aa r X i v : . [ m a t h . A P ] F e b The system of ionized gas dynamics
Fumioki ASAKURA ∗ Andrea CORLI † July 21, 2018
Abstract
The aim of this paper is to study a system of three equations for ionized gas dynamics at hightemperature, in one spatial dimension. In addition to the mass density, pressure and particle velocity,a further quantity is needed, namely, the degree of ionization. The system is supplemented by thefirst and second law of thermodynamics and by an equation of state; all of them involve the degreeof ionization. At last, under the assumption of thermal equilibrium, the system is closed by requiringSaha’s ionization equation.The geometric properties of the system are rather complicated: in particular, we prove the loss ofconvexity (genuine nonlinearity) for both forward and backward characteristic fields, and hence theloss of concavity of the physical entropy. This takes place in a small bounded region, which we areable to characterize by numerical estimates on the state functions. The structure of shock waves isalso studied by a detailed analysis of the Hugoniot locus, which will be used in a forthcoming paperto study the shock tube problem.
Key words and phrases:
Systems of conservation laws, ionized gas, Hugoniot locus.
The thermodynamical variables temperature, specific entropy, specific internal energy, pressure, specificvolume and the velocity of the gas are denoted in this paper by T , S , e , p , v and u , respectively. Forbrevity we simply refer to S and e as the entropy and internal energy, respectively. In the simplest (non-ionized) thermodynamical system, an equation of state relating the variables p , v and T , for example, isassigned. Then, the knowledge of any two of them allows to fully determine the state of the system, sincethe other variables are determined by the first and second law of thermodynamics T d S = d e + p d v. (1.1)However, if the gas is partly ionized as in this paper, further details must be taken into account. Ionizationis a process where an atom or a molecule becomes charged due to the loss or gain of electrons. When agas is heated to a high temperature, at first almost all of its molecules dissociate and it behaves like amonatomic gas. However, if the gas is heated to an even higher temperature, of the order of thousands ofdegrees Kelvin, some of its atoms become ionized according to the first ionization reaction X → X + + e − ,where X is an atom of the monatomic gas, X + an ion and e − an electron. In the case of hydrogen, atthis level the gas is made up by atoms, protons and electrons; in heavier atoms the first ionization canbe followed by a second ionization and so on. For simplicity we focus on the case of a gas that undergoesonly one ionization and denote the concentration (number per unit volume) of atoms, ions and electronsby n a , n i and n e , respectively. ∗ Research Center for Physics and Mathematics, Osaka Electro-Communication University, Neyagawa, Osaka, Japan,e-mail: [email protected] † Department of Mathematics and Computer Science, University of Ferrara, Via Machiavelli 30, 40121 Ferrara, Italy,e-mail: [email protected]
1t any given (high) temperature T, this ionization reaction reaches a state of thermal equilibriumanalogous to the chemical equilibrium for ordinary chemical reaction. We may assume that the ratio n i n e n a depends only on the temperature T ; an actual formula was derived in 1920 by M. Saha [9], see also[2, 4, 11, 12]. The state of an ionized gas depends on a further variable which measures the level ofionization, namely, the degree of ionization α = n e n a + n i . However, by means of Saha’s formula, the degree of ionization can be expressed as a function of thedensity ρ and T , and this makes consistent the formulation of the model. Of course, the expressions ofthe thermodynamical variables depend on the degree of ionization; for instance, in the case of a monatomicgas, the pressure of the ionized gas can be written as p = (1 + α ) Rm ρT, where R is the universal gas constant and m the molecular mass of the gas. In the non-ionized case α = 0we