Explicit fluxes and productions for large systems of the moment method based on extended thermodynamics

The moment method of kinetic theory solves Boltzmann's equation approximately via an infinite hierarchy of transfer equations for the moments of the distribution function. Extended thermodynamics furnishes the moment method with a rational constitutive theory. Since more and more moment equations are needed to describe extreme non-equilibrium processes, there is need for an algorithmical derivation of large explicit moment equations. This paper presents detailed techniques and formulas which are needed to implement a numerical equation generator. This includes tensorial conversion formulas as well as the core equations of the constitutive theory. In the last part of the paper, the special case of a one dimensional process is discussed. In such a case, only one generic polynomial evaluation needs to be implemented, whereas the coefficients may be easily calculated a priori. Received June 4, 2002 / Published online November 6, 2002 RID="a" ID="a" e-mail: manuel@math.ethz.ch RID="b" ID="b" e-mail: J.Au@vt.tu-berlin.de RID="c" ID="c" e-mail: struchtr@me.uvic.ca Communicated by Gilberto M. Kremer, Curitiba     

Penetrative convection in a horizontally isotropic porous layer

Penetrative convection in a horizontally isotropic porous layer is investigated primarily using an internal heat sink model and alternatively, a quadratic density temperature law. Employing the heat sink model, we show that the temporal growth rate for the linearised system is real, which allows us to perform a linear instability analysis. A nonlinear energy analysis is also presented, yielding a global stability threshold. We compare the heat sink model to the quadratic density model and find that the linearised systems are adjoint, implying that the instability boundaries which can be derived from the two models are the same. Received April 12, 2002 / Published online September 4, 2002 RID="a" ID="a" e-mail: Magdalen.Carr@durham.ac.uk RID="b" ID="b" e-mail: s.d.putter@tue.nl Communicated by Brian Straughan, Durham     

Mathematical analysis of the Armstrong-Frederick model from the theory of inelastic deformations of metals. First results and open problems

This paper tries to analyse, from the mathematical point of view, the system of equations proposed by P.J. Armstrong and C.O. Frederick and used in the theory of inelastic deformations of metals. For the quasistatic Armstrong-Frederick model an approximation procedure is presented which limit functions satisfy the so called ”reduced energy inequality”. This nonmonotone model is here written as a model of pre-monotone type with a nonassociated flow rule. Moreover, a monotone model is presented, which has a structure that is very similar to the Armstrong-Frederick model. Received February 28, 2002 / Published online February 17, 2003 / Kolumban Hutter RID="a" ID="a" Supported by Sonderforschungsbereich 298: Deformation und Versagen bei metallischen und granularen Strukturen RID="b" ID="b" 2. Affiliation: Cardinal Stefan Wyszynski University, Warsaw RID="c" ID="c" e-mail: chelminski@mathematik.tu-darmstadt.de     

Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in a porous medium

The nonlinear stability of the conduction-diffusion solution of a fluid mixture heated and salted from below (and of a homogeneous fluid heated from below) and saturating a porous medium is studied with the Lyapunov direct method. Both Darcy and Brinkman models have been used. The porous medium is bounded by two horizontal parallel planes and is rotating about a vertical axis. Necessary and sufficient conditions of unconditional stability are proved (i.e., the critical linear and nonlinear stability Rayleigh numbers coincide). Our results generalize those given by Straughan [21] for a homogeneous fluid in the Darcy regime. In the case of a mixture two stabilizing effects act: that of the rotation and of the concentration of the solute. Received March 05, 2002 / Published online June 4, 2002 RID="a" ID="a" e-mail: lombardo@dmi.unict.it RID="b" ID="b" e-mail: mulone@dmi.unict.it Communicated by Brian Straugham, Durham     

Note on Slemrod's universal relation $\omega_3+\omega_4+\theta_3 = 0$ for the Burnett coefficients

