An analytical study of linear and non-linear convection in Boussinesq-Stokes suspensions

The Rayleigh-Benard situation in Boussinesq-Stokes suspensions is investigated using both linear and non-linear stability analyses. The linear and non-linear analyses are based on a normal mode solution and minimal representation of double Fourier series, respectively. The effect of suspended particles on convection is delineated against the background of the results of the clean fluid. The realm of non-linear convection warrants the quantification of heat transfer and this has been achieved on the Rayleigh-Nusselt plane. Possibility of aperiodic convection is discussed.     

The stability of finite-amplitude interfacial solitary waves

The linear stability of finite-amplitude interfacial gravity solitary waves propagating in a two-layer fluid is investigated analytically focusing on the occurrence of an exchange of stability. We make an asymptotic analysis for small growth rates of infinitesimal disturbances, and explicitly obtain their growth rates near an exchange of stability. The result indicates that an exchange of stability occurs at every stationary value of the total energy of the solitary waves. It also gives us information whether the number of growing modes increases or decreases after experiencing the exchange of stability. We apply these analytical results to specific interfacial solitary waves, and find various features on their stability that are not seen in the case of surface solitary waves.     

理想导体中具有两个松弛时间的电磁热弹性波

研究处于均布磁场中的理想导体的二维电磁热弹性耦合问题,引入势函数使控制方程转化为3个偏微分方程．运用Laplace变换和Fourier变换得到该问题在变换域内的精确表达式,再通过级数展开和Laplace逆变换法求得在时间较短时的逆变换,得到时间-空间域内问题的解．运用此方法研究了表面受到热冲击的半无限空间问题．给出了电磁热弹性波、膨胀波和横向波传播的速度,并通过数值计算,给出了各个场量的分布图．所得结论与已有的结论一致．     

Fourier analysis of the Eulerian–Lagrangian least squares collocation method

A Fourier analysis was performed in order to study the numerical characteristics of the effective Eulerian–Lagrangian least squares collocation (ELLESCO) method. As applied to the transport equation, ELLESCO requires a C¹-continuous trial space and has two degrees of freedom per node. Two coupled discrete equations are generated for a typical interior node for a one-dimensional problem. Each degree of freedom is expanded separately in a Fourier series and is substituted into the discrete equations to form a homogeneous matrix equation. The required singularity of the system matrix leads to a ‘physical’ amplification factor that characterizes the numerical propagation of the initial conditions and a ‘computational’ one that can affect stability. Unconditional stability for time-stepping weights greater than or equal to 0-5 is demonstrated. With advection only, ELLESCO accurately propagates spatial wavelengths down to 2Δx. As the dimensionless dispersion number becomes large, implicit formulations accurately propagate the phase, but the higher-wave-number components are underdamped. At large dispersion numbers, phase errors combined with underdamping cause oscillations in Crank–Nicolson solutions. These effects lead to limits on the temporal discretization when dispersion is present. Increases in the number of collocation points per element improve the spectral behaviour of ELLESCO.     

Collision of dissimilar blunt elastic bodies: a plane problem

The paper deals with a direct central impact of two infinite cylindrical bodies having differently shaped cross sections and made of different materials. A nonstationary plane problem of elasticity is solved. The contact boundary is moving and determined during the solution. A mixed boundary-value problem is formulated. Its solution has the form of Fourier series. Satisfying mixed boundary conditions gives an infinite system of Volterra equations of the second kind for the unknown coefficients of the series. The basic characteristics of the impact process and their dependence on the physical and mechanical properties of the bodies are determined numerically Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 36–45, February 2009.     

多层前向网络研究进展及若干问题

本文概述了多层前向网络研究的发展历史,对其中有代表性的若干成就进行了较为系统的介绍和评论,分析了当前研究工作中存在的一些问题,提出了解决这些问题的几种可行方案。在对多层前向网络的有效逼近机理进行深入剖析的基础上,提出了合理的有限规模多层前向网络应当遵循的若干构造原则。     

Some exact expressions for the temporal evolution of long Rossby waves on a beta-plane

Some exact expressions are derived to describe the temporal evolution of forced Rossby waves in a two-dimensional beta-plane configuration where the background flow has constant zonal-mean velocity. The meridional length scale of the problem is assumed to be small relative to the zonal length scale and so the long-wave limit of zero aspect ratio is taken. In the case where the background flow velocity is zero, an exact solution is obtained in terms of generalized hypergeometric functions. A late-time asymptotic approximation is obtained and it shows that the solution oscillates with time and its amplitude goes to zero in the limit of infinite time. In the case of a non-zero background flow velocity, the solution is evaluated using two different procedures which give two equivalent expressions in terms of different generalized hypergeometric functions. The late-time asymptotic behaviour is investigated and it is found that the solution approaches a steady state in the limit of infinite time.We also derive a solution in the form of an asymptotic series expansion for the more general situation where a Rossby wave packet is generated by a zonally-localized boundary condition comprising a continuous spectrum of wavenumbers or Fourier modes. The exact solutions found here can be used as leading-order solutions in weakly-nonlinear analyses and other studies involving more realistic configurations for time-dependent Rossby waves or wave packets.     

