Low Mach number limit for the non-isentropic Navier-Stokes equations

We study the low Mach number limit of the local in time solutions to the compressible Navier-Stokes equations with zero heat conductivity coefficient as the Mach number tends to zero. A uniform existence result for the one-dimensional initial-boundary value problem is proved provided that the initial data are “well-prepared” in the sense that the temporal derivatives up to order two are bounded initially.     

Low Mach number limit of inviscid Hookean elastodynamics

The low Mach number limit of inviscid Hookean elastodynamic equations is rigorously proved in bounded domain, whole space and periodic domain, respectively. The uniform existence of smooth solutions and convergence results as the Mach number tends to zero are obtained in three different domains.     

On the incompressible limit of inviscid compressible fluids

We consider the Euler equations of barotropic inviscid compressible fluids in a bounded domain. 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. Here we discuss, for the boundary case, the different kinds of convergence under various assumptions on the data, in particular the weak convergence in the case of uniformly bounded initial data and the strong convergence in the norm of the data space.

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Sunto Nel presente lavoro consideriamo le equazioni di Eulero per un fluido inviscido comprimibile barotropico in un dominio limitato. è ben noto che, quando il numero di Mach tende a zero, i moti comprimibili approssimano le soluzioni delle equazioni del moto relative ad un fluido inviscido incomprimibile. Qui discutiamo, per il problema al contorno, i diversi tipi di convergenza sotto differenti ipotesi sui dati, in particolare la convergenza debole nel caso di dati iniziali uniformemente limitati e la convergenza forte nella norma dello spazio dei dati.

     

Incompressible limits of the Navier-Stokes equations for all time

We study the asymptotic behaviors of the regular solutions to the compressible Navier-Stokes equations for “well-prepared” initial data for all time as the Mach number tends to zero, by deriving a differential inequality with certain decay property. The estimates obtained in this paper are uniform both in time and Mach number.     

The Zero Mach Number Limit of the Three-Dimensional Compressible Viscous Magnetohydrodynamic Equations

This paper is concerned with the zero Mach number limit of the three-dimension- al compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equations, first the convergence-stability principle is established. Then it is shown that, when the Mach number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equations towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.     

Diffusive wave in the low Mach limit for non-viscous and heat-conductive gas

The low Mach number limit for one-dimensional non-isentropic compressible Navier–Stokes system without viscosity is investigated, where the density and temperature have different asymptotic states at far fields. It is proved that the solution of the system converges to a nonlinear diffusion wave globally in time as Mach number goes to zero. It is remarked that the velocity of diffusion wave is proportional with the variation of temperature. Furthermore, it is shown that the solution of compressible Navier–Stokes system also has the same phenomenon when Mach number is suitably small.     

Low Mach number limit of global solutions to 3‐D compressible nematic liquid crystal flows with Dirichlet boundary condition

In this paper, we consider the uniform estimates of strong solutions in the Mach number ? and t ∈ [0,∞) for the compressible nematic liquid crystal flows in a 3‐D bounded domain , provided the initial data are small enough and the density is close to the constant state. Here, we consider the case that the velocity field satisfies the Dirichlet boundary condition. Based on the uniform estimates, we obtain the global convergence of the compressible nematic liquid crystal system to the incompressible nematic liquid crystals system as the Mach number tends to zero.     

A note on the zero Mach number limit of compressible Euler equations

This note presents a short and elementary justification of the classical zero Mach number limit for isentropic compressible Euler equations with prepared initial data. We also show the existence of smooth compressible flows, with the Mach number sufficiently small, on the (finite) time interval where the incompressible Euler equations have smooth solutions.

     

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

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 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.     

Exterior problem for the two-dimensional compressible Euler equation

In this paper we show the existence of a solution, on any time intervals, in suitable Sobolev spaces for the 2D compressible Euler equation in the exterior domain, when the Mach number is sufficiently small. We shall extend results obtained in [8] in the case of an infinity velocity different from zero.

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Sunto In questo lavoro proviamo l'esistenza di una soluzione, su ogni intervallo temporale, in opportuni spazi di Sobolev per le equazioni di Eulero 2D comprimibili in un dominio esterno, quando il numero di Mach è sufficientemente piccolo. Estenderemo i risultati ottenuti in [8] al caso di velocità all'infinito diversa da zero.

     

Low Mach number limit of the full compressible Navier–Stokes–Maxwell system

This paper deals with the low Mach number limit of the full compressible Navier–Stokes–Maxwell system. It is justified rigorously that, for the well-prepared initial data, the solutions of the full compressible Navier–Stokes–Maxwell system converge to that of the incompressible Navier–Stokes–Maxwell system as the Mach number tends to zero.     

