The Rotation of a Gravitating Sphere in a Monatomic Gas,II

The purpose of this work is to study the fluid motion caused by the high speed rotation of a gravitating sphere in a monatomic gas. It has been possible to find a stable steady solution only for very small Prandtl number, which can be interpreted to mean an optically thick gas. The flow is characterized by a flat radial jet in the equatorial plane and a viscous boundary layer on the spherical surface which, in some cases, lies beneath a thermal boundary layer. That the outer region must be hydrostatic puts very stringent constraints on the associated velocity field which necessitate still another boundary layer on the sphere. This last layer is shown to be unstable to small disturbances in certain temperature ranges. Finally, a similar solution that exists for order one Prandtl number must be disregarded because this last boundary layer is always unstable.     

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.     

Spatial eigenmodes of a boundary layer with a triple-deck velocity field

The beforehand unclear relation between the viscous-inviscid interaction and the instability of viscous gas flows is illustrated using three-dimensional boundary-layer perturbations in the case of sub- and supersonic outer flows. The assumptions are considered under which asymptotic boundary layer equations with self-induced pressure are derived and the excitation mechanisms of eigenmodes (i.e., Tollmien-Schlichting waves) are described. The resulting dispersion relations are analyzed. The boundary layer in a supersonic flow is found to be stable with respect to two-dimensional perturbations, whereas, in the three-dimensional case, the modes become unstable. The increment of growth is investigated as a function of the Mach number and the orientation of the front of a three-dimensional Tollmien-Schlichting wave.     

Acoustic gravity waves: a computational approach

This paper discusses numerical solutions of a hyperbolic initial boundary value problem that arises from acoustic wave propagation in the atmosphere. Field equations are derived from the atmospheric fluid flow governed by the Euler equations. The resulting original problem is nonlinear. A first-order linearized version of the problem is used for computational purposes. The main difficulty in the problem as with any open boundary problem is in obtaining stable boundary conditions. Approximate boundary conditions are derived and shown to be stable. Numerical results are presented to verify the effectiveness of these boundary conditions.     

Wave motions in a spatial boundary layer

The propagation of perturbations in a boundary layer under conditions when the velocity of the approaching stream may be both subsonic and supersonic is considered. With regard to the initial flow in the boundary layer it is assumed that it is stationary and possesses a spatial character which is caused by the external pressure gradient and not by the curvature of the body around which the flow occurs (boundary layers of this kind are extensively used in experiments at the present time). The linearized equations describing waves of vanishingly small amplitude are studied in detail. An analysis of the dispersion relation which links the frequency of the free oscillations with the components of the wave vector reveals a number of special features which are only present in motions with a three-dimensional velocity field. In particular, it is established that the Cauchy problem for the system of linear equations is ill-posed.     

Superharmonic instability of stokes waves

A stability of nearly limiting Stokes waves to superharmonic perturbations is considered numerically in approximation of an infinite depth. Investigation of the stability properties can give one an insight into the evolution of the Stokes wave. The new, previously inaccessible branches of superharmonic instability were investigated. Our numerical simulations suggest that eigenvalues of linearized dynamical equations, corresponding to the unstable modes, appear as a result of a collision of a pair of purely imaginary eigenvalues at the origin, and a subsequent appearance of a pair of purely real eigenvalues: a positive and a negative one that are symmetric with respect to zero. Complex conjugate pairs of purely imaginary eigenvalues correspond to stable modes, and as the steepness of the underlying Stokes wave grows, the pairs move toward the origin along the imaginary axis. Moreover, when studying the eigenvalues of linearized dynamical equations we find that as the steepness of the Stokes wave grows, the real eigenvalues follow a universal scaling law, that can be approximated by a power law. The asymptotic power law behavior of this dependence for instability of Stokes waves close to the limiting one is proposed. Surface elevation profiles for several unstable eigenmodes are made available through  http://stokeswave.org website.     

