About the Stability of Gas Balls

We prove the nonlinear stability of the motionless, spherically symmetric equilibrium states of barotropic, self-gravitating viscous fluids with respect to perturbations of zero total angular momentum. These equilibrium states as well as the non-stationary solutions occupy part of space, and a constant pressure is assumed on the free surface, but no surface tension.     

Stability Balls in the Rotating Bénard System

Nonlinear stability in the standard rotating Bénard system with free boundaries is investigated. The perturbations are assumed to be three-dimensional and to be periodic in the horizontal directions. Below the critical value of the Rayleigh numberRa there is conditional stability, i.e. there are nonvanishing stability balls such that perturbations with initial values (measured in a suitable norm) in these balls decay exponentially in time. We give here explicit bounds to these stability balls from below in terms of the parameters of the system, i.e.Ra, the Taylor numberT and the Prandtl numberPr as well as the size of the periodicity cell of the perturbation. The bounds are valid in the entire parameter space; in particular, forPr<1 and for arbitrarily large values ofT. They provide a qualitative explanation for the experimental observation of subcritical instabilities in the rangePr<1. The method is based on a mode expansion of the perturbation equations and explicit estimates of the semigroup operator as well as of the nonlinearity.     

Linear Stability Analysis of a Viscous Liquid Sheet in a High-Speed Viscous Gas

Air-assisted atomizers in which a thin liquid sheet is deformed under the action of a high-speed air flow are extensively used in industrial applications, e.g., in aircraft turbojet injectors. Primary atomization in these devices is a consequence of the onset and growth of instabilities on the air/liquid interfaces. To better understand this process, a temporal linear instability analysis is applied to a thin planar liquid sheet flowing between two semi-infinite streams of a high-speed viscous gas. This study includes the full viscous effects both in the liquid and gas basic states and perturbations. The relevant dimensionless groups entering the non-dimensional Orr–Sommerfeld equations and boundary conditions are the liquid and gas stream Reynolds numbers, the gas to liquid momentum flux ratio, the gas/liquid velocity ratio, the Weber number and the equivalent gas boundary layer to liquid sheet thickness ratio. Growth rates and temporal frequencies as a function of the wave number, varying the different dimensionless parameters are presented, together with neutral stability curves. From the results of this parametric study it is concluded that when the physical properties of gas and liquid are fixed, the momentum flux ratio is especially relevant to determine the instability conditions. It is also observed that the gas boundary layer thickness strongly influences the wave propagation, and acts by damping sheet oscillation frequency and growth. This is especially important because viscosity in the basic gas velocity profile has always been ignored in instability analysis applied to the geometry under study. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stability Conditions for Linear Impulsive Systems with Delay

Sufficient asymptotic-stability conditions for a linear impulsive system with delay are established on the basis of Lyapunov—Razumikhin functions. A scalar equation is considered as an example __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 130–138, June 2005.     

框架结构的线性欧拉稳定理论

框架结构的稳定分析一直采用特征值稳定理论.近来研究发现该理论是错误的.本文分别对两个文献中用特征值稳定理论进行稳定分析的例题,用线性欧拉理论进行再分析.结果显示:在第一个例题中,二者相差太大,特征值理论得出的临界荷载过大;得出的稳定约束下的优化截面-稳定临界解过小,为危险设计.说明该理论是错误的.在第二个例题中,却出现了相反的情况,具有无效的特征值稳定约束的目标函数却大于具有有效的欧拉稳定约束的目标函数.这是因为第二个文献作者的设计思想违背了常规设计的原则,得到的是非优解.鉴于在国际文献中应用该理论研究框架结构稳定问题的论文屡见不鲜,应引起特别关注.     

About Clapeyron's Theorem in Linear Elasticity

We examine some elementary interpretations of the classical theorem of Clapeyron in linear elasticity theory. As we show, a straightforward application of this theorem in the purely mechanical setting leads to an apparent paradox which can be resolved by referring either to dynamics or to thermodynamics. These richer theories play an essential part in understanding the physical significance of this theorem.     

Stability for Nonautonomous Linear Differential Systems with Infinite Delay

Journal of Dynamics and Differential Equations - We study the stability of general n-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as...     

About the Linear Stability of the Spherically Symmetric Solution for the Equations of a Barotropic Viscous Fluid under the Influence of Self-Gravitation

This paper is concerned with the question of linear stability of motionless, spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. We prove the linear asymptotic stability of such equilibria with respect to perturbations which leave the angular momentum, momentum, mass and the position of the center of gravity unchanged. We also give some decay estimates for such perturbations, which we derive from resolvent estimates by means of analytic semigroup theory.     