recover the usual expression for the pressure; if α = 0 the gas is not ideal [10].The aim of this paper is to study the mathematical properties of the gasdynamics system ρ t + ( ρu ) x = 0 , ( ρu ) t + ( ρu + p ) x = 0 , ( ρE ) t + ( ρuE + pu ) x = 0 , (1.2)for an ionized gas; here, E = u + e is the (specific) total energy. To the best of our knowledge such astudy has never been done previously in spite of the fact that the system of gasdynamics attracted theinterest of several researchers in the last decade, however mostly for ideal gases [10]; we quote [8] forthe case of real gases. The main motivation for our study was provided by the paper [5] , where theauthors made laboratory experiments about the reflection at an interface of a shock wave in an ionizedgas. In this paper we assume that the gas is monatomic , differently from [5], where a two-component gasis considered. This choice simplifies the analysis of the system while catching however the main featuresof ionization; more general cases (non-monatomic or multicomponent gases) can be considered as wellbut at the price of much heavier computations.After a short review of basic thermodynamics in Section 2 and a first study of the thermodynamicvariables of the model in Section 3, we show in Section 4 that system (1.2) is strictly hyperbolic, as inthe non-ionized case; however, differently from the latter case, the genuine non-linearity of the extremeeigenvalues of the system is lost and we study the set where this happens. We refer to [3, 10] for moreinformation on systems of conservation laws. Our analysis is completed by a detailed investigation of theHugoniot locus of the system in Section 5 and by a study of the entropy increase in Section 6. Moreover,a study of the integral curves in provided in Section 7. At last, in Section 8 we study a related model,which is deduced by an approximation of Saha’s law at very high temperatures; in this case, the systemdoes not loose the genuine nonlinearity property.In a second forthcoming paper we exploit the results obtained in this paper to study the shock reflectionin an electromagnetic shock tube (T-tube) with a reflector. We thus provide a rigorous mathematicalbasis to the physical phenomena observed in [5]. The equation of state of an ideal gas can be written as [4, (6)] p = ρm p kT = n p kT, (2.1) An English translation of [5] is available upon request to F. Asakura. m p is the particle mass, n p the total number of particles per unit volume and k is Boltzmann’sconstant; we clearly have ρ = m p n p . The molecular mass is defined as m = N m p , (2.2)where N is Avogadro number. If R = kN is the universal gas constant, then (2.3) becomes [4, (8)] p = Rm ρT, (2.3)or p = M RρT by introducing the number M = m of moles. We shall often use p and T as independentthermodynamical variables; in order to compute the other variables we introduce the (specific) enthalpy H = e + pv . Using the enthalpy equation (1.1) becomes dH = T dS + vdp. (2.4)We also introduce the Gibbs function G = H − T S , see [4, (111)].By (1.1) we deduce dG = v dp − S dT ,whence (cid:18) ∂G∂p (cid:19) T = v, (cid:18) ∂G∂T (cid:19) p = − S. (2.5)As usual, a subscript as T or p above means that the derivative is computed when that variable is fixed.By (2.5) we deduce the compatibility condition (cid:18) ∂v∂T (cid:19) p = − (cid:18) ∂S∂p (cid:19) T , (2.6)which is one of the so-called Maxwell reciprocity relations. In turn, by (2.5) and (2.6) we obtain (cid:18) ∂H∂p (cid:19) T = T (cid:18) ∂S∂p (cid:19) T + v = − T (cid:18) ∂v∂T (cid:19) p + v, (2.7) (cid:18) ∂H∂T (cid:19) p = T (cid:18) ∂S∂T (cid:19) p . (2.8) Example 2.1 (Polytropic gas) . In the case of a polytropic gas, i.e., an ideal gas whose specific heat isconstant, we have [4, (29)] e = C v T, (2.9) where C v is the specific heat at constant volume [4, (34)]. We also define the specific heat at constantpressure C p = Rm + C v . Then, e = C v T and H = C p T . The adiabatic constant is γ = C p C v = 1 + RmC v [4,(37)]. We conclude by computing the entropy S . By (1.1) we deduce (cid:18) ∂S∂e (cid:19) v = 1 T , (cid:18) ∂S∂v (cid:19) e = pT , and then S = C p log T − Rm log p + Const by (2.3) and (2.9) . By introducing the non-dimensional entropy η = mR S we easily find p = a γ e ( γ − η v − γ , v = a p − γ e γ − γ η , T = a p γ − γ e γ − γ η (2.10) where a = Rm . (2.11)
