On a boundary layer problem for the nonlinear Boltzmann equation

This article deals with a boundary-layer problem arising in the kinetic theory of gases when the mean free path of molecules tends to zero. The model considered here is the stationary, nonlinear Boltzmann equation in one dimension with a slightly perturbed reflection boundary condition. We restrict our attention to the case of hard spheres collisions, with Grad's cutoff assumption. Existence, uniqueness and asymptotic behavior are derived by means of energy estimates.     

Decay and Continuity of the Boltzmann Equation in Bounded Domains

Boundaries occur naturally in kinetic equations, and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: in-flow, bounce-back reflection, specular reflection and diffuse reflection. We establish exponential decay in the L ^∞ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set at the boundary. Our contribution is based on a new L ² decay theory and its interplay with delicate L ^∞ decay analysis for the linearized Boltzmann equation in the presence of many repeated interactions with the boundary.     

Die Differentialgleichungen der Bewegung für eine Klasse von Systemen starrer Körper im Gravitationsfeld

Übersicht Mit Hilfe der Lagrangeschen Gleichungen werden für eine weite Klasse von Systemen starrer Körper die Differentialgleichungen der Drehbewegungen im Gravitationsfeld entwickelt. Ihre Formulierung mit Hilfe von Matrizen macht sie geeignet für den Einsatz von digitalen Rechenmaschinen. Die Wahl der generalisierten Koordinaten und die Form der Gleichungen legen es nahe, von verallgemeinerten Eulerschen Bewegungsgleichungen zu sprechen.

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Summary Using Lagrange's equations the differential equations of rotational motions in a gravitational field are derived for a broad class of systems of interconnected rigid bodies. The matrix formulation of these equations makes them suitable for use on digital computers. The choice of the generalized coordinates and the structure of the equations suggest referring to them as generalized Eulerian equations of motion.

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Auszug aus der Dissertation des Autors. Die Arbeit entstand auf Anregung von Prof. Dr. R. E. Roberson (University of California, San Diego) unter der Leitung von Prof. Dr.-Ing. E. Pestel (Techn. Hochschule Hannover).     

Interatomic forces and gas theory from newton to Lennard-Jones

A recurrent theme in the physical science of the past three centuries has been provided by the program attributed to Isaac Newton: from the phenomena of nature to find the forces between particles of matter, and from these forces to explain and predict other phenomena. The success or failure of this program as a guide for scientific research can be assessed by considering some of the cases in which it has been applied: Newton's own theory of gas pressure, the Boscovich theory of interatomic forces, the Laplace theory (short-range attractive forces and long-range repulsive forces), the billiard-ball model used in the elementary kinetic theory of gases, the Maxwell r ^–5 repulsive force, and the van der Waals equation. A more detailed examination is presented of the rise and fall of the Lennard-Jones potential in relation to calculations and experimental data on virial coefficients and transport properties of gases, solid state properties, and the quantum theory of interatomic forces.The history of the subject suggests that the hypothetico-deductive model of scientific method has not been followed in practice, since the reasons for adopting or rejecting new interatomic force laws are often not simply related to the success or failure of the force law in calculations of gas properties.At present there is serious doubt as to whether it is worthwhile trying to establish a single realistic force law for the interaction between two atoms or molecules. It may be more fruitful to abandon this program and to choose force laws instead on the basis of their convenience in a particular mathematical theory of the properties of matter.Expository Memoir, invited by the EditorsA preliminary version of this paper was presented at a seminar of the Mechanics Department, Johns Hopkins University, 16 January 1970. The author thanks Professor C. Truesdell for several suggestions which were used in preparing the final version. This research was supported by the National Science Foundation, Grant GS-2475.     

Uniqueness of positive solutions of Δu−u+up=0 in Rn

We establish the uniqueness of the positive, radially symmetric solution to the differential equation u–u+u^p=0 (with p>1) in a bounded or unbounded annular region in R ⁿ for all n1, with the Neumann boundary condition on the inner ball and the Dirichlet boundary condition on the outer ball (to be interpreted as decaying to zero in the case of an unbounded region). The regions we are interested in include, in particular, the cases of a ball, the exterior of a ball, and the whole space. For p=3 and n=3, this a well-known result of Coffman, which was later extended by McLeod & Serrin to general n and all values of p below a certain bound depending on n. Our result shows that such a bound on p is not needed. The basic approach used in this work is that of Coffman, but several of the principal steps in the proof are carried out with the help of Sturm's oscillation theory for linear second-order differential equations. Elementary topological arguments are widely used in the study.     

