一类非线性双曲型发展方程的孤子解

研究了一类非线性发展方程.首先在无扰动情形下,利用待定函数和泛函同伦映射方法得到了非扰动发展方程的孤子精确解和扰动方程的任意次近似行波孤子解.接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出非线性双曲型发展扰动方程孤子解的各次近似解析解.再利用摄动理论举例说明了用该方法得到的近似解析解的有效性和各次近似解的近似度.最后,简述了用同伦映射方法得到的近似解的意义,指出了用上述方法得到的各次近似解具有便于求解、精度高等优点.     

一神非定常振荡流的稳定性分析

本文分析了一种非定常振荡的不稳定性问题.其特点是.应用偏微分方程特征理论以及O-S方程特征值的展开,求解扰动波的相函数而不是预先给定扰动波的波动形式.本文研究平面Poiseuille流与其垂向振荡流的组合流动系统.对于连续振荡源导致的波包演化,该系统存在不稳定性.     

扰动Boussinesq方程的近似守恒律

构造了具有扰动项的Boussinesq方程的近似守恒向量和近似守恒律.在方程允许拉格朗日函数的情况下,利用欧拉方程的部分拉格朗日函数方法,研究了含有一阶线性组合扰动项的Boussineq方程的近似守恒律.给出了该方程的近似守恒向量及近似守恒律的分类结果.     

拟流函数法——叶轮机械内三维流动分析问题和设计问题的一种求解方法

本文从动量方程的特点出发并满足连续方程,引入了两个拟流函数.每个拟流函数的主方程均只包含其自身的二阶偏导数,而不包含另一个拟流函数的二阶偏导数.这样,完全的三维解便可通过两个拟流函数主方程的分别单独求解和它们之间的相互迭代来获得.文中给出了在任意非正交坐标系中拟流函数的主方程和相应的边界条件.对叶轮机械内部三维气动分析问题和设计问题的求解进行了讨论和计算,并与解析解、其他数值解法做了比较.结果表明:这种拟流函数方法计算准确而简便,易于得到收敛结果,是求解三维无粘流动的好方法.     

非线性扰动方程的紊动性态

在这篇文章中,首先,我们用Melnikov函数确定自治扰动方程双曲不动点的稳定和不稳定流形的相对位置,从而给出存在极限环的参数范围和极限环的一些性质. 其次,我们给出周期扰动方程存在紊动解和m阶次谐波解的某些充分条件.     

一类非线性强阻尼扰动发展方程的解

研究了在数学、力学中广泛出现的一类三阶非线性强阻尼发展扰动偏微分方程,并求其近似解析解.首先,构造一个泛函同伦映射,将方程的解表示以人工参数的幂级数形式,代入同伦映射,得到一个非线性扰动方程解的逐次迭代关系式,并考虑对应的一个无扰动项情形下的强阻尼发展方程,利用Fourier变换理论,求出其精确解.其次,以得到的精确解为同伦映射迭代式的初始函数,通过非线性扰动方程解的迭代关系式,再用Fourier变换法求解对应的方程.最后,便依次地得到了非线性强阻尼发展扰动偏微分方程的各次近似解析解.用上述方法得到的各次近似解,具有便于求解、精度高等特点.     

流面上的Navier-Stokes方程及其维数分裂方法

该文首先提出了流面和流层的概念,然后推导出了半测地坐标系下流层内的三维NS (Navier-Stokes)方程,以及流面上的二维NS方程.通过引入流面上的流函数,得到了流函数方程的非线性初边值问题,并讨论了方程解的存在性和唯一性.基于以上讨论,提出了求解三维NS方程的维数分裂方法, 并给出了算例.     

N-S方程的拟序扰动序列与改善的渐近展开匹配法

提出了以拟序扰动序列逼近N-S方程定解问题渐近解的一种方法。对N-S方程及其边界条件的渐近拟序扰动序列解进行了讨论,并应用此方法对球坐标系中的圆球绕流进行求解,改善了渐近展开匹配方法,使匹配函数更容易确定。改善后的阻力曲线与实测资料相比在雷诺数小于等于4×104以前完全吻合。     

一类带扰动风险模型的Gerber-Shiu函数(英文)

本文研究了一类带扰动风险模型, 得到了此过程下Gerber-Shiu函数的微分积分方程, 并得到了推广Erlang(2)情形下Gerber-Shiu函数满足的更新方程.     

