Wave angle for oblique detonation waves

The flow field associated with a steady, planar, oblique detonation wave is discussed. A revision is provided for- diagrams, where is the wave angle and is the ramp angle. A new solution is proposed for weak underdriven detonation waves that does not violate the second law. A Taylor wave, encountered in unsteady detonation waves, is required. Uniqueness and hysteresis effects are also discussed.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.     

Dynamic-mechanical properties of a bitumen-silica composite

Summary The dynamic-mechanical behaviour of bitumensilica composites is described by a linear biparabolic model. Its mathematical expression allows the calculation of the mean relaxation times () either at different temperatures and given filler contents or for diverse filler contents () at imposed temperatures. At fixed filler concentration and within restricted temperature domains, obeys Arrhenius' law. The activation energies are respectively close to 10 kcal/mole (creep) and 30 kcal/mole (glass-transition). varies exponentially with. The mathematical treatment of the expressions ofE , as a function of temperature and of, leads to a general equation relating the complex modulus to temperature, frequency and filler content. A unique master curve, accounting for the viscoelastic behaviour of the composites, in limited ranges, can thus be constructed.

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Zusammenfassung Das dynamisch-mechanische Verhalten von Bitumen-Siliziumdioxyd-Zusammensetzungen kann durch ein lineares biparabolisches Modell beschrieben werden. Sein mathematischer Ausdruck erlaubt die Ausrechnung der mittleren Relaxationszeiten () entweder für verschiedene Temperaturen bei gegebenem Füllstoffgehalt oder für unterschiedliche Siliziumdioxydmengen () bei bekannter Temperatur. Für einen bestimmten Füllstoffgehalt folgt in einem beschränkten Temperaturbereich dem Arrheniusschen Gesetz. Die Aktivierungsenergien betragen näherungsweise 10 kcal/Mol (Fließprozeß) bzw. 30 kcal/Mol (Glasübergang). ändert sich exponentiell mit. Die mathematische Umformung der Ausdrücke fürE und als Funktion der Temperatur und des Parameters ergibt eine allgemeine Gleichung, die den komplexen Modul mit der Temperatur, der Frequenz und dem Füllstoffgehalt verknüpft. Man kann eine einzige Masterkurve bilden, die das viskoelastische Verhalten der Zusammensetzungen zumindest in begrenzten Bereichen beschreibt.

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Résumé Le comportement mécanique dynamique des composites à base de bitume et de silice peut être décrit par un modèle biparabolique linéaire. L'expression mathématique permet le calcul des temps moyens () de relaxation d'une part aux différentes températures, à taux de charge donné, et d'autre part pour diverses valeurs des taux de charge (paramètre) à température imposée. A taux de charge donné, et pour des domaines de température restreints, suit la loi d'Arrhénius. Les énergies apparentes d'activation sont respectivement voisines de 10 kcal/mole (processus de fluage) et de 30 kcal/mole (passage à l'état vitreux). Avec, varie exponentiellement. L'évaluation mathématique deE , de en fonction deT et de conduit à une expression générale du module complexe en fonction de la température, de la fréquence et du taux de charge. On peut donc construire une courbe maitresse unique qui décrit entièrement, mais dans des domaines restreints, le comportement viscoélastique des composites.

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With 6 figures     

The Global Structure of Traveling Waves in Spatially Discrete Dynamical Systems

We obtain existence of traveling wave solutions for a class of spatially discrete systems, namely, lattice differential equations. Uniqueness of the wave speed c, and uniqueness of the solution with c0, are also shown. More generally, the global structure of the set of all traveling wave solutions is shown to be a smooth manifold where c0. Convergence results for solutions are obtained at the singular perturbation limit c 0.     

Construction of exact numerical solutions of the stationary traveling wave type for viscous thin films

An effective numerical procedure, based on the Galerkin method, for finding solutions of the stationary traveling wave type in the complete formulation is proposed for the case of viscous liquid films. Examples of a viscous film flowing freely down a vertical surface have been calculated. The calculations have been made for various values of the dimensionless surface tension , including =0. The method makes it possible to predict a number of bifurcations that occur as decreases. The existence of numerous families of stationary traveling waves when 1 was demonstrated in [6]. The present study shows that as 1 all but one of these families of wave solutions disappear. The shape of the periodic and solitary waves and the pressure distribution in the film are found for various . When =0 and the wave number is fairly small, the periodic solution has a singularity, as predicted in [14]: at the crest of the wave a corner point appears; the angle between the tangents at this point =140–150. The method proposed can be used to calculate other wavy film flows.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 94–100, May–June, 1990.     

