Entropy and causality as criteria for the existence of shock waves in low temperature heat conduction

A set of symmetric hyberbolic field equations, describing heat conduction in dielectric solids at low temperatures, is studied with respect to the propagation of temperature shock waves. The field equations have been derived from the Boltzmann-Peierls equation and include the phenomenon of second sound, a special form of wavelike energy transport occuring in some crystals in a temperature range close to absolute zero.Two physical criteria, an entropy shock condition and the Lax condition, which is based on a causality argument, are applied to study the existence of so called hot and cold shocks. These are characterized by a temperature rise or fall across the shock respectively, and it turns out that the only possible solution to the problem is a hot shock, predicted by either one of the criteria.In the recent literature, however, a similar case was treated revealing a partial contradiction between the two criteria. Regarding the fact that there exists a proof of equivalence for small shocks, we were thus led to investigate this equivalence in the general case, which is illustrated by means of a simple example.     

Positively invariant regions for a problem in phase transitions

Positively invariant regions for the system v _t + p(W) _x = V _xx , W _t –V _x = W _xx are constructed where p < 0, w < , w > , p(w) = 0, w , > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide independent L bounds on the solution (w, v).Dedicated to Professor James Serrin on the occasion of his sixtieth birthday     

Diffusion in anisotropic porous media

An experimental system was constructed in order to measure the two distinct components of the effective diffusivity tensor in transversely isotropic, unconsolidated porous media. Measurements were made for porous media consisting of glass spheres, mica particles, and disks made from mylar sheets. Both the particle geometry and the void fraction of the porous media were determined experimentally, and theoretical calculations for the two components of the effective diffusivity tensor were carried out. The comparison between theory and experiment clearly indicates that the void fraction and particle geometry are insufficient to characterize the process of diffusion in anisotropic porous media. Roman Letters A interfacial area between - and -phases for the macroscopic system, m² - A _e area of entrances and exits of the -phase for the macroscopic system, m² - A interfacial area contained within the averaging volume, m² - a characteristic length of a particle, m - b average thickness of a particle, m - c _A concentration of species A, moles/m³ - c _o reference concentration of species A, moles/m³ - c _A intrinsic phase average concentration of species A, moles/m³ - c _a c _A–c _A, spatial deviation concentration of species A, moles/m³ - C c _A/c ₀, dimensionless concentration of species A - binary molecular diffusion coefficient, m²/s - D _eff effective diffusivity tensor, m²/s - D _xx component of the effective diffusivity tensor associated with diffusion parallel to the bedding plane, m²/s - D _yy component of the effective diffusivity tensor associated with diffusion perpendicular to the bedding plane, m²/s - D _eff effective diffusivity for isotropic systems, m₂/s - f vector field that maps c _A on to c _a , m - h depth of the mixing chamber, m     

An example of a quasiconvex function that is not polyconvex in two dimensions

We study the different notions of convexity for the function f () = ||² (||² – 2 det ) where ^2×2, introduced by Dacorogna & Marcellini. We show that f is convex, polyconvex, quasiconvex, rank-one convex, if and only if ¦¦ 2/3 2, 1, 1+ (for some >0), 2/3, respectively.     

Shock Waves in a Fluid with Bubbles Containing Evaporating Drops of a Liquefied Gas

The propagation and reflection of one-dimensional plane unsteady waves and pulses in a mixture of a fluid with two-phase bubbles containing evaporating drops is investigated. A significant effect of unsteady evaporation of the drops in the zone ahead of the shock wave on the wave propagation is demonstrated. The evaporation of the drops results in a pressure increase ahead of the wave and the shock wave as it were climbs to increasing pressure level. In contrast to bubbly fluids with single-phase bubbles, in a fluid with two-phase bubbles, at a fixed phase volume fraction, a decrease in bubble size results in an increase rather than a decrease of the oscillation amplitude. The wave reflection from a solid wall is essentially nonlinear and the maximum pressure attained at the wall is several times greater than the incident-wave intensity.     

Investigation of stationary flow of viscous gas in a thin three-dimensional shock layer

A three-dimensional shock layer near the blunt surface of a fairly smooth body is analyzed asymptotically. Equations of the first approximation are obtained and justified in various cases of the limit 1, 0, ( – 1)^–1M ^-2 0. These equations are simplified for the flow near the stagnation point of a body with double curvature and near the blunt leading edge of a sweptback wing. The results of some calculations are given and compared with the results of [17, 18] in the case of axisymmetric flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 115–126, September–October, 1980.     

