On the Rayleigh-Taylor Instability for Two Uniform Viscous Incompressible Flows

The authors study the Rayleigh-Taylor instability for two incompressible immis- cible fluids with or without surface tension, evolving with a free interface in the presence of a uniform gravitational field in Eulerian coordinates. To deal with the free surface, instead of using the transformation to Lagrangian coordinates, the perturbed equations in Eule- rian coordinates are transformed to an integral form and the two-fluid flow is formulated as a single-fluid flow in a fixed domain, thus offering an alternative approach to deal with the jump conditions at the free interface. First, the linearized problem around the steady state which describes a denser immiscible fluid lying above a light one with a free interface separating the two fluids, both fluids being in （unstable） equilibrium is analyzed. By a general method of studying a family of modes, the smooth （when restricted to each fluid domain） solutions to the linearized problem that grow exponentially fast in time in Sobolev spaces are constructed, thus leading to a global instability result for the linearized problem. Then, by using these pathological solutions, the global instability for the corresponding nonlinear problem in an appropriate sense is demonstrated.     

Numerical solution of a Rayleigh-Taylor instability problem

The numerical solution of a Rayleigh-Taylor instability problem where an inviscid liquid of finite depth is accelerated into a gas of semi-infinite extent is obtained by transforming the irregular flow domain into a rectangular domain by a coordinate transformation. The free surface equation is solved by a Crank-Nicolson procedure. The boundary condition at the free surface for the velocity potential ø which contains the time derivative of ø is also treated by an implicit scheme. The numerical results agree well with those obtained by higher order perturbation analysis.     

Rayleigh-Taylor instability and Rayleigh-type waves on a Maxwell-fluid

The maximum growth rate for Rayleigh-Taylor instability on a Maxwell-fluid has been found. The Rayleigh-Taylor instability mode exists for all wave-numbers and dimensionless time numbers. Further it was found that a propagating wave mode may exist at the same time. It was shown that this mode indeed was a Rayleigh-wave mode ask. A cut-off range may exist for the propagating wave-mode.

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Zusammenfassung Die maximale Wachstumsrate für Rayleigh-Taylor-Instabilitäten auf einer Maxwell-Flüssigkeit wurde berechnet. Die Rayleigh-Taylor-Instabilität existiert für alle Wellenzahlen und dimensionslosen (Memory)-Zeitzahlen. Gleichzeitig kann eine sich fortpflanzende Welle bestehen. Es wurde gezeigt, daß dies eine Rayleigh-Welle ist, wennk geht. Für die sich fortpflanzende Welle kann ein cut-off-Bereich existieren.

     

On integral stability and Lipschitz stability of motion

We establish new conditions of integral stability and uniform Lipschitz stability based on the use of the comparison principle and a matrix-valued Lyapunov function. Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal Vol. 49, No. 1, pp. 77–83, January, 1997.     

On stability of isometric transformations

The research was supported by the Russian Foundation for Fundamental Research (Grant 93-011-228).     

On q-analogues and stability theorems

In this survey recent results about q-analogues of some classical theorems in extremal set theory are collected. They are related to determining the chromatic number of the q-analogues of Kneser graphs. For the proof one needs results on the number of 0-secant subspaces of point sets, so in the second part of the paper recent results on the structure of point sets having few 0-secant subspaces are discussed. Our attention is focussed on the planar case, where various stability results are given.     

On stability ofm-variateC 1 interpolation

A simplicial mesh (triangulation) is constructed that generalizes the two-dimensional 4-direction mesh to ℝ ^m . This mesh, with symmetric, shift-invariant values at the vertices, is shown to admit a bounded C¹ interpolant if and only if the alternating sum of the values at the vertices of any 1-cube is zero. This implies that interpolation at the vertices of an m-dimensional, simplicial mesh by a C¹ piecewise polynomial of degree m+1 with one piece per simplex is unstable.     

On thermomechanics of elastic stability

Zusammenfassung Es wird die Thermomechanik des Stabilitätsphenomens elastischer Körper unter äusseren Belastungen betrachtet, welche auf beliebige Weise von der Deformation und der Bewegung benachbarter Materialbereiche, an denen sie angreifen, abhängen können; diese Kräfte werden bewegungsabhängige Belastungen genannt. Eine allgemeine Ungleichung, welche für alle Materialien unter bewegungsabhängigen Belastungen gilt, wird mittels der Erhaltungssätze und der Entropiebilanz hergeleitet; diese Ungleichung wird auf elastische Körper, welche jedoch noch Energie auf allgemeine Weise dissipieren können, spezialisiert. Mittels einer Energienorm werden hinreichende Bedingungen für die örtliche Stabilität endlich deformierter Gleichgewichtskonfigurationen solcher Körper und gewisse notwendige Bedingungen hergeleitet. Schliesslich wird das allgemeine (nicht notwendig lokale) Problem der Stabilität eines Gleichgewichtszustandes kurz untersucht und gewisse in diesem Zusammenhang notwendige und hinreichende Bedingungen werden diskutiert.

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Dedicated to Professor Ziegler on his sixtieth birthday

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This research was supported by the National Science Foundation under Grant GK-10161.     

Large-eddy-simulation of 3-dimensional Rayleigh-Taylor instability in incompressible fluids

The 3-dimensional incompressible Rayleigh-Taylor instability is numerically studied through the large-eddy-simulation ( LES) approach based on the passive scalar transport model. Both the instantaneous velocity and the passive scalar fields excited by sinusoidal perturbation and random perturbation are simulated. A full treatment of the whole evolution process of the instability is addressed. To verify the reliability of the LES code, the averaged turbulent energy as well as the flux of passive scalar are calculated at both the resolved scale and the subgrid scale. Our results show good agreement with the experimental and other numerical work. The LES method has proved to be an effective approach to the Rayleigh-Taylor instability.     

On stability of sampling-reconstruction models

A useful sampling-reconstruction model should be stable with respect to different kind of small perturbations, regardless whether they result from jitter, measurement errors, or simply from a small change in the model assumptions. In this paper we prove this result for a large class of sampling models. We define different classes of perturbations and present a way of quantifying the robustness of a model with respect to them. We also use the theory of localized frames to study the dual frame method for recovering the original signal from its samples.     

On stability in finite models

We search for a set-up in which results from the theory of infinite models hold for finite models. As an example we prove results from stability theory. Received: 23 February 1998     

On stability of -semigroups

We prove that if is a -semigroup on a Hilbert space , then (a) if and only if , for all , and (b) is exponentially stable if and only if , for all . Analogous, but weaker, statements also hold for semigroups on Banach spaces.

     

Nonuniform Approach to Terminal Velocity for Single Mode Rayleigh-Taylor Instability

Abstract The temporal development of a single mode Rayleigh-Taylor instability consists of three stages:the linear, free fall and terminal velocity regimes. The purpose of this paper is to report on new phenomenaobserved in the approach to terminal velocity. Our numerical study shows an unexpected nonuniform approachto terminal velocity. The nonuniformity applies especially to the spikes, which are fingers of heavy fluid fallinginto the light fluid, but it also applies to the rising bubbles of light fluid. For spikes especially our results callinto question the meaningfulness of a terminal velocity for moderate values of the Atwood number A. After ashort time period of pseudo-terminal plateau, the spike velocity increases to a significantly higher maximum,followed by a decrease. This phenomena appears to be due to a slow evolution in the shape of the spike andbubble. We find a relation between the spike (bubble) acceleration and the tip curvature. In correlation with anincrease in the spike velocity, th