Detailed computational and experimental fluid dynamics of fluidized beds

To describe the hydrodynamic phenomena prevailing in large industrial scale fluidized beds continuum models are required. The flow in these systems depends strongly on particle–particle interaction and gas–particle interaction. For this reason, proper closure relations for these two interactions are vital for reliable predictions on the basis of continuum models. Gas–particle interaction can be studied with the use of the lattice Boltzmann model (LBM), while the particle–particle interaction can suitably be studied with a discrete particle model. In this work it is shown that the discrete particle model, utilizing a LBM based drag model, has the capability to generate insight and eventually closure relations in processes such as mixing, segregation and homogeneous fluidization.     

Equations of motion of rotating bodies in general relativity in the post-Newtonian approximation

Equations of the translational and rotational motion of two bodies possessing intrinsic angular momentum are obtained by the Einstein—Infeld—Hoffmann method in the post-Newtonian approximation. The results agree with the Kerr metric expressed in a harmonic system of coordinates with symmetry of the spatial components of the metric with respect to its indices and with a conservation law for the total angular momentum that is the sum of the orbital and spin angular momenta, and they give the correct passage to the limit to the equation of motion of a test particle with spin.All-Russia Institute of Experimental Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 123–135, October, 1994.     

Hopf Bifurcation Calculations in Delayed Systems with Translational Symmetry

The Hopf bifurcation of an equilibrium in dynamical systems consisting of n equations with a single time delay and translational symmetry is investigated. The Jacobian belonging to the equilibrium of the corresponding delay-differential equations always has a zero eigenvalue due to the translational symmetry. This eigenvalue does not depend on the system parameters, while other characteristic roots may satisfy the conditions of Hopf bifurcation. An algorithm for this Hopf bifurcation calculation (including the center-manifold reduction) is presented. The closed form results are demonstrated for a simple model of cars following each other along a ring.     

Prediction of microstructure in a Cosserat continuum using relaxed energies

We develop a nonlinear incompressible multiphase material model in a Cosserat continuum with microstructure. The free energy of the material is enriched with an interaction potential taking into account the intergranular kinematics at the continuum scale. As a result the total energy becomes non-convex, thus giving rise to the development of microstructural phases. To guarantee the existence of minimizers an exact quasi-convex envelope of the corresponding energy functional is derived. As a result a three phase material energy appears, among them two of the phases are with microstructure in the translational motion (displacment field) and micromotion (microrotation field), whereas the third phase is without internal structure. The corresponding relaxed energy is then used for finding the minimizers of the two field minimization problem corresponding to a Cosserat continuum. Results from a numerical example predicting the development of microstructure in the material are presented. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical investigation of blood flow. Part I: In microvessel bifurcations

In some diseases there is a focal pattern of velocity in regions of bifurcation, and thus the dynamics of bifurcation has been investigated in this work. A computational model of blood flow through branching geometries has been used to investigate the influence of bifurcation on blood flow distribution. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier–Stokes equations for Newtonian fluids. The governing equations of mass and momentum conservation were solved to calculate the pressure and velocity fields. Movement of blood flow from an arteriole to a venule via a capillary has been simulated using the volume of fluid (VOF) method. The proposed simulation method would be a useful tool in understanding the hydrodynamics of blood flow where the interaction between the RBC deformation and blood flow movement is important. Discrete particle simulation has been used to simulate the blood flow in a bifurcation with solid and fluid particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, and translational and angular momenta. These kinds of simulations open a new way for modeling the dynamics of complex, viscoelastic fluids at the micro-scale, where both liquid and solid phases are treated with discrete particles.     

Implicit formulation for advection–diffusion simulation based on particle distribution moments

This paper introduces an implicit method for advection–diffusion equations called Implicit DisPar, based on particle displacement moments applied to uniform grids. The present method tries to solve constraints associated with explicit methods also based on particle displacement methods, in which diffusivity-dominated situations can only be handled by considerably increasing the associated computational costs. In fact, a higher particle destination nodes number allows the use of higher diffusion coefficients for the transport simulation without instabilities. The average was evaluated by an analogy between the Fokker–Planck and the transport equations. The variance is considered to be Fickian. The particle displacement distribution is used to predict deterministic mass transfers between domain nodes. Mass conservation was guaranteed by the distribution concept. In the truncation error analysis, it was shown that the linear Implicit DisPar formulation does not have numerical error up to v − 1 order, if the first v particle moments are forced by the Gaussian moments. It was shown by theoretical tests for linear conditions that the model accuracy level is proportional to the number of particle destination nodes.     