Die näherungsweise Berücksichtigung der mittragenden Breite bei Spannungs-, Stabilitäts- und Schwingungsproblemen

Übersicht Um die Schubverformungen in den Deckplatten von Plattenbalken und Hohlkästen zu berücksichtigen, wird mit Hilfe eines Variationsprinzips vom Minimum der potentiellen Energie ein Näherungsverfahren entwickelt. Bei der Annahme über den Formänderungszustand wird die Bernoullische Annahme vom Ebenbleiben der Querschnitte verlassen. Die Euler-Gleichungen des Variationsproblems ergeben einfache gewöhnliche Differentialgleichungen, die eine Behandlung von Spannungs-, Stabilitäts- und Schwingungsproblemen ermöglichen. Es zeigt sich, daß die Spannungen, Knicklasten und Eigenfrequenzen ganz erheblich von den bekannten Werten der elementaren Stabtheorie abweichen können. Die Genauigkeit der Lösung wurde durch Vergleich mit der exakten Scheibentheorie überprüft, wobei sich eine gute Übereinstimmung ergab.

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Summary A method of determining the shear deformations of the cover plates of T- and box-beams is developed, employing a variational principle of minimum potential energy. In assuming a convenient shape of deformation Bernoulli's hypothesis is rendered invalid. The Euler equations yield ordinary differential equations for the solution of stress, stability, and vibration problems. It is shown that the stresses, the buckling loads and the characteristic frequencies may differ significantly from the values obtained with the elemantary theory of elasticity. The accuracy of the solution is investigated and shown to be of a high degree by comparing the results with those obtained with the exact theory of disks.

     

Long-Time Convergence of an Adaptive Biasing Force Method: The Bi-Channel Case

We present convergence results for an adaptive algorithm to compute free energies, namely the adaptive biasing force (ABF) method (Darve and Pohorille in J Chem Phys 115(20):9169–9183, 2001; Hénin and Chipot in J Chem Phys 121:2904, 2004). The free energy is the effective potential associated to a so-called reaction coordinate ξ(q), where q = (q ₁, … , q _3N ) is the position vector of an N-particle system. Computing free energy differences remains an important challenge in molecular dynamics due to the presence of metastable regions in the potential energy surface. The ABF method uses an on-the-fly estimate of the free energy to bias dynamics and overcome metastability. Using entropy arguments and logarithmic Sobolev inequalities, previous results have shown that the rate of convergence of the ABF method is limited by the metastable features of the canonical measures conditioned to being at fixed values of ξ (Lelièvre et al. in Nonlinearity 21(6):1155–1181, 2008). In this paper, we present an improvement on the existing results in the presence of such metastabilities, which is a generic case encountered in practice. More precisely, we study the so-called bi-channel case, where two channels along the reaction coordinate direction exist between an initial and final state, the channels being separated from each other by a region of very low probability. With hypotheses made on ‘channel-dependent’ conditional measures, we show on a bi-channel model, which we introduce, that the convergence of the ABF method is, in fact, not limited by metastabilities in directions orthogonal to ξ under two crucial assumptions: (i) exchange between the two channels is possible for some values of ξ and (ii) the free energy is a good bias in each channel. This theoretical result supports recent numerical experiments (Minoukadeh et al. in J Chem Theory Comput 6:1008–1017, 2010), where the efficiency of the ABF approach is demonstrated for such a multiple-channel situation.     

Eigenfunction expansions for singular Schrodinger operators

We extend Ikebe's theory of eigenfunctions to a class of Schrodinger operators H = - + V on L ² (IR³), where the potential V is replaced by a measure which need not be absolutely continuous with respect to Lebesgue measure. Applications include the proof of the existence and completeness of the wave operators for a free particle which is partly reflected and partly transmitted by a compact 2-dimensional surface.     

Hopf Bifurcation in the presence of symmetry

Using group theoretic techniques, we obtain a generalization of the Hopf Bifurcation Theorem to differential equations with symmetry, analogous to a static bifurcation theorem of Cicogna. We discuss the stability of the bifurcating branches, and show how group theory can often simplify stability calculations. The general theory is illustrated by three detailed examples: O(2) acting on R ², O(n) on R ⁿ , and O(3) in any irreducible representation on spherical harmonics.The work of second author was also supported by a visiting position in the Department of Mathematics, University of Houston     

On the equations of age-dependent population dynamics

The aim of this work is to give a direct and constructive proof of existence and uniqueness of a global solution to the equations of age-dependent population dynamics introduced and considered by M. E. Gurtin & R. C. MacCamy in [3]. The linear theory was developed by F. R. Sharpe & A. J. Lotka [10] and A. G. McKendrick [8] (see also [1], [9]) and extended to the nonlinear case by M. E. Gurtin & R. C. MacCamy in [3] (see also [4] [5] [6]). In [3], the key of the proof of existence and uniqueness was to reduce the problem to a pair of integral equations. In fact, as we shall see, the problem can also be solved by a simple fixed point argument. To outline more clearly the ideas of the proof, we will first discuss the setting and the resolution of the linear case, and then we will generalize the results of [3].     

