrarily given equilibrium of positive constant density and zero momentum. Time decay properties in L2 norms for solutions of the linearized problem are investigated to obtain the rate of convergence in L2 norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in \(\). Some lower bounds are derived for solutions to the linearized problem, one of which indicates a nonlinear phenomenon not appearing in the case of the Cauchy problem on the whole space.

Averaged Lagrangians and the mean effects of fluctuations in ideal fluid dynamics

We begin by placing the generalized Lagrangian mean (GLM) equations for a compressible adiabatic fluid into the Euler–Poincaré (EP) variational framework of fluid dynamics, for an averaged Lagrangian. We then state the EP Averaging Result—that GLM equations arise from GLM Hamilton’s principles in the EP framework. Next, we derive a new set of approximate small-amplitude GLM equations ( gℓ m equations) at second order in the fluctuating displacement of a Lagrangian trajectory from its mean position. These equations express the linear and nonlinear back-reaction effects on the Eulerian mean fluid quantities by the fluctuating displacements of the Lagrangian trajectories in terms of their Eulerian second moments. The derivation of the gℓ m equations uses the linearized relations between Eulerian and Lagrangian fluctuations, in the tradition of Lagrangian stability analysis for fluids. The gℓ m derivation also uses the method of averaged Lagrangians, in the tradition of wave, mean flow interaction (WMFI). The gℓ m EP motion equations for compressible and incompressible ideal fluids are compared with the Euler-alpha turbulence closure equations. An alpha model is a GLM (or gℓ m) fluid theory with a Taylor hypothesis closure (THC). Such closures are based on the linearized fluctuation relations that determine the dynamics of the Lagrangian statistical quantities in the Euler-alpha closure equations. We use the EP Averaging Result to bridge between the GLM equations and the Euler-alpha closure equations. Hence, combining the small-amplitude approximation with THC yields in new gℓm turbulence closure equations for compressible fluids in the EP variational framework.

Convective heat transport in compressible fluids.

We present hydrodynamic equations of compressible fluids in gravity as a generalization of those in the Boussinesq approximation used for nearly incompressible fluids. They account for adiabatic processes taking place throughout the cell (the piston effect) and those taking place within plumes (the adiabatic temperature gradient effect). Performing two-dimensional numerical analysis, we reveal some unique features of plume generation and convection in transient and steady states of compressible fluids. As the critical point is approached, the overall temperature changes induced by plume arrivals at the boundary walls are amplified, giving rise to overshoot behavior in transient states and significant noise in the temperature in steady states. The velocity field is suggested to assume a logarithmic profile within boundary layers. Random reversal of macroscopic shear flow is examined in a cell with unit aspect ratio. We also present a simple scaling theory for moderate Rayleigh numbers.

Life span of 2‐D irrotational compressible fluids in the halfplane

AbstractWe consider the Euler equations of barotropic inviscid compressible fluids in the half plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension two such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial velocity. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small.We consider smooth irrotational solutions. First, we study the life span, i.e. the largest time interval T(ε) of existence of classical solutions, when the initial data are a small perturbation of size εfrom a constant state. For the proof of this result we use a combination of energy and decay estimates. Then, the estimate of the life span allows to show, by a suitable scaling of variables, the existence of irrotational solutions on any arbitrary time interval, for any small enough Mach number. Copyright © 2002 John Wiley & Sons, Ltd.

On Slightly Compressible Ideal Flow¶in the Half-Plane

We consider the Euler equations of barotropic inviscid compressible fluids in the half-plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In 2D (two dimensions) such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small.

On local-in-time existence for the Dirichlet problem for equations of compressible viscous fluids

The local existence of solutions for the compressible Navier–Stokes equations with the Dirichlet boundary conditions in the $L_p$-framework is proved. Next an almost-global-in-time existence of small solutions is shown. The considerations are made i

Remarks on weak solutions to the Navier-Stokes equations for 2-D compressible isothermal fluids with spherically symmetric initial data

We prove the existence of global weak solutions to the Cauchy problem for the Navier-Stokes equations of 2-dimensional compressible isothermal fluids when ρ 0 and m 0 are spherically symmetric, ρ 0 ∈ L 1 n L M , and m 0 /ρ 0 ∈ L 2 , where ρ 0 and m 0 are the initial density and momentum respectively, L M is the Orlicz space over R 2 with M = M(s) = (1 + s )ln(1 + s)-s. The proof is based on a uniform L 2 ([0, T], L∞ loc (R 2 )) estimate of the velocity in the spherically symmetrical approximate solutions and a compactness lemma which gives a compactness result concerning H n/2 and L M in R n .