Long-wave oscillatory Marangoni instability in a horizontal liquid layer

The long-wave instability in the problem of thermocapillary convection in a horizontal layer with a free deformable boundary and a solid bottom is investigated. The transcendental equation for the main asymptotic term of the spectral parameter is written in explicit form. The main attention is paid to investigating oscillatory instability. For the frequency of neutral oscillations, simple transcendental equations are obtained that contain the Prandtl and Biot numbers. In a number of cases, exact solutions are indicated. Explicit formulae are given for the main asymptotic term of the Marangoni number. In the case of a non-heat-conducting solid wall, the relation between the critical values of the parameters for inverse Prandtl numbers is found. It is shown that, for different Prandtl numbers, the asymptotic values are in good agreement with the numerical values.     

The flow pattern in a liquid layer and the spectrum of the boundary-value problem when the surface tension depends non-linearly on the temperature

The flow of a liquid in a plane channel on the bottom of which a specified temperature distribution is maintained while the free surface is thermally isolated is considered. The surface tension of the liquid depends quadratically on the temperature. The system of Navier-Stokes and heat conduction equations possess a self-similar solution which leads to the non-linear eigenvalue problem of finding the flow temperature fields in the channel. The spectrum of this problem is investigated analytically for low Marangoni numbers (the second approximation) and numerically in the limiting case of an ideally heat conducting liquid for any Marangoni number. The pattern of the thermocapillary flow in the layer is analysed as a function of the parameter values. The non-uniqueness of the solution, which is typical for problems of this kind, is established. The results are compared with those obtained previously in the first approximation with respect to the Marangoni number.     

Thermal convection problem of micropolar fluid subjected to hall current

Thermal instability of a micropolar fluid layer heated from below in the presence of hall currents is investigated. Using the appropriate boundary conditions on the boundary surfaces of the fluid layer, the frequency equation is derived and then critical Rayleigh number is determined. It is found that hall current parameter has destabilizing effect on the system. For specific values of parameters, oscillatory convection in observed in the system. The behavior of Rayleigh number with wavenumber is also computed for different values of various parameters. The results of some earlier workers have been reduced as a special case from the present problem.     

基于非局部理论的黏弹性纳米杆轴向振动与波传播研究

根据非局部理论和Kelvin黏弹性理论，针对黏弹性纳米杆自由振动和波传播的轴向动力学问题进行研究.首先，推导了黏弹性纳米杆的轴向动力学微分控制方程.然后，通过无量纲化讨论了3种典型边界纳米杆的前三阶振动特性.最后，研究黏弹性纳米杆波的传播问题，导出了圆频率、波速与波数之间的关系.数值结果表明，非局部效应使第一、二阶固有频率持续减小，第三阶频率先增大再减小，出现结构刚度削弱和增强两种趋势.特别地，对于自由端存在集中质量的情形，第二阶频率随着黏性系数增大出现了多值情况，易导致杆件失稳.数值算例还说明了非局部效应的增强可有效降低黏性材料的阻尼效应，产生逃逸频率，使得纵波能够在高波数段传播.另外，黏性系数在低波数段对阻尼比影响可忽略不计，而在高波数段下，黏性系数越大则阻尼比越大.     

非均匀变厚度梁的动力响应的一般解

在本文中提出一个新方法——阶梯折算法来研究在任意载荷下任意非均匀和任意变厚度伯努利-欧拉梁的动力响应问题.研究了自由振动和强迫振动.新方法需要将区间离散为一定数目的元素,每个元素可看作是均匀和等厚度的.因此均匀、等厚度梁的一般解可在每个元素上应用.然后用初参数表示的整个梁的一般解使之满足相邻二元素间的物理和几何连续条件,这样就可以得到解析形式的自由振动的频率方程和解析形式的强迫振动的最终解,它化为求解二元线性代数方程,与离散元素的数目无关.现在的方法可推广应用至任意非均匀及任意变厚度有粘滞性和其他种类的梁以及其他结构元件问题上去.     

Self-similar solutions describing unsteady thermocapillary fluid flows

Unsteady thermocapillary flows in thin layers and layers of infinite thickness with non-uniform heating of the free boundary are investigated at high Marangoni numbers. In the plane and axially symmetric cases, self-similar solutions of the non-linear boundary-layer equations are constructed and asymptotic formulae are presented. It is shown that the self-similar solutions may be non-unique for certain values of the parameters of the problem. The branching points are calculated numerically and the branched solutions are investigated.     

