非线性发展方程的自相似解

本文着力于给出非线性发展方程的自相似解的一些最新的研究进展．借助于调和分析的方法(特别是利用Littlewood-Paley理论、时空估计等)，通过非线性发展方程的Cauchy问题的研究来获得自相似解．主要技术是将初始状态空间X推广到非自反的Banach空间(使得X包含那些具自相似结构的初始函数)，相应地将适定性中解在t=0处的连续性放宽成弱连续．另一方面，用Scaling的方法来分析时空可积空间的形式、非线性增长与空间X的选取等．这对非线性发展方程Cauchy问题的研究是至关重要的，它本质上给出了研究非线性发展方程Cauchy问题的工作空间．进而，对于自相似解的结构、自相似解作用(可以是某些整体解的大尺度极限)亦给出了一些具体的分析．     

对冲圆射流生成径向扩张液膜研究

采用两股互相冲击的圆射流可以形成环形的液体薄膜,液膜在径向扩展到一定的临界半径距离会破碎．数值模拟了液膜在周围气体中形成和破碎的非定常过程．考虑了液体和气体都是不可压缩Newton流体的轴对称问题．液体和气体的界面采用Level set函数来跟踪,Navier-Stokes 控制方程和物理边界条件采用有限差分格式离散求解．计算结果给出了环形液体薄膜形成并在其环形边缘处破碎,并缓慢运动的过程．液膜的厚度随着液膜在轴向的扩展会逐渐变薄,因此定义的局部Weber数会在径向逐渐减小,这里的局部Weber数定义为ρu2h/σ,其中ρ和σ分别为液体的密度和界面的张力,u和h分别为在径向某个位置的液膜的平均径向速度和半液膜厚度．数值结果表明就像实验中所观察到的那样,液膜径向扩展的过程的确会在局部Weber数趋向于1的时候终结而停止扩张．根据空间-时间线性稳定性理论,液膜的破碎最初是由正弦模式在临界局部Weber数Wec=1引起的,在临界局部Weber数小于1时会发生绝对不稳定性．在线性理论中另一个独立的模式,所谓的余弦模式,则增长比正弦模式要慢,从而会推测到正弦模式主导破碎的结论．然而,这里的数值结果却表明,余弦模式在界面波的非线性发展阶段实质的超越了正弦模式的增长,并对液膜的最终阶段的破碎起主导作用．这验证了线性理论只能够对触发时扰动波的性质进行预测,而对失稳后情况和结果的预测则不一定正确．     

圆薄膜在集中力作用下大变形基本方程的精确解

利用一种新型的简易方法,获得圆薄膜在中心集中力作用下的基本方程与边界条件下非线性边值问题的精确解;并利用现代不动点定理讨论了该问题解的存在唯一性．虽然求解的是圆薄膜在中心集中力作用下的非线性问题,但此原理亦可应用在其它类似的非线性问题．     

关于薄膜外延生长模型隐式全离散格式的误差分析

1引言分子薄膜的外延增长模型是由四阶非线性扩散方程描述的:设区域Ω=[O,L]~2,外延表面高度函数h(x,t)满足如下方程     

分数阶非线性薄膜方程的不变子空间和精确解

方程的精确解是方程的非线性现象以及它本身所蕴含的物理意义的一种具体表现,而不变子空间则可以理解为方程的稳定区域.在给出一类非线性薄膜方程所对应的几种不变子空间后,通过两边系数比较得到了其相应的有限维动力系统.再借助Mittag-Leffler函数的一些基本性质以及拉普拉斯变换等求解该系统,从而构造得到了分数阶非线性薄膜...     

动载荷下不可压缩超弹性球形薄膜的若干定性性质

对于由横观各向同性不可压缩的Rivlin-Saunders材料组成的球形薄膜,研究了薄膜的内、外表面在周期阶梯载荷作用下的轴对称变形的非线性动力学特性．通过令球形结构的厚度趋近于1,得到了近似描述薄膜径向对称运动的二阶非线性常微分方程．详细讨论了解的定性性质．特别地,给出了球形薄膜随时间的运动产生非线性周期振动的可控性条件,证明了在某些情形下周期振动的振幅会出现“∞”型同宿轨道以及周期振动的振幅会出现不连续增长现象,并给出了相应的数值模拟．     

三维半导体器件问题在时空局部加密复合网格上的有限差分格式

半导体器件的瞬时状态由三个方程组成的非线性偏微分方程组的初边值问题决定,电子位势方程是椭圆型的,电子和空穴浓度方程是抛物型的．依据实际数值模拟的需要,提出了一类三维半导体问题在时间和空间上进行局部加密的复合网格上的有限差分形式,并给出了电子和空穴浓度的最大模误差估计,最后给出了数值算例．     

圆薄膜在集中力作用下的大变形

利用圆薄膜在中心集中力作用下大变形的基本方程、边界条件和Hercky变换,求解了非线性边值问题,推广了Hencky变换,得到了集中力作用下圆薄膜大变形问题的精确解.     

