Construction of Dynamically Equivalent Time-Invariant Forms for Time-Periodic Systems

In this study dynamically equivalent time-invariant forms are obtained for linear and non-linear systems with periodically varying coefficients via Lyapunov–Floquet (L–F) transformation. These forms are equivalent in the sense that the local stability and bifurcation characteristics are identical for both systems in the entire parameter space. It is well known that the L–F transformation converts a linear periodic first order system into a time-invariant one. In the first part of this study a set of linear second order periodic equations is converted into an equivalent set of time-independent second order equations through a sequence of linear transformations. Then the transformations are applied to a time-periodic quadratic Hamiltonian to obtain its equivalent time-invariant form. In the second part, time-invariant forms of nonlinear equations are studied. The application of L–F transformation to a quasi-linear periodic equation converts the linear part to a time-invariant form and leaves the non-linear part with time-periodic coefficients. Dynamically equivalent time-invariant forms are obtained via time-periodic center manifold reduction and time-dependent normal form theory. Such forms are constructed for general hyperbolic systems and for some simple critical cases, including that of one zero eigenvalue and a purely imaginary pair. As a physical example of these techniques, a single and a double inverted pendulum subjected to periodic parametric excitation are considered. The results thus obtained are verified by numerical simulation.     

Stability of a Liquid Film Flowing Down an Oscillating Inclined Surface

The stability of flow of a liquid film along an inclined plate subject to periodic oscillations under the action of the gravity force is investigated with allowance for the surface tension. An equation of the Orr-Sommerfeld type with time-periodic coefficients is used. A method for determining the eigenvalues of the linear stability problem is developed on the basis of Floquet theory, spectral representation of the variables, and multistep methods of integration of ordinary differential equations. The bifurcation spectrum of the resonance modes is investigated, and the amplification coefficients and phase velocities are calculated for the surface waves, Tollmien-Schlichting waves, and resonance waves. The influence of external parameters, namely, the inclination, the surface tension, and the layer thickness, on the resonance modes and the steady-state flow modes is studied.     

Nonlinear Stability of Time-Periodic Viscous Shocks

In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green’s distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.     

Effect of Thermal Modulation on the Onset of Convection in Walters B Viscoelastic Fluid-Saturated Porous Medium

The linear stability of Walters B viscoelastic fluid-saturated horizontal porous layer is examined theoretically when the walls of the porous layer are subjected to time-periodic temperature modulation. Three types of boundary temperature modulations are considered namely, symmetric, asymmetric, and only the lower wall temperature is modulated while the upper wall is held at constant temperature. A regular perturbation method based on small amplitude of applied temperature field is used to compute the critical values of Rayleigh number and the corresponding wave number. The shift in critical Rayleigh number is calculated as a function of modulation frequency, viscoelastic parameter, and Prandtl number. The effect of all three types of modulations is found to be destabilizing as compared to the unmodulated system. This result is in contrast to the system with other types of fluids. Besides, the influence of physical parameters on the control of convective instability of the system is discussed.     

The Onset of Convection in a Strongly Heterogeneous Porous Medium with Transient Temperature Profile

The effect of heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a nonuniform basic temperature gradient resulting from transient heating, is studied analytically using linear stability theory for the case of strong heterogeneity. Two particular situations, corresponding to instantaneous bottom heating and constant-rate bottom heating, are studied. Estimates of the timescale for the development of convection instability are obtained. The case of a strongly nonlinear temperature gradient is studied with the help of a computer package.     