Stability of thermocapillary-driven flow in rectangular cavities

The flow in an infinitely extended cavity with rectangular cross section is considered. The upper boundary is a free surface where thermocapillary forces are driving a circulation within the cavity (Marangoni effect). If the driving force (temperature difference) exceeds a critical threshold, the two dimensional flow becomes unstable towards an infinitesimal three-dimensional perturbation. The critical mode depends on the aspect ratio and may be steady or oscillatory. Results are presented for vanishing Prandtl number and varying aspect ratios of the cavity. These are compared to the classical lid-driven cavity problem and the mechanism leading to instability is discussed. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bending and vibration of an electrostatically actuated circular microplate in presence of Casimir force

The pull-in instability and the vibration for a prestressed circular electrostatically actuated microplate are investigated in consideration of the Casimir force. Based on von Kármán’s nonlinear bending theory of thin plates, the governing equations for the whole analysis are decomposed into two two-point boundary value problems. For static deformation of the plate, the geometric nonlinearity is involved and the pull-in parameters are obtained by using the shooting method through taking the applied voltage or Casimir parameter as an unknown. This algorithm is also used to study the small amplitude free vibration about the predeformed bending configuration following an assumed harmonic time mode, and the variation of the prestress and Casimir parameters dependent fundamental natural frequency with the applied voltage is presented. Several case studies are compared with available published simulations to confirm the proposed method. The influences of various parameters, such as the initial gap-thickness ratio, Casimir effect, prestress on the pull-in instability behavior and the natural frequency are examined.     

Periodic motions of a reversible second-order mechanical system: Application to the Sitnikov problem

A theory of the symmetric periodic motions (SPMs) of a reversible second-order system is presented which covers both oscillations and rotations. The structural stability property of the generating autonomous reversible system, which lies in the fact that the presence or absence of SPMs in a perturbed system is independent of the actual form of the “reversible” perturbations, is established. Both the case of the generation of SPMs from the family of SPMs of the generating system and birth cycle from the equilibrium state are investigated. Criteria of Lyapunov stability in a non-degenerate situation are obtained for the SPMs which are generated (in case of small values of the parameter). A method is proposed for constructing and investigating the Lyapunov stability of all the SPMs. The conditions for the existence of a cycle (symmetric and asymmetric) in the neighbourhood of a support “almost” resonance SPM are established for all cases of resonances. The theoretical results are applied to a study of the motion of a particle along a straight line which passes through the centre of mass of the system perpendicular to the plane of the identical attracting and simultaneously radiating main bodies (an extension of the Sitnikov problem) in the photogravitational version of the three-body problem. The circular problem is analysed and two different series of families of SPMs are found in the weakly elliptic problem. The instability of the equilibrium state is proved in the case of parametric resonance and the stability (and instability) domains are distinguished for arbitrary values of the eccentricity. All the SPMs with a period of 2π are constructed and the property of Lyapunov stability is investigated for these motions.     

The dynamic stability of an elastic column

Lyapunov's second method is used to investigate the stability of the rectilinear equilibrium modes of a non-linearly elastic thin rod (column) compressed at its end. Stability here is implied relative to certain integral characteristics, of the type of norms in Sobolev spaces; the analysis is carried out for all values of the problem parameter except the bifurcation values.

The realm of problems connected with the Lagrange-Dirichlet equilibrium stability theorem and its converse involves specific difficulties when considered in the infinite-dimensional case: stability in infinite-dimensional systems is investigated relative to certain integral characteristics such as norms /1/, and as the latter may be chosen with a certain degree of arbitrariness, different choices may result in different stability results. On the other hand, there is no relaxation of any of the difficulties encountered in the case of a finite number of degrees of freedom.

We shall consider a certain natural mechanical system with a finite number of degrees of freedom. If the first non-trivial form of the potential energy expansion is positive-definite, the equilibrium position is stable. A similar statement has been proved for infinitely many dimensions as well /1–3/, using Lyapunov's direct method, and the total energy may play the role of the Lyapunov function.

The situation with respect to instability is more complex. In the finite-dimensional case, if the first non-trivial form of the potential energy expansion may take negative values, instability may be demonstrated in many cases by means of a function proposed by Chetayev in /4/. A general theorem has been proved /1/ for instability in infinitely many dimensions, relying on an analogue of Chetayev's function. Such functions have also been used /5, 6/ to prove the instability of equilibrium in specific linear systems with an infinite number of degrees of freedom.