二维Prandtl方程单调剪切流在Sobolev空间中的整体稳定性

考虑初值是某个关于y变量单调且长时间衰减得充分快的剪切流附近的小扰动,本文证明二维Prandtl方程在Sobolev空间中的整体适定性.本文证明的主要思路是将Masmoudi和Wong (2015)指出的非线性对消性质与Paicu和Zhang (2021)引入的能够证明解有更快长时间衰减估计的好函数结合起来.本文中需要在剪切流满足的方程中添加一个外力项,否则可以证明不存在长时间衰减足够快的单调剪切流.     

A Mean Flow First Harmonic Theory for Hydrodynamic Instabilities

A general theory is presented for nonlinear instabilities arising in steady hydrodynamic motions. For quasiparallel flows at high values of the Reynolds number it is found that for relatively small disturbance levels the usual ideas concerning the generation of higher harmonics and the subsequent modification of the fundamental may be overwhelmed by three dimensional interactions between the evolving mean flow and the first harmonic wave. The differences from and similarities to existing asymptotic and numerical studies are discussed. The theory developed applies to a variety of flow configurations. Numerical results are given for Poiseuille flow and the Blasius boundary layer. In addition the theory developed here is applied to simulate the instabilities produced in a boundary layer due to the presence of free stream disturbances.     

Parabolic flow over a flat plate with wave disturbance in the main stream

In this paper the effects of unsteady flow past a flat plate, which is at rest, are studied when the steady parabolic flow outside the boundary layer is in a wave disturbance. Solutions are developed for large and small times. The behaviour of the amplitudes and phase leads of skin friction components along chordwise and spanwise directions are studied in detail. Two cases, one when the main stream lines are straight and yawed, and the other when the main stream lines are parabolic and concave with respect to the points on the chordwise direction are studied separately and the results are interpreted graphically.     

Transient Nonlinear Disturbances and Shelves in a Stratified Fluid Layer

We revisit the classical problem of internal wave propagation in a stratified fluid layer bounded by rigid walls and point out a mechanism by which unsteady locally confined disturbances generate far-field shelves. Carrying the standard expansion procedure to fourth order in the wave amplitude reveals that weakly nonlinear long waves of a certain mode shed, in general, lower- and higher-mode shelves, which propagate upstream and downstream with the corresponding long-wave speeds. This phenomenon is brought about by the combined effect of nonlinear interactions and the presence of transience in the main disturbance. While the shelves accompanying small-amplitude waves are relatively weak, numerical solutions of the full Euler equations indicate that shelves induced by unsteady disturbances of finite amplitude close to breaking can be quite significant.     

Large-Amplitude Long-Wave Instability of a Supersonic Shear Layer

For sufficiently high Mach numbers, small disturbances on a supersonic vortex sheet are known to grow in amplitude because of slow nonlinear wave steepening. Under the same external conditions, linear theory predicts slow growth of long-wave disturbances to a thin supersonic shear layer. An asymptotic formulation that adds nonzero shear-layer thickness to the weakly nonlinear formulation for a vortex sheet is given here. Spatial evolution is considered for a spatially periodic disturbance having amplitude of the same order, in Reynolds number, as the shear-layer thickness. A quasi-equilibrium inviscid nonlinear critical layer is found, with effects of diffusion and slow growth appearing through a nonsecularity condition. Other limiting cases are also considered, in an attempt to determine a relationship between the vortex-sheet limit and the long-wave limit for a thin shear layer; there appear to be three special limits, corresponding to disturbances of different amplitudes at different locations along the shear layer.     

