Second-order diffraction by a bottom-seated compound cylinder

The second-order diffraction potential around a bottom-seated compound cylinder is considered. The solution method is based on a semi-analytical formulation for the double-frequency diffraction potentials, which are properly decomposed into a rational number of components in order to satisfy all boundary conditions involved in the problem. The solution process results in two different Sturm–Liouville problems which are treated separately in the ring-shaped fluid regions defined by the geometry of the structure. The matching of the potentials on the boundaries of adjacent fluid regions is established using the ‘free’ wave components of the potentials. Different Green's functions are constructed for each of the fluid regions surrounding the body. The calculation of integrals of the pressure distribution on the free surface is carried out using an appropriate Gauss–Legendre numerical technique. The efficiency of the method described in the present work is validated through comparative calculations. Finally, extensive numerical predictions are presented concerning the complete second-order excitation and the nonlinear wave elevation for various configurations of vertical axisymmetric bodies.     

Global dynamics behaviors for new delay SEIR epidemic disease model with vertical transmission and pulse vaccination

A robust SEIR epidemic disease model with a profitless delay and verti- cal transmission is formulated,and the dynamics behaviors of the model under pulse vaccination are analyzed.By use of the discrete dynamical system determined by the stroboscopic map,an‘infection-free’periodic solution is obtained,further,it is shown that the‘infection-free’periodic solution is globally attractive when some parameters of the model are under appropriate conditions.Using the theory on delay functional and impulsive differential equatibn,the sufficient condition with time delay for the perma- nence of the system is obtained,and it is proved that time delays,pulse vaccination and vertical transmission can bring obvious effects on the dynamics behaviors of the model. The results indicate that the delay is‘profitless’.     

Behaviour of a simple crystallisation model in tube and channel flow

A formulation describing the rheology of crystallising polymers is discussed. For some semi-crystalline polymers where spherulites form as part of the crystallisation process, the use of a suspension-type model is appropriate. Whilst it is possible to so describe simple shearing and elongational rheology during on-going crystallisation with such models, the flow through a capillary tube is much more complex and numerical solution is usually necessary. To give some insight into this complex flow, a ‘step function’ or ‘amorphous-frozen’ model of the viscosity changes due to crystallisation has been devised so that a semi-analytical approach is feasible. We use this simple model and compare the results with recently published experiments in tubes and channels at high (O(10³ s^− 1)) shear rates using poly(butene-1). A direct correlation between simple shear and tube flow crystallisation onset times is found.     

Nonlinear free surface action with an array of vertical cylinders

Nonlinear diffraction of regular waves by an array of bottom-seated circular cylinders is investigated in frequency domain, based on a Stokes expansion approach. A complete semi-analytical solution is developed which allows an efficient evaluation of the second-order potentials in the entire fluid domain, and the wave forces on the structure. Expressions are derived for the second-order potential in the vicinity of individual cylinders. These expressions have a simple form, thus providing an effective means for investigating the wave enhancement due to nonlinear interactions with multiple cylinders. Based on the present method, the wave run-up and free-surface elevations around an array of two, three and four cylinders are investigated numerically.     

Smart Baffle Placement for Chaotic Mixing

It is well known that fluid mixing can often be improved by the introduction of ‘baffles’ into the flow – the problem of baffle placement is examined here for chaotic fluid mixing of a highly viscous fluid. A simple model for a planetary mixer, with one stirring element, is modified by the introduction of one or more stationary baffles. Regular regions of poor mixing in the unbaffled flow are shown to be significantly reduced in size if the location of the baffles is chosen so that the flow necessarily generates ‘topological chaos’. By contrast, the positioning of baffles in superficially similar ways that do not generate such ‘topological chaos’ fails to provide a similar improvement.     

Singular solutions for an anisotropic plate with an elliptical hole

A solution of the bending problem for a plate with an elliptical hole subjected to a point force (a singular solution) is obtained using the engineering theory of thin anisotropic plates and Lekhnitskiis complex potentials. The solution is constructed by conformal mapping of the exterior of the elliptical hole onto the exterior of a unit circle with evaluation of the Cauchy-type integrals over closed contours. Different versions of the boundary conditions on the holw contour are considered. In the limiting case where the ellipse becomes a slot, the solution describes the bending of a plate with a rectilinear crack or a rigid inclusion.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 144–152, January–February, 2005.     

