An analytical study of linear and nonlinear double diffusive convection in a fluid saturated anisotropic porous layer with Soret effect

The double diffusive convection in a horizontal anisotropic porous layer saturated with a Boussinesq fluid, which is heated and salted from below in the presence of Soret coefficient is studied analytically using both linear and nonlinear stability analyses. The normal mode technique is used in the linear stability analysis while a weak nonlinear analysis based on a minimal representation of double Fourier series method is used in the nonlinear analysis. The generalized Darcy model including the time derivative term is employed for the momentum equation. The critical Rayleigh number, wavenumber for stationary and oscillatory modes and frequency of oscillations are obtained analytically using linear theory. The effect of anisotropy parameters, solute Rayleigh number, Soret parameter and Lewis number on the stationary, oscillatory, finite amplitude convection and heat and mass transfer are shown graphically.     

Hydromagnetic stability of convective flow of variable viscosity fluids generated by internal heat sources

The stability of convective motion of a variable viscosity fluid contained in a vertical layer generated by uniformly distributed internal heat sources in the presence of a transverse magnetic field is studied. The viscosity of the fluid is assumed to depend on the temperature. The undisturbed steady state motion is assumed to consist of purely vertical motion with a nonlinear temperature distribution across the layer. The equations were solved by the spectral collocation method. The results show that thermal running waves are the most unstable modes and dominate the shear modes when the viscosity decreases.     

Hydromagnetic stability of convective flow of variable viscosity fluids generated by internal heat sources

The stability of convective motion of a variable viscosity fluid contained in a vertical layer generated by uniformly distributed internal heat sources in the presence of a transverse magnetic field is studied. The viscosity of the fluid is assumed to depend on the temperature. The undisturbed steady state motion is assumed to consist of purely vertical motion with a nonlinear temperature distribution across the layer. The equations were solved by the spectral collocation method. The results show that thermal running waves are the most unstable modes and dominate the shear modes when the viscosity decreases.     

Long-Wave Instability in a Three-Layer Stratified Shear Flow

In inviscid fluid flows, instability can occur because of a resonance between two wave modes. For the case when the modes remain distinct at the critical point where the wave phase speeds coincide, then in the weakly nonlinear, long-wave limit, there is an expectation that the generic outcome is a model consisting of two coupled Korteweg–deVries equations. This situation is examined for a certain three-layer stratified shear flow.     

The Evolution of Nonlinear Wave Trains in Stratified Shear Flows

The propagation of an internal wave train in a stratified shear flow is investigated for a Boussinesq fluid in a horizontal channel. Linear effects are primarily reflected in the dispersion relation for the various modes. The phenomenon of Eckart resonance occurs for more realistic stratification profiles. The evolution of nonlinear internal wave packets is studied through a systematic perturbation analysis. A nonlinear Schrodinger equation for the envelope of the internal wave train is derived. Depending on the relative sign of the dispersive and nonlinear terms, a wave train may disperse or form an envelope soliton. The analysis demonstrates the existence of two types of critical layers: one the ordinary critical point where ū=c, while the other occurs where ū=c_g. In order to calculate the coefficients of the nonlinear Schrodinger equation a numerical code has been developed which computes the second-harmonic and induced mean motions. The existence of these envelope solitons and their dependence on environmental conditions are discussed.     

A model equation for nonlinear wavelength selection and amplitude evolution of frontal waves

Summary We study a model equation describing the temporal evolution of nonlinear finite-amplitude waves on a density front in a rotating fluid. The linear spectrum includes an unstable interval where exponential growth of the amplitude is expected. It is shown that the length scale of the waves in the nonlinear situation is determined by the linear instabilities; the effect of the nonlinearities is to limit the amplitude's growth, leaving the wavelength unchanged. When linearly stable waves are prescribed as initial data, a short interval of rapid decrease in amplitude is encountered first, followed by a transfer of energy to the unstable part of the spectrum, where the fastest growing mode starts to dominate. A localized disturbance is broken up into its Fourier components, the linearly unstable modes grow at the expense of all other modes, and final amplitudes are determined by the nonlinear term. Periodic evolution of linearly unstable waves in the nonlinear situation is also observed. Based on the numerical results, the existence of low-order chaos in the partial differential equation governing weakly nonlinear wave evolution is conjectured.     

