On the comparison of two different solutions in the form of series of the governing equation of an unsteady flow of a second grade fluid

In this paper, two different solutions in the form of series of the governing equation of unsteady flow of a second grade fluid are considered. These are series expansions with respect to inverse power of time and a perturbation expansion. Two illustrative examples are given. One of them is the unsteady flow of a second grade fluid over a plane wall suddenly set in motion and the other is the diffusion of a line vortex in a fluid of second grade. It is a remarkable fact that the expression of the series expansion with respect to inverse power of time is exactly in the same form as that of the perturbation expansion. Thus, it is possible to replace a series expansion with respect to inverse power of time with a perturbation expansion.     

THE BNARD CONVECTION IN A LAYER OF FLUID WITH A TIME-DEPENDENT MEAN TEMPERATURE

The onset of Bénard convection, or the critical Rayleigh number in a layer of fluid with a time-dependent mean temperature has been investigated theoretically. The critical Rayleigh number is regarded as a function of time and is expanded in series of a small parameter. Up to second approximation a simple expression of critical Rayleigh number is obtained for the time region for away from the point of zero.     

Formulation of a semi-analytical approach based on Gurtin variational principle for dynamic response of general thin plates

4 semi-analytical approach for the dynamic response of general thin plates which employes finite element discretization in space domain and a series of representation in time domain is developed on the basis of Gurtin variational principles. The formulation of time series is also investigated so that the dynamic response of plates with arbitrary shape and boundary constraints can be achieved with adequate accuracy.Project supported by the National Natural Science Foundation of China     

A computational method for simulating transient motions of an incompressible inviscid fluid with a free surface

A new numerical method has been developed for the analysis of unsteady free surface flow problems. The problem under consideration is formulated mathematically as a two-dimensional non-linear initial boundary value problem with unknown quantities of a velocity potential and a free surface profile. The basic equations are discretized spacewise with a boundary element method and timewise with a truncated forward-time Taylor series. The key feature of the present paper lies in the method used to compute the time derivatives of the unknown quantities in the Taylor series. The use of the Taylor series expansion has enabled us to employ a variable time-stepping method. The size of time increment is determined at each time step so that the remainders of the truncated Taylor series should be equal to a given small error limit. Such a variable time-stepping technique has made a great contribution to numerically stable computations. A wave-making problem in a two-dimensional rectangular water tank has been analysed. The computational accuracy has been verified by comparing the present numerical results with available experimental data. Good agreement is obtained.     

Impact of an elastic rod on a deformable barrier: analytical and numerical investigations on models of a valve and a rod-shaped stamping tool

The longitudinal motion of an elastic rod is studied for the case that the rod is suddenly elastically fixed at one end and is hit by a mass at its other end. This configuration represents real settings e.g. as a valve impacts an elastic valve-seat or as a stamping device used in forging is hit by a large mass. The solution of the problem is formulated in the Laplace transformation space. The inverse transformation into the time domain is performed by engaging the so-called Laguerre polynomial technique. This method allows to calculate exact solutions for finite times from a finite number of series elements. Rigorous mathematical proofs not established up to now are given with respect to the convergence of the series encountered and the validity of exchanging the order of inversion of the Laplace transformation and summation of the established series. For comparison also a numerical solution of the problem is presented. An analysis of the energy transfer between rod, impacting mass and elastic barrier elucidates the marked influence of the deformability of the elastic barrier on the stress state in the rod.     

Effect of regularized functions on the dynamic response of a clutch system using a high-order algorithm

The main objective of this work is to propose some regularization techniques for modeling contact actions in a clutch system and to solve the obtained nonlinear dynamic problem by a high-order algorithm. This device is modeled by a discrete mechanical system with eleven degrees of freedom. In several works, the discontinuous models of the contact actions are replaced by the smoothed functions using the hyperbolic tangent. We propose, in this work, to replace the discontinuous model by a regularized model with new continuous functions that permit us to search the solution under Taylor series expansion. This regularized model approaches better the discontinuous model than the model based on the smoothing functions, especially in the vicinity of the zone of singularities. To solve the equations of motion of discrete mechanical systems, we propose to use a high-order algorithm combining a time discretization, a change of variable based on the previous time, a homotopy transformation and Taylor series expansion in the continuation process. The results obtained by this modeling are compared with those computed by the Newton–Raphson algorithm.     

