Conservation laws and reduction to quadratures of the generalized time-dependent duffing equation

The field-momentum method and the method of Hamilton and Jacobi are applied for finding the conservation laws and reduction to quadratures of the non-linear time-dependent equation

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where Q_i are constant parameters and f_i(t) are supposed to be power-type functions of time, that is, f_i(t) = t^k_i.     

Flow in a cylinder driven by rotating split-disk endwalls

Flow of an incompressible viscous fluid contained in a cylindrical vessel (radius R, height H) is considered. Each of the cylinder endwalls is split into two parts which rotate steadily about the central axis with different rotation rates: the inner disk (r < r₁) rotating at Ω₁, and the outer annulus (r₁ < r < R) rotating at Ω₂. Numerical solutions to the axisymmetric Navier-Stokes equations are secured for small system Ekman numbers E ( v/(ΩH²)). In the linear regime, when the Rossby number Ro

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, the numerical results are shown to be compatible with the theoretical prediction as well as the available experimental measurements. Emphasis is placed on the results in the nonlinear regime in which Ro is finite. Details of the structures of azimuthai and meridional flows are presented by the numerical results. For a fixed Ekman number, the gross features of the flow remain qualitatively unchanged as Ro increases. The meridional flows are characterized by two circulation cells. The shear layer is a region of intense axial flow toward the endwall and of vanishing radial velocity. The thicknesses of the shear layer near r = r₁ and the Ekman layer on the endwall scale with E and E , respectively. The numerical results are consistent with these scalings.     

On the stability of solutions of a non-linear, non-autonomous equation

Sufficient conditions for the stability of solutions of the equation

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are deduced. These conditions depend on coefficient ratios a/c and b/c as well as on initial conditions. As an example the pendulum of variable length is discussed.     

A general correlation for the local loss coefficient in Newtonian axisymmetric sudden expansions

Results from numerical simulations and guidance from an approximated corrected-theory, developed by Oliveira and Pinho (1997), (Oliveira, P.J. and Pinho, F.T. 1997. Pressure drop coefficient of laminar Newtonian flow in axisymmetric sudden expansions. Int. J. Heat and Fluid flow 18, 518–529) have been used to arrive at a correlation expressing the irreversible loss coefficient for laminar Newtonian flow in axisymmetric sudden expansions. The correlation is valid for the ranges 1.5 < D₂/D₁ < 4 and 0.5 < Re < 200 with errors of less than 5%, except for 25 < Re < 100 where the error could be as much as 7%. The recirculation bubble length is also presented for the same range of conditions and the pressure recovery coefficient was calculated for Reynolds numbers above 15.     

Numerical prediction of locally forced turbulent separated and reattaching flow

An unsteady numerical simulation was performed for locally forced separated and reattaching flow over a backward-facing step. The local forcing was given to the separated and reattaching flow by means of a sinusoidally oscillating jet from a separation line. A version of the k––f_μ model was employed, in which the near-wall behavior without reference to distance and the nonequilibrium effect in the recirculation region were incorporated. The Reynolds number based on the step height (H) was fixed at Re_H=33 000, and the forcing frequency was varied in the range 0St_H2. The predicted results were compared and validated with the experimental data of Chun and Chun. It was shown that the unsteady locally forced separated and reattaching flows are predicted reasonably well with the k––f_μ model. To characterize the large-scale vortex evolution due to the local forcing, numerical flow visualizations were carried out.     

A study of two alternate tangent modulus formulations and attendant implicit algorithms for creep as well as high-strain-rate plasticity

Two alternate tangent modulus formulations based on the use of material characteristics expressed as

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, respectively, are prese for the analysis of material response under conditions such as high temperature creep and high strainrate dynamic plasticity. In each formulation, implicit algorithms of generalized midpoint radial mapping are presented to compute stress histories at a material point. Several examples to illustrate the stability, accuracy, and convergence of the presented computational methodologies are included. In each instance, the two alternate tangent modulus approaches lead to results of comparable accuracy and compare excellently with each other as well as other available independent solutions.     

On the Fremond’s constitutive model for shape memory alloys

The remarkable properties of shape memory alloys have increasing the interest in applications in different areas varying from biomedical to aerospace hardware. Despite the large number of applications, the modeling of SMA is the objective of many researches developed in order to describe all details of its thermomechanical behavior. The present contribution revisits a constitutive model presented by Savi et al. (2002), which is built up on the classical Fremond’s model, in order to contemplate the horizontal enlargement of the stress–strain hysteresis loop. Numerical simulations present qualitative agreement with experimental data, showing pseudoelastic, one-way and two-way shape memory effects.     

Numerical simulation of the noise generated by a low Mach number, low Reynolds number jet

In this paper we study the sound field produced by a turbulent round jet with a Mach number of 0.6 based on the centerline velocity and the ambient speed of sound c_∞. The turbulent flow field is found by solving the fully compressible Navier–Stokes equations with help of high-order compact finite difference schemes. It is shown that the simulated flow field is in good agreement with experiments. The corresponding sound field has been obtained with help of the Lighthill equation using two different formulations for the Lighthill stress tensor T_ij. In the first formulation of T_ij the fluctuating density is taken into account. In the second formulation the density is assumed to be constant. As an additional check we have also performed an acoustic calculation using a formulation in which a homogeneous wave equation is solved. The boundary conditions for this homogeneous wave equation are obtained from the numerical simulation of the Navier–Stokes equation. The results obtained with both formulations of the Lighthill stress tensor are nearly identical. This implies that an incompressible formulation of the conservations laws could be used to predict jet noise at low Mach numbers.     

