不可压缩流中二元翼运动的余维二分叉

考虑二元机翼在不可压缩流作用下运动的余维二分叉问题,弹簧的非线性刚度考虑为立方型。化方程为四维一阶微分方程,解析求出了系统发生叉式分叉与HOPF分叉的临界流速的表达式和发生余维二分叉的临界流速和一次刚度。在两个参数平面上研究了系统在余维二分叉点附近的不同区域内平衡点和极限环的稳定性及其个数。最后用数值积分方法给出相应的结果。     

Multistability, oscillations and chaos in an exothermic branched-chain reaction model

The behaviour of a model branched-chain reaction with radical termination is investigated for a well-stirred flow reactor. The termination step is assumed to be exothermic and the chain-branching or autocatalysis has an Arrhenius temperature dependence. Under adiabatic conditions, ignition and extinction phenomena are exhibited in the form of a hysteresis loop. For a non-adiabatic reactor more complex stationary state bifurcation diagrams, including isola responses, are found. Dynamic instabilities also arise, with simple periodic oscillations emerging from Hopf bifurcations. These periodic solutions may undergo secondary bifurcations, typically through period-doubling: a full cascade to chaotic responses is found. A comparison of this model with simpler schemes and its implications for various combustion reactions are discussed.     

Axial drive to nonlinear flow between rotating cylinders

Stability of pseudoplastic rotational flow between cylinders in presence of an independent axial component is investigated. The fluid is assumed to follow the Carreau model and mixed boundary conditions are imposed. The conservation of mass and momentum equations give rise to a four-dimensional low-order dynamical system, including additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. In absence of the axial flow, as the pseudoplasticity effects increases, the purely-azimuthal base flow loses its stability to the vortex structure at a lower critical Taylor number. Emergence of the vortices corresponds to the onset of a supercritical bifurcation also present in the flow of a linear fluid. However, unlike the Newtonian case, pseudoplastic Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Existence of an axial flow induced by a pressure gradient appears to further advance each critical point on the bifurcation diagram. In continuation, complete flow field together with viscosity maps is analyzed for different flow scenarios. Through evaluation of the Lyapunov exponent, flow stability and temporal behavior of the system for cases with and without axial flow are brought to attention.     

Fluid dynamics of a tubular polymerizer

The flow field of a non-Newtonian fluid in a conduit undergoing exothermal chemical reactions is strongly determined by heat and mass transfer processes predominatly via the molecular weight dependence of the fluid properties. The specific example in this study is polymerization of styrene in a steady, laminar flow tubular reactor. Avoiding the region of thermal instability, nonlinear transport and conversion mechanisms leading to characteristically elongated, jet-like velocity profiles are discussed. Specifically, the newly developed distributed parameter model has been used to investigate the coupled effects of strong temperatures, gradients and local polymerization processes on the tubular fluid dynamics. For extreme operational conditions either the highly viscous polymer almost plugs the reactor or most of the incoming monomer solution leaves the tube unreacted.     

热塑性装饰材料点燃性能研究

利用锥形量热仪研究了典型热塑性装饰材料的点燃性能,并利用数值模拟结合线性回归方法导出了适合热中型材料的点燃模型。利用此模型对实验得到的点燃时间进行了分析,得到了材料的临界热辐射通量、点燃温度与热物性参数,结果表明研究得到的材料热辐射通量与点燃温度、点燃时间和文献中给出的值比较吻合,导出的模型具有较好的适用性。     

Tracing the equilibrium path beyond simple critical points

A unified presentation of some popular continuation procedures used in the non-linear finite element analysis of structural mechanics is introduced. An extension of the elliptical constraint equation proposed by Crisfield is given. It is shown that in the proposed procedure real roots can always be obtained in solving the iterative change of the load parameter. Updated weighting factors are introduced in the constraint equation in order to get better convergence characteristics in the case when localized deformations occur. For bifurcation points a modification of Rheinboldt's branching procedure is presented. Post critical response after limit and bifurcation points is determined in some numerical examples.     