In the case of a monatomic gas, we have C v = a , C p = a and γ = [4, (34) and (37)]. Then, e = 32 a T, H = 52 a T, S = 52 a log T − a log p + Const . (2.12)3ow, we briefly discuss ionization; to avoid multiple references we mainly refer to [2] but analogousexpressions can be easily found in [4, 11, 12]. Saha’s equation provides the degree of ionization of an atomicelement as a function of temperature and electron density, under assumptions of thermal equilibrium. Ina general form it can be written [2, (4.15)] n r +1 n e n r = G r +1 g e G r (2 πm e kT ) h e − χrkT . (2.13)Here n r and n r +1 are the number densities of atoms in the ionization state r (where r electrons aremissing) and the ionization state r + 1 (where r + 1 electrons are missing) of a given element, n e isthe electron number density, G r and G r +1 are the partition functions of the two states, g e = 2 is thestatistical weight of the electron, m e is the electron mass, h is the Planck constant, k is the Boltzmannconstant, χ r is the ionization potential from state r to state r + 1. We also denote the degree of ionizationby α = n e n a + n i . (2.14)As we mentioned in the Introduction, we consider the case r = 0 of a single ionization from a neutralstate of a monatomic gas; then (2.13) becomes n i n e n a = 2 Z i Z a (2 πm e kT ) h e − T i T , (2.15)where we denoted by Z a = G and Z i = G the partition functions of the neutral state and 1-ionizedstate, respectively; T i = χ k is the dissociation energy expressed by the temperature. Moreover, n e = n i and then α = n i n a + n i . The total number of atoms and ions is n p = n a + n i and we can assume that n p = ρ/m p . As a consequence, by (2.1) we deduce the pressure law for an ionized gas, p = ( n a + n i + n e ) kT = (1 + α ) n p kT = (1 + α ) ρm p kT, (2.16)or better, by using the molecular mass (2.2), p = (1 + α ) Rm ρT. (2.17)The equation of state (2.17) expresses α as a function of p , ρ and T . Indeed, as we show in the nextLemma 3.1, α does not depend on ρ and p independently but solely on one of these two variables. As aconsequence, an ionized gas is not an ideal gas, which is characterized instead by the equation of state(2.3). The specific volume v is deduced as v = (1 + α ) RTmp . (2.18)In the current case of an ionized gas, the specific internal energy should include not only kinetic, but alsoionization energy; then e = pρ + χ m p α. But χ m p = kT i N m = Rm T i and then, since the gas is monatomic, e = 32 Rm (1 + α ) T + RT i m α, H = 52 Rm (1 + α ) T + RT i m α. (2.19)We stress that, differently from ideal gases where e = e ( T ), for an ionized gas the specific internal energy e depends on both p and T . The above expressions (2.19) slightly change for non-monatomic gases; weemphasize that most numerical coefficients that occur in the following (as 5 / /
3) are merely due tothis assumption.
In this section we provide several explicit expressions for the thermodynamical variables of an ionizedgas. In the proofs we use from time to time the shorthand notation τ = T i T and q = 52 + τ. (3.1)4irst, we study the properties of the degree α of ionization. By exploiting the notation introduced inthe previous section, we denote κ = h (2 πm e ) k Z a Z i and ¯ κ = m p Z i Z a (2 πm e k ) h . We notice that ¯ κ = m p k κ = mR κ . Lemma 3.1 (Degree of ionization) . The degree α of ionization can be expressed as a function of ρ and T through the formula α − α = ¯ κρ T e − T i T . (3.2) The degree of ionization can also be expressed as a function of p and T by the formula α = (cid:18) κpT − e T i T (cid:19) − . (3.3) Moreover, we have (cid:18) ∂α∂p (cid:19) T = − p α (1 − α ) and (cid:18) ∂α∂T (cid:19) p = 12 T α (1 − α ) (cid:18)
52 + T i T (cid:19) . (3.4) At last, − Tp (cid:18) ∂α∂T (cid:19) p = (cid:18)
52 + T i T (cid:19) (cid:18) ∂α∂p (cid:19) T . (3.5) Proof.