时滞速度反馈对强迫自持系统动力学行为的影响

研究强迫自持振动系统因时滞反馈产生的主共振解及其分岔．通过对强迫非自治系统的时滞反馈控制,得到所要研究的数学模型．讨论对应的线性化系统使平凡平衡态失稳出现周期解的稳定性临界条件．特别关注主共振及分岔．结果表明,稳定的主共振解随着时滞的变化周期性地出现在系统中．同时,也给出了不稳定的主共振关于时滞变化的区域,在理论方面给出了系统出现概周期运动的时滞区域．数据模拟证实了理论结果．     

Continuum models in problems of the hypersonic flow of a rarefied gas around blunt bodiest

All possible continuum (hydrodynamic) models in the case of two-dimensional problems of supersonic and hypersonic flows around blunt bodies in the two-layer model (a viscous shock layer and shock-wave structure) over the whole range of Reynolds numbers, Re, from low values (free molecular and transitional flow conditions) up to high values (flow conditions with a thin leading shock wave, a boundary layer and an external inviscid flow in the shock layer) are obtained from the Navier-Stokes equations using an asymptotic analysis. In the case of low Reynolds numbers, the shock layer is considered but the structure of the shock wave is ignored. Together with the well-known models (a boundary layer, a viscous shock layer, a thin viscous shock layer, parabolized Navier-Stokes equations (the single-layer model) for high, moderate and low Re numbers, respectively), a new hydrodynamic model, which follows from the Navier-Stokes equations and reduces to the solution of the simplified (“local”) Stokes equations in a shock layer with vanishing inertial and pressure forces and boundary conditions on the unspecified free boundary (the shock wave) is found at Reynolds numbers, and a density ratio, k, up to and immediately after the leading shock wave, which tend to zero subject to the condition that (k/Re)^1/2 → 0. Unlike in all the models which have been mentioned above, the solution of the problem of the flow around a body in this model gives the free molecular limit for the coefficients of friction, heat transfer and pressure. In particular, the Newtonian limit for the drag is thereby rigorously obtained from the Navier-Stokes equations. At the same time, the Knudsen number, which is governed by the thickness of the shock layer, which vanishes in this model, tends to zero, that is, the conditions for a continuum treatment are satisfied. The structure of the shock wave can be determined both using continuum as well as kinetic models after obtaining the solution in the viscous shock layer for the weak physicochemical processes in the shock wave structure itself. Otherwise, the problem of the shock wave structure and the equations of the viscous shock layer must be jointly solved. The equations for all the continuum models are written in Dorodnitsyn--Lees boundary layer variables, which enables one, prior to solving the problem, to obtain an approximate estimate of second-order effects in boundary-layer theory as a function of Re and the parameter k and to represent all the aerodynamic and thermal characteristic; in the form of a single dependence on Re over the whole range of its variation from zero to infinity.

An efficient numerical method of global iterations, previously developed for solving viscous shock-layer equations, can be used to solve problems of supersonic and hypersonic flows around the windward side of blunt bodies using a single hydrodynamic model of a viscous shock layer for all Re numbers, subject to the condition that the limit (k/Re)^1/2 → 0 is satisfied in the case of small Re numbers. An aerodynamic and thermal calculation using different hydrodynamic models, corresponding to different ranges of variation Re (different types of flow) can thereby, in fact, be replaced by a single calculation using one model for the whole of the trajectory for the descent (entry) of space vehicles and natural cosmic bodies (meteoroids) into the atmosphere.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.     

Hopf bifurcations of travelling waves in a brake-like system

We consider a system composed of an elastic tube, which is fixed at the outer boundary and in frictional contact with a rigid cylinder, rotating inside the tube about the common axis. Under the assumption of Coulomb's friction law at the contact surface between the two bodies several types of rotating slip-stick and also slip-stick-separation travelling waves with different wave numbers can be observed. For a wide range of parameters the linearized system has unstable complex eigenvalues, which cause high frequency oscillations and unpleasant squeal. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

弱粘性流体中垂直激励表面波的阻尼效应

在竖直振动的圆柱形容器中,将Navier-Stokes方程线性化,利用两时间尺度奇异摄动展开法研究了弱粘性流体的单一自由面驻波运动．整个流场被分为外部势流区和内部边界层区两部分,对两部分区域分别求解,得到包含阻尼项和外驱动影响的线性振幅方程．利用稳定性分析,得到形成稳定表面波的条件,给出了临界曲线．此外,还获得了阻尼系数的解析表达式．最后,将线性阻尼加到理想流体条件下所得到的色散关系中对其进行修正,理论结果证明修正后的驱动频率更加接近实验的结果．通过计算发现,当驱动的频率较低时,流体的粘性对表面波模式选择有重要影响,而表面张力的影响不明显;但当驱动频率较高时,流体的表面张力起主要作用,而流体的粘性影响甚小．     

