General relations for waves in one-dimensional elastic systems

Common features inherent in waves propagating in one-dimensional elastic systems are pointed out. Local laws of energy and wave momentum transfer when the Lagrangian of an elastic system depends on the generalized coordinates and their derivatives up to the second order inclusive are presented. It is shown that in a reference system moving with the phase velocity, the ratio of the energy flux density to the wave momentum flux density is equal to the phase velocity. It is established that for systems, the behaviour of which is described by linear equations or by nonlinear equations in the unknown function, the ratio of the mean values of the energy flux density to the wave momentum density is equal to the product of the phase and group velocities of the waves.     

Global solution to the full one-dimensional Frémond model for shape memory alloys

In this paper, we prove the existence and uniqueness of the solution to the one-dimensional initial-boundary value problem resulting from the Frémond thermomechanical model of structural phase transitions in shape memory materials. In this model, the free energy is assumed to depend on temperature, macroscopic deformation and phase fractions. The resulting equilibrium equations are the balance laws of (linear) momentum and energy, coupled with an evolution variational inequality for the phase fractions. Fourth-order regularizing terms in the quasi-stationary momentum balance equation are not necessary, and, as far as we know for the first time, all the non-linear terms of the energy balance equation are taken into account.     

On the bending and tension of thermoelastic shells undergoing phase transitions

Applying the general non-linear theory of shells undergoing phase transitions, we derive the balance equations along the singular surface curve modelling the phase interface in the shell. From the integral forms of balance laws of linear momentum, angular momentum, and energy as well as the entropy inequality we obtain the local static balance equations along the curvilinear phase interface. We also derive the thermodynamic condition allowing one to determine the interface position on the deformed shell midsurface. The theoretical model is illustrated by the example of thin circular cylindrical shell made of two-phase material subjected to tensile forces and bending couples at the shell boundary. The elastic solution reveals the existence of the hysteresis loop whose size depends upon values of several loading parameters. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

CFD methodology for sedimentation tanks: The effect of secondary phase on fluid phase using DPM coupled calculations

Computational Fluid Dynamics (CFD) methods are employed in order to simulate the 3D hydrodynamics and flow behaviour in a sedimentation tank. Unlike most of the previous numerical investigations, in the present paper the momentum exchange between the primary and the secondary phase is taken into account, using a Lagrangian method (discrete phase model) with two-way coupled calculations. By computing particle trajectories the proposed numerical model can track the momentum gained or lost by the particle stream that follows that trajectory and these quantities can be incorporated in the subsequent continuous phase calculations. Thus, while the continuous phase always impacts the discrete phase, the effect of the discrete phase trajectories on the continuum can be incorporated. This interchange affects fluid velocity, especially in the case of large particles sizes, which have a greater relaxation time in relation to the characteristic time of the tank. The present investigation compares a series of numerical simulations for a sedimentation tank with varying particle diameters and volume fractions, in order to identify the influence of the secondary phase to the primary phase and vice-versa and the way that this influence affects the efficiency of the tank.     

Development of boundary transfer method in simulation of gas–solid turbulent flow of a riser

Computational fluid dynamics (CFD) is used to simulate the behavior of two phase gas solid in a fluid catalytic cracking (FCC) riser. Gas and particle phases are considered as separate fully interpenetrating continuous media within each control volume. Each phase described in terms of its own separate mass and momentum conservation equations. Simple k–epsilon (k_g–?_g) turbulence model is used for the gas phase and the solid phase is handled with the kinetic theory of granular flows. Source terms are used to account for the influence of hydrodynamic drag on the production, dissipation and exchange of turbulent kinetic energy between the phases. For the particles partial slip condition is considered at the wall.     

Analysis of a thermomechanical phase transition model

Subject of this work is a macroscopic thermomechanical model of phase transitions in steel. Effects like transformation strain and transformation plasticity induced by the phase transitions are considered and used to formulate a consistent thermomechanical model. The resulting system of state equations consists of a quasistatic momentum balance coupled with a nonlinear stress-strain relation, a nonlinear energy balance equation and a system of ODEs for the phase volume fractions. We prove the existence of a unique weak solution using fixed-point arguments. A key issue is a regularity analysis for the mechanical subsystem to obtain continuity of the stress tensor. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation

A fundamental class of solutions of symmetric Hamiltonian systems is relative equilibria. In this paper the nonlinear problem near a degenerate relative equilibrium is considered. The degeneracy creates a saddle-center and attendant homoclinic bifurcation in the reduced system transverse to the group orbit. The surprising result is that the curvature of the pullback of the momentum map to the Lie algebra determines the normal form for the homoclinic bifurcation. There is also an induced directional geometric phase in the homoclinic bifurcation. The backbone of the analysis is the use of singularity theory for smooth mappings between manifolds applied to the pullback of the momentum map. The theory is constructive and generalities are given for symmetric Hamiltonian systems on a vector space of dimension (2n+2) with an n-dimensional abelian symmetry group. Examples for n=1,2,3 are presented to illustrate application of the theory.     

Reduced spaces and their relation to relative equilibria

Summary It is known that the Hamiltonian motion of a mechanical system with symmetry induces Hamiltonian flows on reduced phase spaces. In this paper we apply Morse theory to study the relationship between the topology of the reduced space and the number of relative equilibria in the corresponding momentum level set. Our attention is restricted to simple mechanical systems with compact configuration space and compact symmetry group. We begin by showing that the set of relative equilibria in a level set of the momentum map is compact. We then employ techniques from Morse theory to prove that the number of orbits of relative equilibria with momentum in the coadjoint orbit of a given regular momentum value is bounded below by the the sum of Betti numbers of the corresponding reduced space when the Hamiltonian is fibre quadratic and the reduced Hamiltonian is nondegenerate. In addition, for a certain class of group actions on the configuration manifold, it is shown that the above result extends to Hamiltonians of the form potential plus kinetic.     