A nonequilibrium theory of a slurry

A nonequilibrium theory of a slurry is developed and its practical use is illustrated by a simple stability analysis. Here a slurry is defined as a deformable continuum consisting of a liquid phase containing in suspension a large number of small solid particles which have formed by solidification from the liquid. The liquid is assumed to consist of two components and the solid to contain only one of the two. Consequently, the process of change of phase requires redistribution of material on the scale of the solid particles. This process is assumed to take a finite amount of time, requiring a nonequilibrium macroscopic theory. This theory contains four thermodynamic variables, three to represent the equilibrium state of the binary system and a fourth measuring the departure from thermodynamic equilibrium. The process of microscale diffusion of material is parameterized in the macroscale theory, leading to a Landau-type relaxation term in the equation of evolution of the fourth variable. The theory is simplified to yield a Boussinesq-like set of governing equations. Their practical use is illustrated by analyzing the stability of a simple steady solution of the equations and the effects of a non-zero relaxation time are discussed. A novel instability mechanism involving sedimentation of particles, previously found to occur in the equilibrium case, is found to persist in nonequilibrium, but disappears in the limit of no change of phase.Key to symbols a, b, c thermodynamic coefficients; see (3.36)–(3.38) - sedimentation coefficient; see (5.18) - C _p specific heat; see (3.24) - C _p ^de specific heat of the slurry; see (3.28) and (3.30) - c radius of solid particle (in §4) - D, D diffusive coefficients; see (3.40) and (3.41) - material diffusivity in liquid phase - D ^* modified diffusion coefficient; see (5.15) - d thermodynamic coefficient; see (3.39) - E specific internal energy - f, g, h thermodynamic coefficients; see (3.36)–(3.38) - g acceleration of gravity - reduced gravity; see (5.10) - i total diffusive flux vector of constituent 1 - i diffusive flux vector of constituent 1 in the liquid phase - j diffusive flux vector of solid phase - k thermal conductivity - k entropy flux vector - k _T, k_T thermodiffusion coefficients; see (3.40) and (3.41) - L latent heat of solidification per unit mass; see (3.7) and (3.24) - m wave number - m ^s rate of creation of mass of solid per unit volume through solidification - m ₁ ^s rate of creation of mass of solid constituent 1 per unit volume through solidification - mass rate of freezing per unit area per unit time - N number of solid particles per unit volume - p pressure - p _H hydrostatic component of pressure - p _m mechanical pressure - p ₁ dynamic component of pressure - q heat flux vector - Q _D rate of regeneration of heat through diffusive fluxes - Q _M rate of regeneration of heat through phase-change processes - Q _v rate of regeneration of heat through viscosity - Q vector defined by (3.16) - r heat externally supplied per unit mass (in §3); spherical radial coordinate (in §4) - S specific entropy of slurry - change of specific entropy with mass fraction of constituent 1; also change of chemical potential of liquid phase with temperature barring change of phase - change of chemical potential of liquid phase with temperature in phase equilibrium; see (3.28) and (3.30) - T temperature - t time - t ₀ relaxation time; see (5.30) - u barycentric velocity - u _H horizontal perturbation velocity - V sedimentation speed - w _a upward speed of simple state; see (6.5) and (6.12) - z upward vertical coordinate - upward unit vector - thermal expansion coefficient barring change of phase; see (3.23) - > ^* thermal expansion coefficient in phase equilibrium; see (3.27) and (3.30) - modified thermal expansion coefficient; see (5.1) and (5.4) - isothermal compressibility of slurry barring change of phase; see (3.23) - ^* isothermal compressibility of slurry in phase equilibrium; see (3.27) and (3.30) - dimensionless measure of departure from liquidus equilibrium; see (5.2) - _a deviation from phase equilibrium in simple state; see (6.6) and (6.13) - vertical wave number - volume expansion per unit mass upon melting; see (3.6) - change of chemical potential of liquid phase with pressure; see (3.25) - change of chemical potential of liquid phase with pressure for slurry; see (3.29) and (3.30) - compositional gradient in the static state; see (6.15) - vector defined by (3.35) - constant of integration; see (6.7) and (6.8) - coefficient defined by (6.23) - nonequilibrium expansion coefficient; see (5.1) and (5.4) - thermal diffusivity; =k/C _p - modified thermal diffusivity; see (5.33) - relaxation rate to phase equilibrium; see (2.2) - ₁ relaxation rate to solid-composition equilibrium; see (2.3) - sedimentation coefficient; see (4.29) - horizontal wave number vector - sedimentation coefficient; see (4.30) - ^L , ^s chemical potential of constituent 1 relative to constituent 2 in liquid and solid phase per unit mass; see (2.6) - change of chemical potential of liquid with liquid composition; see (3.8) - coefficient defined by (3.10) - kinematic shear viscosity - total mass fraction of constituent 1 (i.e., solute) - ^L, ^s mass fraction of constituent 1 in liquid and solid phases - density of slurry - _s density of solid phase - - - , growth rate of disturbance - stress tensor - deviatoric stress tensor - dimensionless temperature; see (5,3) - _a constant of integration; see (6.7) - mass fraction of solid phase in slurry - _b vertical gradient of mass fraction of solid; see (6.1) - dimensionless measure of _b; see (6.22) - _c temporal gradient of mass fraction of solid; see (6.1) - specific Gibbs free energy; see (3.13) - _L,_s specific Gibbs free energy of liquid and solid phases; see (2.12) - measure of departure from liquidus equilibrium; see (2.14) - measure of departure from solidus equilibrium; see (2.5) - spherical polar coordinate (in §4); see (4.20); wave angle (in §6); see (6.38)     

