On the Singular 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. In this paper 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.     

Incompressible Limit for the Compressible Flow of Liquid Crystals

The connection between the compressible flow of liquid crystals with low Mach number and the incompressible flow of liquid crystals is studied in a bounded domain. In particular, the convergence of weak solutions of the compressible flow of liquid crystals to the weak solutions of the incompressible flow of liquid crystals is proved when the Mach number approaches zero; that is, the incompressible limit is justified for weak solutions in a bounded domain.     

Global Solutions for Incompressible Viscoelastic Fluids

We prove the existence of both local and global smooth solutions to the Cauchy problem in the whole space and the periodic problem in the n-dimensional torus for the incompressible viscoelastic system of Oldroyd-B type in the case of near- equilibrium initial data. The results hold in both two- and three-dimensional spaces. The results and methods presented in this paper are also valid for a wide range of elastic complex fluids, such as magnetohydrodynamics, liquid crystals, and mixture problems.     

The Compressible to Incompressible Limit of One Dimensional Euler Equations: The Non Smooth Case

We prove a rigorous convergence result for the compressible to incompressible limit of weak entropy solutions to the isothermal one dimensional Euler equations.     

On the Capillary Problem for Compressible Fluids

Classical capillarity theory is based on a hypothesis that virtual motions of fluid particles distinct from those on a surface interface have no effect on the form of the interface. That hypothesis cannot be supported for a compressible fluid. A heuristic reasoning suggests that even small amounts of compressibility could have significant effect on surface behavior. In an earlier work, Finn took a partial account of compressibility, and formulated a variant of the classical capillarity equation for fluid surface height in a vertical capillary tube; he was led to a necessary condition for existence of a solution with prescribed mass in a tube closed at the bottom. For a circular tube, he proved that the condition also suffices, and that solutions are uniquely determined for any contact angle γ. Later Finn took more complete account of compressibility and obtained a new equation of highly nonlinear character but for which the same necessary condition holds. In the present work we consider that equation for circular tubes. We prove that the necessary condition again suffices for existence when 0 ≤ γ < π, and we establish uniqueness when 0 ≤ γ ≤ π/2. Our result is put into relief by the observation that for the unconstrained problem of a tube dipped into an infinite liquid bath, solutions do not in general exist when γ > π/2. Presumably an actual fluid would in that case descend to the bottom of the tube. This kind of singular behavior does not occur for the equation previously considered, nor does it occur in the present case under the presence of a mass constraint.     

Global Well-Posedness for Compressible Viscoelastic Fluids near Equilibrium

In this paper, we prove local well-posedness for compressible viscoelastic fluids of the Oldroyd model under the assumption that the initial density is bounded away from zero and global well-posedness near equilibrium. The proof of global well-posedness relies on some intrinsic properties of viscoelastic fluids and on a uniform estimate for a linearized hyperbolic–parabolic system with convection terms.     

Existence of Strong Solutions for Incompressible Fluids with Shear Dependent Viscosities

Certain rheological behavior of non-Newtonian fluids in engineering sciences is often modeled by a power law ansatz with p ∈ (1, 2]. In the present paper the local in time existence of strong solutions is studied. The main result includes also the degenerate case (δ = 0) of the extra stress tensor and thus improves previous results of [L. Diening and M. Růžička, J. Math. Fluid Mech., 7 (2005), pp. 413–450].     

Density-Dependent Incompressible Fluids in Bounded Domains

This paper is devoted to the study of the initial value problem for density dependent incompressible viscous fluids in a bounded domain of with boundary. Homogeneous Dirichlet boundary conditions are prescribed on the velocity. Initial data are almost critical in term of regularity: the initial density is in W^1,q for some q > N, and the initial velocity has fractional derivatives in L^r for some r > N and arbitrarily small. Assuming in addition that the initial density is bounded away from 0, we prove existence and uniqueness on a short time interval. This result is shown to be global in dimension N = 2 regardless of the size of the data, or in dimension N ≥ 3 if the initial velocity is small. Similar qualitative results were obtained earlier in dimension N = 2, 3 by O. Ladyzhenskaya and V. Solonnikov in [18] for initial densities in W^1,∞ and initial velocities in with q > N.     

