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Asymptotic Behavior of Riemann and Riemann with Structure Problems for a 2x2 Hyperbolic Dissipative System
Abstract
We present a simple 2 × 2 hyperbolic dissipative system that has many features in common with systems of Extended Thermodynamics. Trough numerical experiments we validate the Brini–Ruggeri conjecture, according to which the Riemann and the Riemann with structure problems converge, for large time, to a combination of shock structures (with or without sub-shocks) and rarefactions of the equilibrium subsystem.
... In this section, we solve the system (3) numerically to check the theoretical consideration classified into four regions. By using the Uniformly accurate Central Scheme of order 2 [30], we perform numerical calculations on the shock structure obtained after long time for the Riemann problem in which two equilibrium states u 0 and u I satisfying (10) and (11) are connected at x ¼ 0. This strategy to solve PDE system (3) by using a sort of the Riemann solver [31] instead of the ODE system (9) is based on the conjecture about the large-time behavior of the Riemann problem and the Riemann problem with structure [32,33] for a system of balance laws proposed by Ruggeri and coworkers [34][35][36] -following an idea of Liu [37]. For more details of the conjecture, please see the papers [34][35][36] and the survey [27]. ...
... By using the Uniformly accurate Central Scheme of order 2 [30], we perform numerical calculations on the shock structure obtained after long time for the Riemann problem in which two equilibrium states u 0 and u I satisfying (10) and (11) are connected at x ¼ 0. This strategy to solve PDE system (3) by using a sort of the Riemann solver [31] instead of the ODE system (9) is based on the conjecture about the large-time behavior of the Riemann problem and the Riemann problem with structure [32,33] for a system of balance laws proposed by Ruggeri and coworkers [34][35][36] -following an idea of Liu [37]. For more details of the conjecture, please see the papers [34][35][36] and the survey [27]. This conjecture was tested numerically for a Grad 13-moment system and a mixture of fluids [22,34,38] and was verified its usefulness in a simple 2 Â 2 dissipative models [19,20,36]. ...
... For more details of the conjecture, please see the papers [34][35][36] and the survey [27]. This conjecture was tested numerically for a Grad 13-moment system and a mixture of fluids [22,34,38] and was verified its usefulness in a simple 2 Â 2 dissipative models [19,20,36]. All numerical simulations confirm the theoretical results and will be published elsewhere. ...
... Dissipative quasi-linear hyperbolic systems compatible with an entropy principle have been widely studied and, recently, important results have been obtained on existence and uniqueness of smooth solutions for systems of hyperbolic-parabolic type [9][10][11]. Moreover, the large-time behavior of the solution of the Riemann problem for such systems has been investigated numerically [12][13][14]. In particular, thanks to the general 0096-3003/$ -see front matter Ó 2008 Elsevier Inc. ...
... 10.007 results due to Liu [15,16] for systems of conservation laws, in [12] a conjecture states, and some numerical results confirm it in the context of the extended thermodynamics, that the solution of the Riemann problem for such systems converges in time to the one of their equilibrium subsystems. In [13,14] different cases confirming this conjecture have been studied. In [17], the steady detonation problem for slow and fast chemical reactions has been faced and, by numerical simulations, it was shown that the solution in the case of slow reaction converges to the one for fast reaction. ...
... which gives the mass action law at chemical equilibrium. Now there is no trace at all of collision contributions in the hydrodynamic equations, and the physical scenario at the kinetic level is more restrictive, in the sense that velocity distribution functions are still Maxwellians, but parameters are additionally bound together by the mass action law (13), which reduces independent variables from 8 to 7, in three space dimensions, and thus from 6 to 5 in our one-dimensional (in space) setting. It is immediately clear that one can perform in turn asymptotic analysis of the set (6)-(9) when its collision operator becomes dominant (limiting case c 34 12 ! ...
This paper deals with shock propagation features in a gas mixture undergoing reversible bimolecular reactions, governed by suitable closures at Euler level of Boltzmann-type equations. Slow and fast chemical processes are considered. At macroscopic level, the slow case is described by a set of balance laws, whereas the fast one yields a set of conservation equations. Within the framework of hierarchies of hyperbolic systems, it is possible to prove that the system governing fast reactions is an equilibrium subsystem of the one describing slow reactions, and then to show how the solutions of the slow system converge to those of the fast system, in case of steady shock problems as well as of Riemann problems.
... By extending the conjecture of Liu [43] given for a 2 ⇥ 2 system, Ruggeri and coworkers [44][45][46] proposed a conjecture on the large-time behavior of the Riemann problems with and without structure for a system of balance laws: First, we study the Riemann problem for the equilibrium subsystem and obtain the usual result, i. e., a combination of ideal shocks, contact shocks and rarefaction waves. According to this conjecture, the solutions of both Riemann problems with and without structure converge to the solutions for large time in which the ideal shock waves are replaced by the shock wave structures (with and without sub-shocks) of the full system while the rarefaction waves and contact shocks are the same as the ones of the equilibrium subsystem. ...
... According to this conjecture, the solutions of both Riemann problems with and without structure converge to the solutions for large time in which the ideal shock waves are replaced by the shock wave structures (with and without sub-shocks) of the full system while the rarefaction waves and contact shocks are the same as the ones of the equilibrium subsystem. The numerical tests support this conjecture for several physical systems of balance laws [44,45,47] and in particular was numerically proved in a toy model proposed by Mentrelli and Ruggeri [46]. ...
... Starting from a toy model proposed in [46], in order to understand these problematic results, it was considered the following 2 ⇥ 2 system [52]: ...
We present the state of the art of the mathematical theory of shock waves for hyperbolic systems. We start with a brief review of ideal shock waves discussing, in particular, the Riemann problem and the phase transition induced by shock waves in real gases. Then we consider dissipative systems and summarise the results concerning the behaviour of the shock thickness for increasing Mach number. In the last part, we present the framework of Rational Extended Thermodynamics (RET) of nonequilibrium rarefied gas and its theoretical predictions of shock waves in cases of both monatomic and polyatomic gases. Particular emphasis will be given to subshock formation and the related open problem.
... The field equations derived from the framework of 2 × 2 dissipative hyperbolic system of balance laws proposed by Mentrelli and Ruggeri [19] are given by ...
... It should be emphasized that the system (1) satisfies all the requirement of rational extended thermodynamics. In fact, any solution of the balance equations (1) satisfies the entropy inequality [19]: ...
... As we see from (3) 1 , the convexity of the entropy density h with respect to the field (u, v) T is automatically satisfied. The sub-characteristic conditions [20] also holds [18,19]. Moreover, the Shizuta-Kawashima condition [21,22] is satisfied when [18,19] ...
For a generic hyperbolic system of balance laws, the shock-structure solution is not continuous and a discontinuous part (sub-shock) arises when the velocity of the front s is greater than a critical value. In particular, for systems compatible with the entropy principle, continuous shock-structure solutions cannot exist when s is larger than the maximum characteristic velocity evaluated in the unperturbed state \(s >\lambda ^{\max }_0 \). This is the typical situation of systems of Rational Extended Thermodynamics (ET). Nevertheless, in principle, sub-shocks may exist also for s smaller than \(\lambda ^{\max }_0 \). This was proved with a simple example in a recent paper by Taniguchi and Ruggeri (Int J Non-Linear Mech 99:69, 2018). In the present paper, we offer another simple case that satisfies all requirements of ET, that is, the entropy inequality, convexity of the entropy, sub-characteristic condition and Shizuta-Kawashima condition, however, there exists a sub-shock with s slower than \(\lambda ^{\max }_0 \). Therefore there still remains an open question which other property makes the systems coming from ET have this beautiful property that the sub-shock exists only for s greater than the unperturbed maximum characteristic velocity.
... In order to obtain the shock-structure solution also for large Mach number, in the present analysis, instead of solving the ODE system (3), we use a different procedure solving ad hoc Riemann problem for the PDE system (14) according with the conjecture about the largetime behavior of the Riemann problem and the Riemann problem with structure [26,27] for a system of balance laws proposed by Ruggeri and coworkers [28][29][30]-following an idea of Liu [31]. According to this conjecture, the solutions of both Riemann problems with and without structure, for large time, instead to converge to the corresponding Riemann problem of the equilibrium sub-system (i.e combination of shock and rarefaction waves), converge to solutions that represent a combination of shock structures (with and without sub-shocks) of the full system and rarefactions waves of the equilibrium subsystem. ...
... This strategy was adopted in several shock phenomena of ET [2]. In particular the conjecture was tested numerically for a Grad 13-moment system and a mixture of fluids [28,33] and was verified in a simple 2 ù 2 dissipative model considered by Mentrelli and Ruggeri [30] for which it is possible to calculate the shock structures of the full system and the rarefactions of the equilibrium subsystem analytically. ...
... Let us consider the following 2 ù 2 dissipative hyperbolic system of balance laws proposed by Mentrelli and Ruggeri [30]: ...
In hyperbolic dissipative systems, the solution of the shock structure is not always continuous and a discontinuous part (sub-shock) appears when the velocity of the shock wave is greater than a critical value. In principle, the sub-shock may occur when the shock velocity $s$ reaches one of the characteristic eigenvalues of the hyperbolic system. Nevertheless, Rational Extended Thermodynamics (ET) for a rarefied monatomic gas predicts the sub-shock formation only when $s$ exceeds the maximum characteristic velocity of the system evaluated in the unperturbed state $\lambda^{\max}_0$. This fact agrees with a general theorem asserting that continuous shock structure cannot exist for $s >\lambda^{\max}_0 $. In the present paper, first, the shock structure is numerically analyzed on the basis of ET for a rarefied polyatomic gas with $14$ independent fields. It is shown that, also in this case, the shock structure is still continuous when $s$ meets characteristic velocities except for the maximum one and therefore the sub-shock appears only when $s >\lambda^{\max}_0 $. This example reinforces the conjecture that, the differential systems of ET theories have the special characteristics such that the sub-shock appears only for $s$ greater than the unperturbed maximum characteristic velocity. However, in the second part of the paper, we construct a counterexample of this conjecture by using a simple $2 \times 2$ hyperbolic dissipative system which satisfies all requirements of ET. In contrast to previous results, we show the clear sub-shock formation with a slower shock velocity than the maximum unperturbed characteristic velocity.
... Therefore, the shock structure solution for a chosen value of s may exhibit more than one stable sub-shock; the number of sub-shocks arising within the shock structure depends on how many characteristic speeds λ (k) + , or critical values s cr k are exceeded by s. In [3][4][5][6][7], Ruggeri and coauthors studied the Riemann problem, classical and with structure, for a dissipative hyperbolic system focusing on the long time behaviour of the solution [8]. Those investigations led to the conjecture that the solution converges to a combinations of shock-structures (with or without sub-shocks), contact waves and rarefactions of the equilibrium sub-system [4]. ...
... A dissipative system of the form (1) of wide interest as that modelling a one-dimensional multi-temperature inert binary mixture of gases in the framework of Extended Thermodynamics [9,10] will be discussed in the remaining of this paper. This choice is motivated by the simplicity of the system, which allows for a theoretical treatment in addition to the numerical one, as well as by the fact that many results concerning shock wave structure solutions for this system exist [4][5][6][7][11][12][13][14][15][16][17]. Among the others, substantial contributions to the study of shock structures in hyperbolic dissipative systems may be found in [18][19][20][21][22]; recent results for a reacting gas mixture are presented in [23][24][25]. ...
... Since we will analyze the sub-shock formation for s > μ (4) + , it can be easily seen that the only characteristic velocities of interest in the following are λ (5) + and λ (6) + . ...
The problem of sub-shock formation within a shock structure solution of hyperbolic systems of balance laws is investigated for a binary mixture of multi-temperature Eulerian fluids. The main purpose of this work is the analysis of the ranges of Mach numbers characterizing shock-structure solutions with different features, continuous or not, and to show the existence of ranges, below the maximum unperturbed characteristic velocity, for which each constituent of the mixture may develop a sub-shock within a smooth shock structure profile. The theoretical results are supported by numerical calculations.
... A general theory of the RP for hyperbolic homogeneous 19 systems was developed by Lax in the fundamental paper [5]. He proved that, under 20 the hypotesis of "small" jump for the initial states, an unique solution of the RP exists 21 and it is determined by constant states separated by shock waves, rarefaction waves 22 and/or contact discontinuities. In particular, rarefaction waves are smooth solutions with 23 discontinuities in the first derivatives characterised by the well known exact solutions 24 called simple waves. ...
In this paper, within the framework of the Method of Differential Constraints, the celebrated p-system is studied. All the possible constraints compatible with the original governing system are classified. In solving the compatibility conditions between the original governing system and the appended differential constraint, several model laws for the pressure p(v) are characterised. Therefore, the analysis developed in the paper has been carried out in the case of physical interest where p=p0v−γ, and an exact solution that generalises simple waves is determined. This allows us to study and to solve a class of generalised Riemann problems (GRP). In particular, we proved that the solution of the GRP can be discussed in the (p,v) plane through rarefaction-like curves and shock curves. Finally, we studied a Riemann problem with structure and we proved the existence of a critical time after which a GRP is solved in terms of non-constant states separated by a shock wave and a rarefaction-like wave.
... In Section 3.3, the general Riemann problem is discussed and its general solution is determined making use of the Hugoniot loci and of the integral curves calculated, respectively, in Section 3.1 and Section 3.2. In order to check the theoretical results, the numerical solution of the Riemann problem is numerically calculated both in the case of water and in the incompressible limit with the aid of a general purpose code developed by the Authors [14]. The numerical results are in perfect agreement with the theoretical ones. ...
The aim of the present paper is to investigate shock and rarefaction waves in a hyperbolic model of incompressible fluids. To this aim, we use the so-called extended-quasi-thermal-incompressible (EQTI) model, recently proposed by Gouin and Ruggeri (H. Gouin, T. Ruggeri, International Journal of Non-Linear Mechanics 47 (2012) 688–693). In particular, we use as constitutive equation a variant of the well-known Boussinesq approximation in which the specific volume depends not only on the temperature but also on the pressure, leading to a hyperbolic system of differential equations. The limit case of ideal incompressibility, namely when the thermal expansion coefficient and the compressibility factor vanish, is also considered.
The results show that the propagation of shock waves in an EQTI fluid is characterized by small jump in specific volume and temperature, even when the jump in pressure is relevant, and rarefaction waves originating from a general Riemann problem are characterized by a very steep profile. The knowledge of the loci of the states that can be connected to a given state by a shock wave or a rarefaction wave allows also to completely solve the Riemann problem. The obtained results are confirmed by means of numerical calculations.
The shock structure in a binary mixture of polyatomic Eulerian gases with different degrees of freedom of a molecule is studied based on the multi-temperature model of rational extended thermodynamics. Since the system of field equations is hyperbolic, the shock-structure solution is not always regular, and discontinuous parts (sub-shocks) can be formed. For given values of the mass ratio and the specific heats of the constituents, we identify the possible sub-shocks as the Mach number $M_0$ of the shock wave and the concentration $c$ of the constituents change. In the plane $(c,M_0)$, we identify the possible regions for the sub-shock formation. The analysis is obtained to verify when the velocity of the shock wave meets a characteristic velocity in the unperturbed or perturbed equilibrium states that gives a necessary condition for the sub-shock formation. The condition becomes necessary and sufficient when the velocity of the shock becomes greater than the maximum characteristic velocity in the unperturbed state. Namely, the regions with no sub-shocks, a sub-shock for only one constituent, or sub-shocks for both constituents are comprehensively classified. The most interesting case is that the lighter molecule has more degrees of freedom than that of the heavy one. In this situation, the topology of the various regions becomes different. We also solve the system of the field equations numerically using the parameters in the various regions and confirm whether the sub-shocks emerge or not. Finally, the relationship between an acceleration wave in a constituent and the sub-shock in the other constituent is explicitly derived.
We test for a 2 × 2 hyperbolic dissipative system, by numerical experiments, the conjecture according to which the solutions
of Riemann problem and Riemann problem with structure converge, for large time, to a combination of shock structures (with
or without subshocks) and rarefactions of the equilibrium subsystem.
A complete classification of shock waves in a van der Waals fluid is undertaken. This is in order to gain a theoretical understanding of those shock-related phenomena as observed in real fluids which cannot be accounted for by the ideal gas model. These relate to admissibility of rarefaction shock waves, shock-splitting phenomena, and shock-induced phase transitions. The crucial role played by the nature of the gaseous state before the shock (the unperturbed state), and how it affects the features of the shock wave are elucidated. A full description is given of the characteristics of shock waves propagating in a van der Waals fluid. The strength of these shock waves may range from weak to strong. The study is carried out by means of the theory of hyperbolic systems supported by numerical calculations.
We study the interaction between a shock and an acceleration wave in an Euler fluid satisfying the ideal gas law, following the general theory developed by Boillat and Ruggeri [G. Boillat, T. Ruggeri, Reflection and transmission of discontinuity waves through a shock wave. General theory including also the case of characteristic shocks, Proc. Roy. Soc. Edinburgh 83A (1979) 17–24]. Special attention is devoted to analyzing the effects of varying the shock strength on the jump in the shock acceleration and on the amplitudes of the reflected and/or transmitted waves, for both weak and strong shock conditions. Our analysis confirms that, for a weak shock, the jump in the shock acceleration vanishes only when the incident wave belongs to the same family as the shock. Numerical calculations have also been performed and the numerical results are in perfect agreement with those obtained by application of the theory. Moreover, the numerical results, at variance with the theory, allow to gather information about the evolution of the solution after the impact time.
This book is dedicated to the recent developments in RET with the aim to explore polyatomic gas, dense gas and mixture of gases in non-equilibrium. In particular we present the theory of dense gases with 14 fields, which reduces to the Navier-Stokes Fourier classical theory in the parabolic limit. Molecular RET with an arbitrary number of field-variables for polyatomic gases is also discussed and the theory is proved to be perfectly compatible with the kinetic theory in which the distribution function depends on an extra variable that takes into account a molecules internal degrees of freedom. Recent results on mixtures of gases with multi-temperature are presented together with a natural definition of the average temperature. The qualitative analysis and in particular, the existence of the global smooth solution and the convergence to equilibrium are also studied by taking into account the fact that the differential systems are symmetric hyperbolic. Applications to shock and sound waves are analyzed together with light scattering and heat conduction and the results are compared with experimental data. Rational extended thermodynamics (RET) is a thermodynamic theory that is applicable to non-equilibrium phenomena. It is described by differential hyperbolic systems of balance laws with local constitutive equations. As RET has been strictly related to the kinetic theory through the closure method of moment hierarchy associated to the Boltzmann equation, the applicability range of the theory has been restricted within rarefied monatomic gases. The book represents a valuable resource for applied mathematicians, physicists and engineers, offering powerful models for potential applications like satellites reentering the atmosphere, semiconductors and nano-scale phenomena. © Springer International Publishing Switzerland 2015. All rights reserved.
The structure of a shock wave in a rarefied polyatomic gas is studied on the basis of the theory of extended thermodynamics. Three types of the shock wave structure observed in experiments, that is, the nearly symmetric shock wave structure (type A, small Mach number), the asymmetric structure (type B, moderate Mach number), and the structure composed of thin and thick layers (type C, large Mach number), are explained by the theory in a unified way. The theoretical prediction of the profile of the mass density agrees well with the experimental data. The well-known Bethe-Teller theory of the shock wave structure in a polyatomic gas is reexamined in the light of the present theory.