A new class of particle methods

Summary. Particle methods are numerical methods designed to solve problems in fluid mechanics and related problems in continuum mechanics. A general approach to the construction of such particle methods is presented in this article. The particles are no mass points but possess a finite extension. They can rotate in space and have a spin. The conservation of mass is automatically guaranteed by the ansatz. The forces of interaction between the particles are derived in a canonical way from the force laws of continuum mechanics and are directly based on a regularized stress tensor. In the absence of external forces and of heat sources and sinks, momentum, angular momentum, and energy are conserved as in the continuum case. Received February 17, 1995 / Revised version received December 28, 1995     

A lattice model for the fermionic projector in a static and isotropic spacetime

We introduce a lattice model for a static and isotropic system of relativistic fermions. An action principle is formulated, which describes a particle‐particle interaction of all fermions. The model is designed specifically for a numerical analysis of the nonlinear interaction, which is expected to lead to the formation of a Dirac sea structure. We discuss basic properties of the system. It is proved that the minimum of the variational principle is attained. First numerical results reveal an effect of spontaneous symmetry breaking. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Eulerian and Lagrangian predictions of particulate two-phase flows: a numerical study

To predict particulate two-phase flows, two approaches are possible. One treats the fluid phase as a continuum and the particulate second phase as single particles. This approach, which predicts the particle trajectories in the fluid phase as a result of forces acting on particles, is called the Lagrangian approach. Treating the solid as some kind of continuum, and solving the appropriate continuum equations for the fluid and particle phases, is referred to as the Eulerian approach.Both approaches are discussed and their basic equations for the particle and fluid phases as well as their numerical treatment are presented. Particular attention is given to the interactions between both phases and their mathematical formulations. The resulting computer codes are discussed.The following cases are presented in detail: vertical pipe flow with various particle concentrations; and sudden expansion in a vertical pipe flow. The results show good agreement between both types of approach.The Lagrangian approach has some advantages for predicting those particulate flows in which large particle accelerations occur. It can also handle particulate two-phase flows consisting of polydispersed particle size distributions. The Eulerian approach seems to have advantages in all flow cases where high particle concentrations occur and where the high void fraction of the flow becomes a dominating flow controlling parameter.     

Localization of light particles on fermion-induced domain walls

We offer a possible explanation for the appearance of light composite fermions and Higgs bosons on a four-dimensional domain wall. The pattern of light particle trapping is attributed to a strong self-interaction of five-dimensional pre-quarks. We calculate the low-energy effective action which manifests the invariance under the so-called τ-symmetry. After that, we find a set of vacuum solutions that break the symmetry and the five-dimensional translational invariance. The induced relations between low-energy couplings for Yukawa and scalar field interactions allow us to make certain predictions for light particle masses and couplings themselves, which may provide a signature of the higher-dimensional origin of particle physics in the forthcoming experiments. Bibliography: 53 titles. Dedicated to our old friend Petr P. Kulish, in honor of his 60th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 317, 2004, pp. 11–42.     

Linear stability and instability of relativistic Vlasov‐Maxwell systems

We consider the linear stability problem for a symmetric equilibrium of the relativistic Vlasov‐Maxwell (RVM) system. For an equilibrium whose distribution function depends monotonically on the particle energy, we obtain a sharp linear stability criterion. The growing mode is proved to be purely growing, and we get a sharp estimate of the maximal growth rate. In this paper we specifically treat the periodic 1½D case and the 3D whole‐space case with cylindrical symmetry. We explicitly illustrate, using the linear stability criterion in the 1½D case, several stable and unstable examples. © 2006 Wiley Periodicals, Inc.     

Elastic lattices: equilibrium, invariant laws and homogenization

In the recent years, lattice modelling proved to be a topic of renewed interest. Indeed, fields as distant as chemical modelling and biological tissue modelling use network models that appeal to similar equilibrium laws. In both cases, obtaining an equivalent continuous model allows to simplify numerical procedures. We define the basic properties of lattices: elasticity, frame-indifference, hyperelasticity. We determine rigorously the form that constitutive laws undertake under frame-indifference and hyperelasticity assumptions. Finally, we describe an homogenization technique designed for discrete structures that provides a limit continuum mechanics model and, in the special case of hexagonal lattices, we investigate the symmetry properties of the limit constitutive law.      

