A method for Lyapunov spectrum estimation using cloned dynamics and its application to the discontinuously-excited FitzHugh–Nagumo model

This work presents a new method to calculate the Lyapunov spectrum of dynamical systems based on the time evolution of initially small disturbed copies (“clones”) of the motion equations. In this approach, it is not necessary to construct the tangent space associated with the time evolution of linearized versions of motion equations, being the Lyapunov exponents directly estimated in terms of the rate of convergence or divergence of these disturbed clones with respect to the fiducial trajectory, there being periodic correction via the Gram–Schmidt Reorthonormalization procedure. The proposed method offers the possibility of partial estimation of the Lyapunov spectrum and can also be applied to nonsmooth dynamics, since the linearization procedure is no longer required. The idea is tested for representative continuous- and discrete-time dynamical systems and validated by means of comparison with the classical method to perform this calculation. To illustrate its applicability in the nonsmooth context, the largest Lyapunov exponent of the FitzHugh–Nagumo neuronal model under discontinuous periodic excitation is calculated taking the amplitude of stimulation as control parameter. This analysis reveals some complex behaviours for this simple neuronal model, which motivates relevant discussions about the possible role of chaos in the cognitive process.     

An improved theoretical model for the dynamics of a free–clamped cylinder in axial flow

A new, improved linear analytical model is presented in this paper for the dynamics of a slender cantilevered cylinder with an ogival free end, subjected to axial flow directed from its free end towards the clamped one. In the present model the fluid-dynamic forces at the free end of the cylinder are analysed in a meticulous manner. The model predicts that the cylinder loses stability at relatively low flow velocity by flutter, and then at higher flow velocity by static divergence. This agrees with the dynamical behaviour observed in experiments. Moreover, quantitative agreement in the critical flow velocities for flutter and divergence between this improved theory and experiment is fairly good.     

An approximate solution for the Couette–Poiseuille flow of the Giesekus model between parallel plates

An approximate analytical solution is derived for the Couette–Poiseuille flow of a nonlinear viscoelastic fluid obeying the Giesekus constitutive equation between parallel plates for the case where the upper plate moves at constant velocity, and the lower one is at rest. Validity of this approximation is examined by comparison to the exact solution during a parametric study. The influence of Deborah number (De) and Giesekus model parameter (α) on the velocity profile, normal stress, and friction factor are investigated. Results show strong effects of viscoelastic parameters on velocity profile and normal stress. In addition, five velocity profile types were obtained for different values of α, De, and the dimensionless pressure gradient (G).     

A multi-scale architecture for multi-scale simulation and its application to gas–solid flows

A multi-scale hardware and software architecture implementing the EMMS (energy-minimization multi-scale) paradigm is proven to be effective in the simulation of a two-dimensional gas–solid suspension. General purpose CPUs are employed for macro-scale control and optimization, and many integrated cores (MICs) operating in multiple-instruction multiple-data mode are used for a molecular dynamics simulation of the solid particles at the meso-scale. Many cores operating in single-instruction multiple-data mode, such as general purpose graphics processing units (GPGPUs), are employed for direct numerical simulation of the fluid flow at the micro-scale using the lattice Boltzmann method. This architecture is also expected to be efficient for the multi-scale simulation of other complex systems.     

An integrated 2-D Navier–Stokes equation and its application to 3-D internal flows

An efficient two-dimensional (2-D) analytical and numerical procedure has been proposed to investigate three-dimensional (3-D) internal flows through a passage with a spatially variable depth, in which the viscous forces act significantly on both upper and lower walls. The integral 2-D version of the Navier–Stokes equation was obtained by integrating the full Navier–Stokes equation in a 3-D form over the depth of the passage. In order to examine the validity of the integrated momentum equations, fully-developed flows in straight noncircular ducts were investigated analytically prior to numerical investigations. It has been shown that the exact solutions for circular, elliptical and equilateral triangular ducts are obtainable from the integrated Navier–Stokes equation. Having confirmed its wide applicability to internal flows, numerical computations were conducted to investigate the oscillation mechanism of a fluidic oscillator. Comparison of the present prediction and experiment reveals the validity of the present treatment.     

A unified model for polystyrene–nanorod and polystyrene–nanoplatelet melt composites

We present a unified constitutive model capable of predicting the steady shear rheology of polystyrene (PS)–nanoparticle melt composites, where particles can be rods, platelets, or any geometry in between, as validated against experimental measurements. The composite model incorporates the rheological properties of the polymer matrix, the aspect ratio and characteristic length scale of the nanoparticles, the orientation of the nanoparticles, hydrodynamic particle–particle interactions, the interaction between the nanoparticles and the polymer, and flow conditions of melt processing. We demonstrate that our constitutive model predicts both the steady rheology of PS–carbon nanofiber composites and the steady rheology of PS–nanoclay composites. Along with presenting the model and validating it against experimental measurements, we evaluate three different closure approximations, an important constitutive assumption in a kinetic theory model, for both polymer–nanoparticle systems. Both composite systems are most accurately modeled with a quadratic closure approximation.     

