On the convexity of a nonequilibrium entropy and shock waves

Acta Mathematicae Applicatae Sinica, English Series, 2003

In shock wave theory there are two considerations in selecting the physically relevant shock waves.

Archive for Rational Mechanics and Analysis, 2015

ABSTRACT We consider by a combination of analytical and numerical techniques some basic questions regarding the relations between inviscid and viscous stability and existence of a convex entropy. Specifically, for a system possessing a convex entropy, in particular for the equations of gas dynamics with a convex equation of state, we ask: (i) can inviscid instability occur? (ii) can there occur viscous instability not detected by inviscid theory? (iii) can there occur the ---necessarily viscous--- effect of Hopf bifurcation, or "galloping instability"? and, perhaps most important from a practical point of view, (iv) as shock amplitude is increased from the (stable) weak-amplitude limit, can there occur a first transition from viscous stability to instability that is not detected by inviscid theory? We show that (i) does occur for strictly hyperbolic, genuinely nonlinear gas dynamics with certain convex equations of state, while (ii) and (iii) do occur for an artifically constructed system with convex viscosity-compatible entropy. We do not know of an example for which (iv) occurs, leaving this as a key open question in viscous shock theory, related to the principal eigenvalue property of Sturm Liouville and related operators. In analogy with, and partly proceeding close to, the analysis of Smith on (non-)uniqueness of the Riemann problem, we obtain convenient criteria for shock (in)stability in the form of necessary and sufficient conditions on the equation of state.

Proceedings of the American Mathematical Society, 2014

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable L 2 dependence on initial data of Lax 1-or n-shock solutions of an n×n system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freistühler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small BV or H s perturbations.

Proceedings, 2012

For a class of nonconservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to nonconservative systems of a similar concept introduced by Abeyaratne, Knowles, and Truskinovsky for subsonic phase transitions and by LeFloch for nonclassical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for nonconservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch, and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase plane analysis of traveling solutions associated with an augmented version of the nonconservative system. We illustrate with several examples that nonconservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics, we provide a detailed analysis of the existence and properties of traveling waves which yields the corresponding kinetic function.

Physics of Fluids, 2004

Insights into symmetric and asymmetric vortex mergers using the core growth model Phys. Fluids 24, 073101 (2012) Nonlinear finite amplitude torsional vibrations of cantilevers in viscous fluids J. Appl. Phys. 111, 124915 (2012) A subgrid-scale model for large-eddy simulation based on the physics of interscale energy transfer in turbulence Phys. Fluids 24, 065104 (2012) An efficient approach for eigenmode analysis of transient distributive mixing by the mapping method Phys. Fluids 24, 053602 (2012) Dynamics of freely swimming flexible foils Phys. Fluids 24, 051901 (2012)

International Journal of Thermophysics, 1993

The objective of this paper is twofold: first, to examine how the concepts of extended irreversible thermodynamics are related to the notion of accompanying equilibrium state introduced by Kestin; second, to compare the behavior of both the classical local equilibrium entropy and that used in extended irreversible thermodynamics. Whereas the former does not show a monotonic increase, the latter exhibits a steady increase during the heat transfer process; therefore it is more suitable than the former one to cope with the approach to equilibrium in the presence of thermal waves.

Continuum Mechanics and Thermodynamics, 1998

In this paper we give a brief survey of the problem of shock structure solutions in fluid dynamics. For a generic system of balance laws compatible with an entropy principle and a convex entropy we prove that C 1 solutions cannot exist when the shock velocity exceeds the maximum characteristic velocity in the equilibrium state in front of the shock. This is in agreement with a conjecture of Extended Thermodynamics.

Pramana-journal of Physics, 1999

It is reiterated that without a Gibbs-Duhem equation no thermodynamic description of irreversible and reversible processes exists. It is shown with the help of Gibbs-Duhem equation of extended irreversible thermodynamics that the physical contents of intensive quantities, the temperature and the pressure, do not change in going from reversible to irreversible processes. This confirms well with the earlier demonstrations of Eu and García-Colín.

Condensed Matter Physics, 1998

A small viscosity approach to discontinuous flows is discussed in relativistic hydrodynamics with a general (possibly, non-convex) equation of state that typically occurs in the domains of phase transitions. Different forms of criteria for the existence and stability of relativistic shock waves, such as evolutionarity conditions, entropy criterion and corrugation stability conditions are compared with the requirement of the existence of shock viscous profile. The latter is shown to be most restrictive in case of a single-valued shock adiabat expressed as a function of pressure. One-dimensional numerical simulations with artificial viscosity for a simple piecewise-linear equation of state are carried out to illustrate the criteria in the case of planar and spherical shock waves. The effect of a phase transition domain on the shock amplitude in the process of a hydrodynamical spherical collapse is demonstrated.

Communications in Partial Differential Equations, 2010