Nonlinear wave propagation in binary mixtures of Euler fluids

In the present paper, we study the propagation of acceleration and shock waves in a binary mixture of ideal Euler fluids, assuming that the difference between the atomic masses of the constituents is negligible. We evaluate the characteristic speeds, proving that they can be separated into two groups: one is related to the case of a single Euler fluid, provided that an average ratio of specific heats is introduced; the other is new and related to the propagation speed due to diffusion. We evaluate the critical time for sound acceleration waves and compare its value to that of a single fluid. We then study shock waves, showing that three types of shock waves appear: sonic and contact shocks, which have counterparts in the single fluid case, and the diffusive shock, which is peculiar to the mixture. We discuss the admissibility of the shock waves using the Lax-Liu conditions and the entropy growth criterion. It is proved that the sonic and the characteristic shock obey the same properties as in the single fluid case, while for the diffusive shock there exists a locally exceptional case that is determined by a particular value of the concentration of the constituents, for which the genuine nonlinearity is lost and no shocks are admissible. For other values of the unperturbed concentration, the diffusive shock is stable in a bounded interval of admissibility.Received: 15 December 2002, Accepted: 28 June 2003 Correspondence to: T. RuggeriS. Simi: On leave from the Department of Mechanics, Faculty of Technical Sciences, University of Novi Sad, Serbia     

Generation of axisymmetric magnetosonic shock waves: Applications to static equilibria and accretion disks

Magnetosonic waves traveling in a MHD equilibrium may evolve into shocks. We develop a criterion for the creation of fast shocks in the equatorial plane of axisymmetric equilibria and analyze the influence of the most important parameters. The results are applied to Grad–Shafranov equilibria and accretion disks.     

Entropy in self-similar shock profiles

In this paper, we study the structure of a gaseous shock, and in particular the distribution of entropy within, in both a thermodynamics and a statistical mechanics context. The problem of shock structure has a long and distinguished history that we review. We employ the Navier–Stokes equations to construct a self-similar version of Becker’s solution for a shock assuming a particular (physically plausible) Prandtl number; and that solution reproduces the well-known result of Morduchow & Libby that features a maximum of the equilibrium entropy inside the shock profile. We then construct an entropy profile, based on gas kinetic theory, that is smooth and monotonically increasing. The extension of equilibrium thermodynamics to irreversible processes is based in part on the assumption of local thermodynamic equilibrium. We show that this assumption is not valid except for the weakest shocks. We conclude by hypothesizing a thermodynamic nonequilibrium entropy and demonstrating that it closely estimates the gas kinetic nonequilibrium entropy within a shock.     

On the response of rubbers at high strain rates—II. Shock waves

In this series of papers, we examine the propagation of waves of finite deformation in rubbers through experiments and analysis; in the present paper, Part II, attention is focused on the propagation of one-dimensional tensile shock waves in strips of latex and nitrile rubber. Tensile wave propagation experiments were conducted at high strain rates by holding one end fixed and displacing the other end at a constant velocity. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in rubber strips. Shock waves have been generated under tensile impact in prestretched rubber strips; analysis of the response yields the tensile shock adiabat for rubbers. The propagation of shocks is analyzed by developing an analogy with the theory of detonation; it is shown that the condition for shock propagation can be determined using the Chapman-Jouguet shock condition.     

Modeling plastic shocks in periodic laminates with gradient plasticity theories

Steady plastic shocks generated by planar impact on metal-polymer laminate composites are analyzed in the framework of gradient plasticity theories. The laminate material has a periodic structure with a unit cell composed of two layers of different materials. First- and second-order gradient plasticity theories are used to model the structure of steady plastic shocks. In both theories, the effect of the internal structure is accounted for at the macroscopic level by two material parameters that are dependent upon the layer's thickness and the properties of constituents. These two structure parameters are shown to be uniquely determined from experimental data. Theoretical predictions are compared with experiments for different cell sizes and for various shock intensities. In particular, the following experimental features are well-reproduced by the modeling:

•

the shock width is proportional to the cell size;

•

the magnitude of strain rate is inversely proportional to cell size and increases with the amplitude of applied stress following a power law.

While these results are equally described by both the plasticity theories, the first gradient plasticity approach seems to be favored when comparing the structure of the shock front to the experimental data.     