超音速流中结构非线性二元机翼的复杂响应研究

基于活塞理论计算作用在二元机翼上的气动力，采用拉格朗日方程建立系统的运动微分方程。通过平衡点的Jacobi矩阵的特征方程求出了系统的Hopf分叉点，研究了带有立方非线性俯仰刚度二自由度机翼系统在典型参数下的稳定极限环颤振和混沌响应。结果表明，在超过一定的流体速度后，系统平衡点的个数及稳定性均发生了变化；随着流速的增大，在积分初始值较小时，系统出现混沌等极为复杂的响应。     

Linear stability analysis of spherically propagating thermal frontal polymerization waves

Spherically propagating frontal polymerization (FP) waves were observed for the first time in condensed media. After a brief period of ignition in a spherical domain by an external UV source, the front began to expand radially. Once the front attained a critical size, it became unstable, resulting in so-called ‘spin modes’. These spin modes are manifested as slightly raised regions that travel on the surface of the expanding spherical front. The onset of these instabilities from a stable, uniformly propagating spherical front can be described by a linear stability analysis. The bifurcation parameter is the Zeldovich number which is related to the activation energy of the reaction. A basic solution was constructed which describes a spherically symmetric outward propagating front of radius R. An asymptotic analysis was then employed under the assumption that R is large. This corresponds to the case where the conditions of ignition do not affect front propagation. It was found to leading order that the front propagates at a constant velocity and corrections to velocity due to curvature have been determined. The linear stability analysis shows that for the Zeldovich number in a certain range, there exists a situation in which the sphere will be unstable but will recover its stability in time as it expands.     

Creation of spatial structure by an electric field applied to an ionic cubic autocatalator system

The effects of applying electric fields to a reactor with kinetics based on an ionic version of the cubic autocatalator are considered. Three types of boundary condition are treated, namely (constant) prescribed concentration, zero flux and periodic. A linear stability analysis is undertaken and this reveals that the conditions for bifurcation from the spatially uniform state are the same for both the prescribed concentration and zero-flux boundary conditions, suggesting bifurcation to steady structures, whereas, for periodic boundary conditions, the bifurcation is essentially different, being of the Hopf type, leading to travelling-wave structures. The various predictions from linear theory are confirmed through extensive numerical simulations of the initial-value problem and by determining solutions to the (non-linear) steady state equations. These reveal, for both prescribed concentration and zero-flux boundary conditions, that applying an electric field can change the basic pattern form, give rise to spatial structure where none would arise without the field, can give multistability and can, if sufficiently strong, suppress spatial structure entirely. For periodic boundary conditions, only travelling waves are found, their speed of propagation and wavelength increasing with increasing field strength, and are found to form no matter how strong the applied field.     

Mixed convection around a horizontal circular cylinder immersed in a Darcy flow

The forced and free mixed convection problem around a horizontal circular cylinder in a fluid-saturated porous medium is investigated at small Peclet number with Gr/Re = O(1). The Darcy flow model is used for the velocity field. The method of matched asymptotic expansions is used to obtain asymptotic solutions for small Peclet number. It is shown that, up to the present order of approximation, natural convection has no effects on the temperature field, and the effects of the parameter Gr/Re on the velocity field are examined in detail.     

Stability of Ferrofluid Flows in a Horizontal Channel Subjected to a Longitudinal Temperature Gradient and an Oblique Magnetic Field

The stability of thermoconvective flows of a ferrofluid in a horizontal channel subjected to a longitudinal temperature gradient and an oblique strong magnetic field is studied. The flows are governed by the mass, momentum, heat balance and Maxwell equations, in the Boussinesq approximation. Strong magnetic fields are characterized by a small parameter measuring the deviation of the magnetization across the layer from the external magnetic field. An approximate solution of the stationary hydrodynamic problem was found in analytical form in Hennenberg et al. (Phys Fluids 18:093602, 2006). The flow depends on the thermal (gravitational) and magnetic Rayleigh numbers. In the present paper the linear stability of that basic flow is studied by the Galerkin method. The analysis shows that for large Prandtl numbers, typical for ferrofluids, and relatively small magnetic Rayleigh numbers, only oscillatory instability can appear. For a given magnetic Rayleigh number, the critical wavenumber does not depend on the inclination of the magnetic field, while the critical thermal Rayleigh number slightly changes. For horizontal and vertical magnetic fields, both critical numbers decrease when increasing the magnetic Rayleigh number.     

