圆柱形容器中竖直激励表面波的毛细影响

在竖直振动的圆柱形容器中,利用理想流体中两时间尺度奇摄动展开法,研究了包括表面张力影响的自由面单一表面驻波的运动．通过求解势流方程,获得了一个包含三阶非线性项、外激励及表面张力影响的非线性振幅方程．结果表明当驱动频率较低时,表面张力对表面波模式选择不重要;然而,当驱动频率较高时,表面张力的影响是不可忽略的．说明表面张力具有使得自由面返回到平衡位置的作用．另外,由于考虑了表面张力的影响,使得理论结果比无表面张力时更加接近先前的实验结果．     

重力场中Navier-Stokes方程的线性不稳定性

研究了三维不可压粘性Navier-Stokes方程的线性不稳定性.即当流体处在重力场时,若稳态密度有一定的连续性并随着高度的增加而增加时(即Rayleigh-Taylor密度),重力场中的Navier-Stokes流体是线性不稳定的.利用变分法解出线性方程的线性不稳定解.     

两个模态耦合的Ginzburg-Landau方程的时空混沌同步化

根据数值计算的结果提出了模态耦合的条件,两个方程在高频模态上是耦合的,而在低频模态上是不耦合的．利用了无穷维动力系统理论,证明了两个高频模态耦合的Ginzburg-Landau方程在函数空间中存在吸引域,因而存在连通的、有限维的紧的整体吸引子．驱动方程存在时空混沌．将方程组联系一个截断形式,得到的修正方程组将保持原方程组的动力学行为．高频模态耦合的两个方程在一定的条件下具有挤压性质,证明了可达到完全的时空混沌同步化．在数学上定性解释了无穷维动力系统的同步化现象．研究方法不同于有限维动力系统中通常使用的Liapunov函数方法与近似线性方法．     

多孔介质平板通道发展传热中非局部热平衡时的温度分布特征

研究了多孔介质平板通道中,Darcy流体发展传热强迫对流非局部热平衡下,固相骨架和孔隙流体的温度分布特征．考虑流体流动方向的热传导以及固相和流相相互作用的粘性耗散,根据非局部热平衡的两能量方程模型,得到了常壁温度时多孔介质固相骨架温度和孔隙流体温度的解析解．证明了当两相间的热交换系数趋于无穷大时,两能量方程的温度解趋于局部热平衡时一能量方程的温度解．针对不同的无量纲参数,给出了固相和流相的温度分布状态,通过参数研究,揭示了非局部热平衡强迫对流时温度对无量纲参数的依赖关系．     

参数激励圆柱形容器中的非线性Faraday波

在柱坐标系下,通过奇异摄动理论的多尺度展开法求解势流方程,研究了垂直强迫激励圆柱形容器中的单一模式水表面驻波模式。假设流体是无粘、不可压且运动是无旋的,在忽略了表面张力的影响下,用两变量时间展开法得到一个具有立方项以及底部驱动项影响的非线性振幅方程。对上述方程进行了数值计算,计算的结果显示了在不同驱动振幅和驱动频率下,会激发不同自由水表面驻波模式,从等高线的图像来看,和以往的实验结果相当吻合。     

微通道周期流动电位势及电粘性效应

求解了双电层的Poisson-Boltzmann方程和流体运动的Navier-Stokes方程,得到在周期压差作用下,二维微通道的周期流动电位势,流动诱导电场和液体流动速度的解析解．量纲分析表明,流体电粘性力与以下3个参数有关:1) 电粘性数,它表示定常流动时,通道最大电粘性力与压力梯度的比;2) 形状函数,它表示电粘性力在通道横截面的分布形态; 3) 耦合系数,它表示电粘性力的振幅衰减特征和相位差．分析结果表明,微通道周期流动诱导电场、流动速度与频率Reynolds数有关．在频率Reynolds数小于1时,流动诱导电场随频率Reynolds数变化很慢．在频率Reynolds数大于1时,流动诱导电场随频率Reynolds数的增加快速衰减．在通道宽度与双电层厚度比值较小情况下,电粘性效应对周期流动速度和流动诱导电场有重要影响．     

非轴对称模型Hagen-Poiseuille流的不稳定性

本文讨论流体通过圆管的运动不稳定性问题.作为流体运动所受的扰动波,我们考虑一个三维非线性模型.它的相关振幅函数满足扩散方程,当流体的雷诺数增大时,由于复杂的分子扩散和流体粘性的相互作用.该方程的扩散系数会出现负值.在"负扩散"现象出现时.在流体运动中出现的"湍流段"内部会引起能量的集中和使流体的阻尼减少.文中所得结果对说明圆管流中出现湍流段的实验现象是有价值的.     

