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Global solutions for a hyperbolic model of multiphase flow
No full textWe study a strictly hyperbolic system of three balance laws arising in the modelling of fluid flows, in one space dimension. The fluid is a mixture of liquid and vapor, and pure phases may exist as well. The flow is driven by a reaction term depending either on the deviation of the pressure p from an equilibrium value pe and on the mass density fraction of the vapor in the fluid; this makes possible for metastable regions to exist. A relaxation parameter is also involved in the model.
First, for the homogeneous system, we review a result about the global existence of weak solutions to the initial-value problem, for initial data with large variation. Then we focus on the inhomogeneous case. For initial data sufficiently close to the stable liquid phase we prove, through a fractional step algorithm, that weak global solutions still exist. At last, we study the relax- ation limit under such assumptions, and prove that the solutions previously constructed converge to weak solutions of the homogeneous system for the pure liquid phase
Diffusion–dispersion limits for multidimensional scalar conservation laws with source terms
No full textAbstractIn this paper we consider conservation laws with diffusion and dispersion terms. We study the convergence for approximation applied to conservation laws with source terms. The proof is based on the Hwang and Tzavaras's new approach [Seok Hwang, Athanasios E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Application to relaxation and diffusion–dispersion approximations, Comm. Partial Differential Equations 27 (5–6) (2002) 1229–1254] and the kinetic formulation developed by Lions, Perthame, and Tadmor [P.-L. Lions, B. Perthame, E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1) (1994) 169–191]
Wave Structure Induced By Fluid Dynamic Limits In The Broadwell Model
No full text. Consider the fluid dynamic limit problem for the Broadwell system of the kinetic theory of gases, for Riemann, Maxwellian 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. x1. Introduction Being among the simplest models in the kinetic theory of gases, the Broadwell model has served as a paradigm to understand the phenomenon o..
Nonlinear Analysis Techniques for Shear Band Formation At High Strain-Rates
No full textthis article, we focus on a theory suggested for explaining the formation of shear bands during high speed, plastic deformations of metals. It was recognized by Zener and Hollomon [34] that, at high speed processes, the effect of the deformation speed is twofold: First, an increase in the deformation speed changes the deformation conditions from isothermal to nearly adiabatic. Second, strain rate has an effect per se and needs to be included in the constitutive modeling. Under isothermal conditions, metals in general strain harden and exhibit a stable response. As the deformation speed increases, the heat produced by the plastic work triggers thermal effects. In particular, thermal-softening properties of metals may outweigh the tendency of the material to harden, so that the combined outcome results to (effective) softening. A destabilizing feedback mechanism is then induced, operating according to the following scenario ( Clifton, Duffy, Hartley and Shawki [11]): Nonuniformities in the strain rate result in nonuniform heating. Since the material is softer at the hotter spots and harder at the colder spots, if heat diffusion is too weak to equalize the temperatures, the initial nonuniformities in the strain rate are, in turn, amplified. This mechanism tends to localize the total deformation into narrow regions. On the other hand, there is opposition to this process by "viscous effects" induced by strain-rate sensitivity. The two effects are competing and which one prevails depends on the relative weights of thermal softening, strain hardening and strain-rate sensitivity, as well as the loading circumstances. Experimental, numerical and linearized analysis studies indicate that, at least when the degree of thermal softening is large, the competition results to instabi..
Wave Interactions and Variation Estimates For Self-Similar Viscous Limits in Systems of Conservation Laws.
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