Weakly Nonlinear Evolution of a Wave Packet in a Zonal Mixing Layer

A nonlinear integrodifferential equation governing the amplitude evolution of a wavepacket near the critical value of the beta parameter is derived. The basic velocity profile is a hyperbolic tangent shear layer and although the neutral eigensolution is regular, all higher-order terms in the expansion of the stream function are singular at the critical point. The analysis is inviscid and in the critical layer both wave packet effects and nonlinearity are present, but the former are taken to be slightly larger. Unlike the Stuart–Watson theory, the critical layer analysis dictates the form of the amplitude equation, the outer expansion being relatively passive. A secondary instability analysis shows that the packet is unstable to sideband perturbations, but the instability is weak so its main consequence would be to produce some modulation of the packet without destroying its coherence.     

The Interaction of a Strong Bore with Small Disturbances in Shallow Water

This paper gives an extension of previous work [2] on weakly nonlinear shallow water waves over a variable bottom to include the effects of strong bores and small surface disturbances. We first consider the interaction of a strong bore with quiescent water over an isolated bottom disturbance to highlight some of the modifications that are introduced in our results for both noncritical and transcritical Froude numbers. We also exhibit the secular effect on the bore trajectory of a bottom disturbance that has a nonzero average. In a second example, we consider the interaction of a strong bore with a small amplitude periodic surface disturbance upstream. We show that downstream of the bore, the wave length of this disturbance increases, whereas its amplitude increases (decreases) depending on whether the bore speed is larger (smaller) than a critical value. We also use this example to illustrate the derivation of the solution and bore trajectory to second order accuracy. All our asymptotic results, obtained in the form of multiple scale expansions, are compared with numerical solutions for a number of illustrative cases.     

On spatial instability of electrically forced axisymmetric jets with variable applied field

This paper considers the problem of instability of electrically forced axisymmetric jets with respect to spatially growing disturbances and in the presence of a variable applied electric field. A mathematical model, which is developed for the dependent variables of such disturbances, is based on the relevant approximated versions of the equations of the electrohydrodynamics for an electrically forced jet flow. The approximations include the assumptions that the length scale along the axial direction of the jet is much larger than that in the radial direction of the jet and the disturbances are axisymmetric and infinitesimal in amplitude. For neutral temporal stability boundary, we find, in particular, two new spatial modes of instabilities under certain conditions. Both modes are found to be enhanced with increasing the strength of the field. The more dominant instability mode is found to exist for a wider range of values of the wave number in the axial direction. The effect of variable applied electric field is found to increase the growth rates of the disturbances but operate over a more restricted domain in the axial wave number.     

Disturbance Propagation in the Melt Spinning Process

This work deals with the propagation of a disturbance acting on a fibre in the melt-spinning process, where molten polymer is ejected vertically downwards from an orifice before being drawn onto a wind-up spool. The disturbance may be produced by a sudden horizontal draught of air impinging on a part of the fibre. The flow is modeled as a wave propagating on a moving string under the influence of damping and variable tension. The amplitude of the disturbance is obtained in closed-form along the characteristics which emanate from the boundaries of the localised initial disturbance; the general solution of the damped disturbance is determined numerically. An important aspect of the model is the ability to predict the magnitude of this disturbance close to the orifice, where the extruded polymer is molten and therefore extremely sensitive to disturbances. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modified function projective lag synchronization of chaotic systems with disturbance estimations

This paper addresses the modified function projective lag synchronization (MFPLS) for a class of chaotic systems with unknown external disturbances. The disturbances are supposed to be generated by the exogenous systems. By using the disturbance-observer-based control and the linear matrix inequality approach, the disturbance observers are developed to ensure the boundedness of the disturbance error dynamics. Then by employing the sliding mode control (SMC) technique, an active SMC law is established to guarantee the disturbance rejection and realize MFPLS between the master and slave systems. And the corresponding numerical simulation is provided to illustrate the effectiveness of the proposed method.     

Control synthesis for a system with non-linear resistance

A dynamic controlled system with one degree of freedom is considered. The system is acted upon by a non-linear resistance, a controlling force and a disturbance. The control and the disturbance are of bounded magnitude. A feedback control is designed which takes the system to a prescribed state. The proposed control law has certain advantages compared with the alternative approach which totally ignores the disturbances. The control is synthesized for an arbitrary non-linear resistance. It is time-suboptimal and robust to variations of the parameters and disturbances.