Wave interaction in a nonequilibrium gas flow

By employing the method of multiple time scales, we derive here the transport equations for the primary amplitudes of resonantly interacting high-frequency waves propagating into a non-equilibrium gas flow. Evolutionary behavior of non-resonant wave modes culminating into shocks or no shocks, together with their asymptotic decay behavior, is studied. Effects of non-linearity, which are noticeable over times of order O(ε^-1), are examined, and the model evolution equations for resonantly interacting multi-wave modes are derived.     

A geometric study of shocks in equations that change type

In this paper we validate the generalized geometric entropy criterion for admissibility of shocks in systems which change type. This condition states that a shock between a state in a hyperbolic region and one in a nonhyperbolic region is admissible if the Lax geometric entropy criterion, based on the number of characteristics entering the shock, holds, where now the real part of a complex characteristic replaces the characteristic speed itself. We test this criterion by a nonlinear inviscid perturbation. We prove that the perturbed Cauchy problem in the elliptic region has a solution for a uniform time if the data lie in a suitable class of analytic functions and show that under small perturbations of the data a perturbed shock and a perturbed solution in the hyperbolic region exist, also for a uniform time.     

Erratum to “Bistable Waves in an Epidemic Model” [<Emphasis Type="Italic">J. Dynam. Diff. Eq.</Emphasis> 16, 679--707 (2004)]

The existence, uniqueness up to translation and global exponential stability with phase shift of bistable travelling waves are established for a quasi- monotone reaction–diffusion system modelling man–environment–man epidemics. The methods involve phase space investigation, monotone semiflows approach and spectrum analysis.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday; Supported in part by the NSERC of Canada.     

Bistable Waves in an Epidemic Model

The existence, uniqueness up to translation and global exponential stability with phase shift of bistable travelling waves are established for a quasimonotone reaction–diffusion system modelling man–environment–man epidemics. The methods involve phase space investigation, monotone semiflows approach and spectrum analysis.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.Supported in part by the NSERC of Canada     

Shocks versus kinks in a discrete model of displacive phase transitions

We consider dynamics of phase boundaries in a bistable one-dimensional lattice with harmonic long-range interactions. Using Fourier transform and Wiener–Hopf technique, we construct traveling wave solutions that represent both subsonic phase boundaries (kinks) and intersonic ones (shocks). We derive the kinetic relation for kinks that provides a needed closure for the continuum theory. We show that the different structure of the roots of the dispersion relation in the case of shocks introduces an additional free parameter in these solutions, which thus do not require a kinetic relation on the macroscopic level. The case of ferromagnetic second-neighbor interactions is analyzed in detail. We show that the model parameters have a significant effect on the existence, structure, and stability of the traveling waves, as well as their behavior near the sonic limit.     

Wave structure induced by fluid-dynamic limits in the Broadwell model

Consider the fluid-dynamic limit problem for the Broadwell system of the kinetic theory of gases, for Maxwellian Riemann initial data. The formal limit is the Riemann problem for a pair of conservation laws and is invariant under dilations of coordinates. The approach of self-similar fluid-dynamic limits consists in replacing the mean free path in the Broadwell model so that the resulting problem preserves the invariance under dilations. The limiting procedure was justified in [ST]. Here, we study the structure of the emerging solutions. We show that they consist of two wave fans separated by a constant state. Each wave fan is associated with one of the characteristic fields and is either a rarefaction wave or a shock wave. The shocks satisfy the Lax shock conditions and have the internal structure of a Broadwell shock profile.     

Solution of a Riemann problem for elasticity

This paper describes a numerical algorithm for the Riemann solution for nonlinear elasticity. We assume that the material is hyperelastic, which means that the stress-strain relations are given by the specific internal energy. Our results become more explicit under further assumptions: that the material is isotropic and that the Riemann problem is uniaxial. We assume that any umbilical points lie outside the region of physical relevance. Our main conclusion is that the Riemann solution can be obtained by the iterative solution of functional equations (Godunov iterations) each defined in one- or two-dimensional spaces.Supported in part by AFOSR-88-0025.     