A survey of weakly-nonlinear acoustic models: 1910–2009

A chronological survey of the weakly-nonlinear acoustic equations put forth between the years 1910–2009 is presented. The derivation, properties, and historical development of these models, along with the approximation scheme on which they are based, are discussed. Connections between the models are also discussed and errors/omissions in the literature are noted and, in some cases, corrected. Brief chronologies of the weakly-nonlinear models that describe propagation in relaxing fluids, bubbly liquids, and fluids that saturate porous solids are also presented.     

A new method for Fourier series expansions: Applications in rotor-seal systems

Complex nonlinearities of rotor-seal systems make it difficult to implement some widely used techniques for nonlinear vibrations. This is partly due to the fact that it is rather cumbersome to expand the nonlinear terms into Fourier series. In this paper, a novel Fourier series expansion method is proposed to circumvent this difficulty. The incremental harmonic balance method is utilized to obtain the solutions of a rotor-seal system, where the complicated nonlinearities are handled by the expansion method. Periodic, double-periodic and triple-periodic solutions are obtained in excellent agreement with numerical results, which shows the validity and efficacy of the proposed solution procedures.     

层合圆柱三维温度场分析的半解析-精细积分法

针对圆柱体的三维温度场分析,提出了一种高效的半解析-精细积分法。将温度场展开为环向坐标的Fourier级数,并对径向坐标进行差分离散,从而把三维热传导方程简化为一系列二阶常微分方程;将这些二阶常微分方程转化为哈密顿体系下的一阶状态方程,并利用两点边值问题的精细积分法求解。由于该方法仅对径向坐标进行差分离散,故相对于传统的数值方法离散规模大幅度减少,不仅提高了计算效率、降低了存贮量,而且缓解了代数方程的病态问题。此外,针对Fourier半解析解,根据热平衡原理推导出了两种材料衔接面的半解析差分方程,从而为求解复合材料层合柱问题打下了基础。算例结果表明,即使对于细长比高达400的圆柱杆件,此方法仍然可以给出精度较高的解答。     

基于Fourier级数的时变周期系数Riccati微分方程精细积分

结合Fourier级数展开方法,本文提出了基于精细积分的时变周期系数Riccati微分方程求解高效算法.首先,利用Fourier级数展开方法将周期系统表示成三角级数形式,在一个积分步内使用精细积分方法得到对应Hamilton系统状态转移矩阵的表达式.然后,通过Riccati变换的方法,得到含有状态转移矩阵的时变周期系数Riccati微分方程解的递推格式.本文方法充分利用了方程本身的周期性特点,文中的数值算例表明算法具有计算效率高、结果可靠等优势.     

Obtaining Fourier series graphically from large amplitude oscillatory shear loops

Large amplitude oscillatory shear (LAOS) flow has been used to characterize the nonlinear viscoelasticity of polymer melts and solutions. Results are frequently reported with shear stress versus strain loops, or with shear stress versus shear rate loops. A Fourier analysis of the stress response to LAOS is often desired for comparison with theory, or for quantitative comparison between resins. A method is presented which employs the discrete Fourier transform to obtain the Fourier series coefficients from LAOS loops.     

Quadratically nonlinear torsional hyperelastic waves in a transversely isotropic cylinder: Primary analysis of evolution

Nonlinear torsional waves propagating along an infinite transversely isotropic cylinder are considered. The hyperelasticity of the cylinder is described by Guz’s generalization of the Murnaghan potential to quasi-isotropic materials. The evolution of the initial waveprofile in cylinders made of composite materials reinforced with micro-and nanoscale fibers is modeled numerically. It is shown that the transverse isotropy of this class of composites has a weak effect on the wave phenomena accompanying the propagation of a torsional wave. Three-dimensional plots of evolution are presented __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 32–44, May 2008.     

Localization of wave motions in a fluid-filled cylinder

The properties of harmonic surface waves in a fluid-filled cylinder made of a compliant material are studied. The wave motions are described by a complete system of dynamic equations of elasticity and the equation of motion of a perfect compressible fluid. An asymptotic analysis of the dispersion equation for large wave numbers and a qualitative analysis of the dispersion spectrum show that there are two surface waves in this waveguide system. The first normal wave forms a Stoneley wave on the inside surface with increase in the wave number. The second normal wave forms a Rayleigh wave on the outside surface. The phase velocities of all the other waves tend to the velocity of the shear wave in the cylinder material. The dispersion, kinematic, and energy characteristics of surface waves are analyzed. It is established how the wave localization processes differ in hard and compliant materials of the cylinder __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 4, pp. 72–86, April 2008.     

EXACT ANALYSIS OF WAVE PROPAGATION IN AN INFINITE RECTANGULAR BEAM

The Fourier series method was extended for the exact analysis of wave propagation in an infinite rectangular beam. Initially, by solving the three-dimensional elastodynamic equations a general analytic solution was derived for wave motion within the beam. And then for the beam with stress-free boundaries, the propagation characteristics of elastic waves were presented. This accurate wave propagation model lays a solid foundation of simultaneous control of coupled waves in the beam.