In this paper we investigate the basic features of shock waves propagation in freshwater in the framework of a hyperbolic model consisting of the one-dimensional Euler equations closed by means of polynomial equations of state extracted from experimental tabulated data available in the literature (Sun et al. in Deep-Sea Res. I 55:1304–1310, 2008). The Rankine–Hugoniot equations are numerically solved in order to determine the Hugoniot locus representing the set of perturbed states that can be connected through a k-shock to an unperturbed state.
The results are found to be consistent with those previously obtained in the framework of the EQTI model by means of a modified Boussinesq equation of state.
The aim of the present paper is to investigate shock and rarefaction waves in a hyperbolic model of incompressible materials. To this aim, we use the so-called extendedquasi- thermal-incompressible (EQTI) model, recently proposed by Gouin and Ruggeri [H. Gouin, T. Ruggeri, Internat. J. Non-Linear Mech. 47 688-693 (2012)]. In particular, we use as constitutive equation a variant of the well-known Bousinnesq approximation in which the specific volume depends not only on the temperature but also on the pressure. The limit case of ideal incompressibility, namely when the thermal expansion coefficient and the compressibility factor vanish, is also considered.
Wave propagation phenomena give us an important mean to check the validation of the nonequilibrium thermodynamics theory. In this chapter, we present a short review on the modern theory of wave propagation for hyperbolic systems. Firstly, we present the theory of linear waves emphasizing the role of the dispersion relation. The high frequency limit in the dispersion relation is also studied. Secondly, nonlinear acceleration waves are discussed together with the transport equation and the critical time. Thirdly we present the main results concerning shock waves as a particular class of weak solutions and the admissibility criterion to select physical shocks (Lax condition, entropy growth condition, and Liu condition). The chapter finishes with the discussion of traveling waves, in particular, shock waves with structure. The sub-shock formation is particularly interesting. The Riemann problem and the large time asymptotic behavior are also discussed.
The problem of sub-shock occurrence within a shock structure solution is investigated for an inert binary mixture of monoatomic gases, modelled by a Grad 10-moment approximation of the Boltzmann equations. The main purpose of this paper is to show by numerical simulations the existence of discontinuous shock structure solutions for values of the shock speed below the maximum unperturbed characteristic velocity. Moreover, for suitable concentrations of the two species, and for shock velocities beyond the maximum unperturbed characteristic velocity, each constituent of the mixture generates a jump discontinuity, and the shock structure solution exhibits two sub-shocks.
Wave propagation phenomena give us an important mean to check the validation of a nonequilibrium thermodynamics theory. In this chapter, we present a short review on the modern theory of wave propagation for hyperbolic systems. Firstly, we present the theory of linear waves emphasizing the role of the dispersion relation. The high frequency limit in the dispersion relation is also studied. Secondly, nonlinear acceleration waves are discussed together with the transport equation and the critical time. Thirdly, we present the main results concerning shock waves as a particular class of weak solutions and the admissibility criterion to select physical shocks (Lax condition, entropy growth condition, and Liu condition). Fourth, we discuss traveling waves, in particular, shock waves with structure. The subshock formation is particularly interesting. The Riemann problem and the problem of the large-time asymptotic behavior are also discussed. Lastly, we present toy models to show explicitly some interesting features obtained here.
Shock waves and shock-induced phase transitions are theoretically and numerically studied on the basis of the system of Euler equations with caloric and thermal equations of state for a system of hard spheres with internal degrees of freedom. First, by choosing the unperturbed state (the state before the shock wave) in the liquid phase, the Rankine-Hugoniot conditions are studied and their solutions are classified on the basis of the phase of the perturbed state (the state after the shock wave), being a shock-induced phase transition possible under certain conditions. With this regard, the important role of the internal degrees of freedom is shown explicitly. Second, the admissibility (stability) of shock waves is studied by means of the results obtained by Liu in the theory of hyperbolic systems. It is shown that another type of instability of a shock wave can exist even though the perturbed state is thermodynamically stable. Numerical calculations have been performed in order to confirm the theoretical results in the case of admissible shocks and to obtain the actual evolution of the wave profiles in the case of inadmissible shocks (shock splitting phenomena).
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