On a hidden symmetry of quantum harmonic oscillators

We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg uncertainty principle is presented.     

On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system

In this paper, we study the relativistic Vlasov-Fokker-Planck-Maxwell system in one space variable and two momentum variables. This non-linear system of equations consists of a transport equation for the phase space distribution function combined with Maxwell's equations for the electric and magnetic fields. It is important in modelling distribution of charged particles in the kinetic theory of plasma. We prove the existence of a classical solution when the initial density decays fast enough with respect to the momentum variables. The solution which shares this same decay condition along with its first derivatives in the momentum variables is unique.     

Global existence result for thermoviscoelastic problems with hysteresis

We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a three-dimensional setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouët. Finally a global existence result is derived.     

Flow Structure in a Technical Scale Reactor with an Internal Reboiler

In this contribution, results of experimental flow investigations in a technical scale cylindrical reactor with centrally located internal reboiler are presented. The fluid motion in the kettle is induced by natural circulation. Two different geometrical configurations are considered. First, the liquid level is set below the evaporator's outlet. Thus, only the liquid phase penetrates the liquid surface. The second configuration characterizes the a direct injection of the two phase mixture into bulk liquid. In both cases the liquid leaves the evaporator radially and horizontally towards the cylindrical wall. Gathered results reveal the presence of the big, toroidal vortex in the kettle annulus. Independently on the boundary conditions, self similarity of velocity profiles in the near wall region is observed. In contrast, investigations of the momentum transfer between the jet and the bulk liquid reveal strong dependency on the geometrical configuration of the setup. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of cavitation dynamics using a cavitation-induced-momentum-defect (CIMD) correction approach

A new unsteady cavitation event tracking model is developed for predicting vapor dynamics occurring in multi-dimensional incompressible flows. The procedure solves incompressible Navier–Stokes equations for the liquid phase supplemented with an additional vapor transport equation for the vapor phase. The novel cavitation-induced-momentum-defect (CIMD) correction methodology developed in this study accounts for cavitation inception and collapse events as relevant momentum-source terms in the liquid phase momentum equations. The model tracks cavitation zones and applies compressibility effects, employing homogeneous equilibrium model (HEM) assumptions, in constructing the source term of the vapor transport model. Effects of vapor phase accumulation and diffusion are incorporated by detailed relaxation models. A modified RNG k–ε model, including the effects of compressibility in the vapor regions, is employed for modeling turbulence effects. Numerical simulations are carried out using a finite volume methodology available within the framework of commercial CFD software code Fluent v.6.2. Simulation results are in good qualitative agreement with experiments for unsteady cloud cavitation behavior in planar nozzle flows. Multitude of mechanisms such as formation of vortex cavities, vapor cluster shedding and coalescence, cavity pinch off are sharply captured by the CIMD approach. Our results indicate the profound influence of re-entrant jet motion and adverse pressure gradients on the cavitation dynamics.     

A continuum model for transport phenomena in convective flow of solid–liquid phase change material suspensions

Heat and mass transport is modeled in convective flow of a dilute binary mixture of a continuous fluid with mono-dispersed particles (PCM suspensions), in which solid–liquid phase change can take place. The model is based on the mixture continuum approach together with an approximate enthalpy formulation, in which the temporal and spatial variations of phase change fraction in the particles are considered explicitly. Derivations are given for a set of equations governing conservation of mass, momentum, species, and energy of the suspensions, as well as the evolution of phase change fraction of the dispersed particles.     

Generalized Weyl–Wigner–Moyal–Ville Formalism and Topological Groups

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal, and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on, we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of ‘Umklapp–Prozessen’. Copyright © 2011 John Wiley & Sons, Ltd.     

CFD simulation of gas–solid flow with a cluster structure-dependent drag coefficient model in circulating fluidized beds

To model the effect of clusters on hydrodynamics of gas and particles phases in risers, the interfacial drag coefficient is taken into account in computational fluid dynamic simulations by means of a two-fluid model. The momentum and energy balances that characterize the clusters in the dense phase and dispersed particles in the dilute phase are described by the multi-scale resolution approach. The model of cluster structure-dependent (CSD) drag coefficient is proposed on the basis of the minimization of energy dissipation by heterogeneous drag (MEDHD) in the full range of Reynolds number. The model of CSD drag coefficient is then incorporated into the two-fluid model to simulate flow behavior of gas and particles in a riser. The distributions of volume fraction and velocity of particles are predicted. Simulated results are in agreement with experimental data published in the literature.     

Non-relativistic global limits of entropy solutions to the extremely relativistic Euler equations

We are concerned in this paper with the non-relativistic global limits of the entropy solutions to the Cauchy problem of 3 × 3 system of relativistic Euler equations modeling the conservation of baryon numbers, momentum, and energy respectively. Based on the detailed geometric properties of nonlinear wave curves in the phase space and the Glimm’s method, we obtain, for the isothermal flow, the convergence of the entropy solutions to the solutions of the corresponding classical non-relativistic Euler equations as the speed of light c → +∞.     

Phase-field modeling of martensitic phase transformations in polycrystals coupled with crystal plasticity – A spectral-based approach

The purpose of this work is the phase-field modeling of fcc-to-bcc martensitic phase transformations in polycrystals and the coupling with crystal plasticity. Assuming microscopic periodic fields, Green-function- and fast Fourier transform (FFT)-methods are used to solve the quasi-static balance of linear momentum. The Allen-Cahn evolution equation is discretized based on a semi-implicit time integration scheme in Fourier space. Two-dimensional results are presented and the interplay between martensitic phase transformation and plastic slip is studied at different stages of the deformation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)