Fast Diffusion to Self-Similarity: Complete Spectrum,Long-Time Asymptotics,and Numerology

The complete spectrum is determined for the operator on the Sobolev space W^1,2(Rⁿ) formed by closing the smooth functions of compact support with respect to the norm Here the Barenblatt profile is the stationary attractor of the rescaled diffusion equation in the fast, supercritical regime m the same diffusion dynamics represent the steepest descent down an entropy E(u) on probability measures with respect to the Wasserstein distance d₂. Formally, the operator H=HessE is the Hessian of this entropy at its minimum , so the spectral gap H:=2–n(1–m) found below suggests the sharp rate of asymptotic convergence: from any centered initial data 0u(0,x)L¹(Rⁿ) with second moments. This bound improves various results in the literature, and suggests the conjecture that the self-similar solution u(t,x)=R(t)^–n(x/R(t)) is always slowest to converge. The higher eigenfunctions – which are polynomials with hypergeometric radial parts – and the presence of continuous spectrum yield additional insight into the relations between symmetries of Rⁿ and the flow. Thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass-preserving dilation (or still better, affine image). The strange numerology of the spectrum is explained in terms of the number of moments of .Dedicated to Elliott H. Lieb on the occasion of his 70th birthday.     

Variational formulation of ideal fluid flows according to gauge principle

On the basis of the gauge principle of field theory, a new variational formulation is presented for flows of an ideal fluid. The fluid is defined thermodynamically by mass density and entropy density, and its flow fields are characterized by symmetries of translation and rotation. The rotational transformations are regarded as gauge transformations as well as the translational ones. In addition to the Lagrangians representing the translation symmetry, a structure of rotation symmetry is equipped with a Lagrangian Λ_A including the vorticity and a vector potential bilinearly. Euler's equation of motion is derived from variations according to the action principle. In addition, the equations of continuity and entropy are derived from the variations. Equations of conserved currents are deduced as the Noether theorem in the space of Lagrangian coordinate a. Without Λ_A, the action principle results in the Clebsch solution with vanishing helicity. The Lagrangian Λ_A yields non-vanishing vorticity and provides a source term of non-vanishing helicity. The vorticity equation is derived as an equation of the gauge field, and the Λ_A characterizes topology of the field. The present formulation is comprehensive and provides a consistent basis for a unique transformation between the Lagrangian a space and the Eulerian x space. In contrast, with translation symmetry alone, there is an arbitrariness in the transformation between these spaces.     

Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks

A numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a ‘universal approximator’ based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of ‘finite element‐type’ discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re = 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of , Re = 400 for the driven cavity flow with a density of and Rayleigh number Ra = 1 000 000 for the natural convection flow with a density of . Copyright © 2001 John Wiley & Sons, Ltd.     

Experimental investigation of the grid-generated turbulence features in a free surface flow

The present study aims to investigate the features of a grid-generated turbulence occurring in a current flow with a free surface flow. The interest is focused on the length and time scales of the turbulence. These are the macro, the micro and the Kolmogorov scales. To analyze the flow, a 2D LDV system has been used to measure , , u′ and w′. This non-intrusive and optical technique is really accurate (in terms of space and time resolution). Furthermore, it does not disturb the flow and provides a high data rate. Both horizontal and vertical velocities are recorded at the same time according to a coincidence window (τ_cw). Bias measurements are avoided by using a filtering technique during data processing. The improved homogeneity and isotropy of the turbulence downstream of the grid allows the use of the Taylor hypothesis. Thus, all length and time scales of the flow can be estimated. Results are discussed as well as the influence of the upcoming mean velocity on the turbulence properties.     

The proof of Fermat's Last Theorem

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Separation of variables in one problem of motion of the generalized Kowalevski top

In the problem of motion of the Kowalevski top in a double force field the new case of reduction to a Hamiltonian system with two degrees of freedom was pointed out by Kharlamov [Kharlamov, M.P., 2004. Mekh. Tverd. Tela 34, 47–58]. We show that the equations of motion in this case can be separated by the appropriate change of variables, the new variables U,V being hyperelliptic functions of time. The natural phase variables (components of the angular velocity and the direction vectors of the forces with respect to the movable basis) are expressed via U,V explicitly in elementary algebraic functions.     

A study of electromechanical switching in ferroelectric single crystals

This article documents both modeling and experimental studies developed to investigate the switching behavior of ferroelectric single crystals. The theoretical model makes a priori ansatz that switching follows the evolution of a particular domain pattern. The choice of this configuration is dictated by the requirement that domains remain compatible during evolution, giving rise to a low-energy path for the overall switching. The construction of this pattern is achieved using multirank laminates. It offers an advantage of specifying different types of domain wall movements, leading to a distinction for the switching types. A loading experiment is performed on a barium titanate (BaTiO₃) single crystal with a constant compressive stress and a cyclic electric field. Both 180 and 90 coercive fields are measured as input parameters required for the theoretical framework. The simulation results show good agreement with the observed strains measured by the present and other available experiments. It is found that depolarization has a non-trivial influence on attainable actuation strains.     

Experimental study of liquid sloshing dynamics in a barge carrying tank

An experimental work has been carried out to study the phenomena of sloshing of liquid in partially filled tanks mounted on a barge exposed to regular beam waves. Three liquid fill levels with liquid depth, h_s to length of tank, l ratio (h_s/l) of 0.163, 0.325 and 0.488, are studied. The time histories of sloshing oscillation are measured along the length of container at predefined locations. The nonlinear behaviour of sloshing oscillation is observed for the regular wave excitation. The spectra of the sloshing oscillation and their qualitative assessment are reported. Attempts are made to evaluate the harmonics present in the sloshing oscillation and compare with the results of earlier studies. The effects of wave excitation frequency and wave height on the sloshing oscillation as well as on the response of the barge are studied.