In this note we comment on the universal relation for the Burnett coefficients, which was proved by Slemrod [1] using a result valid for a Maxwellian potential as a hypothesis for general spherical potentials. It is also shown that this relationship is not valid for relativistic gases. Received October 22, 2002 - Published online April 23, 2003 RID="a" ID="a" e-mail: carcer@mate.polimi.it RID="b" ID="b" kremer@fisica.ufpr.br ID="Communicated by Kolumban Hutter, Darmstadt"     

Maximum entropy principle in relativistic radiation hydrodynamics II: Compton and double Compton scattering

Recently [1], a procedure has been proposed in order to close the set of the moment equations of relativistic radiative fluid dynamics. In particular explicit expressions for the moments of the bremsstrahlung and Thomson scattering source terms have been given. In this work, as anticipated in [1], we shall treat in a systematic way Comptonization and double Compton scattering too. Numerical results relative to the Compton cooling of hot electrons are shown. Received November 14, 2001 / Published online June 4, 2002 RID="a" ID="a" e-mail: mascali@dmi.unict.it Communicated by Ingo Müller, Berlin     

Maximum entropy moment systems and Galilean invariance

Maximum entropy moment closure systems of gas dynamics are investigated. It is shown that polynomial weight functions growing super-quadratically at infinity lead to hyperbolic systems with an unpleasant state space: equilibrium states are boundary points with possibly singular fluxes. This in its generality previously unknown result applies to any moment system including, for example, the 26 or 35 moment case. One might try to avoid singular fluxes by choosing non-polynomial weight functions which grow sub-quadratically at infinity. This attempt, however, is shown to be incompatible with the Galilean invariance of the moment systems because rotational and translational invariant, finite dimensional function spaces necessarily consist of polynomials. Received December 14, 2001 / Published online June 4, 2002 RID="a" ID="a" e-mail:junk@mathematik.uni-kl.de RID="b" ID="b" e-mail: unterreiter@math.tu-berlin.de Communicated by Ingo Müller, Berlin     

Global nonlinear stability in the Bénard problem for a mixture near the bifurcation point

The nonlinear stability of the motionless state of a binary fluid mixture heated and salted from below, in the Oberbeck-Boussinesq scheme, for stress-free and rigid-rigid boundary conditions and Schmidt numbers P_C greater than Prandtl numbers P_T, is studied in the region around the bifurcation point of linear instability. An improvement of the results in Mulone [11] is found for small values of p = P _C /P _T and P_T. For p sufficiently large the critical nonlinear Rayleigh number is very close to the linear one (with relative difference less than in the sea water case) Received December 12, 2002 / Published online April 23, 2003 RID="a" ID="a" e-mail: mbasurto@dmi.unict.it RID="b" ID="b" e-mail: lombardo@dmi.unict.it ID="Communicated by Brian Straugham, Durham"     

A model for convection in the evolution of under-ice melt ponds

A model for convection in the evolution of under-ice melt ponds is presented. The system exhibits two competing effects namely, a temperature gradient which is destabilising and a salt gradient which is stabilising. Density is assumed to have a dependence quadratic in temperature and linear in concentration. A linear instability analysis and a nonlinear stability analysis are performed. The standard energy method does not yield unconditional stability so a weighted energy analysis is employed to achieve global results. The global stability bound is found to be independent of the salt field and a presentation of the region of possible subcritical instabilities is given. Received May 16, 2002 / Published online September 4, 2002 RID="a" ID="a" e-mail: Magdalen.Carr@durham.ac.uk Communicated by Brian Straughan, Durham     

Relaxation analysis and linear stability vs. adsorption in porous materials

The paper presents a linear stability analysis of a 1D stationary flow through a poroelastic medium. This base flow is perturbed in four ways: by longitudinal (1D) disturbances without and with mass exchange and by transversal (2D) disturbances without and with mass exchange. The eigenvalue problem for the first step field equations is solved using a finite-difference-scheme. For both disturbances without mass exchange results are confirmed by an analytical solution. We present the stability and relaxation properties in dependence on the two most important model parameters, namely the bulk and surface permeability coefficients. Received May 17, 2002 / Published online October 15, 2002 RID="*" ID="*" e-mail: albers@wias-berlin.de, web: http://www.wias-berlin.de/private/albers Communicated by Brian Straughan, Durham     