Static collapse: A survey of methods and modes of behavior

The phenomenon of static collapse, henceforth called ‘buckling’, is first illustrated by the behavior of a fairly thick cylindrical shell, which under axial compression deforms at first axisymmetrically and later nonaxisymmetrically. Thus, static buckling encompasses two modes of behavior, nonlinear collapse at the maximum point in a load versus deflection curve and bifurcation buckling. Accurate prediction of critical loads corresponding to either mode in the plastic range of material behavior requires a simultaneous accounting for moderately large deflections and nonlinear, irreversible, path-dependent material. A survey is given of plastic buckling, which spans three areas: asymptotic analysis of postbifurcation behavior of perfect and imperfect simple structures, general nonlinear analysis of arbitrary structures, and nonlinear analysis for collapse at a maximum load and bifurcation buckling of shells of revolution. In the survey of general nonlinear structural analysis, some emphasis is given to strategies for solving the governing nonlinear equations incrementally. Numerous examples, generated primarily with the STAGS computer program, which was developed by Almroth and his colleagues, reveal many complex modes of buckling behavior.     

Resonant three-wave interactions in non-linear hyper-elastic fluid-filled tubes

A multiple-scales method is used to derive the Three-Wave Interaction (TWI) equations describing resonantly interacting triads in nonlinear hyper-elastic fluid-filled tubes. The tube wall is assumed to be an axially-tethered nonlinear membraneous cylindrical shell for which the resultant stresses can be determined by a strain-energy functional. The fluid within the tube is assumed to be two-dimensional, axi-symmetric and inviscid. We show that small-but-finite amplitude strongly dispersive pressure wave packets can continuously exchange energy in a resonant triad while conserving total energy. For a Mooney-Rivlin shell wall the theory presented predicts a short wavelength cutoff on the order of the tube radius. Thus pressure pulses containing wavelengths on the order of the tube radius and longer may contain resonantly interacting modes. Special solutions are presented: temporally developing modes, pump-wave approximations and explosively unstable steadily-traveling wave packets.     

Permanent Wave Structures and Resonant Triads in a Layered Fluid

A closed three layer fluid with small density differences between the layers has two closely related modes of gravity wave propagation. The nonlinear interactions between the wave modes are investigated, particularly the nearly resonant or significant interactions. Permanent wave solutions are calculated, and it is shown that a permanent wave of the slower mode can generate resonantly a wave harmonic of the faster mode. The equations governing resonant triads of the two modes are derived, and solutions having a permanent structure are calculated from them. It is found that some resonant triad solutions vanish when the triad is embedded in the set of all harmonics with wavenumbers in its neighborhood     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Couple stress fluid improve rub-impact rotor-bearing system – Nonlinear dynamic analysis

This study performs a dynamic analysis of the rub-impact rotor supported by two couple stress fluid film journal bearings. The strong nonlinear couple stress fluid film force, nonlinear rub-impact force and nonlinear suspension (hard spring) are presented and coupled together in this study. The displacements in the horizontal and vertical directions are considered for various non-dimensional speed ratios. The numerical results show that the dynamic behaviors of the system vary with the dimensionless speed ratios, the dimensionless unbalance parameters and the dimensionless parameter, l^∗. Inclusive of the periodic, sub-harmonic, quasi-periodic and chaotic motions are found in this analysis. The results of this study contribute to a further understanding of the nonlinear dynamics of a rotor-bearing system considering rub-impact force existing between rotor and stator, nonlinear couple stress fluid film force and nonlinear suspension. We also prove that couple stress fluid used to be lubricant do improve dynamics of rotor-bearing system.     

Linear and nonlinear stability analysis of binary viscoelastic fluid convection

The linear and weakly nonlinear stability analysis of the quiescent state in a viscoelastic fluid subject to vertical solute concentration and temperature gradients is investigated. The non-Newtonian behavior of the viscoelastic fluid is characterized using the Oldroyd model. Analytical expressions for the critical Rayleigh numbers and corresponding wave numbers for the onset of stationary or oscillatory convection subject to cross diffusion effects is determined. A stability diagram clearly demarcates non-overlapping regions of finger and diffusive instabilities. A Lorenz system is obtained in the case of the weakly nonlinear stability analysis. The effect of Dufour and Soret parameters on the heat and mass transports are determined and discussed. Due to consideration of dilute concentrations of the second diffusing component the route to chaos in binary viscoelastic fluid systems is similar to that of single-component (thermal) viscoelastic fluid systems.     

On the Generation of Mean Flows by the Interaction of Görtler Vortices and Tollmien-Schlichting Waves in Curved Channel Flows

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien-Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. Essentially it is found that the mean state adjusts itself so that any modes present are neutrally stable even at finite amplitude. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien-Schlichting wave is considered. It is shown that for a given channel curvature and vortex wavelength there is an upper limit to the mass flow rate which the channel can support as the pressure gradient is increased. When Tollmien-Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. At sufficiently high Tollmien-Schlichting amplitudes it is found that the vortex flows are destroyed, but there is a range of amplitudes where a fully nonlinear mixed vortex-wave state exists and indeed drives a mean state having little similarity with the flow which occurs without the instability modes. The vortex and Tollmien-Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien-Schlichting wave amplitude.     