轻气炮冲击实验中的一些改进措施及在爆炸硬化上的应用

本文介绍了在高锰钢的轻气炮冲击实验中的一些改进措施和参数的选取，并介绍了对实验后的试件进行硬度测量的结果在爆炸硬化上的应用及对爆炸硬化的指导作用，此外，本文还测量出高锰钢的冲击波速度随所受击压力变化而变化的规律。     

Asymptotic Linear Stability of Solitary Water Waves

In this paper, we study the well-posedness of Cahn–Hilliard equations with degenerate phase-dependent diffusion mobility. We consider a popular form of the equations which has been used in phase field simulations of phase separation and microstructure evolution in binary systems. We define a notion of weak solutions for the nonlinear equation. The existence of such solutions is obtained by considering the limits of Cahn–Hilliard equations with non-degenerate mobilities.     

On Stability with Respect to Linear Approximation

We prove new propositions on stability with respect to linear approximation.     

Experimental Study of the Linear Stability of a Falling Liquid Film in the Presence of a Turbulent Gas Stream

Using the local electrical conductivity method, the parameters of the linear waves generated on the surface of a falling liquid film in the presence of a co- or counter-current gas stream are measured. The Reynolds numbers of the fluid and the gas were varied from 24 to 125 and from 0 to 8000, respectively. The results are presented in the form of dispersion relations. In the case of the absence of a gas stream, the results are compared with calculations based on a linear integral theory. It is shown that a gas stream increases the instability of the film and a counter-current gas flow has a greater effect on the wave phase velocity than a co-current flow.     

Stability Analysis for Flows in Rotating Spherical Layers (Linear Theory)

We present a technique and the results of linear stability analysis for the viscous incompressible flows in an annulus between two concentric coaxial spheres, of which the outer remains stationary while the inner is rotated. We show that the characteristics of the main flow instability strongly depend on the layer thickness, qualitatively as well quantitatively. The critical perturbation for the main flow in a thin layer is monotonic and has the form of a system of ring vortices covering the entire flow region from one pole to the other; the vortices are axisymmetric but nonsymmetric about the equatorial plane. The critical perturbation in a thick layer is nonmonotonic, three-dimensional, and nonsymmetric about the equatorial plane. The shape of the critical perturbation depends on the layer thickness. A comparison with the experimental data is carried out.     

Linear Stability of Steady States for Thin Film and Cahn-Hilliard Type Equations

We study the linear stability of smooth steady states of the evolution equation

Stability of Solutions of Linear Impulsive Differential Inclusions

We establish conditions for the uniqueness of R-solutions of linear differential inclusions with pulses at fixed moments of time. We prove theorems on the stability of solutions and R-solutions for homogeneous inclusions.     

二自由度耦合线性随机系统的最大Lyapunov指数和稳定性

利用摄动方法和Fokker-Planck算子及其伴随算子的特征函数展开法,讨论了两个模态都处于临介状态的耦合二自由度振动系统,在小强度的非高斯噪声参数激励下系统运动的稳定性,获得了系统扩散过程的稳态概率密度的渐近表达式,建立了系统最大Lyapunov指数的渐近表达式,由此获得了系统运动模态几乎必然稳定的充分必要条件。     

Linear Stability of Traveling Waves in First-Order Hyperbolic PDEs

In the analysis of traveling waves it is common that coupled parabolic-hyperbolic problems occur, where the hyperbolic part is not strictly hyperbolic. For example, this happens whenever a reaction diffusion equation with more than one non-diffusing component is considered in a co-moving frame. In this paper we analyze the stability of traveling waves in nonstrictly hyperbolic PDEs by reformulating the problem as a partial differential algebraic equation (PDAE). We prove uniform resolvent estimates for the original PDE problem and for the PDAE by using exponential dichotomies. It is shown that the zero eigenvalue of the linearization is removed from the spectrum in the PDAE formulation and, therefore, the PDAE problem is better suited for the stability analysis. This is rigorously done via the vector valued Laplace transform which also leads to optimal rates. The linear stability result presented here is a major step in the proof of nonlinear stability.     

Evolution Characteristics of Gas Permeability Under Multiple Factors

Transport in Porous Media - Permeability, one of the most significant parameters for CBM, is affected by many influencing factors. In this paper, an improved fully coupled permeability model was...     

Stability of Relaxation Waves in a Non-Equilibrium Gas

A correct formulation of the problem of the structure and stability of relaxation waves is given for arbitrary Lewis (Le) and Prandtl (Pr) numbers. The high activation-energy approximation in which the reaction to heating zone length ratio is assumed to be small is analyzed in detail.It is shown that the case Le = 1 is a singular point and that only at that point is the calculation method used in the paper of O. Yu. Travnikov et al. (Phys. Fluids, 1997, 9, 3935–3937) applicable. For Le 1 and small wavelengths, the Landau-Darrieus mechanism gives way to another mechanism of perturbation amplification in which the amplification factor strongly depends not only on the degree of non-equilibrium but also on the Lewis and Prandtl numbers. It is shown that with decrease in the thermal expansion coefficient of the medium the results go over into the well-known results obtained for diffusion-thermal instability.