Asymptotic Modeling of Nonlinear Wave Processes in Shock‐Loaded Elastoplastic Materials

Nonlinear wave processes in shockloaded elastoplastic materials are modeled. A comparison of the results obtained with experimental data and numerical solutions of exact systems of dynamic equations shows that the model equations proposed qualitatively describe the stressdistribution evolution in both the elasticflow and plasticflow regions and can be used to solve one and twodimensional problems of pulsed deformation and fracture of elastoplastic media.     

Optical co-ordinates in general relativity 2 — the problem of distance

Summary Let denote the congruence of null geodesics associated with a given optical observer inV ₄. We prove that determines a unique collection of vector fieldsM₍₎ ( =1, 2, 3) and ₍₀₎ overV ₄, satisfying a weak version of Killing's conditions.This allows a natural interpretation of these fields as the infinitesimal generators of spatial rotations and temporal translation relative to the given observer. We prove also that the definition of the fieldsM₍₎ and ₍₀₎ is mathematically equivalent to the choice of a distinguished affine parameter f along the curves of, playing the role of a retarded distance from the observer.The relation between f and other possible definitions of distance is discussed.

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Sommario Sia la congruenza di geodetiche nulle associata ad un osservatore ottico assegnato nello spazio-tempoV ₄. Dimostriamo che determina un'unica collezione di campi vettorialiM₍₎ ( =1, 2, 3) e ₍₀₎ inV ₄ che soddisfano una versione in forma debole delle equazioni di Killing. Ciò suggerisce una naturale interpretazione di questi campi come generatori infinitesimi di rotazioni spaziali e traslazioni temporali relative all'osservatore assegnato. Dimostriamo anche che la definizione dei campiM₍₎, ₍₀₎ è matematicamente equivalente alla scelta di un parametro affine privilegiato f lungo le curve di, che gioca il ruolo di distanza ritardata dall'osservatore. Successivamente si esaminano i legami tra f ed altre possibili definizioni di distanza in grande.

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Work performed in the sphere of activity of: Gruppo Nazionale per la Fisica Matematica del CNR.     

The dynamic performance of the Weissenberg rheogoniometer III. Large amplitude oscillatory shearing — harmonic analysis

The harmonic content of the nonlinear dynamic behaviour of 1% polyacrylamide in 50% glycerol/water was studied using a standard Model R 18 Weissenberg Rheogoniometer. The Fourier analysis of the Oscillation Input and Torsion Head motions was performed using a Digital Transfer Function Analyser.In the absence of fluid inertia effects and when the amplitude of the (fundamental) Oscillation Input motion _I is much greater than the amplitudes of the Fourier components of the Torsion Head motion _Tn empirical nonlinear dynamic rheological propertiesG _n (, ⁰),G _n (, ⁰) and/or _n (, ⁰), _n (, ⁰) may be evaluated without a-priori-knowledge of a rheological constitutive equation. A detailed derivation of the basic equations involved is presented.Cone and plate data for the third harmonic storage modulus (dynamic rigidity)G ₃ (, ⁰), loss modulusG ₃ (, ⁰) and loss angle ₃ (, ⁰) are presented for the frequency range 3.14 × 10^–2 1.25 × 10² rad/s at two strain amplitudes, _CP ⁰ = 2.27 and 4.03. Composite cone and plate and parallel plates data for both the third and fifth harmonic dynamic viscosities ₃ (, ⁰), _S (, ⁰) and dynamic rigiditiesG ₃ (, ⁰),G ₅ (, ⁰) are presented for strain amplitudes in the ranges 1.10 _CP ⁰ 4.03 and 1.80 _PP ⁰ 36 for a single frequency, = 3.14 × 10^–1 rad/s. Good agreement was obtained between the results from both geometries and the absence of significant fluid inertia effects was confirmed by the superposition of the data for different gap widths.     

Chaos and integrability in a nonlinear wave equation

We consider the parametrized family of equations _tt ,u- _xx u-au+u ₂ ² u=O,x(0,L), with Dirichlet boundary conditions. This equation has finite-dimensional invariant manifolds of solutions. Studying the reduced equation to a four-dimensional manifold, we prove the existence of transversal homoclinic orbits to periodic solutions and of invariant sets with chaotic dynamics, provided that =2, 3, 4,.... For =1 we prove the existence of infinitely many first integrals pairwise in involution.     