无穷区间上一类不连续非线性积分方程的唯一解

在一般Banach空间中研究了一类无穷区间上不连续非线性积分方程的唯一解．在非常弱的条件下证明了非线性积分方程的唯一解可以由迭代序列的一致极限得到，并给出了逼近解的迭代序列的误差估计式，然后应用到无穷区间一阶微分方程的终值问题，本质改进（将紧型条件删去）并推广了一些结果．     

圆柱形容器中自由表面波的三波内共振相互作用

由Luke变分原理导出了圆柱形容器中自由毛细重力表面波的基本方程.首先对速度势和自由面波高作Galerkin展开,用多重尺度法导出了控制方程前二阶摄动方程,在此基础上讨论了三个波二阶内共振的非线性相互作用,导出了三波内共振的非线性耦合作用方程和守恒律.针对非退化的作用方程,分析了相平面上平衡点的位置,研究了参数对应共振与非共振的各种情况,在不同参数情况下求出二阶作用方程的稳态解并分析了解的稳定性态,还讨论了只在有限时间内有效的解.分析表明,在非退化情况下,由于初始条件不同,3个波之间能量传递的模式不尽相同,有可能能量在3波之间周期性传递,亦可能单波的能量有衰减或增长.     

An interior layer in the thermal power-law blown film model

Film blowing is a highly complex industrial process used to manufacture thin sheets of polymer. Models that describe this process are highly nonlinear and numerical instabilities often occur when solving the highly nonlinear differential equations. This paper investigates the structure of typical solutions that arise when the polymer is assumed to be described by a power-law fluid operating under nonisothermal conditions. We consider both a shear-thinning and shear-thickening polymer and use a balance of orders argument to identify the structure of a region of rapid expansion in the radial profile of the film. A mixture of heuristic and singular perturbation techniques is applied to obtain a closed form approximate expression for the radial profile of the film which displays the interior layer phenomenon. We demonstrate how approximate solutions to the highly nonlinear two-point boundary value problem describing this process may be constructed using this expression as an initial estimate in an iterative scheme. Numerical solutions for the radial temperature, velocity and thickness profiles of the film are subsequently obtained by iteration.     

Dynamic method of determining the mechanical characteristics of the surface layer of polymers

A thin polymer film is treated as a three-layered plate, the inner layer being characterized by the bulk and the outer layers by the surface properties of the polymer. Using the equations describing the vibrations of a cantilever element of rectangular section, the authors propose formulas for the thickness, modulus of elasticity, and mechanical loss factor of the surface layer. The mechanical characteristics of the surface layers of PMM and nitrocellulose have been determined by electromagnetically exciting and photometrically recording vibrations in film specimens of varying thickness.Institute of the Chemistry of High-Molecular Compounds, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 548–550, May–June, 1969.     

Nonlinear stability characterization of thin Newtonian film flows traveling down on a vertical moving plate

The paper presents both the linear and nonlinear stability theories for the characterization of thin Newtonian film flows traveling down along a vertical moving plate. The linear model is first developed to characterize the flow behavior. After showing the inadequacy of the linear model in representing certain flow characteristics, the nonlinear kinematics model is then developed to represent the system. The long-wave perturbation method is employed to derive the generalized kinematic equations with free film surface condition. The linear model is solved by using the normal mode method for three different, namely, the quiescent, up-moving and down-moving, moving conditions. Subsequently, the elaborated nonlinear film flow model is solved by the method of multiple scales. The modeling results clearly indicate that both subcritical instability and supercritical stability conditions are possible to occur in the film flow system. The effect of the down-moving motion of the vertical plate tends to enhance the stability of the film flow.     

Numerical methods for fourth order nonlinear degenerate diffusion problems

Numerical schemes are presented for a class of fourth order diffusion problems. These problems arise in lubrication theory for thin films of viscous fluids on surfaces. The equations being in general fourth order degenerate parabolic, additional singular terms of second order may occur to model effects of gravity, molecular interactions or thermocapillarity. Furthermore, we incorporate nonlinear surface tension terms. Finally, in the case of a thin film flow driven by a surface active agent (surfactant), the coupling of the thin film equation with an evolution equation for the surfactant density has to be considered. Discretizing the arising nonlinearities in a subtle way enables us to establish discrete counterparts of the essential integral estimates found in the continuous setting. As a consequence, the resulting algorithms are efficient, and results on convergence and nonnegativity or even strict positivity of discrete solutions follow in a natural way. The paper presents a finite element and a finite volume scheme and compares both approaches. Furthermore, an overview over qualitative properties of solutions is given, and various applications show the potential of the proposed approach.     

Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film

The problem of hydrodynamic instability of a thin condensate viscoelastic liquid film flowing down on the outer surface of an axially moving vertical cylinder is investigated. In order to improve the accuracy of numerical results, the viscoelastic and heat transfer parameters have been included into the governing equations. Also, the analytical solutions are obtained by utilizing the long-wave perturbation method. The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem. It has been revealed that the stability of the film flow is weakened when the radius of cylinder and the temperature difference are reduced. Moreover, it is found that the increment of down-moving motion of the cylinder can enhance the flow stability. Further, the thin film flow can be destabilized by the viscoelastic property. The results show that both supercritical stability and subcritical instability can take place within the film flow system given appropriate conditions. Moreover, the absence of Reynolds number leads to an obvious difference in the behavior of some physical parameters.     

Breakup of finite fluid films

We study the dewetting process of thin fluid films that partially wet a solid surface. Using long wave (lubrication) approximation, we formulate a nonlinear partial differential equation governing the evolution of the film thickness, h. This equation includes the effects of capillarity, gravity, and additional conjoining/disjoining pressure term to account for intermolecular forces. We perform standard linear stability analysis of an infinite flat film, and identify the corresponding stable, unstable and metastable regions. Within this framework, we analyze the evolution of a semi-infinite film of length L in one direction. The numerical simulations show that for long and thin films, the dewetting fronts of the film generate a pearling process involving successive formation of ridges at the film ends and consecutive pinch-off behind these ridges. On the other hand, for shorter and thicker films, the evolution ends up by forming a single drop. The time evolution as well as the final drops pattern shows a competition between the dewetting mechanisms caused by nucleation and by free surface instability. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Predictions of the gas–liquid flow in wet electrostatic precipitators

Based on Computational Fluid Dynamics (CFD), the present paper aims to simulate several important phenomena in a wet type ESP from the liquid spray generation to gas-droplet flow in electric field. A single passage between the adjacent plates is considered for the simulation domain. Firstly, the electric field intensity and ion charge density are solved locally around a corona emitter of a barbed wire electrode, which are applied to the entire ESP using periodic conditions. Next, the Euler–Lagrange method is used to simulate the gas-droplet flow. Water droplets are tracked statistically along their trajectories, together with evaporation and particle charging. Finally, the deposition density on the plate is taken as the input for the liquid film model. The liquid film is simulated separately using the homogenous Eulerian approach in ANSYS-CFX. In the current case, since the free surface of the thin water film is difficult to resolve, a special method is devised to determine the film thickness.As parametric study, the variables considered include the nozzle pressure, initial spray spreading patterns (solid versus hollow spray) and plate wettability. The droplet emission rate and film thickness distribution are the results of interest. Main findings: electric field has strong effect on the droplet trajectories. Hollow spray is preferred to solid spray for its lower droplet emission. The liquid film uniformity is sensitive to the plate wettability.     

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.     

Thin-film flow over a nonlinear stretching sheet

A thin viscous liquid film flow is developed over a stretching sheet under different nonlinear stretching velocities. An evolution equation for the film thickness, is derived using long-wave approximation of thin liquid film and is solved numerically by using the Newton–Kantorovich method. A comparison is made with the analytic solution obtained in [B. S. Dandapat, A. Kitamura, B. Santra, “Transient film profile of thin liquid film flow on a stretching surface”, ZAMP, 57, 623-635 (2006)]. It is observed that all types of stretching produce film thinning but non-monotonic stretching produces faster thinning at small distance from the origin. The velocity u along the stretching direction strongly depends on the distance along the stretching direction and the Froude number.     

Homogenization of a Class of Nonlinear Variational Inequalities with Applications in Fluid Film Flow

The authors consider the homogenization of a class of nonlinear variational inequalities, which include rapid oscillations with respect to a parameter. The homogenization of the corresponding class of differential equations is also studied. The results are applied to some models for the pressure in a thin fluid film fluid between two surfaces which are in relative motion. This is an important problem in the lubrication theory. In particular, the analysis includes the effects of surface roughness on both faces and the phenomenon of cavitation. Moreover, the fluid can be modeled as Newtonian or non-Newtonian by using a Rabinowitsch fluid model.