Die Reduktion von nichteuklidischen geometrischen Objekten in eine euklidische Form und physikalische Deutung der Reduktion durch Eigenspannungszustände in Kristallen

Summary Self-stresses in crystals are closely related to tearing processes. These processes have a meaning only in connection with geometrical objects. The present paper deals with tearing processes connected with the geometrical objects metric and connexion. A tearing process is commonly associated with either one of these processes. It is therefore of interest to investigate the question whether a tearing process which is connected with both objects exists. The investigation of this problem leads to the subdivision of the self-stresses according to the metric properties of the connexion: metrical stresses caused by dislocations and disclinations and nonmetrical stresses caused by non-mechanical sources and by extra-matter.It should be mentioned that the mathematical part of this paper has an independent interest for the understanding of the structures associated with higher differential geometry.

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Vorgelegt von T. Meixner

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Herrn Professor Dr. A. Seeger danke ich für die verständnisvolle Förderung dieser Arbeit. Herrn Dr. C. Teodosiu bin ich für zahlreiche Diskussionen sehr dankbar.

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Supersonic air flow past a sphere with account for vibrational relaxation

A numerical solution is obtained for the problem of air flow past a sphere under conditions when nonequilibrium excitation of the vibrational degrees of freedom of the molecular components takes place in the shock layer. The problem is solved using the method of [1]. In calculating the relaxation rates account was taken of two processes: 1) transition of the molecular translational energy into vibrational energy during collision; 2) exchange of vibrational energy between the air components. Expressions for the relaxation rates were computed in [2]. The solution indicates that in the state far from equilibrium a relaxation layer is formed near the sphere surface. A comparison is made of the calculated values of the shock standoff with the experimental data of [3].Notation uV_max, vV_max velocity components normal and tangential to the sphere surface - V_max maximal velocity - P V _max ² pressure - density - TT temperature - e_viRT vibrational energy of the i-th component per mole (i=–O₂, N₂) - =rb–1 shock wave shape - a _f the frozen speed of sound - HRT/m gas total enthalpy     

Shear softening and thixotropic properties of wheat flour doughs in dynamic testing at high shear strain

Shear softening and thixotropic properties of wheat flour doughs are demonstrated in dynamic testing with a constant stress rheometer. This behaviour appears beyond the strictly linear domain (strain amplitude ₀ 0.2%),G,G and |^*| decreasing with ₀, the strain response to a sine stress wave yet retaining a sinusoidal shape. It is also shown thatG recovers progressively in function of rest time. In this domain, as well as in the strictly linear domain, the Cox-Merz rule did not apply but() and | ^*())| may be superimposed by using a shift factor, its value decreasing in the former domain when ₀ increases. Beyond a strain amplitude of about 10–20%, the strain response is progressively distorted and the shear softening effects become irreversible following rest.     

New integral estimates for deformations in terms of their nonlinear strains

If u is a bi-Lipschitzian deformation of a bounded Lipschitz domain in ⁿ (n2), we show that the L ^P norm (p1, pn) of a certain nonlinear strain function e(u) associated with u dominates the distance in L ^q (q= np/(n–p) if p if p>n) from u to a suitably chosen rigid motion of ⁿ . This work extends that of F. John, who proved corresponding estimates for p}>1 under the hypothesis that u has uniformly small strain. We also obtain a bound for the oscillation of Du in L ². These estimates are apparently the first to apply with no a priori pointwise hypotheses upon the strain of u. In ³ the integral e(u) ² d³ is dominated by typical hyperelastic energy functionals proposed in the literature for modeling the behavior of rubber; thus the case n=3, p=2 gives the first general bound for the deformations of such materials in terms of the associated nonlinear elastic work.     