Exact Solutions of the Hydrodynamic Equations Derived from Partially Invariant Solutions

The paper proposes a heuristic approach to constructing exact solutions of the hydrodynamic equations based on the specificity of these equations. A number of systems of hydrodynamic equations possess the following structure: they contain a reduced system of n equations and an additional equation for an extra function w. In this case, the reduced system, in which w = 0, admits a Lie group G. Taking a certain partially invariant solution of the reduced system with respect to this group as a seed:rdquo; solution, we can find a solution of the entire system, in which the functional dependence of the invariant part of the seed solution on the invariants of the group G has the previous form. Implementation of the algorithm proposed is exemplified by constructing new exact solutions of the equations of rotationally symmetric motion of an ideal incompressible liquid and the equations of concentrational convection in a plane boundary layer and thermal convection in a rotating layer of a viscous liquid.     

Stochastic-Perturbation Analysis of a One-Dimensional Dispersion-Reaction Equation: Effects of Spatially-Varying Reaction Rates

We carry out a stochastic-perturbation analysis of a one-dimensional convection–dispersion-reaction equation for reversible first-order reactions. The Damköhler number, Da, is distributed randomly from a distribution that has an exponentially decaying correlation function, controlled by a correlation length, . Zeroth- and first-order approximations of the dispersion coefficient, D are computed from moments of the residence-time distribution obtained by solving a one-dimensional network model, in which each unit of the network represents a Darcy-level transport unit, and the solution of the transfer function in zeroth- and first-order approximations of the transport equation. In the zeroth-order approximation, the dispersion coefficient is calculated using the convection–dispersion-reaction equation with constant parameters, that is, perturbation corrections to the local equation are ignored. This zeroth-order dispersion coefficient is a linear function of the variance of the Damköhler number, (Da)². A similar result was reported in a two-dimensional network simulation. The zeroth-order approximation does not give accurate predictions of mixing or spreading of a plume when Damköhler numbers, Da 1 and its variance, (Da)² > 0.25 Da². On the other hand, the first-order theory leads to a dispersion coefficient that is independent of the reaction parameters and to equations that do accurately predict mixing and spreading for Damköhler numbers and variances in the range (Da)²/Da0.3     

Evolution of the Diffusion Mixing Layer of Two Gases upon Interaction with Shock Waves

A mathematical model of mechanics of a twovelocity twotemperature mixture of gases is developed. Based on this model, evolution of the mixing layer of two gases with different densities under the action of shock and compression waves is considered by methods of mathematical simulation in the onedimensional unsteady approximation. In the asymptotic approximation of the full model, a solution of an initialboundary problem is obtained, which describes the formation of a diffusion layer between two gases. Problems of interaction of shock and compression waves with the diffusion layer are solved numerically in the full formulation. It is shown that the layer is compressed as the shock wave traverses it; the magnitude of compression depends on shockwave intensity. As the shock wave passes from the heavy gas to the light gas, the mixing layer becomes overcompressed and expands after shockwave transition. The wave pattern of the flow is described in detail. The calculated evolution of the mixinglayer width is in good agreement with experimental data.     

Nearly Spherically Symmetric Expanding Fronts in a Bistable Reaction-Diffusion Equation

We consider nearly spherically symmetric expanding fronts in the scalar bistable reaction-diffusion equation on R^N. As t, the front is known to look more and more like a sphere under the rescaling of the radius to unity. In this paper we prove that, if the initial state is spherically symmetric and approximated by a one-dimensional traveling wave with a sufficiently large radius, then the solution is approximated uniformly for all t0 without the rescaling of the radius by the one-dimensional traveling wave with the speed of V=c–(N–1), where c>0 is the speed of the one-dimensional traveling wave solution and the mean curvature of the sphere. We further show that, if the initial state is a slightly perturbed one from the spherical front, the difference between the actual front and the expanding sphere hardly grows or decays for all t0, although the relative magnitude of the perturbation to the radius of the sphere decreases to zero.     

Protective properties of hot‐air screens

An analytical and numerical study of protective properties of hotair screens is performed for a plane wave with a triangular profile. The reduction of the shockwave pressure behind the screen is only observed at temperatures higher a certain critical value; at lower temperatures, the pressure is higher than that in a shock wave traveling in an isothermal gas. With allowance for real properties of air, an analytical relation between the critical temperature of the hot screen and the incidentwave intensity is obtained.     

Shear Instability at the “Explosion Product–Metal” Interface for Sliding Detonation of an Explosive Charge

Periodic perturbations at the explosion product–metal interface were studied experimentally. Experiments were performed for both spherical and plane geometry. Critical conditions of wave formation (detonation velocity of an explosive charge D 6.9 mm/sec) are determined, and an explanation of this effect is given. It is found experimentally that a dynamic pulse causes intense plastic strains at the explosion products–metal interface, leading to thermal softening of the steel boundary layer. In this layer, Kelvin–Helmholtz instability occurs. Calculationanalytical estimates of the critical boundary unstable wavelength agree satisfactorily with experimental results.     