流过半无限竖直可渗透壁面时的磁流体动力学自由对流

研究不可压缩粘性导电流体,流过半无限竖直可渗透平板时,将其偏微分形式的流动和传热的基本控制方程,应用适当的相似变换,简化为非线性的常微分方程组．对两种抽吸参数:大的和小的抽吸参数,采用摄动法得到变换后方程的近似解．数值结果表明,随着磁场参数和抽吸参数的增大,任意点的速度场在减小;磁场参数的影响,引起热边界层厚度的增大;速度和温度场随着热汇参数的增大而减小．     

柱状纤维上超薄液膜的线性稳定性分析

运用线性理论分析了粘性超薄液膜沿柱状纤维垂直下落的稳定性特征,研究了厚度低于100 nm的薄膜在外力驱动下的流动以及van der Waals力的影响．结果表明随着薄膜相对厚度的下降,纤维表面的曲率将使得线性扰动的发展得到抑制,而van der Waals力促进扰动的增长,这一竞争机制导致了增长率随薄膜相对厚度非单调的变化．还得到了流动的绝对和对流不稳定分区．结果表明van der Waals力扩大绝对不稳定流动区域,表面张力也会有利于绝对不稳定的发展,而外驱动力正好起到相反的作用．     

剪切流中表面重力波的四阶Zakharov方程及其应用

当二维剪切流沿垂向线性变化时,剪切流中的二维重力波仍满足无旋条件.本文首先推导了线性剪切流中二维表面重力波的三阶和四阶Zakharov方程,它也适用于无剪切流的三维重力表面波场.用Zakharov方程研究了剪切流中波列的第一类不稳定性问题,给出了不稳定判别式,详细讨论了剪切流对不稳定区域和增长率的影响,得到了由于流的存在而出现的新的不稳定区.     

Asymptotic behavior of solutions to a class of nonlinear wave equations of sixth order with damping

We investigate the initial value problem for a class of nonlinear wave equations of sixth order with damping. The decay structure of this equation is of the regularity‐loss type, which causes difficulty in high‐frequency region. By using the Fourier splitting frequency technique and energy method in Fourier space, we establish asymptotic profiles of solutions to the linear equation that is given by the convolution of the fundamental solutions of heat and free wave equation. Moreover, the asymptotic profile of solutions shows the decay estimate of solutions to the corresponding linear equation obtained in this paper that is optimal under some conditions. Finally, global existence and optimal decay estimate of solutions to this equation are also established. Copyright © 2016 John Wiley & Sons, Ltd.     

Painlev\'{e} Analysis and Auto-B\"{a}cklund Transformation for a General Variable Coefficient Burgers Equation with Linear Damping Term

This paper investigates a general variable coefficient (gVC) Burgers equation with linear damping term. We derive the Painlev\''{e} property of the equation under certain constraint condition of the coefficients. Then we obtain an auto-B\"{a}cklund transformation of this equation in terms of the Painlev\''{e} property. Finally, we find a large number of new explicit exact solutions of the equation. Especially, infinite explicit exact singular wave solutions are obtained for the first time. It is worth noting that these singular wave solutions will blow up on some lines or curves in the $(x,t)$ plane. These facts reflect the complexity of the structure of the solution of the gVC Burgers equation with linear damping term. It also reflects the complexity of nonlinear wave propagation in fluid from one aspect.     

The effects of viscosity on the linear stability of damped Stokes waves,downshifting, and rogue wave generation

We investigate a higher order nonlinear Schrödinger equation with linear damping and weak viscosity, recently proposed as a model for deep water waves exhibiting frequency downshifting. Through analysis and numerical simulations, we discuss how the viscosity affects the linear stability of the Stokes wave solution, enhances rogue wave formation, and leads to permanent downshift in the spectral peak. The novel results in this work include the analysis of the transition from the initial Benjamin–Feir instability to a predominantly oscillatory behavior, which takes place in a time interval when most rogue wave activity occurs. In addition, we propose new criteria for downshifting in the spectral peak and determine the relation between the time of permanent downshift and the location of the global minimum of the momentum and the magnitude of its second derivative.     

