Compressible Barotropic Fluid Research Articles

Global wellposedness of the 3D compressible Navier–Stokes equations with free surface in the maximal regularity class

This paper concerns the global well posedness issue of the compressible Navier–Stokes equations (CNS) describing barotropic compressible fluid flow with free surface occupied in the three dimensional exterior domain. Combining the maximal L p -L q estimate and the L p -L q decay estimate of solutions to the linearized equations, we prove the unique existence of global in time solutions in the time weighted maximal L p -L q regularity class for some p > 2 and q > 3. Namely, the solution is bounded as L p in time and L q in space. Compared with the previous results of the free boundary value problem of (CNS) in unbounded domains, we relax the regularity assumption on the initial states, which is the advantage by using the maximal L p -L q regularity framework. On the other hand, the equilibrium state of the moving boundary of the exterior domain is not necessary the sphere. To our knowledge, this paper is the first result on the long time solvability of the free boundary value problem of (CNS) in the exterior domain.

Global Well-posedness of the free boundary problem for the compressible Euler equations with damping and gravity

We consider the free boundary problem for a compressible barotropic fluid lying above a rigid bottom and below the air, in a horizontally periodic setting. The fluid dynamics is governed by the compressible Euler equations with damping and gravity, and the effect of surface tension is neglected on the upper free surface. We prove the global well-posedness of the problem for the initial data around the equilibrium and that the solution decays to the equilibrium at an almost exponential rate. The difficulties caused by the less regularity of the free surface and the troublesome linear effects induced by the nontrivial equilibrium density are overcome by using certain good unknowns.

On numerical approximations to fluid–structure interactions involving compressible fluids

In this paper we introduce a numerical scheme for fluid–structure interaction problems in two or three space dimensions. A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions.

Vanishing viscosity limit of the three-dimensional barotropic compressible Navier–Stokes equations with degenerate viscosities and far-field vacuum

We are concerned with the inviscid limit of the Navier–Stokes equations to the Euler equations for barotropic compressible fluids in \mathbb{R}^3 . When the viscosity coefficients obey a lower power law of the density (i.e., \rho^\delta with 0<\delta<1 ), we identify a quasi-symmetric hyperbolic–singular elliptic coupled structure of the Navier–Stokes equations to control the behavior of the velocity of the fluids near a vacuum. Then this structure is employed to prove that there exists a unique regular solution to the corresponding Cauchy problem with arbitrarily large initial data and far-field vacuum, whose life span is uniformly positive in the vanishing viscosity limit. Some uniform estimates on both the local sound speed and the velocity in H^3(\mathbb{R}^3) with respect to the viscosity coefficients are also obtained, which lead to the strong convergence of the regular solutions of the Navier–Stokes equations with finite mass and energy to the corresponding regular solutions of the Euler equations in L^{\infty}([0, T]; H^{s}_{\mathrm{loc}}(\mathbb{R}^3)) for any s\in [2, 3) . As a consequence, we show that, for both viscous and inviscid flows, it is impossible that the L^\infty norm of any global regular solution with vacuum decays to zero asymptotically, as t tends to infinity. Our framework developed here is applicable to the same problem for other physical dimensions via some minor modifications.

Weak-strong uniqueness principle for compressible barotropic self-gravitating fluids

The aim of this work is to prove the weak–strong uniqueness principle for the compressible Navier–Stokes–Poisson system on an exterior domain, with an isentropic pressure of the type p(ϱ)=aϱγ and allowing the density to be close or equal to zero. In particular, the result will be first obtained for an adiabatic exponent γ∈[9/5,2] and afterwards, this range will be slightly enlarged via pressure estimates “up to the boundary”, deduced relaying on boundedness of a proper singular integral operator.

Blow up of classical solutions to the barotropic compressible fluid models of Korteweg type in bounded domains

In this paper, we study the multi-dimensional barotropic compressible Navier–Stokes–Korteweg equations with degenerate coefficients of viscosities and capillary in bounded domains. It is shown that any classical solutions to the initial–boundary-value problem will blow up in finite time, when the initial density admits an isolated mass group. Moreover, a direct application of our result (μ(ρ)=μρ,λ=0,α=0) shows that the weak solution obtained by Bresch et al. (2003) cannot be improved to a classical one as long as there is an isolated mass group initially.