Effects of relaxation time on start-up time for starting flow of Maxwell fluid in a pipe

Although the analytical solution of the starting flow of Maxwell fluid in a pipe has been derived for a long time, the effect of relaxation time λ on start-up time ts of this flow is still not well understood. Especially, there exist a series of jumps on the ts-λ. curve. In this paper we introduce a normalized mechanical energy by mode decomposition and mathematical analogy to describe the start-up process. An improved definition of start-up time is presented based on the normalized mechanical energy. It is proved that the ts-λ. curve contains a series of jumps if λ is larger than a critical value. The exact positions of the jumps are determined and the physical reason of the jumps is discussed.     

Thermal instability in a rotating anisotropic porous layer saturated by a viscoelastic fluid

Linear and non-linear thermal instability in a rotating anisotropic porous medium, saturated with viscoelastic fluid, has been investigated for free-free surfaces. The linear theory is being related to the normal mode method and non-linear analysis is based on minimal representation of the truncated Fourier series analysis containing only two terms. The extended Darcy model, which includes the time derivative and Coriolis terms has been employed in the momentum equation. The criteria for both stationary and oscillatory convection is derived analytically. The rotation inhibits the onset of convection in both stationary and oscillatory modes. A weak non-linear theory based on the truncated representation of Fourier series method is used to find the thermal Nusselt number. The transient behaviour of the Nusselt number is also investigated by solving the finite amplitude equations using a numerical method. The results obtained during the analysis have been presented graphically.     

Broadband energy exchanges between a dissipative elastic rod and a multi-degree-of-freedom dissipative essentially non-linear attachment

We analyze complex, multi-frequency, non-linear modal interactions in the damped dynamics of a viscously damped dispersive finite rod coupled to a multi-degree-of-freedom essentially non-linear attachment. We perform a parametric study to show that the attachment can be an effective broadband energy absorber and dissipater of shock energy from the rod. It is shown that strong targeted energy transfer from the rod to the attachment occurs when there is strong stiffness asymmetry in the attachment. For weak viscous dissipation, a clear understanding of dynamical transitions in the integrated rod-non-linear attachment system can be gained by wavelet transforming the time series and superimposing the resulting wavelet spectra in the frequency-energy plot (FEP) of the periodic orbits of the underlying Hamiltonian system. Two distinct NES configurations are analyzed in detail, and their damped responses are analyzed by the Hilbert-Huang transform (HHT). We show that the HHT is capable of analyzing even complex non-linear damped transitions, by providing the dominant frequency components (or equivalently, time scales) at which the non-linear phenomena take place, and clarifying the series of non-linear resonance captures between the rod and attachment dynamics that are responsible for the broadband energy exchanges in this system.     

Finite memory quadratic Volterra model for the response prediction of a slender marine structure under a Morison load

A quadratic Volterra model with a finite nonlinear memory effect was introduced and applied to the time series prediction of a slender marine structure exposed to the Morison load. First, the unknown nonlinear single-input–single-output dynamic system was identified using the nonlinear autoregressive with exogenous input (NARX) technique based on the prepared datasets of the wave elevation and system response, which was obtained by running nonlinear time domain analysis for a certain short term sea state. The structure of NARX was designed in such a way that the linear part had infinite memory, whereas the nonlinear part had finite memory of a certain length. Second, the frequency domain Volterra kernels, both linear and quadratic, were derived analytically by applying the harmonic probing method to the identified system. To derive the frequency response functions, the sigmoidal function used in NARX to realize the nonlinear relationship between the input and output was expanded to polynomials based on the Taylor series expansion, so that the harmonics of same frequencies were easily matched between the input and output. Finally, the time series of the system response under arbitrarily given short term sea states were predicted using the quadratic Volterra series. The proposed methodology was used to predict the nonlinear dynamic response of a 2-dimentional free standing catenary riser exposed to a random ocean wave load, and the comparison between the prediction and simulation results was made on the probability distribution of the maximum excursion of riser top. The results show that the proposed methodology can successfully capture the nonlinear effects of the dynamic response of a slender marine structure induced by the quadratic term of the Morison formula.     

Diffusion in a nonuniform velocity field

The effect of a nonuniform velocity field on the diffusion process is examined. When the local passive admixture transport equation is averaged over the channel cross section, the differential equation for the average concentration over the cross section is obtained in the form of an infinite asymptotic series whose terms are linear combinations of the derivatives of the average concentration with respect to the coordinate and time, while the coefficients depend on the degree of transverse nonuniformity of the velocity field and the radial Péclet number. Estimates show that in most of the cases encountered in practice to ensure that the calculations have the necessary accuracy the series must include derivatives up to the third order. An approximate solution of the averaged equation is found by the method of asymptotic expansions and the initial moments of the residence time distribution function are determined.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 122–128, September–October, 1989.     