However, Chetayev's functions /4/ are not suitable tools to prove the instability of equilibrium in most non-linear systems. Another “Chetayev function”, which is actually a perturbed form of Chetayev's original function from /4/, has been proposed /7/, and it has been used to prove instability when the equilibrium position is an isolated critical point of the first non-trivial form of the potential energy expansion.

The majority of problems concerning the onset of instability of equilibrium configurations of elastic systems have been considered from a quasistatic point of view (see, e.g., /8, 9/). Problems of elastic stability and instability were considered in a dynamical setting in /2, 5/, where stability was investigated by Lyapunov's direct method. However, most of the results obtained in this branch of the field concern linear systems, and there are extremely few publications dealing with the onset of instability in non-linear elastic systems using Lyapunov's direct method. This is because in an unstable elastic system the quadratic part of the potential energy may change sign, and therefore the analogues of Chetayev's function from /4/ are not usually suitable for solving these problems. Dynamic instability has been studied or a specific non-linearly elastic system /10/, with the fact of instability established by using an analogue of the Chetayev function from /7/.

This paper presents one more example of a study of dynamic instability crried out for a non-linearly elastic system by Lyapunov's direct method.     

On the linear stability analysis of a Maxwell fluid with double-diffusive convection

The problem of double-diffusive convection and cross-diffusion in a Maxwell fluid in a horizontal layer in porous media is re-examined using the modified Darcy–Brinkman model. The effect of Dufour and Soret parameters on the critical Darcy–Rayleigh numbers is investigated. Analytical expressions of the critical Darcy–Rayleigh numbers for the onset of stationary and oscillatory convection are derived. Numerical simulations show that the presence of Dufour and Soret parameters has a significant effect on the critical Darcy–Rayleigh number for over-stability. In the limiting case some previously published results are recovered.     

Unsteady Marangoni convection heat transfer of fractional Maxwell fluid with Cattaneo heat flux

Fractional shear stress and Cattaneo heat flux models are introduced in characterizing unsteady Marangoni convection heat transfer of viscoelastic Maxwell fluid over a flat surface. Governing equations and boundary condition are formulated firstly via the balance between the surface tension and shear stress. Numerical solutions are obtained by new developed numerical technique and some novel phenomena are found. Results shown that the fractional derivative parameters, Marangoni number and power law exponent have significant influence on characteristics velocity and temperature fields. As fractional derivative parameters increase, the temperature profiles rise remarkably and the viscoelastic effects of the fluid enhance with delayed response to surface tension, however the temperature profiles decline significantly with a thinner thickness of thermal boundary layer with the increase of Marangoni number. The average skin friction coefficient increases with the augment of Marangoni number, while the average Nusselt number decreases for larger values of power law exponent.     

轴向变速运动弦线的非线性振动的稳态响应及其稳定性

研究具有几何非线性的轴向运动弦线的稳态横向振动及其稳定性．轴向运动速度为常平均速度与小简谐涨落的叠加．应用Hamilton原理导出了描述弦线横向振动的非线性偏微分方程．直接应用于多尺度方法求解该方程．建立了避免出现长期项的可解性条件．得到了近倍频共振时非平凡稳态响应及其存在条件．给出数值例子说明了平均轴向速度、轴向速度涨落的幅值和频率的影响．应用Liapunov线性化稳定性理论,导出倍频参数共振时平凡解和非平凡解的不稳定条件．给出数值算例说明相关参数对不稳定条件的影响．     

Magnetohydrodynamic stationary and oscillatory convective stability in a mushy layer during binary alloy solidification

For the case of solidification of a bottom cooled binary alloy, the magnetohydrodynamic stationary and oscillatory convective stability in the mushy layer is investigated analytically using normal mode linear stability analysis. In the limit of large Stefan number (S_t), a near–eutectic approximation with large far field temperature is considered in the present research. To ascertain the instability in the mushy layer, the strength of the superimposed magnetic field is so chosen that it corresponds to a given mush Hartmann number (Ha_m) of the problem. The results are presented for various values of mush Hartmann numbers in the range, 0 ≤ Ha_m ≤ 50. The critical Rayleigh number for stationary convection shows a linear relationship with increasing Ha_m. The magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of the stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of β (β = S_t/℘² ϒ², ℘ is the compositional ratio, ϒ = 1 + S_t/℘), and vice versa for lower β values. Analogous to the behavior for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer.     