Gas-dynamic analogy in the theory of stratified liquid flows with a free boundary

Novel approximate mathematical models of long-wave theory describing flows of a density-stratified liquid with a free boundary are proposed. It is shown that in certain cases the equations of the novel models coincide with either the equations of nonisentropic gas dynamics with a polytropic equation of state for γ = 2 or the equations describing the dynamics of a mixture of two perfect gases.     

The Anisotropic Lagrangian Averaged Euler and Navier-Stokes Equations

 The purpose of this paper is twofold. First, we give a derivation of the Lagrangian averaged Euler (LAE-α) and Navier-Stokes (LANS-α) equations. This theory involves a spatial scale α and the equations are designed to accurately capture the dynamics of the Euler and Navier-Stokes equations at length scales larger than α, while averaging the motion at scales smaller than α. The derivation involves an averaging procedure that combines ideas from both the material (Lagrangian) and spatial (Eulerian) viewpoints. This framework allows the use of a variant of G. I. Taylor's ``frozen turbulence' hypothesis as the foundation for the model equations; more precisely, the derivation is based on the strong physical assumption that fluctutations are frozen into the mean flow. In this article, we use this hypothesis to derive the averaged Lagrangian for the theory, and all the terms up to and including order α² are accounted for. The equations come in both an isotropic and anisotropic version. The anisotropic equations are a coupled system of PDEs (partial differential equations) for the mean velocity field and the Lagrangian covariance tensor. In earlier works by Foias, Holm & Titi [10], and ourselves [16], an analysis of the isotropic equations has been given. In the second part of this paper, we establish local in time well-posedness of the anisotropic LANS-α equations using quasilinear PDE type methods. (Accepted September 2, 2002) Published online November 26, 2002 Dedicated to Stuart Antman on the occasion of his 60th birthday Communicated by S. MüLLER     

Phenomenological Modeling of Electrorheological Fluids with an Extended Casson-Model

In this paper we propose a phenomenological theory for electrorheological fluids. In general these are suspensions which undergo dramatic changes in their material properties if they are exposed to an electric field. In the context of continuum mechanics these fluids can be modeled as non-Newtonian fluids. Recalling the governing equations of rational thermodynamics and electrodynamics of moving media (Maxwell-Minkowski-equations), we derive suitable governing equations of electrorheology using essentially two assumptions concerning magnetic quantities. Furthermore we introduce a 3-dimensional nonlinear constitutive equation for the Cauchy stress tensor which is an extension of the model proposed by Ružička (see [14]). Assuming a viscometric flow, we compare the shear stress of our model with other well known models and fit the parameters by using measurements that were obtained in a rotational viscometer. Excellent agreement between model and measurements is achieved. On the basis of these results we propose a 3-dimensional model, the so-called extended Casson -model. This model is investigated further for a channel flow configuration with a homogeneous electric field. We determine analytical solutions for the electric field, the velocity and the volumetric flow rate and illustrate the velocity profiles and the predicted pressure drop. The velocity profiles are flattened compared to parabolic profiles and become more flat if the electric field increases. Received March 21, 2000     

A linear theory of straight elastic rods

This paper is based on the work of Green & Laws who have given a general thermodynamical theory of rods which is valid for any material. Here, starting with the general non-linear theory of elastic rods, we derive a linear theory allowing for thermal effects. The resulting free energy as a quadratic function of kinematic variables is restricted by certain symmetry conditions. The basic equations then separate into four groups, two for flexure, one for torsion and one for extension of the rod with temperature effects occurring only in the latter group. Wave propagation along an infinite rod is considered. There are two wave speeds for each type of flexure, two for torsion and three for isothermal extension and all wave speeds depend on the wave length.     

A survey of exact solutions of inviscid field equations in the theory of shear flow instabilty

This paper presents a survey of undisturbed flows that take one or another of the field equations of inviscid shear flow instability theory (e.g. theRayleigh equation,Taylor-Goldstein-Haurwitz equation or theKuo equation) to a differential equation satisfied by aknown transcendental function forarbitrary complex values of the parameters. Some mean velocity profiles having this feature are well known. Thus, piecewise linear mean velocity profiles take theRayleigh equations to a constant-coefficient differential equation and the exponential mean velocity profile takes theRayleigh equation to theGauss hypergeometric equation. Less well known is the fact that a variety of mean velocity profiles take theRayleigh equation to a differential equation due toKarl Heun. These profiles include: (i) the sinusoidal profile; (ii) the hyperbolic tangent profile (an example pointed out byMiles (1963)); (iii) the profile in the form of the square of a hyperbolic secant (theBickley jet); and, (iv) a skewed velocity profile in which each component has the form of a quadratic function of the variable exp(–z/l) (in whichz is the cross-stream coordinate andl is a length scale). In all of these cases, one or another author has previously identified aregular neutral mode solution of theRayleigh equation and has expressed that solution in the form of elementary functions. Such regular neutral modes apparently represent cases in which the solution ofHeun's equation (which is normally an infinite series) truncates to a single term. The survey concludes by noting that the parabolic mean velocity profile takes theRayleigh equation to thedifferential equation of the spheroidal wave function.Dedicated to Mårten T. Landahl on the occasion of his sixty-fifth birthday by a former apprentice as a token of his respect and gratitude.     

