Undercompressive Shocks Research Articles

Traveling Waves, Riemann Problems and Computations of a Model of the Dynamics of Liquid/Vapor Phase Transitions

A model for the liquid/vapor phase transitions in a shock tube is discussed. Computations for the one-dimensional isothermal case is carried out to show that this model exhibits one-dimensional wave patterns observed in actual experiments. The existence of traveling waves under two different scalings are studied. For the first scaling, where the diffusion of different phases is very small relative to typical reaction time for the growth of phases while the viscosity is comparable to the reaction time, complete phase diagrams of traveling waves are obtained. Many of these traveling waves are undercompressive. Some of compressive traveling waves are unstable. For the second scaling, where viscosity, typical reaction time, and diffusion of different phases are comparable, the existence of traveling wave profiles of liquefaction and evaporation shocks when shock speeds are larger than some number is proved. These shocks are undercompressive. Evaporation shocks are rarefaction shocks. The nonexistence of these undercompressive shocks when the shock speed is smaller than some number is proved. It is observed through numerical computation that most of these traveling waves are unstable while some of them are stable or metastable. Riemann problems are considered. Admissibility of shocks involving phase changes under the second scaling is discussed. Solutions of Riemann problems with wave patterns observed in actual experiments are presented. Nonuniqueness of solutions of some Riemann problems is discussed.

Elastic phase transitions: A new model

In this note we consider a new model for one-dimensional clastic phase transitions. The model is described by two hyperbolic systems: one which propagates information at speeds ± c and the other at speeds ±λ where λ > c > 0. The model supports slow-moving under-compressive shocks which connect separate phases of the material and microstructural equilibrium solutions similar to those predicted by viscosity-capillarity theories for such materials. Our principal results are contained in Section 3 and describe the long-term behavior of the system. Section 4 presents numerical simulations for our model for a variety of initial and boundary conditions. These results confirm the analysis of Sections 2 and 3.

Nonlinear stability of an undercompressive shock for complex Burgers equation

Though there is strong numerical evidence for the stability of undercompressive shocks, their stability has not been verified analytically. In particular, the energy methods used to analyze stability of standard shocks do not apply. Here, we present the first proof of stability for a particular undercompresive shock, the real Burgers shock considered as a solution of the complex Burgers equation. Our analysis is by direct calculation of the Green's function for the linearized equations, combined with pointwise estimates of nonlinear effects. A benefit of this method is to obtain fairly detailed information about the solution, includingL 1 behavior, and rates of decay in different regions of space.

The persistence of ideal shock waves

Local-in-time piecewise smooth solutions to hyperbolic systems of conservation laws are constructed by means of Li-Yu theory. The novelty consists in the application of this approach to shock waves for which the number of outgoing modes is at least as big as the number of incoming modes (undercompressive shocks), the motivation in a possible interpretation from the zero dissipation limit point of view.

Undercompressive shocks for nonstrictly hyperbolic conservation laws

We study 2×2 systems of hyperbolic conservation laws near an umbilic point. These systems have Undercompressive shock wave solutions, i.e., solutions whose viscous profiles are represented by saddle connections in an associated family of planar vector fields. Previous studies near umbilic points have assumed that the flux function is a quadratic polynomial, in which case saddle connections lie on invariant lines. We drop this assumption and study saddle connections using Golubitsky-Schaeffer equilibrium bifurcation theory and the Melnikov integral, which detects the breaking of heteroclinic orbits. The resulting information is used to construct solutions of Riemann problems.