Spallation of polycarbonate under laser-driven shocks

Laser driven shocks have been used to investigate dynamic failure (spallation) of polycarbonate under uniaxial tensile loading at very high strain rate, of the order of 10 s. First, uninstrumented recovery shots have been performed, post-test examination of the fracture damage has been carried out, and the influences of the experimental parameters (loading conditions and target thickness) have been analyzed. Then, an attempt to model the response of polycarbonate to plane shock loading has been made. On one hand, in-situ measurements have been performed in polycarbonate samples submitted to the plane detonation wave of a strong explosive, and the results have led to content with simple constitutive relations. On the other hand, piezoelectric measurements under laser shocks have provided a characterization of the loading pressure pulse, and comparisons of the measured and computed signals have confirmed the ability of the model to describe wave propagation in polycarbonate. Finally, the spallation experiments have been simulated. A spall strength has been estimated, on the basis of the experimental data, and the predictive capability of the model has been tested. Received: 18 February 1997 / Accepted 1 April 1997     

On the response of rubbers at high strain rates—III. Effect of hysteresis

In this series of papers, we examine the propagation of waves of finite deformation in rubbers through experiments and analysis; in the present paper, Part III, the effect of hysteretic material behavior on the free retraction of prestretched rubber is considered. A rubber strip stretched to many times its initial length is released at one end and the resulting unloading is examined. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in rubber strips. Simple waves as well as shock waves are observed in these unloading experiments. The measurements are modeled using a power-law model of the material behavior. The hysteretic material response and the formation of shocks are characterized.     

Numerical simulations of shock wave reflection phenomena in non-stationary flows using regularized smoothed particle hydrodynamics (RSPH)

In this paper we wish to demonstrate to what extent the numerical method regularized smoothed particle hydrodynamics (RSPH) is capable of modelling shocks and shock reflection patterns in a satisfactory manner. The use of SPH based methods to model shock wave problems has been relatively sparse, both due to historical reasons, as the method was originally developed for studies of astrophysical gas dynamics, but also due to the fact that boundary treatment in Lagrangian methods may be a difficult task. The boundary conditions have therefore been given special attention in this paper. Results presented for one quasi-stationary and three non-stationary flow tests reveal a high degree of similarity, when compared to published numerical and experimental data. The difference is found to be below 5, in the case where experimental data was found tabulated. The transition from regular reflection (RR) to Mach reflection (MR) and the opposite transition from MR to RR are studied. The results are found to be in close agreement with the results obtained from various empirical and semi-empirical formulas published in the literature. A convergence test shows a convergence rate slightly steeper than linear, comparable to what is found for other numerical methods when shocks are involved.     

Finite scale theory: Predicting nature’s shocks

In this paper, we describe a continuum model that accurately reproduces the experimentally measured structure of physical shocks in a perfect gas. We begin by presenting a history of shock structure research, theoretical, experimental and numerical, to quantify the significant discrepancies between Navier–Stokes predictions and laboratory measurements. In our first main result, we discuss modifications that generalize the Chapman–Enskog approximation and lead to a new continuum model termed the Finite Scale Equations. In our second main result, we use this continuum model to calculate shock structure with results comparable in accuracy to numerical simulations based on the Boltzmann equation.     

A new,high-resolution shock-capturing hybrid scheme of flux vector splitting-Harten's TVD

The paper presents a new high-resolution hybrid scheme combining implicit flux vector splitting with Harten's TVD, which is proved suitable for shock-capturing calculation in gasdynamics. Fluxsplitting procedures are applied to discretize the implicit part of the Euler equations whereas Harten's numerical fluxes are used to calculate the residual of steady-state solutions. It ensures good shock-capturing properties and produces sharp numerical discontinuities without oscillations. It excludes expansion shocks and leads only to physically relevant solutions. The block-line-Gauss-Seidel relaxation procedure (block-LGS) is used to solve the resulting difference equations. The time step and the CFL number are much larger than those in the linearized block-alternating-direction-implicit approximate factorization method (block-ADI). Numerical experiments suggest that the hybrid scheme not only has a fairly rapid convergence rate, but also can generate a highly resolved approximation to the steady-state solution. Hence scheme seems to lead to an effective nonoscillatory shock capturing method for steady transonic flow. Project Supported by National Natural Science Foundation of China