双自由面溶质?热毛细液层的不稳定性

溶质?热毛细对流是流体界面的浓度和温度分布不均导致的表面张力梯度驱动的流动, 它主要存在于空间微重力环境、小尺度流动等表面张力占主导的情况中, 例如晶体生长、微流控、合金浇筑凝固、有机薄液膜生长等. 对其流动进行稳定性分析具有重要意义. 本文采用线性稳定性理论研究了双自由面溶质?热毛细液层对流的不稳定性, 得到了两种负毛细力比(η)下的临界Marangoni数与Prandtl数(Pr)的函数关系, 并分析了临界模态的流场和能量机制. 研究发现: 溶质?热毛细对流和纯热毛细对流的临界模态有较大的差别, 前者是同向流向波、逆向流向波、展向稳态模态和逆向斜波, 后者是逆向斜波和逆向流向波. 在Pr较大时, Pr增加会降低流动稳定性; 在其他参数下, Pr增加会增强流动稳定性. 在中低Pr, 溶质毛细力使流动更加不稳定; 在大Pr时, 溶质毛细力的出现可能使流动更加稳定; 在其他参数下, 溶质毛细力会减弱流动稳定性. 流动稳定性不随η单调变化. 在多数情况下, 扰动浓度场与扰动温度场都是相似的. 能量分析表明: 扰动动能的主要能量来源是表面张力做功, 但其中溶质毛细力和热毛细力做功的正负性与参数有关.      

The Onset of Convection in a Heterogeneous Porous Medium with Transient Temperature Profile

The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from transient heating or otherwise, is studied analytically using linear stability theory. Two particular situations, corresponding to instantaneous bottom heating and constant-rate bottom heating, are studied. Estimates of the timescale for the development of convection instability are obtained.     

Normal Forms and the Structure of Resonance Sets in Nonlinear Time-Periodic Systems

The structure of time-dependent resonances arising in themethod of time-dependent normal forms (TDNF) for one andtwo-degrees-of-freedom nonlinear systems with time-periodic coefficientsis investigated. For this purpose, the Liapunov–Floquet (L–F)transformation is employed to transform the periodic variationalequations into an equivalent form in which the linear system matrix istime-invariant. Both quadratic and cubic nonlinearities are investigatedand the associated normal forms are presented. Also, higher-orderresonances for the single-degree-of-freedom case are discussed. It isdemonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2,...) solutions. The discussion is limited to the Hamiltonian case (which encompasses allpossible resonances for one-degree-of-freedom). Furthermore, it is alsoshown how a recent symbolic algorithm for computing stability andbifurcation boundaries for time-periodic systems may also be employed tocompute the time-dependent resonance sets of zero measure in theparameter space. Unlike classical asymptotic techniques, this method isfree from any small parameter restriction on the time-periodic term inthe computation of the resonance sets. Two illustrative examples (oneand two-degrees-of-freedom) are included.     

Stability of a layer of viscoelastic fluid heated from below

The polymerization of methyl methacrylate is accompanied by liberation of heat; this results in overheating of the reaction mass during production of plastics. The temperature distribution in the polymerizing layer is complicated by convection, which disrupts the natural temperature field. Thus, in addition to the stress along the sheet, local internal stresses appear that show up in operation of the product. Product quality and intensification of the polymerization process depend on the critical temperature gradient, which determines the stability threshold of the layer of polymerizing methyl methacrylate. The Rayleigh-Jeffrey problem is considered for a weak viscoelastic fluid described by an integral rheological constitutive relationship. The critical Rayleigh numbers are determined for stationary and oscillatory instabilities with free and ideally heat-conducting rigid boundaries.     

Double diffusive convection in a porous medium with modulated temperature on the boundaries

The effect of temperature modulation on the onset of double diffusive convection in a sparsely packed porous medium is studied by making linear stability analysis, and using Brinkman-Forchheimer extended Darcy model. The temperature field between the walls of the porous layer consists of a steady part and a time dependent periodic part that oscillates with time. Only infinitesimal disturbances are considered. The effect of permeability and thermal modulation on the onset of double diffusive convection has been studied using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as a function of frequency and amplitude of modulation, Vadasz number, Darcy number, diffusivity ratio, and solute Rayleigh number. Stabilizing and destabilizing effects of modulation on the onset of double diffusive convection have been obtained. The effects of other parameters are also discussed on the stability of the system. Some results as the particular cases of the present study have also been obtained. Also the results corresponding to the Brinkman model and Darcy model have been compared.     

Instability of Hadley–Prats Flow with Viscous Heating in a Horizontal Porous Layer

The onset of convective rolls instability in a horizontal porous layer subject to a basic temperature gradient inclined with respect to gravity is investigated. The basic velocity has a linear profile with a non-vanishing mass flow rate, i.e., it is the superposition of a Hadley-type flow and a uniform flow. The influence of the viscous heating contribution on the critical conditions for the onset of the instability is assessed. There are four governing parameters: a horizontal Rayleigh number and a vertical Rayleigh number defining the intensity of the inclined temperature gradient, a Péclet number associated with the basic horizontal flow rate, and a Gebhart number associated with the viscous dissipation effect. The critical wave number and the critical vertical Rayleigh number are evaluated for assigned values of the horizontal Rayleigh number, of the Péclet number, and of the Gebhart number. The linear stability analysis is performed with reference either to transverse or to longitudinal roll disturbances. It is shown that generally the longitudinal rolls represent the preferred mode of instability.     

Stability of thermocapillary advective flow in a slowly rotating liquid layer under microgravity conditions

The stability of thermocapillary flow developed in a slowly rotating fluid layer under microgravity conditions is investigated. Both boundaries of the layer are free and assumed to be plane. The tangential thermocapillary Marangoni force exerts on the boundaries, where heat transfer takes place in accordance with the Newton law, the temperature of the medium in the neighborhood of the boundaries being a linear function of the coordinates. The axis of rotation is perpendicular to the liquid layer, rotation is weak so that the centrifugal force can be neglected. Being the solution of the Navier-Stokes equations, the thermocapillary flow in question can be described analytically. The neutral curves which describe the wavenumber dependence of the critical Marangoni number for various Taylor numbers and various directions of the horizontal temperature gradient on the layer boundaries are obtained within the framework of the linear stability theory. The behavior of finite-amplitude perturbations beyond the stability threshold is studied numerically.     

A model for convection in the evolution of under-ice melt ponds

A model for convection in the evolution of under-ice melt ponds is presented. The system exhibits two competing effects namely, a temperature gradient which is destabilising and a salt gradient which is stabilising. Density is assumed to have a dependence quadratic in temperature and linear in concentration. A linear instability analysis and a nonlinear stability analysis are performed. The standard energy method does not yield unconditional stability so a weighted energy analysis is employed to achieve global results. The global stability bound is found to be independent of the salt field and a presentation of the region of possible subcritical instabilities is given. Received May 16, 2002 / Published online September 4, 2002 RID="a" ID="a" e-mail: Magdalen.Carr@durham.ac.uk Communicated by Brian Straughan, Durham     

Stability Analysis of a Falling Film Flow Down a Plane with Sinusoidal Corrugations

The liquid viscous film falling down a vertical wall with sinusoidal relief is considered. The linear stability of steady-state flow with respect to time-periodic disturbances is studied using the Floquet theory. It is shown that in the case of applying corrugations the variation in the disturbance growth rate is proportional to the second power of their undulations. Depending on the relief parameters there exist two possibilities: the instability domain can expand or certain disturbances can be stabilized. The growth rates are obtained numerically and analytically in the approximation of low-amplitude corrugations. The development of waves from small disturbances is simulated within the framework of nonlinear equations and the formation of structures whose wavelength is significantly greater than the space relief period is found out.     

Microscope activation near the wall during explosive boiling

The inception process of nucleation in explosive boiling systems is theoretically investigated. With the effect of pulse heating or sudden cooling, the temperature distribution near the surface during explosive boiling is calculated. The liquid near the wall can maintain a stable layer induced by strong attractive force, and there exists maximum supersaturation beyond this stable layer. As the surface temperature and temperature gradient are high enough, the critical distance of maximum supersaturation can be larger than the radius of critical bubble, and the homogeneous nucleation will dominate the inception boiling process. For explosive boiling induced by pulse heating, homogeneous nucleation forms after a short time; while homogeneous nucleation can dominate the initial explosive boiling induced by sudden cooling.     

Fluid convection in a rotating porous layer under modulated temperature on the boundaries

The linear stability of thermal convection in a rotating horizontal layer of fluid-saturated porous medium, confined between two rigid boundaries, is studied for temperature modulation, using Brinkman’s model. In addition to a steady temperature difference between the walls of the porous layer, a time-dependent periodic perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. The combined effect of rotation, permeability and modulation of walls’ temperature on the stability of flow through porous medium has been investigated using Galerkin method and Floquet theory. The critical Rayleigh number is calculated as function of amplitude and frequency of modulation, Taylor number, porous parameter and Prandtl number. It is found that both, rotation and permeability are having stabilizing influence on the onset of thermal instability. Further it is also found that it is possible to advance or delay the onset of convection by proper tuning of the frequency of modulation of the walls’ temperature.     

Linear stability analysis of supersonic axisymmetric jets

Stabilities of supersonic jets are examined with different velocities, momentum thicknesses, and core temperatures. Amplification rates of instability waves at inlet are evaluated by linear stability theory (LST). It is found that increased velocity and core temperature would increase amplification rates substantially and such influence varies for different azimuthal wavenumbers. The most unstable modes in thin momentum thickness cases usually have higher frequencies and azimuthal wavenumbers. Mode switching is observed for low azimuthal wavenumbers, but it appears merely in high velocity cases. In addition, the results provided by linear parabolized stability equations show that the mean-flow divergence affects the spatial evolution of instability waves greatly. The most amplified instability waves globally are sometimes found to be different from that given by LST.     

Free surface asymptotics for thermocapillary flow at high marangoni numbers

Formal asymptotic expansions of the solution of the steady-state problem of incompressible flow in an unbounded region under the influence of a given temperature gradient along the free boundary are constructed for high Marangoni numbers. In the boundary layer near the free surface the flow satisfies a system of nonlinear equations for which in the neighborhood of the critical point self-similar solutions are found. Outside the boundary layer the slow flow approximately satisfies the equations of an inviscid fluid. A free surface equation, which when the temperature gradient vanishes determines the equilibrium free surface of the capillary fluid, is obtained. The surface of a gas bubble contiguous with a rigid wall and the shape of the capillary meniscus in the presence of nonuniform heating of the free boundary are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1989.     

Effets d'une modulation en phase de température à la frontière sur l'instabilité convective d'une couche liquide viscoélastiqueEffects of a temperature modulation in phase at the frontier on the convective instability of a viscoelastic layer

In this Note we study the effects of the temperature modulation, applied at the horizontal boundaries, on the onset of convection of a horizontal liquid Maxwellian layer. It is assumed that the temperature imposed features a steady component and a time dependent component. To analyse the effect of the temperature modulation, the study is restricted to a linear stability analysis. Thus the Floquet theory and a technique of converting a boundary value problem to an initial value problem are used to solve the system of equations corresponding to the onset of convection. Results obtained may be used to characterize the influence of modulation effects and that of the viscoelastic nature of liquid on the critical Rayleigh number. To cite this article: B. Oukada et al., C. R. Mecanique 334 (2006).     

Local stability effects of plasma actuation on a zero pressure gradient boundary layer

The effects of plasma actuation in a flat plate boundary layer with zero pressure gradient have been simulated. Based on these simulations, non-dimensional parameters and a combined wall jet/boundary layer model of the velocity profile have been developed. A parametric study using local linear stability analysis has been performed to examine the hydrodynamic stability of the velocity profiles created through this model. Convective and absolute instability mechanisms are found to be important, some of which have not been previously documented. Neutral stability curves have been computed for the different instabilities, and when put in terms of the shape factor, they still compare favorably with reported canonical results, indicating that the critical Reynolds number is primarily a function of the shape factor. These results are also discussed in relation to existing experimental results as well as with respect to their implementation.