Weak Nonlinear Analysis of Moderate Stefan Number Oscillatory Convection in Rotating Mushy Layers

We consider the solidification of a binary alloy in a mushy layer subject to Coriolis effects. A near-eutectic approximation and large far-field temperature is employed in order to study the dynamics of the mushy layer with a Stefan number of unit order of magnitude. The weak nonlinear theory is used to investigate analytically the Coriolis effect in a rotating mushy layer for a new moderate time scale proposed by the author. It is found that increasing the Taylor number favoured the forward bifurcation.     

On the Linear Stability of Large Stefan Number Convection in Rotating Mushy Layers for a New Darcy Equation Formulation

The Coriolis effect on a solidifying mushy layer is considered. A near-eutectic approximation and large far-field temperature is employed in the current study for large Stefan numbers. The linear stability theory is used to investigate analytically the Coriolis effect on convection in a rotating mushy layer for a new formulation of the Darcy equation. It was found that a large Stefan number scaling allows for the presence of both the stationary and oscillatory modes of convection. In contrast to the problem of a stationary mushy layer, rotating the mushy layer has a stabilising effect on convection. It was observed that increasing the Taylor number or the Stefan number encouraged the oscillatory mode of convection.     

Moderate Time Scale Finite Amplitude Analysis of Large Stefan Number Convection in Rotating Mushy Layers

We investigate the steady state convection amplitude for solutal convection occurring during the solidification of a rotating mushy layer in a binary alloy system for a new Darcy equation formulation. We adopt a large far field temperature and assume that the initial composition is very close to the eutectic composition. The linear stability analysis showed that rotation stabilised solutal convection. The results of the weak non-linear analysis of stationary convection indicates the presence of Hopf bifurcation, associated with the oscillatory mode, developing at Ta = 3.     

Moderate Stefan Number Convection in Rotating Mushy Layers: A New Darcy Equation Formulation

The coriolis effect on a solidifying mushy layer is considered. A near-eutectic approximation and large far-field temperature is employed in the current study for moderate Stefan numbers. The linear stability theory is used to investigate analytically the Coriolis effect on convection in a rotating mushy layer for a new formulation of the Darcy equation. It was found that only stationary convection is possible for moderate Stefan numbers. In contrast to the problem of a stationary mushy layer, rotating the mushy layer has a stabilizing effect on convection. It was also discovered that fot Taylor numbers larger than three (i.e., Ta > 3),increasing the retardability coefficient (hence increasing the solid fraction) destablished the convection.     

Linear stability of solutal convection in solidifying mushy layers: permeable mush–melt interface

The linear stability theory is used to investigate analytically the effect of a permeable mush–melt boundary condition on the stability of solutal convection in a mushy layer of homogenous permeability at the near eutectic (solid) limit. The results clearly show that, in contrast to the impermeable mush–melt interface boundary condition, the application of the permeable mush–melt interface boundary condition destabilizes the convection in a mushy layer.     

On Three-Dimensional Non-linear Buoyant Convection in Ternary Solidification

We consider the problem of three-dimensional non-linear buoyant convection in ternary solidification. Under the limit of large far-field temperature, the convective flow is modeled to be in a rectangular cube composing of a horizontal liquid layer above a primary mushy layer, which itself is over a secondary mushy layer. We first apply linear stability analysis to calculate the conditions at the onset of motion. Next, we carry out weakly non-linear analyses to determine solutions in the form of hexagons and their possible stability and to obtain information about tendency for chimney formation. We find that if the flow is driven either from both mushy layers with equal critical conditions at the onset of motion or only by the primary mushy layer, then the flow can be in the form of a double-cell structure vertically with down-hexagons below or above up-hexagons. There is tendency for vertically oriented chimney formation at different horizontal locations in each mushy layer. For the cases where only the critical conditions at the onset of motion are equal in both mushy layers and depending on the values of the mush Rayleigh numbers, the flow can be subcritical (or supercritical) in both mushy layers or mixed subcritical in one layer and supercritical in another layer.     

Coriolis effect on convection for a low Prandtl number fluid

The problem of finite-amplitude thermal convection in a horizontal layer of a low Prandtl number heated from below and rotating about a vertical axis is studied. Linear stability and weak non-linear theories are used to investigate analytically the Coriolis effect on gravity-driven convection. The non-linear steady problem is solved by perturbation techniques, and the preferred mode of convection is determined by a stability analysis. Finite-amplitude results, obtained by using a weak amplitude, correspond to both stationary and oscillatory convections. These amplitude equations permit to identify from the post-transient conditions that the fluid is subject to Pitchfork bifurcation in the stationary convection and Hopf bifurcation in the oscillatory convection. Thereafter, in the small perturbations hypothesis, an amplitude solution is evaluated and drawn in time and space scales.     