The propagation of travelling waves in an open cubic autocatalytic chemical system

The reaction-diffusion travelling waves that can be initiatedin an open isothermal chemical system governed by cubic autocatalytickinetics are discussed. The system is shown to be capable ofsustaining up to three spatially uniform steady states, the(trivial) unreacted state, which is always stable (a node),and two nontrivial states, one of which is always unstable (asaddle point). The third state can change its stability throughHopf bifurcation (both subcritical and supercritical). Thisallows the possibility of two sorts of travelling wave beingestablished; there are wave profiles which connect the unreactedstate ahead to the nontrivial state at the rear, and wave profiles(pulse waves) which have the unreacted state at both the frontand rear. The conditions under which a particular wave is initiatedare considered by both a discussion of the (ordinary) differentialequations governing the travelling waves and by numerical integrationsof an initial-value problem. This treatment also reveals thepossibility of a stable travelling wave propagating throughthe system, leaving behind a temporally unstable stationarystate. Under these conditions, spatiotemporal chaotic behaviouris seen to develop after the passage of the wave.     

基于应力敏感的天然裂缝性页岩气藏双孔模型解的结构

针对天然裂缝性页岩气藏,研究了3种外边界条件（无穷大、定压、封闭）及内边界条件下的定产量生产问题,并建立了考虑应力敏感性和解析吸附的不稳定渗流的试井分析模型.先对此模型作线性化处理;然后通过摄动法以及利用Laplace变换,求得线性化后的无因次储层压力的Laplace空间精确解;最后根据解的相似结构理论,给出了求解该模型解结构的步骤,并且定义了3种外边界条件下的相似核函数,发现了此模型在3种外边界条件下精确解之间的相似结构.这项研究不仅为编制试井分析软件提供便利, 提高计算效率,而且对页岩气藏渗流理论的研究具有重要意义,也为页岩气藏渗流模型的求解提供了一种新的方法.     

Oblique wave scattering by an impermeable ocean-bed of variable depth in a two-layer fluid with ice-cover

Within the framework of linearized theory, obliquely incident water wave scattering by an uneven ocean-bed in the form of a small bottom undulation in a two-layer fluid, where the upper layer has a thin ice-cover while the lower one has the undulation, is investigated here. In such a two-layer fluid, there exist two modes of time-harmonic waves—the one with lower wave number propagating just below the ice-cover and the one with higher wave number along the interface. An incident wave of a particular mode gets reflected and transmitted by the bottom undulations into waves of both the modes. Assuming irrotational motion, a perturbation technique is employed to solve the first-order corrections to the velocity potentials in the two-layer fluid by using Fourier transform appropriately and also to calculate the reflection and transmission coefficients in terms of integrals involving the shape function representing the bottom undulation. For a sinusoidal bottom topography, these coefficients are depicted graphically against the wave number. It is observed that when the oblique wave is incident on the ice-cover surface, we always find energy transfer to the interface, but for interfacial oblique incident waves, there are parameter ranges for which no energy transfer to the ice-cover surface is possible.     

Spectrally Stable Encapsulated Vortices for Nonlinear Schr?dinger Equations

Summary. A large class of multidimensional nonlinear Schrodinger equations admit localized nonradial standing-wave solutions that carry nonzero intrinsic angular momentum. Here we provide evidence that certain of these spinning excitations are spectrally stable. We find such waves for equations in two space dimensions with focusing-defocusing nonlinearities, such as cubic-quintic. Spectrally stable waves resemble a vortex (nonlocalized solution with asymptotically constant amplitude) cut off at large radius by a kink layer that exponentially localizes the solution. For the evolution equations linearized about a localized spinning wave, we prove that unstable eigenvalues are zeroes of Evans functions for a finite set of ordinary differential equations. Numerical computations indicate that there exist spectrally stable standing waves having central vortex of any degree.     