Mathematical Perspectives on Large Eddy Simulation Models for Turbulent Flows

The main objective of this paper is to review and report on key mathematical issues related to the theory of Large Eddy Simulation of turbulent flows. We review several LES models for which we attempt to provide mathematical justifications. For instance, some filtering techniques and nonlinear viscosity models are found to be regularization techniques that transform the possibly ill-posed Navier-Stokes equation into a well-posed set of PDEs. Spectral eddy-viscosity methods are also considered. We show that these methods are not spectrally accurate, and, being quasi-linear, that they fail to be regularizations of the Navier-Stokes equations. We then propose a new spectral hyper-viscosity model that regularizes the Navier-Stokes equations while being spectrally accurate. We finally review scale-similarity models and two-scale subgrid viscosity models. A new energetically coherent scale-similarity model is proposed for which the filter does not require any commutation property nor solenoidality of the advection field. We also show that two-scale methods are mathematically justified in the sense that, when applied to linear non-coercive PDEs, they actually yield convergence in the graph norm.     

Relative Permeabilities for Strictly Hyperbolic Models of Three-Phase Flow in Porous Media

Traditional mathematical models of multiphase flow in porous media use a straightforward extension of Darcys equation. The key element of these models is the appropriate formulation of the relative permeability functions. It is well known that for one-dimensional flow of three immiscible incompressible fluids, when capillarity is neglected, most relative permeability models used today give rise to regions in the saturation space with elliptic behavior (the so-called elliptic regions). We believe that this behavior is not physical, but rather the result of an incomplete mathematical model. In this paper we identify necessary conditions that must be satisfied by the relative permeability functions, so that the system of equations describing three-phase flow is strictly hyperbolic everywhere in the saturation triangle. These conditions seem to be in good agreement with pore-scale physics and experimental data.     

Non-isothermal kinetics of a moving phase boundary

In this paper we derive an explicit formula for a kinetic relation governing the motion of a phase boundary in a bilinear thermoelastic material capable of undergoing solid-solid phase transitions. To obtain the relation, we study traveling wave solutions of a regularized problem that includes viscosity, heat conduction and convective heat exchange with an ambient medium. Both inertia and latent heat of transformation are taken into account. We investigate the effect of material parameters on the kinetic relation and show that in a certain range of parameters the driving force becomes a non-monotone function of the interface velocity. The model also predicts a nonzero resistance to phase boundary motion, part of which is caused by the thermal trapping. Received: November 15, 2001 / Published online September 4, 2002 RID="*" ID="*" e-mail: annav@math.pitt.edu Communicated by Lev Truskinovsky, Minneapolis     

From the Highly Compressible Navier–Stokes Equations to Fast Diffusion and Porous Media Equations,Existence of Global Weak Solution for the Quasi-Solutions

We consider the compressible Navier–Stokes equations for viscous and barotropic fluids with density dependent viscosity. The aim is to investigate mathematical properties of solutions of the Navier–Stokes equations using solutions of the pressureless Navier–Stokes equations, that we call quasi solutions. This regime corresponds to the limit of highly compressible flows. In this paper we are interested in proving the announced result in Haspot (Proceedings of the 14th international conference on hyperbolic problems held in Padova, pp 667–674, 2014) concerning the existence of global weak solution for the quasi-solutions, we also observe that for some choice of initial data (irrotationnal) the quasi solutions verify the porous media, the heat equation or the fast diffusion equations in function of the structure of the viscosity coefficients. In particular it implies that it exists classical quasi-solutions in the sense that they are \({C^{\infty}}\) on \({(0,T)\times \mathbb{R}^{N}}\) for any \({T > 0}\). Finally we show the convergence of the global weak solution of compressible Navier–Stokes equations to the quasi solutions in the case of a vanishing pressure limit process. In particular for highly compressible equations the speed of propagation of the density is quasi finite when the viscosity corresponds to \({\mu(\rho)=\rho^{\alpha}}\) with \({\alpha > 1}\). Furthermore the density is not far from converging asymptotically in time to the Barrenblatt solution of mass the initial density \({\rho_{0}}\).     