A semi-analytical solution for linearized multicomponent cation exchange reactive transport in groundwater

Cation exchange in groundwater is one of the dominant surface reactions. Mass transfer of cation exchanging pollutants in groundwater is highly nonlinear due to the complex nonlinearities of exchange isotherms. This makes difficult to derive analytical solutions for transport equations. Available analytical solutions are valid only for binary cation exchange transport in 1-D and often disregard dispersion. Here we present a semi-analytical solution for linearized multication exchange reactive transport in steady 1-, 2- or 3-D groundwater flow. Nonlinear cation exchange mass–action–law equations are first linearized by means of a first-order Taylor expansion of log-concentrations around some selected reference concentrations and then substituted into transport equations. The resulting set of coupled partial differential equations (PDEs) are decoupled by means of a matrix similarity transformation which is applied also to boundary and initial concentrations. Uncoupled PDE’s are solved by standard analytical solutions. Concentrations of the original problem are obtained by back-transforming the solution of uncoupled PDEs. The semi-analytical solution compares well with nonlinear numerical solutions computed with a reactive transport code (CORE^2D) for several 1-D test cases involving two and three cations having moderate retardation factors. Deviations of the semi-analytical solution from numerical solutions increase with increasing cation exchange capacity (CEC), but do not depend on Peclet number. The semi-analytical solution captures the fronts of ternary systems in an approximate manner and tends to oversmooth sharp fronts for large retardation factors. The semi-analytical solution performs better with reference concentrations equal to the arithmetic average of boundary and initial concentrations than it does with reference concentrations derived from the arithmetic average of log-concentrations of boundary and initial waters.     

Singularities in a Solution to a Rotating Orthotropic Disk with Temperature Gradient

A semi-analytical solution is obtained for a rotating stress-free edge disk of constant thickness and density. In the plastic range, the Hill’s quadratic orthotropic yield criterion is adopted. In the elastic range, the Hooke’s law holds with thermal effects included. The analysis of singularities performed may be used for correct implementation of numerical codes and preliminary engineering design.     

An Existence Theorem for Nonlinearly Elastic ‘Flexural’ Shells

The two-dimensional equations of a nonlinearly elastic ‘flexural’ shell have been recently identified and justified by V. Lods and B. Miara, by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. These equations can be recast as a minimization problem for a ‘two-dimensional energy’ over a manifold of ‘admissible deformations’. The stored energy function is a quadratic expression in terms of the exact difference between the curvature tensor of the deformed middle surface and that of the undeformed one; the admissible deformations are those that preserve the metric of the undeformed middle surface and satisfy boundary conditions of clamping or of simple support. We establish here that this minimization problem has at least one solution. This revised version was published online in July 2006 with corrections to the Cover Date.     

Governing equations and numerical solutions of tension leg platform with finite amplitude motion

It is demonstrated that when tension leg platform （TLP） moves with finite amplitude in waves, the inertia force, the drag force and the buoyancy acting on the platform are nonlinear functions of the response of TLP, The tensions of the tethers are also nonlinear functions of the displacement of TLP. Then the displacement, the velocity and the acceleration of TLP should be taken into account when loads are calculated. In addition, equations of motions should be set up on the instantaneous position. A theo- retical model for analyzing the nonlinear behavior of a TLP with finite displacement is developed, in which multifold nonlinearities are taken into account, i.e., finite displace- ment, coupling of the six degrees of freedom, instantaneous position, instantaneous wet surface, free surface effects and viscous drag force, Based on the theoretical model, the comprehensive nonlinear differential equations are deduced. Then the nonlinear dynamic analysis of ISSC TLP in regular waves is performed in the time domain. The degenerative linear solution of the proposed nonlinear model is verified with existing published one. Furthermore, numerical results are presented, which illustrate that nonlinearities exert a significant influence on the dynamic responses of the TLP.     