Theoretical prediction of piloted ignition of polymeric fuels in microgravity at low velocity flows

A numerical model developed for the prediction of the piloted ignition delay of solid polymeric materials exposed to an external radiant heat flux is used to predict the ignition delay and critical heat flux for ignition of solid fuels in microgravity at low velocity flows. The model considers the coupled thermochemical processes that take place in the condensed phase, including oxidative and thermal pyrolysis, phase change, radiation absorption, and heat and mass transfer in a multi-phase and multi-composition medium. Ignition is considered to occur when a critical pyrolysate mass flow rate is reached at the sample surface. Microgravity experimental surface temperature and ignition delay data previously obtained in a KC-135 aircraft are used to infer, in conjunction with the theoretical analysis, the critical mass flow rate for ignition. This value is then used to predict the ignition delay as a function of the external radiant heat flux, and the critical heat flux for ignition. Calculations are made for Polymethylmethacrylate (PMMA) and a Polypropylene/Fiberglass composite at airflows of 0.09 and 0.15 m/s under microgravity conditions and at 1.0, 1.75 and 2.5 m/s under normal gravity. The experiments and theoretical predictions show that the ignition delay and critical heat flux for ignition decrease as the forced airflow velocity decreases. It is predicted that at the tested lower velocities, the critical heat flux for ignition is close to half the value measured in normal gravity. The results have important implications since they indicate that materials could ignite easier under the conditions expected in spacecraft, and consequently stricter design specifications may be needed for fire safety.     

High-temperature processes in a porous medium

A study is made into a nonlinear filtration problem on the flow of gas in a porous weakly conducting medium, which simulates the process of superheating in a reactor. Explicit estimates are obtained of two critical values of the similarity parameter of the problem, which define the condition of existence of the steady-state mode of cooling a system open to the atmosphere and the condition of global superheating. The second one of these critical values is also the bifurcation point for the input set of equations. It is found that a thin layer forms at high temperatures corresponding to this point, in which the thermal energy is concentrated. A simultaneous radical variation of the physical process of heat transfer occurs, namely, convection is replaced by molecular heat conduction.     

Long-wave theory for a new convective instability with exponential growth normal to the wall

A linear stability theory is presented for the boundary-layer flow produced by an infinite disc rotating at constant angular velocity in otherwise undisturbed fluid. The theory is developed in the limit of long waves and when the effects of viscosity on the waves can be neglected. This is the parameter regime recently identified by the author in a numerical stability investigation where a curious new type of instability was found in which disturbances propagate and grow exponentially in the direction normal to the disc, (i.e. the growth takes place in a region of zero mean shear). The theory describes the mechanisms controlling the instability, the role and location of critical points, and presents a saddle-point analysis describing the large-time evolution of a wave packet in frames of reference moving normal to the disc. The theory also shows that the previously obtained numerical solutions for numerically large wavelengths do indeed lie in the asymptotic long-wave regime, and so the behaviour and mechanisms described here may apply to a number of cross-flow instability problems.     

Analysis of extended micro-Graetz problem in a microtube

In this numerical study, hydrodynamically developed but thermally developing forced convection in a microtube subjected to a step change in the wall heat flux is analysed using a finite-volume method. The slip velocity and temperature jump conditions at the wall and the axial conduction in the fluid are included in the analysis. The combined effects of the Peclet number and the Knudsen number on the local Nusselt numbers as well as on the wall and bulk temperatures are determined in the continuum and slip flow regimes (0 ≤ Kn ≤ 0.1). In the entrance region, large reductions are observed in the Nusselt number with decreasing Peclet number or increasing Knudsen number. The results also show that the thermal length increases with decreasing Peclet number.     