Mathematical modeling of the dynamics of a hydroelastic system—A hollow cylinder with inhomogeneous initial stresses and compressible fluid

This work attempts to present mathematical modeling of the dynamics of the hydroelastic system consisting of a hollow cylinder with inhomogeneous initial stresses and a compressible barotropic inviscid fluid which filled the interior of the cylinder. The motion of the cylinder is described with the 3D linearized theory of elastic waves in bodies with initial stresses; however, the flow of the fluid is described with the linearized Euler equations for compressible inviscid fluid. Formulations of the boundary conditions on the outer surface of the cylinder and compatibility conditions between the cylinder and fluid on the interior surface of the cylinder are presented. Concrete formulations are made for the axisymmetric case, and general aspects of the solution methods of the formulated problems are considered. Under these considerations, two types of problem are selected: (1) the axisymmetric longitudinal wave propagation and (2) fluid flow profile on the outer free surface of the cylinder—the “circular” radial moving load. Numerical results are presented and discussed for the first type of problem. In particular, it is established that the character of the influence of the inhomogeneous initial stresses on the wave propagation velocity in the hydroelastic system under consideration depends on the dimensionless wavenumber. The perspective of future investigations is also presented and discussed.

High Mach number limit for Korteweg fluids with density dependent viscosity

The aim of this paper is to investigate the regime of high Mach number flows for compressible barotropic fluids of Korteweg type with density dependent viscosity. In particular we consider the models for isothermal capillary and quantum compressible fluids. For the capillary case we prove the existence of weak solutions and related properties for the system without pressure, and the convergence of the solution in the high Mach number limit. This latter is proved also in the quantum case for which a weak-strong uniqueness analysis is also discussed in the framework of the so-called “augmented” version of the system. Moreover, as byproduct of our results, in the case of a capillary fluid with a special choice of the initial velocity datum, we obtain an interesting property concerning the propagation of vacuum zones.

Long-time behavior of a point mass in a one-dimensional viscous compressible fluid and pointwise estimates of solutions

We consider the motion of a point mass in a one-dimensional viscous compressible barotropic fluid. The fluid–point mass system is governed by the barotropic compressible Navier–Stokes equations and Newton's equation of motion. Our main result concerns the long-time behavior of the fluid and the point mass. Pointwise convergence estimates of the volume ratio and the velocity of the fluid to their equilibrium values are shown, and as a corollary, it is proved that the velocity V(t) of the point mass satisfies a decay estimate |V(t)|=O(t−3/2). The rate −3/2 is optimal and is essentially related to the compressibility and the nonlinearity. The main tool used in the proof is the pointwise estimates of Green's function. It turns out that the understanding of the time-decay properties of the transmitted and reflected waves at the point mass plays an essential role in the proof.

Strong solutions for the compressible barotropic fluid model of Korteweg type in the bounded domain

This paper is concerned with a barotropic model of capillary compressible fluids derived by Dunn and Serrin (1985). We consider the initial boundary value problem in bounded domains of $${\mathbb {R}}^d (d=2,3)$$ with more general pressure including Van der Waals equation of state. First, the local in time existence and uniqueness of strong solutions is proved for initial data belonging to the Sobolev space $$W^{2}_{2}\times W^{1}_{2}$$, relying on a new linearization technique and the contraction mapping principle. Next, we show that there exists a global unique strong solution by the continuation argument of local solution under small initial perturbation. The proof is based on the elementary energy method, but with a new development, where some techniques are introduced to establish the uniform energy estimates and to treat the pressure function with non-increasing property.

Influence of prestress on the natural frequency of a fluid-filled FGM sphere

This paper deals with the study of the influence of the inhomogeneous prestresses (initial stresses) in a hollow sphere made of a functionally graded material (FGM) filled with an inviscid compressible fluid on the natural vibration of this sphere. For this study, the three-dimensional exact field equations of the linearized theory of elastic waves in bodies with initial stresses are employed. The motion of the fluid is described by the linearized Navier–Stokes equations for inviscid barotropic compressible fluids. The discrete-analytical solution method proposed by the first author is employed for the solution to the corresponding mathematical problems. Numerical results related to the natural vibration of the mentioned sphere and the influence of the problem parameters on these frequencies are presented and discussed. In particular, it is established that the existence of the fluid leads to a decrease in the natural vibration frequencies. However, a change in the FGM properties of the sphere’s material can lead to an increase in these frequencies.

Derivation of the Navier–Stokes–Poisson System with Radiation for an Accretion Disk

We study the 3-D compressible barotropic radiation fluid dynamics system describing the motion of the compressible rotating viscous fluid with gravitation and radiation confined to a straight layer. We show that weak solutions in the 3-D domain converge to the strong solution of the rotating 2-D Navier-Stokes-Poisson system with radiation for all times less than the maximal life time of the strong solution of the 2-D system when the Froude number is small or to the strong solution of the rotating pure 2-D Navier- Stokes system with radiation.

Frozen and almost frozen structures in the compressible rotating fluid

We study a possibilityof existence of localized two-dimensional structures, both smooth and non-smooth, that can move without significant change of their shape in a leading stream of compressible barotropic fluid on a rotating plane.