Global dynamics of a special class of nonlinear semelparous Leslie matrix models

ABSTRACT

This paper considers the dynamics of nonlinear semelparous Leslie matrix models. First, a class of semelparous Leslie matrix models is shown to be dynamically consistent with a certain system of Kolmogorov difference equations with cyclic symmetry. Then, the global dynamics of a special class of the latter is fully determined. Combining together, we obtain a special class of semelparous Leslie matrix models which possesses generically either a globally asymptotically stable positive equilibrium or a globally asymptotically stable cycle. The result shows that the periodic behaviour observed in periodical insects can occur as a globally stable phenomenon.     

COMPRESSIBLE NON-ISENTROPIC BIPOLAR NAVIER-STOKES-POISSON SYSTEM IN R~3

The compressible non-isentropic bipolar Navier-Stokes-Poisson (BNSP) system is investigated in R 3 in the present paper, and the optimal time decay rates of global strong solution are shown. For initial data being a perturbation of equilibrium state in H l (R 3 ) ∩ Bs 1,1 (R 3 ) for l ≥ 4 and s ∈ (0, 1], it is shown that the density and temperature for each charged particle (like electron or ion) decay at the same optimal rate (1 + t) 3 4 , but the momentum for each particle decays at the optimal rate (1 + t) 1 4 s 2 which is slower than the rate (1 + t) 3 4 s 2 for the compressible Navier-Stokes (NS) equations [19] for same initial data. However, the total momentum tends to the constant state at the rate (1+t) 3 4 as well, due to the interplay interaction of charge particles which counteracts the influence of electric field.     

Locomotion of Articulated Bodies in a Perfect Fluid

This paper is concerned with modeling the dynamics of N articulated solid bodies submerged in an ideal fluid. The model is used to analyze the locomotion of aquatic animals due to the coupling between their shape changes and the fluid dynamics in their environment. The equations of motion are obtained by making use of a two-stage reduction process which leads to significant mathematical and computational simplifications. The first reduction exploits particle relabeling symmetry: that is, the symmetry associated with the conservation of circulation for ideal, incompressible fluids. As a result, the equations of motion for the submerged solid bodies can be formulated without explicitly incorporating the fluid variables. This reduction by the fluid variables is a key difference with earlier methods, and it is appropriate since one is mainly interested in the location of the bodies, not the fluid particles. The second reduction is associated with the invariance of the dynamics under superimposed rigid motions. This invariance corresponds to the conservation of total momentum of the solid-fluid system. Due to this symmetry, the net locomotion of the solid system is realized as the sum of geometric and dynamic phases over the shape space consisting of allowable relative motions, or deformations, of the solids. In particular, reconstruction equations that govern the net locomotion at zero momentum, that is, the geometric phases, are obtained. As an illustrative example, a planar three-link mechanism is shown to propel and steer itself at zero momentum by periodically changing its shape. Two solutions are presented: one corresponds to a hydrodynamically decoupled mechanism and one is based on accurately computing the added inertias using a boundary element method. The hydrodynamically decoupled model produces smaller net motion than the more accurate model, indicating that it is important to consider the hydrodynamic interaction of the links.     

On the exact solutions,lie symmetry analysis,and conservation laws of Schamel–Korteweg–de Vries equation

In this work, we study the integrability aspects of the Schamel–Korteweg–de Vries equation that play an important role in studying the effect of electron trapping on the nonlinear interaction of ion‐acoustic waves by including a quasi‐potential. Lie symmetry analysis together with the simplest equation method and Kudryashov method is used to obtain exact traveling wave solutions for this equation. In addition, conservation laws are constructed using two different techniques, namely, the multiplier method and the new conservation theorem. Using the conservation laws and symmetries of the underlying equation, double reduction and exact solution were also constructed. Copyright © 2017 John Wiley & Sons, Ltd.     

Symmetry analysis, conservation laws of a time fractional fifth-order Sawada-Kotera equation

In this paper, we intend to study the symmetry properties and conservation laws of a time fractional fifth-order Sawada-Kotera (S-K) equation with Riemann-Liouville derivative. Applying the well-known Lie symmetry method, we analysis the symmetry properties of the equation. Based on this, we find that the S-K equation can be reduced to a fractional ordinary differential equation with Erdelyi-Kober derivative by the similarity variable and transformation. Furthermore, we construct some conservation laws for the S-K equation using the idea in the Ibragimov theorem on conservation laws and the fractional generalization of the Noether operators.     

A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

This paper presents a long-term analysis of one-stage extended Runge–Kutta–Nyström (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both types of integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for explicit integrators, a relationship between ERKN integrators and trigonometric integrators is established. For the long-term analysis of implicit integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion. By taking some adaptations of this technology for implicit methods, we derive the modulated Fourier expansion and show the near energy conservation.