Active control schemes for aseismic base-isolated structures

Summary  Active control schemes are used for the protection of base-isolated and seismically excited buildings. The desired control objective is to keep the whole structure arbitrarily close to its initial configuration prior to the earthquake. The proposed methods require control force application only at the base of the structure. The controllers developed may depend on information obtained from all the floors or just the first (base) floor alone. Received 13 September 2000; accepted for publication 26 June 2001 RID=" ID=" Dedicated to the memory of Professor P.D. Panagiotopoulos with our warmest prayers. RID=" ID=" The first author wishes to thank Prof. G. Leitmann and Prof. E. Papadopoulos for the numerous helpful discussions. The same author is supported in part by the Institute of Communication and Computer Systems, NTUA, under the program Archimedes 65/1017.     

The Physical Origin of Interfacial Coupling in Two-Phase Flow through Porous Media

Recently developed transport equations for two-phase flow through porous media usually have a second term that has been included to account properly for interfacial coupling between the two flowing phases. The source and magnitude of such coupling is not well understood. In this study, a partition concept has been introduced into Kalaydjian's transport equations to construct modified transport equations that enable a better understanding of the role of interfacial coupling in two-phase flow through natural porous media. Using these equations, it is demonstrated that, in natural porous media, the physical origin of interfacial coupling is the capillarity of the porous medium, and not interfacial momentum transfer, as is usually assumed. The new equations are also used to show that, under conditions of steady-state flow, the magnitude of mobilities measured in a countercurrent flow experiment is the same as that measured in a cocurrent flow experiment, contrary to what has been reported previously. Moreover, the new equations are used to explicate the mechanism by which a saturation front steepens in an unstabilized displacement, and to show that the rate at which a wetting fluid is imbibed into a porous medium is controlled by the capillary coupling parameter, . Finally, it is argued that the capillary coupling parameter, , is dependent, at least in part, on porosity. Because a clear understanding of the role played by interfacial coupling is important to an improved understanding of two-phase flow through porous media, the new transport equations should prove to be effective tools for the study of such flow.     

Two-dimensional unstructured elastic model for acoustic pulse scattering at solid-liquid interfaces

Abstract. A numerical model to simulate elastic waves and acoustic scattering in two spatial dimensions has been developed and thoroughly tested. The model universally includes elastic solids and liquids. The equations of motion are written in terms of stresses, displacements and displacement velocities for control volumes constructed about the nodes of a triangular unstructured grid. The latter conveniently supports various geometries with complex external and internal boundaries separating sub-domains of different elastic properties. Theoretical dispersion for zero mode symmetric () and antisymmetric () waves in a plate has been reproduced numerically with high accuracy, thus verifying the method and code. Comparison of simulated acoustic pulse scattering at water-immersed steel plate with the respective experiments reveals a very good agreement in such delicate features as excitation of the surface (A) wave. The numerical results explain the peculiar location of the surface wave relative to the other ones in experimental registrations. Examples of acoustic pulse interactions with curvilinear metallic shells in water demonstrate flexibility of the method with respect to complex geometries. Potential applications as well as some directions for further improvement to the technique are briefly discussed. Received 5 September 2002 / Accepted 25 November 2002 Published online 4 February 2003 RID="*" ID="*"Permanent address: Ioffe Physical-Technical Institute, 26 Polytekhnicheskaya, 194021 St. Petersburg, Russia Correspondence to: P. Voinovich (e-mail: vpeter@scc.ioffe.ru)