A model for extreme plasticity

We present a mathematical model for elastoplasticity in the regime where the applied stress greatly exceeds the yield stress. This scenario is typically found in violent impact testing, where millimetre thick metal samples are subjected to pressures on the order of 10–102GPa, while the yield stress can be as low as 10−2GPa. In such regimes the metal can be treated as a barotropic compressible fluid in which the strength, measured by the ratio of the yield stress to the applied stress, is negligible to lowest order. Our approach is to exploit the smallness of this ratio by treating the effects of strength as a small perturbation to a leading order barotropic model. We find that for uniaxial deformations, these additional effects give rise to features in the response of the material which differ significantly from the predictions of barotropic flow.

A path-conservative Osher-type scheme for axially symmetric compressible flows in flexible visco-elastic tubes

Flexible tubes are widely used in modern industrial hydraulic systems as connections between different components like valves, pumps and actuators. For the design and the analysis of the temporal behavior of a hydraulic system, one therefore needs an accurate mathematical model that describes the fluid flow in a compliant duct. Hence, in this paper we want to model the fluid–structure-interaction (FSI) problem given by the axially symmetric flow of a compressible barotropic fluid that flows through flexible tubes made of vulcanized rubber. The material of the tube can be described by using a visco-elastic rheology, which takes into account the strain relaxation of the material. The resulting mathematical model consists in a one-dimensional system of nonlinear hyperbolic partial differential equations (PDE) with non-conservative products and algebraic source terms. To solve this system numerically, we apply the DOT method, which is a generalized path-conservative Osher-type Riemann solver for conservative and non-conservative hyperbolic PDE recently proposed in [23] and [22].We provide numerical evidence that the proposed DOT Riemann solver is well-balanced for the governing PDE system under consideration. The method is compared to available quasi-exact solutions of the Riemann problem in the case of an elastic wall described by the Laplace law. It is also compared to available experimental data and exact solutions obtained in the frequency domain for a linear visco-elastic wall behavior. In all cases under investigation the proposed path-conservative finite volume scheme based on the DOT Riemann solver is able to produce very accurate results.

Decomposition methods for coupled 3D equations of applied mathematics and continuum mechanics: Partial survey, classification, new results, and generalizations

The present paper provides a systematic treatment of various decomposition methods for linear (and some model nonlinear) systems of coupled three-dimensional partial differential equations of a fairly general form. Special cases of the systems considered are commonly used in applied mathematics, continuum mechanics, and physics. The methods in question are based on the decomposition (splitting) of a system of equations into a few simpler subsystems or independent equations. We show that in the absence of mass forces the solution of the system of four three-dimensional stationary and nonstationary equations considered can be expressed via solutions of three independent equations (two of which having a similar form) in a number of ways. The notion of decomposition order is introduced. Various decomposition methods of the first, second, and higher orders are described. To illustrate the capabilities of the methods, more than fifteen distinct systems of coupled 3D equations are discussed which describe viscoelastic incompressible fluids, compressible barotropic fluids, thermoelasticity, thermoviscoelasticity, electromagnetic fields, etc. The results obtained may be useful when constructing exact and numerical solutions of linear problems in continuum mechanics and physics as well as when testing numerical and approximate methods for linear and some nonlinear problems.

Robustness of strong solutions to the compressible Navier-Stokes system

We consider the Navier-Stokes system describing the time evolution of a compressible barotropic fluid confined to a bounded spatial domain in the 3-D physical space, supplemented with the Navier’s slip boundary conditions. It is shown that the class of global in time strong solutions is robust with respect to small perturbations of the initial data. Explicit qualitative estimates are given also in terms of the shape of the underlying physical domain, with applications to problems posed on thin cylinders.

Various decomposition techniques for linear equations of continuum mechanics

Various methods of decomposition of a sufficiently general linear set of equations, special cases of which are frequently encountered in continuum mechanics, are described. These methods are based on splitting the sets of coupled 3D equations into several simpler independent equations. In addition to the firstorder decomposition, we also considered higher order decompositions. Examples of the decomposition of sets of equations describing slow motions of viscous and viscoelastic incompressible fluids and viscous compressible barotropic fluids and gases are presented. The obtained results can be used for the exact or numerical solution of 3D problems of continuum mechanics and physics.

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\)). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\)–\(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\)-bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\)–\(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin caratheodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Holder spaces, but our approach is different from them.

Existence and uniqueness of steady solutions to the magnetohydrodynamic equations of compressible flows

We establish the existence and uniqueness of a strong solution to the steady magnetohydrodynamic equations for the compressible barotropic fluids in a bounded smooth domain with a perfectly conducting boundary, under the assumption that the external force field is small.