Effects of Radiative Damping on Resonantly Generated Internal Gravity Waves

Internal gravity waves can propagate indefinitely upwards when a fluid is continuously stratified and unbounded vertically. This causes radiative damping of waves which are resonantly generated by the interaction between a mean flow and a topographic feature at the lower boundary. An evolution equation of Benjamin-Davis-Acrivos type, including temporal evolution, detuning, nonlinearity, dispersion, radiative damping, and forcing, is derived to describe the wave behavior. Using a pseudospectral method, we obtain and discuss various numerical results, especially for the resonant cases, covering forcing whose polarity is the same as, or opposite to, that of a free solitary wave, and for weak or strong radiative damping. In practice, weak-radiative cases are likely to be realized when an inversion layer exists at a certain low level over a mountain range.     

The stability of an inverted pendulum with a vibrating suspension point

The problem of stabilizing the upper vertical (inverted) position of a pendulum using vibration of the suspension point is considered. The periodic function describing the vibrations of the suspension point is assumed to be arbitrary but possessing small amplitudes, and slight viscous damping is taken into account. A formula is obtained for the limit of the region of stability of the solutions of Hill's equation with damping in the neighbourhood of the zeroth natural frequency. The analytical and numerical results are compared and show good agreement. An asymptotic formula is derived for the critical stabilization frequency of the upper vertical position of the pendulum. It is shown that the effect of viscous damping on the critical frequency is of the third-order of smallness and, in all the examples considered, when viscous damping is taken into account the critical frequency increases.     

Damping of Periodic Waves in Physically Significant Wave Systems

Damping of periodic waves in the classically important nonlinear wave systems—nonlinear Schrödinger, Korteweg–deVries (KdV), and modified KdV—is considered here. For small damping, asymptotic analysis is used to find an explicit equation that governs the temporal evolution of the solution. These results are then confirmed by direct numerical simulations. The undamped periodic solutions are given in terms of Jacobi elliptic functions. The damping structure is found as a function of the elliptic function modulus, m=m(t) . The damping rate of the maximum amplitude is ascertained and is found to vary smoothly from the linear solution when m= 0 to soliton waves when m= 1 .     

Solitary waves in dusty plasmas with variable dust charge and two temperature ions

Propagation of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions is analyzed. The Kadomtsev–Petviashivili (KP) equation is derived by using the reductive perturbation theory. A Sagdeev potential for this system has been proposed. This potential is used to study the stability conditions and existence of solitonic solutions. Also, it is shown that a rarefactive soliton can be propagates in most of the cases. The soliton energy has been calculated and a linear dispersion relation has been obtained using the standard normal-modes analysis. The effects of variable dust charge on the amplitude, width and energy of the soliton and its effects on the angular frequency of linear wave are discussed too. It is shown that the amplitude of solitary waves of KP equation diverges at critical values of plasma parameters. Solitonic solutions of modified KP equation with finite amplitude in this situation are derived.     

Energy decay for the elastic wave equation with a local time-dependent nonlinear damping

In this paper, we shall investigate the decay property of the solutions to the initial-boundary value problem for the elastic wave equation with a local time-dependent nonlinear damping. We give some decay rate of the energy when the damping term is effective only in a neighborhood of a suitable subset of the boundary. The results obtained in this paper extend, in particular, the known results for the scalar wave equation.     

Propagation of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space

In the present paper, the dispersion equation which determines the velocity of torsional surface waves in a homogeneous layer of finite thickness over an initially stressed heterogeneous half-space has been obtained. The dispersion equation obtained is in agreement with the classical result of Love wave when the initial stresses and inhomogeneity parameters are neglected. Numerical results analyzing the dispersion equation are discussed and presented graphically. The result shows that the initial stresses have a pronounced influence on the propagation of torsional surface waves. It has also been shown that the effect of density, directional rigidities and non-homogeneity parameter on the propagation of torsional surface waves is prominent.