Stability of a 2-dimensional Mathieu-type system with quasiperiodic coefficients

In the following we consider a 2-dimensional system of ODEs containing quasiperiodic terms. The system is proposed as an extension of Mathieu-type equations to higher dimensions, with emphasis on how resonance between the internal frequencies leads to a loss of stability. The 2-d system has two ‘natural’ frequencies when the time-dependent terms are switched off, and it is internally driven by quasiperiodic terms in the same frequencies. Stability charts in the parameter space are generated first using numerical simulations and Floquet theory. While some instability regions are easy to anticipate, there are some surprises: within instability zones, small islands of stability develop, and unusual ‘arcs’ of instability arise also. The transition curves are analyzed using the method of harmonic balance, and we find we can use this method to easily predict the ‘resonance curves’ from which bands of instability emanate. In addition, the method of multiple scales is used to examine the islands of stability near the 1:1 resonance.     

A new look at an old technique for the measurement of fatigue crack-growth rates

An automated technique is described by which a sequence of ‘beachmarks’ (clam-shell markings) can be produced in a specimen during laboratory tests on the propagation of fatigue cracks. The subsequent use of the beachmarks for the measurement of the lengths of the cracks is described, and the intrinsic accuracy of the method is estimated. It is found that, although fewer data points are obtained by this method, they are more accurate and better characterized than those obtained by the more commonly used ‘indirect’ methods. In addition, valuable supplementary information is obtained from which it is possible to assess both the confidence intervals for the results, and the degree of crack front curvature.     

Development of a Generalized Version of the Poisson– Nernst–Planck Equations Using the Hybrid Mixture Theory: Presentation of 2D Numerical Examples

A numerical scheme for the transient solution of a generalized version of the Poisson–Nernst–Planck (PNP) equations is presented. The finite element method is used to establish the coupled non-linear matrix system of equations capable of solving the present problem iteratively. The PNP equations represent a set of diffusion equations for charged species, i.e. dissolved ions, present in the pore solution of a rigid porous material in which the surface charge can be assumed neglectable. These equations are coupled to the ‘internally’ induced electrical field and to the velocity field of the fluid. The Nernst–Planck equations describing the diffusion of the ionic species and Gauss’ law in use are, however, coupled in both directions. The governing set of equations is derived from a simplified version of the so-called hybrid mixture theory (HMT). The simplifications used here mainly concerns ignoring the deformation and stresses in the porous material in which the ionic diffusion occurs. The HMT is a special version of the more ‘classical’ continuum mixture theories in the sense that it works with averaged equations at macroscale and that it includes the volume fractions of phases in its structure. The background to the PNP equations can by the HMT approach be described by using the postulates of mass conservation of constituents together with Gauss’ law used together with consistent constitutive laws. The HMT theory includes the constituent forms of the quasistatic version of Maxwell’s equations making it suitable for analyses of the kind addressed in this work. Within the framework of HTM, constitutive equations have been derived using the postulate of entropy inequality together with the technique of identifying properties by Lagrange multipliers. These results will be used in obtaining a closed set of equations for the present problem.     

On the General Structure of Small on Large Problems for Elastic Deformations of Varga Materials I: Plane Strain Deformations

In three recent papers [6–8], the present authors show that both plane strain and axially symmetric deformations of perfectly elastic incompressible Varga materials admit certain first integrals, which means that solutions for finite elastic deformations can be determined from a second order partial differential equation, rather than a fourth order one. For plane strain deformations there are three such integrals, while for axially symmetric deformations there are two. The purpose of the present papers is to present the general equations for small deformations which are superimposed upon a large deformation, which is assumed to satisfy one of the previously obtained first integrals. The governing partial differential equations for the small superimposed deformations are linear but highly nonhomogeneous, and we present here the precise structure of these equations in terms of a second-order linear differential operator D², which is first defined by examining solutions of the known integrals. The results obtained are illustrated with reference to a number of specific large deformations which are known solutions of the first integrals. For deformations of limited magnitude, the Varga strain-energy function has been established as a reasonable prototype for both natural rubber vulcanizates and styrene-butadiene vulcanizates. Plane strain deformations are examined in this present part while axially symmetric deformations are considered in Part II [16]. This revised version was published online in August 2006 with corrections to the Cover Date.     