Time-dependent flows of rotating and stratified fluids in geometries with non-uniform cross-sections

Unsteady rotating and stratified flows in geometries with non-uniform cross-sections are investigated under Oseen approximation using Laplace transform technique. The solutions are obtained in closed form and they reveal that the flow remains oscillatory even after infinitely large time. The existence of inertial waves propagating in both positive and negative directions of the flow is observed. When the Rossby or Froude number is close to a certain infinite set of critical values the blocking and back flow occur and the flow pattern becomes more and more complicated with increasing number of stagnant zones when each critical value is crossed. The analogy that is observed in the solutions for rotating and stratified flows is also discussed.     

Nonlinear Marangoni instability in dielectric superposed fluids

Summary The nonlinear Marangoni instability of two dielectric superposed fluids is investigated. The system is stressed by a normal electric field such that it allows for the presence of surface charges at the interface. The method of multiple scale perturbations is used in order to obtain uniformly valid expansions. Two nonlinear Schrödinger equations describing the perturbed system are obtained. One of these equations is used to describe analytically and numerically the necessary conditions for stability and instability near the marginal state, while the other equation is used to obtain the nonlinear electrohydrodynamic cutoff wavenumber separating stable and unstable disturbances for the system.     

Justification of the Ginzburg–Landau approximation for an instability as it appears for Marangoni convection

The Ginzburg–Landau equation appears as a universal amplitude equation for spatially extended pattern forming systems close to the first instability. It can be derived via multiple scaling analysis for the Marangoni convection problem that is driven by temperature‐dependent surface tension and is the subject of our interest. In this paper, we prove estimates between this formal approximation and true solutions of a scalar pattern forming model problem showing the same spectral picture as the Marangoni convection problem in case of a thin fluid. The new difficulties come from neutral modes touching the imaginary axis for the wave number k = 0 and from identical group velocities at the critical wave number k = k_c and the wave number k = 0. The problem is solved by using the reflection symmetry of the system and by using the fact that the modes concentrate at integer multiples of the critical wave number k = k_c. The paper presents a method that is applicable whenever this kind of instability occurs. Copyright © 2012 John Wiley & Sons, Ltd.     

A qualitative investigation of the oscillations of a pendulum with a periodically varying length and a mathematical model of a swing

The behaviour of the amplitude-frequency characteristics of families of periodic solutions, produced from the equilibrium position of a system, is established by a qualitative investigation of the equation of the oscillations of a pendulum, the length of which is an arbitrary periodic function of time. The non-local conditions for their stability and instability, expressed in terms of the amplitude and frequency of the oscillations, are obtained. The results are used when discussing the parametric and self-excited oscillatory model of a swing. In the parametric model the length of a swing is a specified periodic function of time, and in the self-excited oscillatory model it is a function of the phase coordinates of the system. For an appropriate choice of these functions, both systems have a common periodic solution. It is shown that the parametric model leads to an erroneous conclusion regarding the instability of the periodic mode, which is in fact realized in the oscillations of a swing, whereas the self-excited oscillatory model indicates its stability.     

Effects of bearing clearance on the chatter stability of milling process

In the present study, the influences of the bearing clearance, which is a common fault for machines, to the chatter stability of milling process are examined by using numerical simulation method. The results reveal that the presence of bearing clearance could make the milling process easier to enter the status of chatter instability and can shift the chatter frequency. In addition, the spectra analysis to vibration signals obtained under the instable milling processes show that the presence of bearing clearance could introduce more frequency components to the vibration responses but, however, under both the stable and instable milling processes, the generated frequency components will not violate the ideal spectra structures of the vibration responses of the milling process, which are usually characterized by the tooth passing frequency and its associated higher harmonics for the stable milling process and by the complex coupling of the tooth passing frequency and the chatter frequency for the instable milling process. This implies that, even under the case with bearing clearance fault, the stability of the milling process can still be determined by viewing the frequency spectra of the vibration responses. Moreover, the phenomena of the chatter frequency shift and the generation of more components provide potential ways to detect the bearing clearance in machines.     

Oscillatory convection and the Cattaneo law of heat conduction

The problem of thermal convection is investigated for a layer of fluid when the heat flux law of Cattaneo is adopted. The boundary conditions are those appropriate to two fixed surfaces. It is shown that for small Cattaneo number the critical Rayleigh number initially increases from its classical value of 1707.765 until a critical value of the Cattaneo number is reached. For Cattaneo numbers greater than this critical value a notable Hopf bifurcation is observed with convection occurring at lower Rayleigh numbers and by oscillatory rather than stationary convection. The aspect ratio of the convection cells likewise changes.