An FMM for waveguide problems of 2-D Helmholtz’ equation and its application to eigenvalue problems

This paper discusses an FMM for solving waveguide problems and associated eigenvalue problems for Helmholtz’ equation in a two dimensional infinite strip with homogeneous Neumann boundary condition on the sides. Layer potentials with Green’s function for this problem are evaluated efficiently with the help of the method of images and FMM. We apply FMM to solve some boundary value problems in waveguides and associated resonance frequency problems using the Sakurai–Sugiura projection method after discussing the required analytic continuation of the solutions to complex frequencies. Some numerical examples show the accuracy and the efficiency of the proposed method.     

K-ε-T model of dense liquid-solid two-phase turbulent flow and its application to the pipe flow

To predict the characteristics of dense liquid-solid two-phase flow, K-ε-T model is established, in which the turbulent flow of fluid phase is described with fluid turbulent kinetic energy K_f and its dissipation rate ε_f, and the particles random motion is described with particle turbulent energy K_p and its dissipation rate ε_p and pseudothermal temperature T_p. The governing equations are also derived. With K-ε-T model, numerical study of dense liquid-solid two-phase turbulent up-flow in a pipe is performed. The calculated results are in good agreement with experimental data of Alajbegovic et al. (1994), and some flow features are captured.     

Development of a hybrid model based on mesh and meshfree methods and its application to fluid–elastic structure interaction for free surface waves

In this paper, a hybrid scheme, Fluid–Fluid–Elastic Structure (FFES) model was developed in the time domain to address the wave breaking impact on the structure. The model is developed based on the partitioned approach with different governing equations that describe various regions of the model domain. The fluid–fluid model denotes that two different fluid models were used to describe fluid in the actual physical domain. The method is a physics-based approximation to reduce the computational time, i.e. in the far-field inviscid fluid (fully nonlinear potential flow theory model), and near to the structure, viscous fluid (Navier Stokes model) is used. The coupled model then interacts with the elastic structure (based on Euler–Bernoulli beam theory). The system of equations is strongly coupled both in space and time. The Fluid–Fluid coupling uses an implicit predictor–corrector scheme, and the fluid–structure coupling works based on an iterative scheme. This approach makes the method more robust and for future extension. Three different possibilities for introducing the coupling was identified and implemented. The model was validated against results from the analytical solution and literature. The method proposed is a reliable, robust, and efficient alternative for simulating fluid–structure interaction problems.     

An empirical model for the thermal conductivity of compacted bentonite and a bentonite–sand mixture

The thermal conductivities of compacted bentonite and a bentonite–sand mixture were measured to investigate the effects of dry density, water content and sand fraction on the thermal conductivity. A single expression has been proposed to describe the thermal conductivity of the compacted bentonite and the bentonite–sand mixture once their primary parameters such as dry density, water content and sand fraction are known.     

On the averaging principle for one-frequency systems. An application to satellite motions

This paper is related to our previous works (Morosi and Pizzocchero in J. Phys. A, Math. Gen. 39:3673–3702, 2006; Nonlinear Dyn., 2008), on the error estimate of the averaging technique for systems with one fast angular variable. In the cited references, a general method (of mixed analytical and numerical type) has been introduced to obtain precise, fully quantitative estimates on the averaging error. Here, this procedure is applied to the motion of a satellite in a polar orbit around an oblate planet, retaining only the J ₂ term in the multipole expansion of the gravitational potential. To exemplify the method, the averaging errors are estimated for the data corresponding to two Earth satellites; for a very large number of orbits, computation of our estimators is much less expensive than the direct numerical solution of the equations of motion.     

A micro-mechanical model of knitted fabric and its application to the analysis of buckling under tension in wale direction： buckling analysis

With the aid of the micro-mechanical model of knitted fabric proposed in Part 1 we analyze the buckling of a knitted fabric sheet when it is subjected to a tension along the wale direction. The large deformation of the fabric sheet in the critical configuration is considered and, to avoid possible deviation due to the approximation of the theory of thin plate, the three-dimensional theory of instability is used. The fabric sheet is considered as a three-dimensional body and all boundary conditions are satisfied. It is shown that the buckling of the fabric sheet is possible, two buckling modes and the corresponding buckling conditions are obtained, but only the flexural mode is physically possible as observed in experiments.The project supported by the National Natural Science Foundation of China (10272079)     

Existence and time-discretization for the finite-strain Souza–Auricchio constitutive model for shape-memory alloys

We prove the global existence of solutions for a shape-memory alloys constitutive model at finite strains. The model has been presented in Evangelista et al. (Int J Numer Methods Eng 81(6):761–785, 2010) and corresponds to a suitable finite-strain version of the celebrated Souza–Auricchio model for SMAs (Auricchio and Petrini in Int J Numer Methods Eng 55:1255–1284, 2002; Souza et al. in J Mech A Solids 17:789–806, 1998). We reformulate the model in purely variational fashion under the form of a rate-independent process. Existence of suitably weak (energetic) solutions to the model is obtained by passing to the limit within a constructive time-discretization procedure.     

