Finite amplitude long-wave instability of a thin viscoelastic fluid during spin coating

This paper investigates the stability of a thin incompressible viscoelastic fluid designated as Walters’ liquid B″ during spin coating. The long-wave perturbation method is proposed to derive a generalized kinematic model of the film flow. The method of normal mode is applied to study the linear stability. The amplitude growth rates and the threshold conditions are characterized subsequently and summarized as the by-products of the linear solutions. Using the multiple scales method, the weakly nonlinear stability analysis is studied for the evolution equation of a film flow. The Ginzburg–Landau equation is determined to discuss the threshold conditions of the various critical flow states. The study reveals that the rotation number and the radius of the rotating circular disk generate the destabilizing effects. Moreover, the viscoelastic parameter k indeed plays a more significant role in destabilizing the film flow than a thin Newtonian fluid during spin coating [27].     

Bifurcation analysis of a viscoelastic fluid heated from below

Convection of a viscoelastic fluid in a square domain heated from below is investigated for the case of nondeformable free surfaces. To describe the rheological behavior of the fluid the generalized Oldroyd model is used. A weakly nonlinear analysis is performed in order to determine the character of branching for both the monotonic and oscillatory modes. We also perform a reduction of the boundary value-problem to the set of nonlinear amplitude equations. The analysis of this dynamic system demonstrates the onset and competition of five convection modes.     

Global stability of centrifugal filtration convection

A nonlinear stability analysis is performed to study the onset of convection in a fluid saturated porous layer subject to alternating direction of the centrifugal body force. By introducing a suitable energy functional, the analysis is carried out for the Darcy and the Brinkman models of flow through porous media. The nonlinear result is unconditional and its sharpest limit is determined and is compared with the corresponding linear limit. The failure of linear theory in describing the instability is established in a certain region of the parameter space where possible subcritical instabilities may arise. The stability boundaries are discussed graphically for various values of the Darcy number and comparison is made with the available known results.     

Asymptotic Analysis and Stability of Inviscid Liquid Sheets

Asymptotic methods are employed to determine the leading-order equations that govern the fluid dynamics of slender, and thin and slender, inviscid, irrotational, planar liquid sheets subject to pressure differences and gravity. Two flow regimes have been identified depending on the Weber number, and analytical solutions to the steady state equations are provided. Linear stability studies indicate that the sinuous mode corresponds to Weber numbers on the order of unity, while the varicose mode is associated with small Weber numbers. For small Weber numbers, the nonlinear stability of liquid sheets is determined analytically in terms of elliptic integrals of the first and second kinds. It is also shown that the sinuous mode of thin and slender liquid sheets is identical to the same mode for slender sheets.     

Mathematical simulation of nonlinear oscillations of viscoelastic pipelines conveying fluid

A mathematical model of the problem of nonlinear oscillations of a viscoelastic pipeline conveying fluid is developed in the paper. The Boltzmann–Volterra integral model with weakly singular kernels of heredity is used to describe the processes of pipeline strain. Using the Bubnov–Galerkin method, the mathematical model of the problem is reduced to the study of a system of ordinary integro-differential equations, where time is an independent variable. The solution of integro-differential equations is determined by a numerical method based on the elimination of the singularity in the relaxation kernel of the integral operator. Using the numerical method for unknowns, a system of algebraic equations is obtained. To solve a system of algebraic equations, the Gauss method is used. A computational algorithm is developed to solve the problems of the dynamics of viscoelastic pipelines with a flowing fluid. The algorithm of the proposed method makes it possible to investigate in detail the effect of rheological parameters on the character of vibrational strength of viscoelastic pipelines with a fluid flow, in particular, in the study of free oscillations of pipelines based on the theory of ideally elastic shells. On the basis of the computational algorithm developed, a set of applied computer programs has been created, which makes it possible to carry out numerical studies of pipeline oscillations. The influence of singularity in the heredity kernels and the geometric parameters of the pipeline on the vibrations of structures possessing viscoelastic properties is numerically investigated. It is shown that an account of viscoelastic properties of pipeline material leads decrease in the amplitude and frequency of oscillation. It is established that to reveal the influence of viscoelastic properties of structure material on the pipeline oscillations, it is necessary to use the Abel-type weakly singular kernels of heredity. The obtained results of numerical simulation can be used in the enterprises of oil and gas industries, as well as in design organizations.     