Nonlinear dynamics of tubes carrying a pulsatile flow

The nonlinear dynamics of a cantilever tube conveying a pulsatile flow and undergoing planar motions is investigated. The mean flow is near its critical value at which the downward vertical position of the tube gets unstable by flutter and executes limit-cycle oscillations. The pulsations in the flow are assumed to be small and harmonic with frequency nearly twice that of the limit cycle. To study the nonlinear dynamics, the method of averaging is utilized and the governing partial differential equation is reduced to a dynamic system on the plane. These two first-order differential equations depend on three parameters and govern the dynamics of the amplitude of motion of the tube. The planar system is studied for its qualitative behaviour using ideas from the local bifurcation theory and a local bifurcation set in the parameter plane is constructed. Using ideas from codimension-two unfolding of singularities, this bifurcation set is further refined. The resulting partial bifurcation set and the associated phase portraits in the various regions of the flowrate-detuning parameter plane show that the averaged equations have constant as well as periodic solutions. The stabifity type and the number of these solutions changes from one region to another. This explains how periodic or 'phase-locked' oscillations of the cantilever tube bifurcate into amplitude-modulated or almost-periodic motions.     

多因素耦合影响亚音速喷口强化气体流场分布研究

为了研究多因素耦合条件下亚音速喷口内气体流场分布演变的作用机制与影响规律,基于标准k-ε湍流模型,对压力-流速-收缩比耦合条件下的亚音速喷口构造均匀气体流场进行了数值模拟,通过定义top-hat流场分布关键参数的方式,定量评估压力-流速-收缩比对强化后流场分布的耦合影响.结果表明:多因素耦合条件下,亚音速喷口通过影响压...     

Stability of Non-axisymmetric Electrolyte Jet in High-frequency AC Electric Field

In the present work linear instability of capillary non-axisymmetric micro-jets of electrolyte solutions in a high-frequency alternating axial electric field is investigated theoretically. The gravity affects are neglected. The problem is described by strongly coupled nonlinear system of PDEs for ion transport, electrical field and fluid flow. Viscous liquid is taken. The problem can be divided into outer and inner ones. Solution for the unsteady double ion layer is obtained in Debye-Huckel approximation provided that the oscillation frequency is sufficiently high while Pecklet number based on the Debye layer thickness is sufficiently small. The unsteady double ion layer produces additional normal and tangential stresses on the liquid–gas interface; the latter can either stabilize or destabilize the flow. It is shown that only axisymmetric mode is unstable while non-axisymmetric perturbations are always stable. It is also shown that in unstable case there is an essential dependence of the main stability characteristics on the parameter proportional to the frequency of external field. There are two threshold values of the parameter at which a bifurcation of stability parameters occurs. In particular, the size of the formed drops suffers a jump at increase of amplitude of fluctuation of an electric field. The problem is solved in a broad region of its parameters. There is a qualitative agreement of the theory developed with the available experimental data.     

Using the Method of Lines to determine critical conditions for thermal ignition

There are many situations in which the critical conditions for thermal ignition cannot be determined analytically. These include cases where the chemistry needs to be properly considered, where the geometry is not just the simplest and where other processes must be included. In these circumstances, numerical (or at least semi-analytical) means are used to determine critical conditions for thermal ignition. Once confronted with a numerical approach to solving a problem, it is necessary to be a little circumspect about the results and seek independent means to corroborate them. For this reason, the present paper reports on the Method of Lines to investigate a recent reactive hotspot problem which has previously been shown to display unexpected behaviour and demonstrates the use of sensitivity analysis to rigourously determine criticality in such a dissipative system.     

Passing of instability points by applying a stabilized Newton–Raphson scheme to a finite element formulation: Comparison to arc-length method

In the vicinity of limit and bifurcation points the global stiffness matrix of a finite element formulation becomes ill-conditioned and at the critical point singular. This disturbs the convergence behavior of the standard Newton–Raphson scheme as well as the arc-length method. The stabilization procedure suggested solves the numerical defects and is thus able to pass critical points. Bifurcation points are passed on the primary path. Branch switching to the secondary path is done automatically. The stabilization procedure and the imperfection force are derived based on the eigenvalues and -vectors of the structure.