Some problems of nonisothermal steady flow of a viscous fluid

The paper presents solutions to the problems of plane Couette flow, axial flow in an annulus between two infinite cylinders, and flow between two rotating cylinders. Taking into account energy dissipation and the temperature dependence of viscosity, as given by Reynolds's relation =₀ exp (–T) (₀, =const). Two types of boundary conditions are considered: a) the two surfaces are held at constant (but in general not equal) temperatures; b) one surface is held at a constant temperature, the other surface is insulated.Nonisothermal steady flow in simple conduits with dissipation of energy and temperature-dependent viscosity has been studied by several authors [1–11]. In most of these papers [1–6] viscosity was assumed to be a hyperbolic function of temperature, viz. =_m 1/1+²(T–T_m.Under this assumption the energy equation is linear in temperature and can he easily integrated. Couette flow with an exponential viscosity-temperature relation. =₀ e ^–T (₀, =const), (0.1) was studied in [7, 8]. Couette flow with a general (T) relation was studied in (9).Forced flow in a plane conduit and in a circular tube with a general (T) relation was studied in [10]. In particular, it has been shown in [10] that in the case of sufficiently strong dependence of viscosity on temperature there can exist a critical value of the pressure gradient, such that a steady flow is possible only for pressure gradients below this critical value.In a previous work [11] the authors studied Polseuille flow in a circular tube with an exponential (T) relation. This thermohydrodynamic problem was reduced to the problem of a thermal explosion in a cylindrical domain, which led to the existence of a critical regime. The critical conditions for the hydrodynamic thermal explosion and the temperature and velocity profiles were calculated.In this paper we treat the problems of Couette flow, pressureless axial flow in an annulus, and flow between two rotating cylinders taking into account dissipation and the variation of viscosity with temperature according to Reynolds's law (0.1). The treatment of the Couette flow problem differs from that given in [8] in that the constants of integration are found by elementary methods, whereas in [8] this step involved considerable difficulties. The solution to the two other problems is then based on the Couette problem.     

Generation of perturbations by a localized vibrator in the boundary layer of a nonswept wing

Origination and development of perturbations generated by a threedimensional vibrating surface in the boundary layer on an airfoil with a zero slip angle is experimentally studied. Surface vibrations were generated by a Mylar membrane. It is shown that highamplitude vibrations of a threedimensional surface lead to simultaneous formation of two types of perturbations in the boundary layer: quasistationary streamwise structures and wave packets accompanying them. The presence of regions with favorable and adverse pressure gradients does not exert a significant effect on evolution of streamwise structures but leads first to attenuation and then to amplification of wave packets.     

Small perturbations of plane unsteady motion of an ideal incompressible fluid with a free boundary in the shape of an ellipse

The problem about small perturbations is solved explicitly. An investigation of the behavior of the solution as t shows its boundedness in a weak potential metric. Meanwhile, the perturbation vector of the free boundary of the ellipse grows without limit with time.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 53–62, July–August, 1971.The author is grateful to L. V. Ovsyannikov and R. M. Garipov for discussing the research.     

The Thermodynamic Significance of the Local Volume Averaged Temperature

Since the temperature is not an additive function, the traditional thermodynamic point of view suggests that the volume integral of the temperature has no precise physical meaning. This observation conflicts with the customary analysis of non-isothermal catalytic reactors, heat pipes, driers, geothermal processes, etc., in which the volume averaged temperature plays a crucial role. In this paper we identify the thermodynamic significance of the volume averaged temperature in terms of a simple two-phase heat transfer process. Given the internal energy as a function of the point temperature and the density

Lösungen und Schmelzen von Hochpolymeren

Zusammenfassung Die Modelle der ideal verdünnten Lösung und des unendlichen Netzes (Bueche) werden besprochen; für praktische Fälle werden die Modelle verdünnte Lösung mit Gelteilchen und unendliches Netz mit Verhängungsbereichen vorgeschlagen und diskutiert. Es wird eine Methode angegeben, aus dem Übergangspunkt verdünnte Lösung unendliches Netz den Verknäuelungsparameter Am auszurechnen.Einleitungsvortrag anläßlich der Arbeitstagung der Sektion Rheologie des Vereins Österreichischer Chemiker am 28. September 1965 in Graz.