A reformulated synthetic turbulence generation method for a zonal RANS–LES method and its application to zero-pressure gradient boundary layers

A synthetic turbulence generation (STG) method for subsonic and supersonic flows at low and moderate Reynolds numbers to provide inflow distributions of zonal Reynolds-averaged Navier–Stokes (RANS) – large-eddy simulation (LES) methods is presented. The STG method splits the LES inflow region into three planes where a local velocity signal is decomposed from the turbulent flow properties of the upstream RANS solution. Based on the wall-normal position and the local flow Reynolds number, specific length and velocity scales with different vorticity content are imposed at the inlet plane of the boundary layer. The quality of the STG method for incompressible and compressible zero-pressure gradient boundary layers is shown by comparing the zonal RANS–LES data with pure LES, pure RANS, and direct numerical simulation (DNS) solutions. The distributions of the time and spanwise wall-shear stress, Reynolds stress distributions, and two point correlations of the zonal RANS–LES simulations are smooth in the transition region and in good agreement with the pure LES and reference DNS findings. The STG approach reduces the RANS-to-LES transition length to less than four boundary-layer thicknesses.     

An application of the k-ɛ model with variable Prandtl number to heat transfer computations in air flows

The two-equation `low Reynolds number' k-? model of turbulence with a set of universal constants suggested by Launder and Sharma is modified in the present paper. The variability of the turbulent Prandtl number Pr_t in the energy equation is assumed along with a change of a constant in the dissipation term of the turbulent kinetic energy equation. The turbulent heat transfer is computed for an air flow in a circular pipe for the Reynolds number within the range of 10⁴?4. The modification considerably improves the agreement between the numerical results and the experiment data published in the available literature.     

An immersed boundary-lattice Boltzmann flux solver and its applications to fluid–structure interaction problems

An immersed boundary-lattice Boltzmann flux solver (IB–LBFS) for the simulation of two-dimensional fluid–structure interaction (FSI) problems is presented in this paper. The IB–LBFS applies the fractional-step method to split the overall solution process into the predictor step and the corrector step. In the predictor step, the intermediate flow field is predicted by applying the LBFS (lattice Boltzmann flux solver) without considering the presence of immersed object. The LBFS applies the finite volume method to solve N–S (Navier–Stokes) equations for the flow variables at cell centers. At each cell interface, the LBFS evaluates its viscous and inviscid fluxes simultaneously through local reconstruction of the LBE (lattice Boltzmann equation) solutions. In the corrector step, the intermediate flow field is corrected by the implicit boundary condition-enforced immersed boundary method (IBM) so that the no-slip boundary conditions can be accurately satisfied. The IB–LBFS effectively combines the advantages of the LBFS in solving the flow field and the flexibility of the IBM in dealing with boundary conditions. Consequently, the IB–LBFS presents a much simpler and more effective approach for simulating complex FSI problems on non-uniform grids. Several test cases, including flows past one and two cylinders with prescribed motions, are firstly simulated to examine the accuracy of present solver. After that, two strongly coupled fluid–structure interaction problems, i.e., particle sedimentations and vortex-induced vibrations of a circular cylinder are investigated. Good agreements between the present results and those in literature verify the capability and flexibility of IB–LBFS for simulating FSI problems.     

An Eulerian–Lagrangian hybrid model for the coarse-grid simulation of turbulent liquid jet breakup

In this paper we present a numerical model for the coarse-grid simulation of turbulent liquid jet breakup using an Eulerian–Lagrangian coupling. To picture the unresolved droplet formation near the liquid jet interface in the case of coarse grids we considered a theoretical model to describe the unresolved flow instabilities leading to turbulent breakup. These entrained droplets are then represented by an Eulerian–Lagrangian hybrid concept. On the one hand, we used a volume of fluid method (VOF) to characterize the global spreading and the initiation of droplet formation; one the other hand, Lagrangian droplets are released at the liquid–gas interface according to the theoretical model balancing consolidating and disruptive energies. Here, a numerical coupling was required between Eulerian liquid core and Lagrangian droplets using mass and momentum source terms. The presented methodology was tested for different liquid jets in Rayleigh, wind-induced and atomization regimes and validated against literature data. This comparison reveals fairly good qualitative agreement in the cases of jet spreading, jet instability and jet breakup as well as relatively accurate size distribution and Sauter mean diameter (SMD) of the droplets. Furthermore, the model was able to capture the regime transitions from Rayleigh instability to atomization appropriately. Finally, the presented sub-grid model predicts the effect of the gas-phase pressure on the droplet sizes very well.     

Darboux transformation,exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

Nonlinear Dynamics - In this paper, we firstly deduce a reverse space-time Fokas–Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of...     

Normal forms for NFDEs with parameters and application to the lossless transmission line

A method for the computation of normal forms for neutral functional differential equations (NFDEs) with parameters is developed by considering an extension of phase space, based on the method of computing normal forms for FDEs with parameters previously introduced by Faria. The Hopf bifurcation of the differential difference equation is considered as an example of a circuit involving a lossless transmission line. The direction and stability of the bifurcating periodic solutions are also determined. Finally, numerical simulations are carried out to support the analytic results. This research is supported by the NNSF of China.