ON THE STABILITY OF DISTORTED LAMINAR FLOW(Ⅰ)—BASIC IDEAS AND THEORY

This paper suggests a hydrodynamic stability theory of distorted laminar flow,andpresents a kind of distortion profile,of mean velocity in parallel shear flow.With suchdistortion profiles,the new theory can be used to investigate the stability behaviour ofparallel shear flow,and thus suggests a new possible approach to instability.     

ON THE STABILITY OF DISTORTED LAMINAR FLOW (I)—BASIC IDEAS AND THEORY

This paper suggests a hydrodynamic stability theory of distorted laminar flow, and presents a kind of distortion profile of mean velocity in parallel shear flow. With such distortion profiles, the new theory can be used to investigate the stability behaviour of parallel shear flow, and thus suggests a new possible approach to instability. The Project Supported by National Natural Science Foundation of China.     

DISCUSSION ON BARENBLATT’S EQUATIONS OF MEAN MOVEMENT OF THE INHOMOGENEOUS FLUID

In this paper,it is proved that Barenblatt’s equation of mean movement of the inhomogeneous fluid is not cor-rect,because his equations are based on the assumption that the characteristics of inhomogeneity of fluid are only af-fected with respect to gravitational terms.The correct form of the equation of mean movement of the inhomogeneous fluid is derived in this paper.     

APPLICATION OF WAVELET THEORY IN RESEARCH ON WEIGHT FUNCTION OF MESHLESS METHOD

IntroductionNeedingonlynodeinformation ,withouttheconnectivityelement,themeshlessmethodsaredevelopingrapidly ,andhavemanykindsofmeshlessmethods,suchasEFEM[1],SPH[2 ],RKPM[3],PU[4 ].Weightfunctionanditsparametersplayanimportantroleinsimulatingdisplacementfiledinmeshlessmethod[1].Togivethereasonableinfluenceradiusofweightfunctionisverydifficult.Theinfluenceradiusofweightfunctionisgivenbythesectioncontinualfunction ,andtheuseofinfluenceradiusortheuseofweightfunctionemphasisonthesimulationde…     

ON THE STABILITY OF DISTORTED LAMINAR FLOW(Ⅲ)—THE NONLINEAR STABILITY ANALYSIS OF DISTORTED PARALLEL SHEAR FLOW

Based on the hydrodynamic stability theory of distorted laminar flow and the kind ofdistortion profiles on the mean velocity in parallel shear flow given in paper[1].this paperinvestigates the nonlinear stability behaviour of parallel shear flow,carries on stabilitycalculation taking account of the perturbations of background turbulence noise undercertain assumption,and obtains some results in accordance qualitatively with those ofexperiment for plane Poiseuille flow and pipe Poiseuille flow.     

ON THE STABILITY OF DISTORTED LAMINAR FLOW(Ⅱ)——THE LINEAR STABILITY ANALYSIS OF DISTORTED PARALLEL SHEAR FLOW

Based on the hydrodynamic stability theory of distorted laminar flow and the kind ofdistortion profiles on the mean velocity in parallel shear flow given in paper[1],this paperinvestigates the linear stability behaviour of parallel shear flow.presents unstable results ofplane Couette flow and pipe Poiseuille flow to two-dimensional or axisymmetricdisturbances for the first time,and obtains neutral curves of these two motions under certaindefinition.     

ON THE STABILITY OF DISTORTED LAMINAR FLOW(Ⅱ)——THE LINEAR STABILITY ANALYSIS OF DISTORTED PARALLEL SHEAR FLOW

Based on the hydrodynamic stability theory of distorted laminar flow and the kind of distortion profiles on the mean velocity in parallel shear flow given in paper [1], this paper investigates the linear stability behaviour of parallel shear flow, presents unstable results of plane Couette flow and pipe Poiseuille flow to two-dimensional or axisymmetric disturbances for the first time, and obtains neutral curves of these two motions under certain definition.     

DISCUSSION ON BARENBLATT’S EQUATIONS OF MEAM MOVEMENT THE INHOMOGENEOUS FLUID

In this paper, it is proved that Barenblatt’s equation of mean movement of the inhomogeneous fluid is not correct, because his equations are based on the assumption that the characteristics of inhomogeneity of fluid are only affected with respect to gravitational terms. The correct form of the equation of mean movement of the inhomogeneous fluid is derived in this paper.     

THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied.This study is a forward development of the study of thestability in the Liapunov sense;furthermore,it is of significance in practice sinceperturbations are often not single in the time domain.Malk in proved a general theoremabout thesubject.To apply the theorem,however,the user has to construct a Liapunovfunction which satisfies specified conditions and it is difficult to find such a function fornonlinear nonautonomous systems.In the light of the principle of Liapunov’s indirectmethod,which is an effective method to decide the stability of nonlinear systems in theLiapunov sense,the authors have achieved several important conclusions expressed in theform of theorems to determine the stability of nonlinear nonautonomous systems under thefrequently-acting perturbation.     

THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov’s indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.     

On the stability of elastico-viscous flows

Summary The stability ofCouette flow and planePoiseuille flow of an elastico-viscous liquid is analyzed. It is found that in both cases the flow is less stable than that of Newtonian liquids. The critical wave number is also changed due to the elasticity of the liquid. These changes depend on the type of flow under consideration.This work forms a part of the D. Sc. thesis of the first author. The work was supported by the United States Office of Naval Research under Contract N 62558-4093 and by the Technion Research Found.     

On the impedance of the pipe in laminar and turbulent pulsating flows

The response of a facility, consisting of a valveless reciprocating pump, a large settling chamber and a long straight smooth pipe, to a periodic change in the volume was analysed. The impedance of the pipe was estimated in both laminar and turbulent flow regimes under otherwise identical flow conditions. A good agreement with theory was obtained for the laminar flow. The estimate of the pipe impedance from the experimental data in turbulent flow was based on the momentum equation as well as on the measured resonant frequency of the system. These independent methods show that the inertance of the pipe has a qualitatively different behavior in laminar and turbulent flow regimes.     

On the stability of shear flows in nematics

A theoretical study of the effect of an applied magnetic field on the stability of the flow of nematic slabs subjected to an arbitrary shear is presented. Homeotropic boundary conditions with strong anchoring and a constant magnetic field applied perpendicular to the plates are considered. We discuss the general conditions on the control parameters under which the flow is stable, for a low molecular weight liquid crystal and for a polymer liquid crystal, and obtain estimations of the critical values.     

Long-wavelength stability of the jet flows of a perfect fluid

The spatial stability of jets whose velocity profiles depend linearly on one coordinate is considered. The influence of the velocity profile and the presence of solid walls on the development of perturbations in the jets is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkostl I Gaza, No. 5, pp. 170–175, September–October, 1979.We thank A. A. Pavel'ev for making available experimental material and for a helpful discussion of the results.     

Geometrical theory of fluid flows and dynamical systems

Various dynamical systems have often common geometrical structures and can be formulated on the basis of Riemannian geometry and Lie group theory, provided that a dynamical system has a group symmetry, namely it is invariant under group transformations, and further that the group manifold is endowed with a Riemannian metric. The basic ideas and tools are described, and their applications are presented for the following five problems: (a) free rotation of a rigid body, which is a well-known system in mechanics and presented as an illustrative example of the geometrical theory; (b) geodesic equation and KdV equation on the group of diffeomorphisms of a circle and its extended group; (c) a self-gravitating system of a finite number of points masses and a geometrical interpretation of chaos of Hénon–Heiles system; (d) geometrical formulation of hydrodynamics of an incompressible ideal fluid on the group of volume preserving diffeomorphisms, where an interpretation of the origin of Riemannian curvatures of the fluid flow is given; (e) geodesic equation on a loop group and the local induction equation for the motion of a vortex filament, where the geodesic equation on its extended group is found to be equivalent to the equation for a vortex filament with an axial flow along it.

It is remarkable that the present geometrical formulations are successful for all the problems considered here and give an insight into the deep background common to the diverse physical systems. Furthermore, the geometrical formulation opens a new approach to various dynamical systems, which is rewarded with new results.