Unsteady Flux-Vector-Based Green Element Method

This article extends the mathematical formulation and solution procedure of the modified ‘q-based’ GEM to unsteady situations, namely to the modified unsteady ‘q-based’ GEM. Solutions that provide information on the evolution of the pressure and the flux over long time intervals are available by incorporating the additional dimension of time into steady problems. This approach is first tested by solving an example for which an analytical solution is available. The numerical results for this example is found to be in excellent agreement with the analytical solution. Several problems involving geological features, such as wells and faults, are then investigated, with different properties applying to the faults. A strong influence of the low permeability faults is in evidence in these problems.     

Asymptotic expansions by Γ-convergence

Our starting point is a parameterized family of functionals (a ‘theory’) for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is ‘small’ but finite. Since Γ-convergence may be non-uniform within the ‘theory’, we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called ‘size’ or ‘scale’ effects.      

A second order stationary solution of the three dimensional wave diffraction potential

A theory on the second order wave diffraction by a three dimensional body fixed in a regular sea has been developed in the present paper. By regarding the sinusoidal disturb potential as a stationary solution of an initial value problem, and using Laplace transformation method and Tauberian theorem, the boundary value problems of stationary solution of the first and second order diffraction potential have been derived in this paper. Furthermore, the explicit solution of the second order stationary diffraction potential has been obtained with the method of the double Fourier transformation. It is found that the asymptotic behaviour of the second order stationary solution at far field is dependent on two wave systems, the first is “free wave”, travelling independently of the first order wave system, the other is “phase locked waves”, which accompany the first order waves. At the same time, the radiation conditions of the second order diffraction problems are derived. We also find that one can still pursue a steady state formulation with the inclusion of Rayleigh damping. Finally, as an example, the second order wave forces upon a fixed vertical circular cylinder have been calculated, and the numerical results agree well with the experimental data.     

From Rate-Dependent to Rate-Independent Brittle Crack Propagation

On the base of many experimental results, e.g., Ravi-Chandar and Knauss (Int. J. Fract. 26:65–80, 1984), Sharon et al. (Phys. Rev. Lett. 76(12):2117–2120, 1996), Hauch and Marder (Int. J. Fract. 90:133–151, 1998), the object of our analysis is a rate-dependent model for the propagation of a crack in brittle materials. Restricting ourselves to the quasi-static framework, our goal is a mathematical study of the evolution equation in the geometries of the ‘Single Edge Notch Tension’ and of the ‘Compact Tension’. Besides existence and uniqueness, emphasis is placed on the regularity of the evolution making reference also to the ‘velocity gap’. The transition to the rate-independent model of Griffith is obtained by time rescaling, proving convergence of the rescaled evolutions and of their energies. Further, the discontinuities of the rate-independent evolution are characterized in terms of unstable points of the free energy. Results are illustrated by a couple of numerical examples in the above mentioned geometries.     

Conduction–radiation interaction in 3D irregular enclosures using the finite volume method

Interaction of conduction and radiation in 3D enclosures is carried out with a gray participating media. Application of block structured grid is shown with the finite volume method (FVM). Radiation modeling is performed with the FVM and is coupled with an ‘in-house’ code to solve the set of transport equations. The detailed numerical results are presented for a cubical and a cylindrical enclosure as these results are not available in the literature. The numerical simulation for the cylindrical enclosure is performed using a block-structured ‘O’ grid. Two additional geometries are considered in order to show the applicability of the present work. Results of temperature, radiative heat flux and total heat flux distributions are presented for different optical thicknesses, scattering albedoes, emissivities and conduction–radiation parameters. The 3D results are validated with the available 2D results or results with pure radiation problems as limiting cases.     

Non-Ergodicity of the Nosé–Hoover Thermostatted Harmonic Oscillator

The Nosé–Hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical Gibbs distribution. Newton’s equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nosé–Hoover dynamics being ergodic. Hoover presented numerical experiments to show that the Nosé–Hoover dynamics are non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nosé–Hoover thermostat does not give an ergodynamical system for the one- dimensional harmonic oscillator when the “mass” of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nosé–Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass.