Nonlinear wave propagation in binary mixtures of Euler fluids

Ruggeri, Tommaso; Simić, Srboljub

doi:10.1007/s00161-003-0146-0

Cited by 9 publications

(4 citation statements)

References 15 publications

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“…is the adiabatic speed of sound. Moreover, it follows straightforward that the eigenvectors of the Jacobian matrix A(U) constitute a complete set and therefore the system of balance equations (1-4) is hyperbolic, as expected when the hydrodynamic equations are closed at the Euler level (see [22]). In this case, the single-temperature model for such Euler fluid can be regarded as a principal subsystem of a multi-temperature model in the sense of extended thermodynamics (see [23,24]).…”

Section: Reactive Euler Equations and Their Hyperbolicitymentioning

confidence: 68%

Equilibrium and stability properties of detonation waves in the hydrodynamic limit of a kinetic model

Marques¹,

Soares²,

Bianchi³

et al. 2015

J. Phys. A: Math. Theor.

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A shock wave structure problem, as the one which can be formulated for the planar detonation wave, is analyzed here for a binary mixture of ideal gases undergoing the symmetric reaction A 1 + A 1 A 2 + A 2 . The problem is studied at the hydrodynamic Euler limit of a kinetic model of the reactive Boltzmann equation. The chemical rate law is deduced in this frame with a second-order reaction rate, in a chemical regime such that the gas flow is not far away from the chemical equilibrium. The caloric and the thermal equations of state for the specific internal energy and temperature are employed to close the system of the balance laws. With respect to other approaches known in the kinetic literature for detonation problems with a reversible reaction, this paper is addressed to improve some aspects of the wave solution. Within the mathematical analysis of the detonation model, the equation of the equilibrium Hugoniot curve of the final states is explicitly derived for the first time and used to define the correct location of the equilibrium Chapman-Jouguet point in the Hugoniot diagram. The parametric space is widened to investigate the response of the detonation solution to the activation energy of the chemical reaction. At last, the mathematical formulation of the linear stability problem is given for the wave detonation structure via a normal-mode approach, when bidimensional disturbances perturb the steady solution. The stability equations with their boundary conditions and the radiation condition of the considered modeling are explicitly derived for small transversal deviations of the shock wave location. The paper shows how a second-order chemical kinetics description, derived at the microscopic level, and an analytic deduction of the equilibrium Hugoniot curve, lead to an accurate picture of the steady detonation with reversible reaction, as well as to a proper bidimensional linear stability analysis.

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Section: Reactive Euler Equations and Their Hyperbolicitymentioning

confidence: 68%

Equilibrium and stability properties of detonation waves in the hydrodynamic limit of a kinetic model

Marques¹,

Soares²,

Bianchi³

et al. 2015

J. Phys. A: Math. Theor.

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show abstract

“…Instead, in the case of a ST system, the evaluation of the velocities is very difficult also in an equilibrium state due to the fact that the characteristic polynomial is, in general, irreducible (see e.g. [3,23]), but thanks to the subcharacteristic property (41) of principal subsystems, we are able now to establish the following lower and upper bound for the characteristic velocities of the ST model: (7) is a particular case of a system of balance laws (8) and it is dissipative due to the presence of the productions that satisfy the entropy principle. Moreover, we have verified that h 0 is a convex function of the densities u ≡ F 0 .…”

Section: Characteristic Velocities and Their Upper Bound In The St Modelmentioning

confidence: 99%

“…T 1 = T 2 = T and = 0, one obtains T (c) ≡ T from (51) 2 , while (51) 3 is reduced to the average ratio of specific heats introduced in [23]. Hence, the mixture now appears as a single fluid but besides the usual fields ( , v, T ), we have also the extended new ones (c, J, ).…”

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confidence: 92%

On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single‐ and multi‐temperature models

Ruggeri

Simić

2006

Math Methods in App Sciences

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SUMMARYThe first rational model of homogeneous mixtures of fluids was proposed by Truesdell in the context of rational thermodynamics. Afterwards, two different theories were developed: one with a single-temperature (ST) field of the mixture and the other one with several temperatures. The two systems are from the mathematical point of view completely different and the relationship between their solutions was not clarified.In this paper, the hyperbolic multi-temperature (MT) system of a mixture of Eulerian fluids will be explained and it will be shown that the corresponding single-temperature differential system is a principal subsystem of the MT one. As a consequence, the subcharacteristic conditions for characteristic speeds hold and this gives an upper-bound esteem for pulse speeds in an ST model. Global behaviour of smooth solutions for large time for both systems will also be discussed through the application of the ShizutaKawashima condition. Finally, as an application, the particular case of a binary mixture is considered.

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“…Instead, in the case of a ST system the evaluation of the velocities is very difficult also in an equilibrium state due to the fact that the characteristic polynomial is, in general, irreducible (see e.g. [2], [15]) but thanks to the subcharacteristic property (18) of principal subsystems we are able now to establish the following lower and upper bound for the characteristic velocities of the ST model:…”

Section: Characteristic Velocities and Their Upper Bound In The St Modelmentioning

confidence: 99%

Multi-Temperature Mixture of Fluids

Ruggeri¹,

Sugiyama²

2015

Rational Extended Thermodynamics Beyond the Monatomic Gas

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We present a survey on some recent results concerning the different models of a mixture of compressible fluids. In particular we discuss the most realistic case of a mixture when each constituent has its own temperature (MT ) and we first compare the solutions of this model with the one with a unique common temperature (ST ). In the case of Eulerian fluids it will be shown that the corresponding (ST ) differential system is a principal subsystem of the (MT ) one. Global behavior of smooth solutions for large time for both systems will also be discussed through the application of the Shizuta-Kawashima condition.Then we introduce the concept of the average temperature of mixture based upon the consideration that the internal energy of the mixture is the same as in the case of a single-temperature mixture. As a consequence, it is shown that the entropy of the mixture reaches a local maximum in equilibrium. Through the procedure of Maxwellian iteration a new constitutive equation for non-equilibrium temperatures of constituents is obtained in a classical limit, together with the Fick's law for the diffusion flux.Finally, to justify the Maxwellian iteration, we present for dissipative fluids a possible approach of a classical theory of mixture with multi-temperature and we prove that the differences of temperatures between the constituents imply the existence of a new dynamical pressure even if the fluids have a zero bulk viscosity.

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Nonlinear wave propagation in binary mixtures of Euler fluids

Cited by 9 publications

References 15 publications

Equilibrium and stability properties of detonation waves in the hydrodynamic limit of a kinetic model

Equilibrium and stability properties of detonation waves in the hydrodynamic limit of a kinetic model

On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single‐ and multi‐temperature models

Multi-Temperature Mixture of Fluids

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