Stability of Solutal Convection in a Gravity Modulated Mushy Layer During the Solidification of Binary Alloys

The linear stability theory is used to investigate analytically the effects of gravity modulation on solutal convection in the mushy layer of solidifying binary alloys. The gravitational field consists of a constant part and a sinusoidally varying part, which is synonymous to a vertically oscillating mushy layer subjected to constant gravity. The linear stability results are presented for both the synchronous and subharmonic solutions. It is demonstrated that up to the transition point between the synchronous and subharmonic regions, increasing the frequency of vibration rapidly stabilizes the solutal convection. Beyond the transition point, further increases in the frequency tend to destabilize the solutal convection, but gradually. It is also demonstrated that the effect of increasing the ratio of the Stefan number and the solid composition (₀) is to destabilize the solutal convection.     

Numerical Investigation on Marginal Stability and Convection with and without Magnetic Field in a Mushy Layer

In this article, an investigation is conducted to analyze the marginal stability with and without magnetic field in a mushy layer. During alloy solidification, such mushy layer, which is adjacent to the solidification front and composed of solid dendrites and liquid, is known to produce vertical chimneys. Here, we carry out numerical investigation for particular range of parameter values, which cover those of available experimental studies, to determine the convective flow at the onset of motion. The governing coupled non-linear partial differential equations are non-dimensionalised and solved to get the steady basic-state solution. The thickness of the mushy layer is determined as a part of the solution. Using multiple shooting technique, we determine the steady-state solutions in a range of critical Rayleigh number. We analyse the effect of several parameters, Chandrasekhar number Q, and Robert’s number τ on the problem. It was found that an increase in Q has a stabilizing effect on solidification and the critical Rayleigh number increases on increasing Q. It was also found that for moderate or small values of Robert’s number τ the critical Rayleigh number is mostly insensitive.     

Center manifold and multivariable approximants applied to non-linear stability analysis

This paper presents a research devoted to the study of instability phenomena in non-linear model with a constant brake friction coefficient. This paper outlines the stability analysis and a procedure to reduce and simplify the non-linear system, in order to obtain limit cycle amplitudes. The center manifold approach, the multivariable approximants theory, and the alternate frequency/time domain (AFT) method are applied. Brake vibrations, and more specifically heavy trucks grabbing are concerned. The modelling introduces sprag-slip mechanism based on dynamic coupling due to buttressing. The non-linearity is expressed as a polynomial with quadratic and cubic terms. This model does not require the use of brake negative coefficient, in order to predict the instability phenomena. Finally, the center manifold approach, the multivariable approximants, and the AFT method are used in order to obtain equations for the limit cycle amplitudes. These methods allow the reduction of the number of equations of the original system in order to obtain a simplified system, without loosing the dynamics of the original system, as well as the contributions of non-linear terms. The goal is the validation of this procedure for a complex non-linear model by comparing results obtained by solving the full system and by using these methods. The brake friction coefficient is used as an unfolding parameter of the fundamental Hopf bifurcation point.     

Bifurcation analysis for thermal convection in a rotating porous layer

The linear and weakly nonlinear thermal convection in a rotating porous layer is investigated by constructing a simplified model involving a system of fifth-order nonlinear ordinary differential equations. The flow in the porous medium is described by Lap wood-Brinkman-extended Darcy model with fluid viscosity different from effective viscosity. Conditions for the occurrence of possible bifurcations are obtained. It is established that Hopf bifurcation is possible only at a lower value of the Rayleigh number than that of simple bifurcation. In contrast to the non-rotating case, it is found that the ratio of viscosities as well as the Darcy number plays a dual role on the steady onset and some important observations are made on the stability characteristics of the system. The results obtained from weakly nonlinear theory reveal that, the steady bifurcating solution may be either sub-critical or supercritical depending on the choice of physical parameters. Heat transfer is calculated in terms of Nusselt number.     

Control of the homoclinic bifurcation in buckled beams: Infinite dimensional vs reduced order modeling

A method for controlling non-linear dynamics and chaos is applied to the infinite dimensional dynamics of a buckled beam subjected to a generic space varying time-periodic transversal excitation. The homoclinic bifurcation of the hilltop saddle is identified as the undesired dynamical event, because it triggers, e.g., cross-well scattered (possibly chaotic) dynamics. Its elimination is then pursued by a control strategy which consists in choosing the best spatial and temporal shape of the excitation permitting the maximum shift of the homoclinic bifurcation threshold in the excitation amplitude-frequency parameters space.The homoclinic bifurcation is detected by the Holmes and Marsden's theorem [A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Ration. Mech. Anal. 76 (1981) 135-165] constituting a generalization of the classical Melnikov's theory. Two classes of boundary conditions (b.c.) are identified: for the first, the Melnikov function is exactly the same as obtained with the reduced order models, while for the second, which is more general, this is no longer true, and the non-linear normal modes theory is used. Based on this distinction, the control method is then separately applied to the two cases, and the optimal spatial and temporal shapes of the excitation are determined.A detailed comparison of the infinite vs finite dimensional models is performed with respect to the control features, and it is shown that, depending on the b.c., the control based on the reduced order model provides either exact or engineering acceptable results, although more systematic investigations are required to generalize the last conclusion.     

多孔介质中二维对流传热的分叉

本文利用分叉理论研究了流体饱和的二维多孔介质从底部加热所引起的自然对流,用有限差分方法确定对流的分叉进程;揭示其模式转换机理及分叉对非正常流动图象形成的影响;同时确定了矩形截面宽高比与临界端利数的关系。还提出了一个判别分支稳定笥的简明方法。     

Low-Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells

This paper discusses the derivation of discrete low-dimensional models for the non-linear vibration analysis of thin shells. In order to understand the peculiarities inherent to this class of structural problems, the non-linear vibrations and dynamic stability of a circular cylindrical shell subjected to dynamic axial loads are analyzed. This choice is based on the fact that cylindrical shells exhibit a highly non-linear behavior under both static and dynamic axial loads. Geometric non-linearities due to finite-amplitude shell motions are considered by using Donnell’s nonlinear shallow shell theory. A perturbation procedure, validated in previous studies, is used to derive a general expression for the non-linear vibration modes and the discretized equations of motion are obtained by the Galerkin method. The responses of several low-dimensional models are compared. These are used to study the influence of the modelling on the convergence of critical loads, bifurcation diagrams, attractors and large amplitude responses of the shell. It is shown that rather low-dimensional and properly selected models can describe with good accuracy the response of the shell up to very large vibration amplitudes.     

Local and Global Transitions to Chaos and Hysteresis in a Porous Layer Heated from Below

The routes to chaos in a fluid saturated porous layer heated from below are investigated by using the weak nonlinear theory as well as Adomian's decomposition method to solve a system of ordinary differential equations which result from a truncated Galerkin representation of the governing equations. This representation is equivalent to the familiar Lorenz equations with different coefficients which correspond to the porous media convection. While the weak nonlinear method of solution provides significant insight to the problem, to its solution and corresponding bifurcations and other transitions, it is limited because of its local domain of validity, which in the present case is in the neighbourhood of any one of the two steady state convective solutions. On the other hand, the Adomian's decomposition method provides an analytical solution to the problem in terms of infinite power series. The practical need to evaluate numerical values from the infinite power series, the consequent series truncation, and the practical procedure to accomplish this task transform the otherwise analytical results into a computational solution achieved up to a finite accuracy. The transition from the steady solution to chaos is analysed by using both methods and their results are compared, showing a very good agreement in the neighbourhood of the convective steady solutions. The analysis explains previously obtained computational results for low Prandtl number convection in porous media suggesting a transition from steady convection to chaos via a Hopf bifurcation, represented by a solitary limit cycle at a sub-critical value of Rayleigh number. A simple explanation of the well known experimental phenomenon of Hysteresis in the transition from steady convection to chaos and backwards from chaos to steady state is provided in terms of the present analysis results.     

Stability properties of the vertical boundary layers in differentially heated cavities

In the present study, the two-dimensional (2-D) stability properties of the vertical boundary layers in a cavity that is differentially heated over two opposing vertical walls is considered. The study is performed by introducing artificial, controlled perturbations at the base of the vertical boundary layer along the hot cavity wall and by following the evolution of these disturbances. For small initial perturbations, the evolution is governed by linear effects. This method accurately predicts the frequency of the bifurcation, which occurs for (much) larger Rayleigh numbers. Convective instability sets in for Rayleigh numbers much smaller than those at which the absolute instability (i.e., the bifurcation) occurs, and these Rayleigh numbers are in reasonable agreement with those for the boundary layer along a plate. The absolute instability does not result from the first wave which becomes unstable. For small Prandtl numbers (≤ 2), the unstable waves which lead to the absolute instability are shear-driven, and a single frequency is introduced in the flow after the bifurcation. For larger Prandtl numbers, the unstable waves are buoyancy driven and no single-frequency unsteady flow is observed after the bifurcation.     