Lattice Boltzmann model for predicting the deposition of inertial particles transported by a turbulent flow

Deposition of inertial solid particles transported by turbulent flows is modelled in a framework of a statistical approach based on the particle velocity Probability Density Function (PDF). The particle-turbulence interaction term is closed in the kinetic equation by a model widely inspired from the famous BGK model of the kinetic theory of rarefied gases. A Gauss-Hermite Lattice Boltzmann model is used to solve the closed kinetic equation involving the turbulence effect. The Lattice Boltzmann model is used for the case of the deposition of inertial particles transported by a homogeneous isotropic turbulent flows. Even if the carrier phase is homogeneous and isotropic, the presence of the wall coupled with particle-turbulence interactions leads to inhomogeneous particle distribution and non-equilibrium particle fluctuating motion. Despite these complexities the predictions of the Lattice Boltzmann model are in very good accordance with random-walk simulations. More specifically the mean particle velocity, the r.m.s. particle velocity and the deposition rate are all well predicted by the proposed Lattice Boltzmann model.     

Simulation of gas–solid particle flows over a wide range of concentration

A two-fluid model of gas–solid particle flows that is valid for a wide range of the solid-phase volume concentration (dilute to dense) is presented. The governing equations of the fluid phase are obtained by volume averaging the Navier–Stokes equations for an incompressible fluid. The solid-phase macroscopic equations are derived using an approach that is based on the kinetic theory of dense gases. This approach accounts for particle–particle collisions. The model is implemented in a control-volume finite element method for simulations of the flows of interest in two-dimensional, planar or axisymmetric, domains. The chosen mathematical model and the proposed numerical method are applied to three test problems and one demonstration problem. © 1998 John Wiley & Sons, Ltd.     

Propagating phase boundaries: Formulation of the problem and existence via the Glimm method

This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL ¹-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.     

Stationary Stokes, Oseen and Navier–Stokes Equations with Singular Data

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case of a bounded open set, connected of class C^1,1{\mathcal{C}^{1,1}} of \mathbbR³{\mathbb{R}^3}. Taking up once again the duality method introduced by Lions and Magenes (Problèmes aus limites non-homogènes et applications, vols. 1 & 2, Dunod, Paris, 1968) and Giga (Math Z 178:287–329, 1981) for open sets of class C^￥{\mathcal{C}^{\infty}} [see also chapter 4 of Necas (Les méthodes directes en théorie des équations elliptiques. (French) Masson et Cie, éd., Paris; Academia, éditeurs, Prague, 1967), which considers the Hilbertian case p = 2 for general elliptic operators], we give a simpler proof of the existence of a very weak solution for stationary Oseen and Navier–Stokes equations when data are not regular enough, based on density arguments and a functional framework adequate for defining more rigourously the traces of non-regular vector fields. In the stationary Navier–Stokes case, the results will be valid for external forces not necessarily small, which lets us extend the uniqueness class of solutions for these equations. Considering more regular data, regularity results in fractional Sobolev spaces will also be discussed for the three systems. All these results can be extended to other dimensions.     

Effect of inelastic collisions on the first-order slip coefficients for a diatomic gas with rotational degrees of freedom

When solving problems of inhomogeneous gas dynamics in the slip regime, it is necessary to know the boundary conditions for the velocity, temperature, heat fluxes, etc., that is, the boundary conditions for the gas macroparameters. In particular, such problems arise in developing the theory of thermophoresis of moderately large aerosol particles [1].The problem of monatomic and molecular (di- and polyatomic) gas slip along a boundary surface is considered in many publications (see, for example, [2–8]). The first-order effects include the isothermal and thermal gas slips characterized by the coefficients C_m and K_TS, respectively.In contrast to a monatomic gas, the molecules of diatomic and polyatomic gases have internal degrees of freedom, which considerably complicates the kinetic equation [9]. The lack of reliable models for the intermolecular interaction potential predetermines the need to construct model kinetic equations [10].In this study, for a diatomic gas whose molecules have rotational degrees of freedom, we propose a model kinetic equation obtained by developing the approach described in [6]. With the use of this model equation, the problem of diatomic gas slip along a plane surface is solved. As a result, for diatomic gases the coefficients C_m and K_TS, which depend on the thermophysical gas parameters and the intensity of inelastic collisions, are obtained.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 176–182. Original Russian Text Copyright © 2004 by Poddoskin.