Entropy Production and Admissibility of Shocks

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.

1989

We consider the limits ofvalidity ofa simple version of extended irreversible thermodynamics as given by the convexity ofthe entropy. In the case of shock waves, the above criterion implies a critical Mach number, whose relation with other theoretical and experimental results is discussed.

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.

Journal of Differential Equations, 1995

In this work we present a new approach to the study of the stability of admissible shock wave solutions for systems of conservation laws that change type. The systems we treat have quadratic ux functions. We employ the fundamental wave manifold W as a global framework to characterize shock waves that comply with the viscosity admissibility criterion. Points of W parametrize dynamical systems associated with shock wave solutions. The region of W comprising admissible shock waves is bounded by the loci of structurally unstable dynamical systems. Explicit formulae are presented for the loci associated with saddle-node, Hopf, and Bogdanov-Takens bifurcation, and with straight-line heteroclinic connections. Using Melnikov's integral analysis, we calculate the tangent to the homoclinic part of the admissibility boundary at Bogdanov-Takens points of W. Furthermore, using numerical methods, we explore the heteroclinic loci corresponding to curved connecting orbits and the complete homoclinic locus. We nd the region of admissible waves for a generic, two-dimensional slice of the fundamental wave manifold, and compare it with the set of shock points that comply with the Lax admissibility criterion, thereby elucidating how this criterion di ers from viscous pro le admissibility.

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)

2001

Library ofCongress Cataloging-in-Publication Data Advances in the theory of shock waves I Heinrich Freistiihler, Anders Szepessy, editors. p. cm. (Progress in nonlinear differential equations and their applications ; v. 47) Includes bibliographical references.

Journal of Differential Equations, 1973

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.

ZAMM - Zeitschrift für Angewandte Mathematik und Mechanik, 1972

Unter Verwendung dea H enc k yachen Verzerrungsmufies im E'alle der Isotropie und eines Pseudo-SpannungstetMors im nichtiaotropen Fall wird die StoJwelle in einem hyperelastischen dfedium in rein raumlicher Weise formuliert, um auj dime Weise die Entropieanderung in schwachen StoJen zu erhalten. Using the H e n c k y strain-measure in the case of isotropy and a pseudo-stress tensor for nonisotropic case. the pnper gices a complete spatial formulation of the shockwave i n a hyperelastic medium. This formulution is used to obtain entropy changes for weak shocks. ~J I H onpenenema M~M~H~H H I I ~HTPOIIMM B cna6b1x m a w a x $op~ynup yeTcH WCTO rrpocTpaircT-BennbiM cnoco60~ ygapHasi Bonna B csepxynpyrofi cpene, npmeena A~F I cnygaa M B O T~V~M H ~e p y neQopMaum reH K M a nnn H~M B O T~O I I H O I ' O caysari rice~no~e113op nanpuwemm.

Journal of Computational Physics, 2008

We consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena. These models admit undercompressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation. Building on earlier work by LeFloch and collaborators, we investigate the numerical approximation of these models by high-order finite difference schemes, and uncover several new features of the kinetic function associated with with physically motivated second and third-order regularization terms, especially viscosity and capillarity terms. On one hand, the role of the equivalent equation associated with a finite difference scheme is discussed. We conjecture here and demonstrate numerically that the (numerical) kinetic function associated with a scheme approaches the (analytic) kinetic function associated with the given model-especially since its equivalent equation approaches the regularized model at a higher order. On the other hand, we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized Camassa-Holm model. Finally, we investigate to what extent a kinetic function can be associated with the equations of van der Waals fluids, whose flux-function admits two inflection points.