Bifurcation phenomena in the rigid inclusion power law matrix composite sphere

Bifurcation of interface separation related to cavity nucleation is analyzed for a radially loaded composite sphere consisting of a rigid inclusion separated from a power law matrix by a uniform, non-linear cohesive zone. Equations for the spherically symmetric and non-symmetric problems are obtained from a hyperelastic finite strain theory by a limiting process that preserves non-linear matrix and interface response at infinitesimal strain. A complete solution to the symmetric problem is presented including bifurcation load, stresses, and evolution of elasto-plastic boundary and interface separation. An analysis of non-symmetric bifurcation, under symmetric conditions of geometry and loading, yields the bifurcation load and first non-symmetric mode shape associated with rigid inclusion displacement. An energy analysis is carried out for both symmetric and non-symmetric problems in order to assess stability of spherically symmetric states to spherically symmetric and non-symmetric “rigid body mode” perturbations.Results are provided for an interface force law that captures interface failure in normal mode and linear response in shear mode. For the symmetric problem, (i) there are threshold parameter values above which bifurcation will generally not occur, (ii) threshold values below which there do not exist equilibria in the post bifurcation regime, (iii) bifurcation occurs after attainment of the maximum interface strength. For the non-symmetric problem, (i) bifurcation always occurs, although it can be delayed by interfacial shear, (ii) for the smooth interface, non-symmetric bifurcation occurs after attainment of the maximum interface strength and always precedes symmetric bifurcation.     

Non-linear vibrations of shallow circular cylindrical panels with complex geometry. Meshless discretization with the R-functions method

Geometrically non-linear forced vibrations of a shallow circular cylindrical panel with a complex shape, clamped at the edges and subjected to a radial harmonic excitation in the spectral neighborhood of the fundamental mode, are investigated. Both Donnell and the Sanders–Koiter non-linear shell theories retaining in-plane inertia are used to calculate the elastic strain energy. The discrete model of the non-linear vibrations is build using the meshfree technique based on classic approximate functions and the R-function theory, which allows for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries; Chebyshev orthogonal polynomials are used to expand shell displacements. A two-step approach is implemented in order to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for expanding the non-linear displacements. Lagrange approach is applied to obtain a system of ordinary differential equations on both steps. Different multimodal expansions, having from 15 up to 35 generalized coordinates associated with natural modes, are used to study the convergence of the solution. The pseudo-arclength continuation method and bifurcation analysis are applied to study non-linear equations of motion. Numerical responses are obtained in the spectral neighborhood of the lowest natural frequency; results are compared to those available in the literature. Internal resonances are also detected and discussed.     

In-plane stability of arches

Classical buckling theory is mostly used to investigate the in-plane stability of arches, which assumes that the pre-buckling behaviour is linear and that the effects of pre-buckling deformations on buckling can be ignored. However, the behaviour of shallow arches becomes non-linear and the deformations are substantial prior to buckling, so that their effects on the buckling of shallow arches need to be considered. Classical buckling theory which does not consider these effects cannot correctly predict the in-plane buckling load of shallow arches. This paper investigates the in-plane buckling of circular arches with an arbitrary cross-section and subjected to a radial load uniformly distributed around the arch axis. An energy method is used to establish both non-linear equilibrium equations and buckling equilibrium equations for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained. Approximations to the symmetric buckling of shallow arches and formulae for the in-plane anti-symmetric bifurcation buckling load of non-shallow arches are proposed, and criteria that define shallow and non-shallow arches are also stated. Comparisons with finite element results demonstrate that the solutions and indeed approximations are accurate, and that classical buckling theory can correctly predict the in-plane anti-symmetric bifurcation buckling load of non-shallow arches, but overestimates the in-plane anti-symmetric bifurcation buckling load of shallow arches significantly.     

Comparative study of amorphous and crystalline Zr-based alloys in response to nanosecond pulse laser ablation

In shock tube experiments, the interaction between the reflected shock and boundary layer can induce shock bifurcation and weak ignition. The weak ignition can greatly affect the ignition delay time measurement in a shock tube experiment. In this work, two-dimensional simulations considering detailed chemistry and transport are conducted to investigate the shock bifurcation and non-uniform ignition behind a reflected shock. The objectives are to interpret the formation of shock bifurcation induced by the reflected shock and boundary layer interaction and to investigate the weak ignition and its transition to strong ignition for both hydrogen and dimethyl ether. It is found that the non-uniform reflection of the incident shock at the end wall produces a wedge-shaped oblique shock foot at the wall. The wedge-shaped structure results in strong interactions between reflected shock and boundary layer, which induces the shock bifurcation. It is demonstrated that the local high-temperature spots at the foot of the bifurcated shock is caused by viscous dissipation and pressure work. As the post-reflected shock temperature increases, the transition from weak ignition to strong ignition in a stoichiometric hydrogen/oxygen mixture is observed. The relative sensitivity of ignition delay time to the post-reflected shock temperature is introduced to characterize the appearance of weak ignition behind the reflected shock. Unlike in the hydrogen/oxygen mixture, weak ignition is not observed in the stoichiometric dimethylether/oxygen mixture since it has a relatively longer ignition delay time and smaller relative sensitivity.