Shock waves and turbulence in a hypersonic inlet

A numerical investigation for an axisymmetric hypersonic turbulent inlet flow field of a perfect gas is presented for a three-shock configuration consisting of a biconic and a cowl. An upwind parabolized Navier-Stokes solver based on Roe's scheme is used to compute an oncoming flow Mach numberM =8, temperatureT =216 K, and pressureP =5.5293×10³ N/m². In order to assess the flow quantities, the interaction between shock and turbulence, and the inlet efficiency, three different flow calculations — laminar, turbulent with incompressible and compressible two-equationk- turbulence models — have been performed in this work.Computational results show that turbulence is markedly enhanced across an oblique shock with step-like increases in turbulence kinetic energy and dissipation rate. This enhancement is at the expense of the mean kinetic energy of the flow. Therefore, the velocity behind the shock is smaller in turbulent flow and hence the shock becomes stronger. The entropy increase through a shock is caused not only by the amplification of random molecular motion, but also by the enhancement of the chaotic turbulent flow motion. However, only the compressiblek- turbulence model can properly predict a decrease in turbulence length scale across a shock. Our numerical simulation reveals that the incompressiblek- turbulence model exaggerates the interaction between shock and turbulence with turbulence kinetic energy and dissipation rate remaining high and almost undissipated far beyond the shock region. It is shown that proper modeling of turbulence is essential for a realistic prediction of hypersonic inlet flowfield. The performed study shows that the viscous effect is not restricted in the boundary layer but extends into the main flow behind a shock wave. The loss of the available energy in the inlet performance therefore needs to be determined from the shock-turbulence interaction. The present study predicts that the inlet efficiency becomes relatively lower when turbulence is taken into account.     

Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentv , which is defined on the scale of measurement support, and a zero mean sub-field-scale componentv _s , which fluctuates randomly on scales smaller than. Without loss of generality, we work formally with unconditional statistics ofv _s and conditional statistics ofv . We then require that, within this (or other selected) working framework,v _s andv be mutually uncorrelated. This holds whenever the correlation scale ofv is large in comparison to that ofv _s . The formalism leads to an integro-differential equation for the conditional mean total concentration c which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration cc. We solve the former, and evaluate the latter, for mildly fluctuatingv by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformv . These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments c and cc of total concentrationc, which are associated with the scale below, cannot be used to estimate the field-scale concentrationc directly. To do so, a spatial average over the field measurement scale is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofc and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.     

The irrotational waves and the convergence of Levi-Civita's series

Summary The paper shows that the existence of irrotational surface waves established by an investigation of Levi-Civita may be extended till the breaking, but that it is valid when is 0,84 p 1 (and when p₁=np with n whole number). No solution is known till to day when is p<0,84 (and when is 1,00<p₁<<1,68).

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Sommario La nota estende la dimostrazione dell'esistenza di onde superficiali irrotazionali, data dal Levi Civita per una ampiezza finita ma sufficientemente piccola dimostrando l'esistenza di queste onde fino al frangimento. Tanto questa dimostrazione che quella originaria di Levi Civita sono valide finchè sia 0,84 p 1 (e quando sia p₁=np con n intero). Nessuna soluzione è nota fino ad oggi quando sia p<0,84 (e quando sia 1 p₁ 1,68).

     

The evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non-conductors

The theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock. To solve these equations, we use perturbation methods with a perturbation parameter characterising the initial shock amplitude. It is shown that for decaying shock waves, if the accompanying second order discontinuity is of order one, the straightforward perturbation procedure yields uniformly valid solutions, but if the accompanying second order discontinuity is of order , the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel. We also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofHunter & Keller's Weakly Nonlinear Geometrical Optics method. The two approaches give exactly the same results within their common range of validity. The explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity, and on what length scale these effects are most pronounced. Communicated by C. C. Wang     