LOW MACH NUMBER LIMIT OF A COMPRESSIBLE NON-ISOTHERMAL NEMATIC LIQUID CRYSTALS MODEL

In this paper, we study the low Mach number limit of a compressible nonisothermal model for nematic liquid crystals in a bounded domain. We establish the uniform estimates with respect to the Mach number, and thus prove the convergence to the solution of the incompressible model for nematic liquid crystals.     

Zero Mach number limit of the compressible Hall-magnetohydrodynamic equations

In this paper, we study the low Mach number limit of the compressible Hall-magnetohydrodynamic equations. It is justified rigorously that, for the well-prepared initial data, the classical solutions of the compressible Hall-magnetohydrodynamic equations converge to that of the incompressible Hall-magnetohydrodynamic equations as the Mach number tends to zero.     

The linear stability of inviscid shear flow of a vibrationally excited diatomic gas

The problem of the linear stability of plane-parallel shear flows of a vibrationally excited compressible diatomic gas is investigated using a two-temperature gas dynamics model. The necessary and sufficient conditions for stability of the flows considered are obtained using the energy integrals of the corresponding linearized system for the perturbations. It is proved that thermal relaxation produces an additional dissipation factor, which enhances the flow stability. A region of eigenvalues of unstable perturbations is distinguished in the upper complex half-plane. Numerical calculations of the eigenvalues and eigenfunctions of the unstable inviscid modes are carried out. The dependence on the Mach number of the carrier stream, the vibrational relaxation time τ and the degree of non-equilibrium of the vibrational mode is analysed. The most unstable modes with maximum growth rate are obtained. It is shown that in the limit there is a continuous transition to well-known results for an ideal fluid as the Mach number and τ approach zero and for an ideal gas when τ → 0.     

The combined quasineutral and low Mach number limit of Euler–Poisson equations coupled to a magnetic field

In this paper, we consider the combined quasineutral and low Mach number limit of compressible Euler–Poisson system coupled to a magnetic field. We prove that, as the Debye length and the Mach number tend to zero simultaneously in some way, the solution of compressible Euler–Poisson system coupled to a magnetic field will converge to that of ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

On Low Mach Number Preconditioning of Finite Volume Schemes

A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments, as well as by analysis, that the preconditioned scheme yields a physically corrected pressure distribution and combined with an explicit time integrator it is stable if the time step Δt satisfies the requirement to be 𝒪(M ²) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Δt = 𝒪(M ),M → 0, though producing unphysical results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations

In this paper, we study the incompressible limit of the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the convergence-stability principle, we show that, when the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient are sufficiently small, the initial-value problem of the model has a unique smooth solution in the time interval where the ideal incompressible magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient go to zero. Moreover, we obtain the convergence of smooth solutions for the model forwards those for the ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

Zero Mach number limit to compressible quantum magnetohydrodynamic equations with vanishing viscosity coefficients

The connection between the compressible viscous quantum magnetohydrodynamic model with low Mach number and the ideal incompressible magnetohydrodynamic equations is studied in a periodic domain. More precisely, for well‐prepared initial data, we prove the convergence of classical solutions of the compressible viscous quantum magnetohydrodynamic model to the classical solutions of the incompressible ideal magnetohydrodynamic equations with a convergence rate when the Mach number, viscosity coefficient, and magnetic diffusion coefficient simultaneously tend to zero.     

一类可压缩流的马赫数极限

该文证明了在二或三维情形下, 当马赫数趋于零时, 一类完全可压缩Navier-Stokes方程的解收敛到相应的完全不可压缩Navier-Stokes方程的解.     

Incompressible limit of all-time solutions to 3-D full Navier–Stokes equations for perfect gas with well-prepared initial condition

In this paper, we prove the incompressible limit of all-time strong solutions to the three-dimensional full compressible Navier–Stokes equations. Here the velocity field and temperature satisfy the Dirichlet boundary condition and convective boundary condition, respectively. The uniform estimates in both the Mach number \({\epsilon\in(0,\overline{\epsilon}]}\) and time \({t\in[0,\infty)}\) are established by deriving a differential inequality with decay property, where \({\overline{\epsilon}\in(0,1]}\) is a constant. Based on these uniform estimates, the global solution of full compressible Navier–Stokes equations with “well-prepared” initial conditions converges to the one of isentropic incompressible Navier–Stokes equations as the Mach number goes to zero.