Static stability analysis of a thin plate with a fixed trailing edge in axial subsonic flow: Possio integral equation approach

In this work, the static stability of a thin plate in axial subsonic airflow is studied using the framework of Possio integral equation. Specifically, we consider the cases when the plate’s leading edge is free and the plate’s trailing edge is either pinned or clamped. We formulate the problem under consideration using a partial differential equations (PDE) model and then linearize the model about the free stream velocity, density, and pressure, to enable analytical treatment. Based on the linearized model, we introduce a new derivation of a Possio integral equation that relates the pressure jump along the thin plate to the plate’s downwash. The steady state solution to the Possio equation is then used to account for the aerodynamic loads in the plate steady state governing equation resulting in a singular differential-integral equation which is transformed to a singular integral equation that represents the static aeroelastic equation of the plate. We verify the solvability of the static aeroelastic equation based on the Fredholm alternative for compact operators in Banach spaces and the contraction mapping theorem. By constructing solutions to the static aeroelastic equation and matching the nonzero boundary conditions at the trailing edge with the zero boundary conditions at the leading edge, we obtain characteristic equations for the free-clamped and free-pinned plates. The minimum solutions to the characteristic equations are the divergence speeds which indicate when static instabilities start to occur. We show analytically that free-pinned plates are statically unstable. We also construct, analytically, flow speed intervals that correspond to static stability regions for free-clamped plates. Furthermore, we resort to numerical computations to obtain an explicit formula for the divergence speed of free-clamped plates. Finally, we apply the obtained results on piezoelectric plates and we show that free-clamped piezoelectric plates are statically more stable than conventional free-clamped plates due to the piezoelectric coupling.     

Asymptotics for a free boundary model in price formation

We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry-Lions on price formation and dynamic equilibria.The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.     

Influence of Finite Amplitude Disturbances on the Nonstationary Modes of a Compressible Boundary Layer Flow

A weakly nonlinear stability analysis is performed to search for the effects of compressibility on a mode of instability of the three-dimensional boundary layer flow due to a rotating disk. The motivation is to extend the stationary work of [ 1 ] (hereafter referred to as S90) to incorporate into the nonstationary mode so that it will be investigated whether the finite amplitude destabilization of the boundary layer is owing to this mode or the mode of S90. Therefore, the basic compressible flow obtained in the large Reynolds number limit is perturbed by disturbances that are nonlinear and also time dependent. In this connection, the effects of nonlinearity are explored allowing the finite amplitude growth of a disturbance close to the neutral location and thus, a finite amplitude equation governing the evolution of the nonlinear lower branch modes is obtained. The coefficients of this evolution equation clearly demonstrate that the nonlinearity is destabilizing for all the modes, the effect of which is higher for the nonstationary waves as compared to the stationary waves. Some modes particularly having positive frequency, regardless of the adiabatic or wall heating/cooling conditions, are always found to be unstable, which are apparently more important than those stationary modes determined in S90. The solution of the asymptotic amplitude equation reveals that compressibility as the local Mach number increases, has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable. Apart from the already unstable positive frequency waves, perturbations with positive frequency are always seen to compete to lead the solution to unstable state before the negative frequency waves do. Also, cooling the surface of the disk will be apparently ineffective to suppress the instability mechanisms operating in this boundary layer flow.     

A Newton-Picard Collocation Method for Periodic Solutions of Delay Differential Equations

This paper presents a collocation method with an iterative linear system solver to compute periodic solutions of a system of autonomous delay differential equations (DDEs). We exploit the equivalence of the linearized collocation system and the discretization of the linearized periodic boundary value problem (BVP). This linear BVP is solved using a variant of the Newton-Picard method [Int. J. Bifurcation Chaos, 7 (1997), pp. 2547–2560]. This method combines a direct method in the low-dimensional subspace of the weakly stable and unstable modes with an iterative solver in the high-dimensional orthogonal complement. As a side effect, we also obtain good estimates for the dominant Floquet multipliers. We have implemented the method in the DDE-BIFTOOL environment to test our algorithm. AMS subject classification (2000) 65J15, 65P30, 65Q05     

Spectral characteristics of atmospheric turbulence model

In this paper, KdV-Burgers equation can be regarded as the normal equation of atmospheric turbulence in the stable boundary layer. On the basis of the travelling wave analytic solution of KdV-Burgers equation, the turbulent spectrum is obtained. We observe that the behavior of the spectra is consistent with actual turbulent spectra of stable atmospheric boundary layer.