Series solutions of unsteady MHD flows above a rotating disk

The series solutions of unsteady flows of a viscous incompressible electrically conducting fluid caused by an impulsively rotating infinite disk are given by means of an analytic technique, namely the homotopy analysis method. Using a set of new similarity transformations, we transfer the Navier–Stokes equations into a pair of nonlinear partial differential equations. The convergent series solutions are obtained, which are uniformly valid for all dimensionless time 0 ≤ τ < ∞ in the whole spatial region 0 ≤ η < ∞. To the best of our knowledge, such kind of series solutions have never been reported. The effect of magnetic number on the velocity is investigated.     

含中心裂纹任意截面柱体扭转时的应力强度因子计算

本文提出了一组应力函数,采用边界配置方法计算了含中心裂纹不同截面形状柱体扭转时的应力强度因子。有关椭圆截面柱体的算例表明,本文方法具有良好的精度。同时,文中给出了圆、椭圆和矩形等不同截面柱体的计算结果。     

A numerical method of K-S entropy calculation for a strange attractor

Based directly on the original definition of K-S entropy, a new algorithm for calculating K-S entropy from chaotic time series is developed by using some techniques of coding and code operation. The project supported by National Natural Science Foundation of China     

Propagation of discontinuities against a static background

The solution of the ideal gasdynamic equations describing propagation of a shock wave initiated, for example, by the motion of a piston against an inhomogeneous static background is considered. The solution is constructed in the form of Taylor series in a special time variable which is equal to zero on the shock wave. In the case of weak shock waves divergence of the series serves as the constraint for such an approach. Then the solution is constructed by linearizing the equations about the solution with a weak discontinuity. In the case of a given background the last solution can be always found exactly by solving successively a set of transport equations, all these equations are reduced to linear ordinary differential equations. The presentation begins from the one-dimensional solutions with plane waves and ends by discussion of spatial problems.     

Mathematical foundation of a new complexity measure

For many continuous bio-medieal signals with both strong nonlinearity and non-stationarity, two criterions were proposed for their complexity estimation ： （1） Only a short data set is enough for robust estimation; （2） No over-coarse graining preproeessing, such as transferring the original signal into a binary time series, is needed. Co complexity measure proposed by us previously is one of such measures. However, it lacks the solid mathematical foundation and thus its use is limited. A modified version of this measure is proposed, and some important properties are proved rigorously. According to these properties, this measure can be considered as an index of randomness of time series in some senses, and thus also a quantitative index of complexity under the meaning of randomness finding complexity. Compared with other similar measures, this measure seems more suitable for estimating a large quantity of complexity measures for a given task, such as studying the dynamic variation of such measures in sliding windows of a long process, owing to its fast speed for estimation.     

Applying local model approach for tidal prediction in a deterministic model

In recent years, a practice of tidal prediction based on a deterministic model or by a time series forecasting model has been established. A deterministic model can predict tidal movement and capture the dynamics of the flow pattern over the entire domain. However, due to the simplification of model settings and near shore effects, the accuracy of the numerical model can diminish. Time series forecasting is capable of capturing the underlying mechanism that may not be revealed in the deterministic model simulation. However, such data‐driven forecast fails to maintain accuracy with the progress of forecast horizon. In this paper, a scheme that combines the advantages of these two methods is introduced. The model errors are forecasted to different time horizons using a data‐driven approach, and are then superimposed on the simulation results in order to correct the model output. Based on the proposed method, it is found that the accuracy is significantly improved with more than 50% of the errors removed on the average. Copyright © 2008 John Wiley & Sons, Ltd.     

Dynamics of a truncated elastic cone

The problem of determining the nonstationary wave field of an elastic truncated cone with nonzero dead weight is formulated in terms of wave functions. The Laplace transform with respect to time and an integral transform with respect to time polar angle are used to reduce the problem to a one-dimensional vector problem in the transform domain. The transforms of the wave functions are expanded into series in inverse powers of the Laplace transform parameter, which makes it possible to study the wave process at the initial instants of interaction. A method is proposed to solve the problem for an elastic cone doubly truncated by spherical surfaces     

General solutions to a class of time fractional partial differential equations

A class of time fractional partial differential equations is considered, which in- cludes a time fractional diffusion equation, a time fractional reaction-diffusion equation, a time fractional advection-diffusion equation, and their corresponding integer-order partial differential equations. The fundamental solutions to the Cauchy problem in a whole-space domain and the signaling problem in a half-space domain are obtained by using Fourier- Laplace transforms and their inverse transforms. The appropriate structures of the Green functions are provided. On the other hand, the solutions in the form of a series to the initial and boundary value problems in a bounded-space domain are derived by the sine- Laplace or cosine-Laplace transforms. Two examples are presented to show applications of the present technique.