Linear and nonlinear response of pipeline suspension bridge due to moving fluid

Basing on the nonlinear dynamic model of flexible pipeline suspended by spatial system of cables, described in Ref. [1], the linear and nonlinear vibrations are investigated in order to estimate the nonlinear effects. The model is based on substructure technique and formulated including features specific to analyzed structure, for example large displacements and time dependent parameters appearing in equations of motion due to fluid flowing inside the pipeline. Due to the fact that modelling problem for the analyzed structure is one's own complicated, a simple case when the conveying fluid is idealized simply as a ballast moving inside the pipe is considered. This paper presents a short numerical analysis of linear and nonlinear, static and dynamic response of exemplary structure for three different cases: during filling the pipe with fluid, when the pipeline is completely filled and during emptying the pipe. Moreover, for the linear problem, the influence of a speed of the fluid on the stability of the pipeline suspension bridge is investigated. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

饮酒驾车

针对酒后驾车问题,根据微分方程理论以及合理的假设,建立了体液中酒精含量随时间变化的数学模型,并求得其特解.再根据给定的数据,运用MATLAB软件确定回归方程的系数.由此,对于不同的喝酒方式以及喝下的不同数量的酒,进行血液中酒精浓度的分析,可以预测喝酒后任意时刻血液中的酒精浓度,并能预测不同喝酒方式以及喝下的不同数量的酒后能否驾车的时间分界点.从而对问题作出科学的解释和证明.     

隔水套管波流联合作用下非线性动力响应

考虑流及波流联合作用,研究了深水套管的涡激非线性振动．将套管简化为梁模型,计及Morison非线性流体动力和涡激荷载,建立套管的涡激振动方程．采用Korolov函数求解套管的固有频率和模态,提出了计算涡激非线性动力响应的Galerkin方法,计算了160 m水深中170 m长套管的固有频率和模态,研究了流引起的主共振和波流联合引起的组合共振．计算结果表明波流联合作用下套管的动力响应明显增大,结果也揭示了波流联合激励下套管复杂的动力响应特性．     

Neural network closures for nonlinear model order reduction

Many reduced-order models are neither robust with respect to parameter changes nor cost-effective enough for handling the nonlinear dependence of complex dynamical systems. In this study, we put forth a robust machine learning framework for projection-based reduced-order modeling of such nonlinear and nonstationary systems. As a demonstration, we focus on a nonlinear advection-diffusion system given by the viscous Burgers equation, which is a prototypical setting of more realistic fluid dynamics applications due to its quadratic nonlinearity. In our proposed methodology the effects of truncated modes are modeled using a single layer feed-forward neural network architecture. The neural network architecture is trained by utilizing both the Bayesian regularization and extreme learning machine approaches, where the latter one is found to be more computationally efficient. A significant emphasis is laid on the selection of basis functions through the use of both Fourier bases and proper orthogonal decomposition. It is shown that the proposed model yields significant improvements in accuracy over the standard Galerkin projection methodology with a negligibly small computational overhead and provide reliable predictions with respect to parameter changes.     

Nonlinear vibration and stability analysis of the curved microtube conveying fluid as a model of the micro coriolis flowmeters based on strain gradient theory

Micro coriolis flowmeters are extensively used in fluidic micro circuits and are of great interest to many researchers. Straight and curved coriolis flowmeters are common types of coriolis flowmeters. Therefore in the present work, the out-of- plane vibration and stability of curved micro tubes are investigated to study the dynamic behavior of curved coriolis flowmeters. The Hamilton principle is applied to derive a novel governing equation based on strain gradient theory for the curved micro tube conveying fluid. Lagrangian nonlinear strain is adopted to take into account the geometric nonlinearity and analyze hardening behavior as a result of the cubic nonlinear terms. Linear stability analysis is carried out to investigate the possibility of linear instabilities. Afterwards, the first nonlinear out-of-plane natural frequency is plotted versus fluid velocity to determine the influence of nonlinear terms and hardening behavior on stability of the system. The influence of the length scale parameter is studied by comparison of the results for classical, coupled stress and strain gradient theory. Finally the phase difference between two points at upstream and downstream is plotted versus fluid velocity. Linear relation between the phase difference and fluid velocity is noticed, thus the curved coriolis flowmeter can be calibrated to measure flow rate by measuring the phase difference between two points.     

非线性振动系统主振型的一种求解方法及稳定性判定

本文提出了一种求解非线性振动系统主振型的新方法,将求解非线性系统主振型的问题化为求解一系列代数方程组的问题。该方法适用于各种多自由度非线性振动系统,计算比较简单。文中还给出了一种判定非线性系统主振型稳定性的方法。