Error Bounds for Semi-Galerkin Approximations of Nonhomogeneous Incompressible Fluids

We consider spectral semi-Galerkin approximations for the strong solutions of the nonhomogeneous Navier–Stokes equations. We derive an optimal uniform in time error bound in the H¹ norm for approximations of the velocity. We also derive an error estimate for approximations of the density in some spaces L^r. P. Braz e Silva was supported for this work by FAPESP/Brazil, #02/13270-1 and is currently supported in part by CAPES/MECD-DGU Brazil/Spain, #117/06. M. Rojas-Medar is partially supported by CAPES/MECD-DGU Brazil/Spain, #117/06 and project BFM2003-06446-CO-01, Spain.     

A Multidimensional Model for the Combustion of Compressible Fluids

We consider a multidimensional model for the combustion of compressible reacting fluids. The flow is governed by the Navier–Stokes in Eulerian coordinates and the chemical reaction is irreversible and is governed by the Arrhenius kinetics. The existence of globally defined weak solutions is established by using weak convergence methods, compactness and interpolation arguments in the spirit of Feireisl [16] and P.L. Lions [24].     

一种可压缩与不可压缩气体流动的统一算法及其应用

本文采用伪时间变化率项及其"预处理"矩阵,并结合LU-SGS离散格式,发展了可压缩与不可压缩气体流动求解的统一算法.该方法有效地消除了采用可压缩方法求解低速流动时容易产生的"刚性"问题,减小了由于压力项在低速情况下产生的舍入误差.同时,在求解低速与高速并存的流场流动时,无需进行预处理矩阵的转换,实现了可压缩与不可压缩气体流动的统一理论求解.作为算法有效性的验证,本文分别计算了低速、高速、高低速混合流动的典型算例.计算值的验证结果比较表明,对求解马赫数大范围变化情况下的流场,具有很好的收敛性与稳定性,而且收敛速度基本不受流动速度的影响.这个算法程序为今后发展用于燃烧反应流动和密度梯度驱动流动的分析建立了方法基础.     

Fluctuation Splitting Schemes for the Compressible and Incompressible Euler and Navier-Stokes Equations

Introduced in the late eighties by Roe, fluctuation splitting (or residual distribution) schemes have recently emerged as a viable alternative to Finite Volume and Finite Element methods for PDE based, fluid dynamics simulations using unstructured meshes. Their application to the numerical approximation of the compressible and incompressible Euler and Navier-Stokes equations is described, emphasizing low Mach number and incompressible applications. The advantages provided by time-preconditioning techniques are discussed and details of the implementation are given.     

Low Mach Number Limits of Compressible Rotating Fluids

We consider the low Mach number limit for a compressible fluid rotating with a constant angular velocity in an exterior domain. The effect of Coriolis and centrifugal forces are taken into account, along with strong stratification due to the effect of gravitation. The anelastic approximation is identified as the limit problem. The main issue addressed in the paper is the interaction of the centrifugal force with acoustic waves.     

Heat Convection of Compressible Viscous Fluids: I

The stationary problem for the heat convection of compressible fluid is considered around the equilibrium solution with the external forces in the horizontal strip domain z ₀ < z < z ₀ + 1 and it is proved that the solution exists uniformly with respect to z ₀ ≥ Z ₀. The limit system as z ₀ → + ∞ is the Oberbeck–Boussinesq equations.     

Heat Convection of Compressible Viscous Fluids. II

We consider steady heat convections of compressible viscous fluids in the horizontal strip domain ${z_0 < z < z_0 + 1}$ under the gravity. Pattern formations are shown uniformly for ${z_0 \geq Z_0}$ . The limit of them as ${Z_0 \rightarrow + \infty}$ is that of Oberbeck-Boussinesq equations.