We first notice that n i n e n a = n n a = n p α − α . Therefore, (2.15) can be written as (see [4, (209)], [11, § V.4, (4.9)]) α − α = m p ρ Z i Z a (2 πm e kT ) h e − T i T , which proves (3.2). By (2.16), we deduce [11, §
5, (4.11)] α − α = 2 Z i Z a (2 πm e ) ( kT ) ph e − T i T , (3.6)whence (3.3). To compute (3.4) it is convenient to introduce the notation β ( p, T ) = κpT − e T i T ; by (3.6)we have α = (1 + β ) − , β = 1 − α α . (3.7)Now, we compute (cid:18) ∂α∂p (cid:19) T = − α κT − e T i T = − α p β, (cid:18) ∂α∂T (cid:19) p = − α (cid:18) ∂β∂T (cid:19) p = α βT (cid:18)
52 + T i T (cid:19) , whence (3.4). Identity (3.5) directly follows by (3.4). However, it can also be obtained without computing(3.4): indeed, by (1.1) and (2.6) we deduce (cid:18) ∂H∂p (cid:19) T = T (cid:18) ∂S∂p (cid:19) T + v = − T (cid:18) ∂v∂T (cid:19) p + v. By the above formula, differentiating (2.19) and using (2.18) we deduce (3.5).5 emark 3.1 (Physical values) . We report here some physical values which are commonly used in thefigures below. As we wrote in the Introduction, we focus on the simple case of a hydrogen gas. In thiscase the specific gas constant a is − K − and κ = 29 . . Moreover T i = 1 . × K . Wefrequently use the following values, where α is computed through (3.3) , see [5]: T = 750K , p = 1466 . , α = 3 . × − ,T = 300K , p = 1466 . , α = 3 . × − . Remark 3.2.
The equation of state (2.17) allows to recover easily the equation of state of the non-ionizedcase α = 0 . Lemma 3.1 shows that this is not the case for other thermodynamics variables: if we let α = 0 in (3.2) or (3.3) , then we formally deduce T = 0 (or T = ∞ , or ρ = ∞ , even less physically meaningful).The reason depends on Saha’s formula (2.15) , which allows ionization at any positive temperature. Thisshows that the non-ionized case cannot be deduced as the limit for α → of the ionized one. T in K × α p=1 Torrp=10 Torrp=100 Torr p in Torr α T=0.05 T i KT=0.08 T i KT=T i K Figure 1: The ionization degree α given by (3.3): as a function of the temperature T , left, and of thepressure, right. We refer to Remark 3.1 for the numerical values used here and in the following. As a consequence of Lemma 3.1, the pressure law (2.17) can be written more carefully as p = p ( ρ, T ) = (cid:0) α ( ρ, T ) (cid:1) Rm ρT, (3.8)or as p = p ( α, T ) = 1 κ − α α T e − T i T . (3.9) Lemma 3.2 (Pressure) . We have p ρ ( ρ, T ) = 22 − α Rm T > and p T ( ρ, T ) = 22 − α " α (1 − α ) (cid:18)
52 + T i T (cid:19) Rm ρ > . (3.10) Proof.
Instead of differentiating directly (3.8), by Lemma 3.1 we understand α as a function of p and T in that expression. We denote F ( ρ, T, p ) = (cid:0) α ( p, T ) (cid:1) Rm ρT − p ; by (3.4) we see that F p = (cid:18) ∂α∂p (cid:19) T Rm ρT − −
12 (2 − α )(1 + α ) < . By the Implicit Function Theorem and (3.4) we deduce F ρ = (1 + α ) Rm T, F T = Rm (1 + α ) ρ (cid:20) α (1 − α ) q (cid:21) . Since p ρ = − F ρ F p and p T = − F T F p , we obtain (3.10). 6 emma 3.3 (Internal energy) . If we express the internal energy by means of ρ and T we have e T ( ρ, T ) > . Proof.
By Lemma 3.2 we may express α as a function of ρ and T ; by (3.4) and (3.10) we compute ∂∂T α (cid:0) p ( ρ, T ) , T (cid:1) = α p p T + α T = α (1 − α )(2 − α ) T ( q − . (3.11)Then by (2.19) and (3.11) we deduce e T ( ρ, T ) = Rm (cid:20)
32 (1 + α ) T + T i α (cid:21) T = Rm α (1 − α )(2 − α ) (cid:20) q + q α (1 − α ) (cid:21) , whence e T >
0. For an alternative proof we can directly use expression (3.2).
Proposition 3.1 (Entropy) . The dimensionless entropy η = mR S can be expressed as a function of p and T through the formula η ( p, T ) = − log p + 2 tanh − α + (cid:18)
52 + T i T (cid:19) α + 52 log T + C, (3.12) where C is a constant; we have (cid:18) ∂η∂p (cid:19) T ≤ − p < and (cid:18) ∂η∂T (cid:19) p ≥ T (1 + α ) > . (3.13) Moreover, the dimensionless entropy η can be expressed as a function of α and T by the formula η ( α, T ) = − − αα + (1 + α ) (cid:18)
52 + T i T (cid:19) + C (3.14) and we have (cid:18) ∂η∂α (cid:19) T = 2 α (1 − α ) + (cid:18)
52 + T i T (cid:19) > and (cid:18) ∂η∂T (cid:19) α = − (1 + α ) T i T < . (3.15) Proof.
By Maxwell relation (2.6), (3.5) and (2.8) we find (cid:18) ∂S∂p (cid:19) T = − (cid:18) ∂v∂T (cid:19) p = − Rm " p (1 + α ) + Tp (cid:18) ∂α∂T (cid:19) p (3.16)= Rm " − p (1 + α ) + q (cid:18) ∂α∂p (cid:19) T , (3.17) (cid:18) ∂S∂T (cid:19) p = 1 T (cid:18) ∂H∂T (cid:19) p = Rm " T (1 + α ) + q (cid:18) ∂α∂T (cid:19) p . (3.18)By integrating (3.17) with respect to p we obtain η = − log p − Z αp d p + qα + f ( T ) , (3.19)where f ( T ) is an arbitrary function of T . To compute the integral we introduce the notation r ( T ) = β ( p,T ) p ,see (3.6) and (3.7). By the change of variables 1 + r ( T ) p = y we deduce Z αp d p = Z p p r ( T ) p d p = 2 Z y − y = log y − y + 1 = log 1 − α α = − − α η = − log p + 2 tanh − α + qα + f ( T ). To compute f , we first differentiate (3.19)with respect to T and use (3.5), (3.1) to deduce (cid:18) ∂η∂T (cid:19) p = − Z p (cid:18) ∂α∂T (cid:19) p d p − T i T α + q (cid:18) ∂α∂T (cid:19) p + f ′ ( T ) = 5 α T + q (cid:18) ∂α∂T (cid:19) p + f ′ ( T ) . (3.20)Now, we compare (3.20) with (3.18) and find that f ′ ( T ) =
52 1 T , so that f ( T ) = log T . Then (3.12)follows. By (3.16), (3.4) and by (3.18), (3.5) we deduce (3.13). To prove (3.14), by (3.12) and (3.9) wecompute η = − log p + log 1 + α − α + qα + 52 log T + C = − log (cid:18) − αα (cid:19) + τ + qα + C, whence (3.14). Then, (3.15) follows.Notice that if α = 0, by (3.12) we recover the entropy (2.12). Proposition 3.2 (Temperature) . The temperature T can be globally expressed as a function of p and η .Moreover, the level curves η ( p, T ) = c lying on the plane ( p, T ) are the graphs of functions T c ( p ) of p satisfying ∂T c ∂p = T c p α (1 − α ) (cid:16) + T i T (cid:17) + α (1 − α ) (cid:16) + T i T (cid:17) . (3.21) Proof.
On the level curves η ( p, T ) = c we have d η = (cid:16) ∂η∂p (cid:17) T d p + (cid:16) ∂η∂T (cid:17) p d T = 0; then, on these curves,we have ∂T∂p = − (cid:16) ∂η∂p (cid:17) T (cid:16) ∂η∂T (cid:17) p =: F ( p, T ) , where we omitted for simplicity the subscript c . We notice that F makes sense and F ( p, T ) > we deduce (cid:18) ∂η∂p (cid:19) T = − αp α − α ) (cid:18)
52 + T i T (cid:19)! , (cid:18) ∂η∂T (cid:19) p = 1 + αT
52 + α − α ) (cid:18)
52 + T i T (cid:19) ! , and then (3.21) follows at least locally and defines a strictly increasing function T ( p ). Indeed, we claimthat T is defined for every p >
0. To prove this claim, we first remark that ∂T∂p ∼ Tp both for T → T → + ∞ , because in those cases α → α →
1, respectively. This means that T ∼ Cp for T →
0+ and T → + ∞ , for C an arbitrary positive constant, which proves our claim. The claimalso implies that for every fixed pair ( p, η ) we uniquely find the relative value of T . This proves theproposition. In this section we study the main properties of system (1.2), where p and e are defined by (2.17), (2.19),respectively, with α as in (3.3). System (1.2) can be written in Lagrangian coordinates as [10] v t − u ξ = 0 ,u t + p ξ = 0 , (cid:0) e + u (cid:1) t + ( pu ) ξ = 0 . (4.1)8or C solutions, equation (4.1) can be written as S t = 0 and we obtain v t − u ξ = 0 ,u t + p ξ = 0 ,S t = 0 . (4.2)We notice that equation (4.2) is equivalent in Eulerian coordinates to( ρS ) t + ( ρuS ) x = 0 . (4.3)In the variables ( p, u, S ) we have v t = v p p t and equation (4.2) becomes p t − v p u ξ = 0. In these variables,the Lagrangian characteristic speeds are λ ± = ± √ − v p and λ = 0 , with corresponding characteristicvectors r ± = (cid:0) ± , √− v p , (cid:1) T and r = (0 , , T . Thus we have r ± ∇ λ ± = v pp − v p ) . (4.4)Now, we use p , u and T as state variables. Since v t − u ξ = v p p t + v T T t − u ξ and η t = η p p t + η T T t , wecan also write system (4.2) under the form p t − η T v p η T − v T η p u ξ = 0 ,u t + p ξ = 0 ,T t + η p v p η T − v T η p u ξ = 0 . (4.5)At last, we exploit the equation of state (2.17). Lemma 4.1.