微分本构粘弹性轴向运动弦线横向振动分析的差分法

给出了微分本构粘弹性轴向运动弦线横向振动数值仿真的一种差分法．文中建立了具有微分本构的粘弹性运动弦线的横向振动模型;通过对系统的控制方程和本构方程在不同的分数节点离散,得到一种新的差分方法．利用这一方法,弦线振动方程的数值计算过程可以交替地显式进行,且有较小的截断误差和好的数值稳定性．与通用的方法比较,新的方法计算简单、方便．文中利用方程的不变量检验了数值结果的可靠性,并利用这一方法给出了一类弦线模型的参数振动分析．     

一类高次自催化耦合反应扩散系统的分歧和斑图

考虑了一类由于自催化剂的耦合而发生的反应扩散系统的空间结构．利用线性化理论讨论了平衡态解的稳定性并且证明了在非耦合系统中空间非一致解出现分歧的必要条件．进一步,利用弱非线性理论讨论了分歧点并且给出了弱耦合系统的图灵分歧解的振幅方程及其性质．     

Linear and weakly nonlinear analysis of the Rosensweig instability

The Rosensweig instability of a layer of magnetic fluid with a free surface manifests itself as a stationary pattern of peaks. The pattern is characterized by a wave vector q whose absolute value gives the wave number q = | q |. Within the frame of a linear stability theory a quantitative analysis of the dependence of the wave number on the strength of the magnetic field is presented. The method of multiple scale analysis is applied for a weakly nonlinear analysis of the Rosensweig instability. For magnetic inductions above the critical value, the stability of different patterns are discussed with respect to the relative permeability of the magnetic fluid and the layer thickness.     

Aggregation and stability analysis of nonlinear complex systems

The problem of stability of large-scale systems in critical cases is investigated. New form of aggregation for essentially nonlinear complex systems is suggested. With the help of this form the sufficient conditions of asymptotic stability are determined. The results obtained are used for the stability analysis of complex systems by the nonlinear approximation and for the investigation of absolute stability conditions for a certain class of nonlinear systems.     

Convection caused by Radiation through the Layer

A nonlinear stability threshold is determined for the problemof convection in a layer of non-Boussinesq fluid with prescribedheat flux on the lower boundary and constant-temperature uppersurface. The convection problem is one in which motion can penetratefrom an unstable layer into a gravitationally stable one andthe unconditional nonlinear analysis necessitates utilizationof a spatially weighted energy.     

具有分数导数型本构关系的粘弹性柱的动力稳定性

研究简支的受轴向周期激励的粘弹性柱动力稳定性,柱的材料满足分数导数型本构关系.建立了描述粘弹性柱动力学行为的弱奇异性Volterra积分-偏微分方程,利用Galerkin方法将其化归为弱奇异性Volterra积分-常微分方程.利用平均化方法的思想给出了粘弹性柱运动稳定状态的存在性条件.给出一种新的计算方法,克服了存储整个响应历史数据的困难,并给出了数值算例,计算结果与解析方法的结论比较吻合.     

静电场驱动下液体薄膜的几何形状

利用六边形-俯视图的弱非线性稳定性分析和数值仿真,在电场作用下,研究高分子薄膜表面静态模式的发展过程．在无限空间域上,空间和高分子薄膜之间的界面,由薄膜方程给出其随时间的演变,综合考虑了电力的驱动和表面张力的传播．非线性界面的增长包括:波幅方程的增长,以及在准对规律方向上,一维结构的叠合．模式的选择由亚临界不稳定性机理确定,高分子薄膜的相对厚度在其中起着决定性的作用．     

Homotopy analysis method for MHD viscoelastic fluid flow and heat transfer in a channel with a stretching wall

In this paper, we analyze the flow and heat transfer characteristics of a magnetohydrodynamic (MHD) viscoelastic fluid in a parallel plate channel with a stretching wall. Homotopy analysis method (HAM) is used to obtain analytical solutions of the governing nonlinear differential equations. The analytical solutions are obtained in the form of infinite series and the convergence of the series solution is discussed explicitly. The obtained results are presented through graphs for several sets of values of the parameters, and the salient features of the solutions are analyzed. A comparison of our HAM results (for a special case of the study) with the available results in the literature (obtained by other methods) shows that our results are accurate for a wide range of parameters. Further, we point that our analysis finds application to the study of the haemodynamic flow of blood in the cardiovascular system subject to external magnetic field.     