Flow of second-order fluids over an enclosed rotating disc

Summary This paper is devoted to a study of the flow of a second-order fluid (flowing with a small mass rate of symmetrical radial outflow m, taken negative for a net radial inflow) over a finite rotating disc enclosed within a coaxial cylinderical casing. The effects of the second-order terms are observed to depend upon two dimensionless parameters ₁ and ₂. Maximum values ₁ and ₂ of the dimensionless radial distances at which there is no recirculation, for the cases of net radial outflow (m>0) and net radial inflow (m<0) respectively, decrease with an increase in the second-order effects [represented by T(=₁+₂)]. The velocities at ₁ and ₂ as well as at some other fixed radii have been calculated for different T and the associated phenomena of no-recirculation/recirculation discussed. The change in flow phenomena due to a reversal of the direction of net radial flow has also been studied. The moment on the rotating disc increases with T.Nomenclature , , z coordinates in a cylindrical polar system - z ₀ distance between rotor and stator (gap length) - =/z ₀, dimensionless radial distance - =z/z ₀, dimensionless axial distance - _s = _s/z₀, dimensionless disc radius - V =(u, v, w), velocity vector - dimensionless velocity components - uniform angular velocity of the rotor - , p fluid density and pressure - P =p/(₂ z ₀₂ ² , dimensionless pressure - ₁, ₂, ₃ kinematic coefficients of Newtonian viscosity, elastico-viscosity and cross-viscosity respectively - ₁, _{2 2}/z ₀ ² , resp. ₃/z ₀ ² , dimensionless parameters representing the ratio of second-order and inertial effects - m = , mass rate of symmetrical radial outflow - l a number associated with induced circulatory flow - Rm =m/(z ₀₁), Reynolds number of radial outflow - R _l =l/(z ₀₁), Reynolds number of induced circulatory flow - Rz =z ₀ ² /₁, Reynolds number based on the gap - ₁, ₂ maximum radii at which there is no recirculation for the cases Rm>0 and Rm<0 respectively - ₁(T), ₂(T) ₁ and ₂ for different T - U _1(T) ⁽⁺⁾ = dimensionless radial velocity, Rm>0 - V _1(T) ⁽⁺⁾ = , dimensionless transverse velocity, Rm>0 - U _2(T) ^(–) = , dimensionless radial velocity, Rm=–Rn<0, m=–n - V _2(T) ^(–) = , dimensionless transverse velocity, Rm<0 - C _m moment coefficient     

“Thermal layer” effect in MHD: Interaction of a parallel shock with a layer of decreased density

Interaction of a parallel fast MHD shock with a layer of decreased density is discussed using ideal MHD approach. This is an extrapolation of gas dynamic thermal layer effect on ideal MHD. Computer simulations show that a magnetic field of a moderate intensity ( 1) may change the character of the flow for intermediate Mach numbers (M 5) and a new raking regime may occur which is not observed in the absence of a magnetic field. Self similar precursor analogous to that in gas dynamics may develop in the case of highM and low density in the layer but magnetic forces essentially decrease its growth rate. This problem appears in connection with cosmical shock propagation where planetary magnetic tails play the role of the thermal layer, and it may also be observed in the laboratory when the shock is strong enough to heat the walls ahead of it.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Nearly viscometric flows in the new rheometers

Summary The linear time-dependent behaviour of elasticoviscous liquids may be determined by conventional rheometers, or by any one of a number of new rheometers which have recently appeared in which the test fluid is contained between two surfaces which rotate with the same angular velocity about axes which are not coincident. (The Orthogonal, Balance and Eccentric-Cylinder rheometers are perhaps the most well known examples of such instruments at the present time.) In this communication, we consider the more general situation in the new rheometers in which the angular velocities of the two surfaces are different.The first part of the paper is concerned with experimental results from the Eccentric-Cylinder Rheometer, with particular reference to the case in which the speeds of the two members are almost the same. The results throw light on the important problem of lag brought about by bearing friction.The main body of work concerns a study of the potential use of the combined steady and oscillatory shear type of flow generated when the instrument members are driven at different speeds. The Balance rheometer is concluded to be of very little use in this connection, and most of the attention is confined to the nearly-viscometric flows generated in the Orthogonal and Eccentric-Cylinder rheometers. It is concluded that the analysis for the Orthogonal Rheometer is too complicated to have any predictive value, but that generalized Eccentric-Cylinder flow may be useful in the context of model assessment.With 4 figures     

The effect of a singular perturbation on nonconvex variational problems

We study the effect of a singular perturbation on certain nonconvex variational problems. The goal is to characterize the limit of minimizers as some perturbation parameter 0. The technique utilizes the notion of -convergence of variational problems developed by De Giorgi. The essential idea is to identify the first nontrivial term in an asymptotic expansion for the energy of the perturbed problem. In so doing, one characterizes the limit of minimizers as the solution of a new variational problem. For the cases considered here, these new problems have a simple geometric nature involving minimal surfaces and geodesics.     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.