Galerkin methods in time for semi-discrete viscoelastodynamics

In semi-discrete nonlinear elastodynamics, higher order energy and momentum conserving time stepping schemes turned out to be well suited for computing long time motions [1]. In comparison to standard ODE integrators, conserving schemes exhibit superior stability properties which are of utmost importance in a nonlinear .nite element framework. We show that conserving schemes are particularly well suited as starting point for the development of energy consistent schemes for dissipative dynamical systems. In particular, viscoelastic material behaviour is considered. A key advantage of energy consistent schemes lies in the fact that the equilibrium state of viscoelastic systems can be de.nitely reached, independent of the material parameters. In the paper, we compare two Galerkin methods for the temporal discretisation of semi-discrete nonlinear viscoelastodynamics: the standard continuous Galerkin (cG) method and an enhanced continuous Galerkin (eG) method which ful.ls the total energy balance exactly. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

轴向运动三参数黏弹性梁弱受迫振动的渐近分析

研究了轴向运动三参数黏弹性梁的弱受迫振动．建立了轴向运动三参数黏弹性梁受迫振动的控制方程．使用多尺度法渐近分析了运动梁的稳态响应,导出了解稳定性边界方程、稳态振幅的表达式以及稳态响应非零解的存在条件．依据Routh-Hurwitz定律决定了非线性稳态响应非零解的稳定性．     

Global existence of solutions to a viscoelastic non-degenerate Kirchhoff equation

ABSTRACT

In this paper, a nonlinear viscoelastic kirchhoff equation in a bounded domain with a time varying delay in the weakly nonlinear internal feedback is considered, where the global existence of solutions in suitable Sobolev spaces by means of the energy method combined with Faedo–Galarkin procedure is proved with respect to the condition of the weight of the delay term in the feedback and the weight of the term without delay and the speed of delay. Furthermore, a general stability estimate using some properties of convex functions is given.     

On the stability of fluid motions

The fundamental ideas concerning the linear and nonlinear stability of fluid motions are presented. The thermal convection in a layer of a dielectric fluid subject to an alternative current is studied.     

Nonlinear Rayleigh-Taylor instability in magnetic fluids with uniform horizontal and vertical magnetic fields

A weakly nonlinear evolution of two dimensional wave packets on the surface of a magnetic fluid in the presence of an uniform magnetic field is presented, taking into account the surface tension. The method used is that of multiple scales to derive two partial differential equations. These differential equations can be combined to yield two alternate nonlinear Schrödinger equations. The first equation is valid near the cutoff wavenumber while the second equation is used to show that stability of uniform wave trains depends on the wavenumber, the density, the surface tension and the magnetic field. At the critical point, a generalized formulation of the evolution equation governing the amplitude is developed which leads to the nonlinear Klein-Gordon equation. From the latter equation, the various stability crteria are obtained.     

An alternative linearization approach applicable to hysteretic systems

In this paper a method is proposed for equivalent linearization of nonlinear restoring forces being governed by differential equations in weakly nonlinear systems. These types of restoring forces cannot be linearized by employing conventional approximate approaches. Two analytical examples are used to show the accuracy of the proposed method. The application of the method to hysteretic systems is examined by constructing equivalent linear representation for Bouc–Wen model in its general formulation. Numerical investigations reveal that the proposed method is efficient in dynamic behavior analysis of weakly nonlinear hysteretic systems.     

Differential equations having rapidly changing solutions: Analytic methods for weakly nonlinear systems

Initial value problems for weakly nonlinear systems of differential equations are studied here. A perturbation result is obtained which is shown to give a valid approximation to the solution. The approximation is determined by solving two associated problems, and it is especially derived for problems which have rapidly oscillating and rapidly decaying solutions.