Under (2.17) we have η T v p η T − v T η p = − p (cid:20) + α (1 − α ) (cid:16) + T i T (cid:17) (cid:21) a T (1 + α ) (cid:20) + α (1 − α ) (cid:16) + T i T + T T (cid:17)(cid:21) < . (4.6) Proof.
By (2.18) we compute v p = − RTmp (1 + α ) (cid:20) α (1 − α ) (cid:21) , v T = Rmp (1 + α ) (cid:20) α (1 − α ) q (cid:21) . Moreover, by (2.6) we have η p = − αp (cid:20) α (1 − α ) q (cid:21) , η T = 1 + αT (cid:20)
52 + 12 α (1 − α ) q (cid:21) . (4.7)Then, we obtain v p η T − v T η p = − a (1 + α ) p "
32 + 12 α (1 − α ) (cid:18) q − q + 52 (cid:19) , whence (4.6) follows.We record here, for future reference, that by (4.7) we deduce − η p η T = T (cid:20) α (1 − α ) (cid:16) + T i T (cid:17)(cid:21) p (cid:20) + α (1 − α ) (cid:16) + T i T (cid:17) (cid:21) . (4.8)9y Lemma 4.1 we can introduce the notation λ = λ ( p, T ) = − η T v p η T − v T η p , (4.9)with λ >
0. Notice that if α = 0 we recover λ = γpρ with γ = , see [10, (18.5), (18.30)]. Under thisnotation system (4.5) becomes p t + λ u ξ = 0 ,u t + p ξ = 0 ,T t − η p η T λ u ξ = 0 . (4.10)The proof of the following lemma is straightforward; for brevity it is omitted. Lemma 4.2 (Eigenvalues and eigenvectors) . The eigenvalues of system (4.10) are λ ± = ± λ, λ = 0 , for λ defined in (4.9) ; the corresponding eigenvectors are R ± = ± λ ∓ η p η T , R = . (4.11) The eigenvalue λ is linearly degenerate; a pair of Riemann invariants for λ is { u, p } . A Riemanninvariant for both λ ± is η . The eigenvalues of system (1.2) are then u ± λ and u . Now, we investigate the genuine nonlinearity of the eigenvalues λ ± ; we refer to [8] for more insightabout the failure of this condition. Notice that, when dealing with functions that only depend on p and T , we have R ± ∇ = ± (cid:18) ∂∂p − η p η T ∂∂T (cid:19) . Lemma 4.3.
We have R ± ∇ α = ± α (1 − α ) T i T p (cid:20) + α (1 − α ) (cid:16) + T i T (cid:17) (cid:21) . (4.12) Proof.
We simply compute ∂α∂p − η p η T ∂α∂T = − α (1 − α )2 p + T (cid:2) α (1 − α ) q (cid:3) p (cid:2) + α (1 − α ) q (cid:3) × α (1 − α )2 T q = − α (1 − α )2 p " + α (1 − α ) q − q + α (1 − α ) q + α (1 − α ) q , whence (4.12).We notice that Lemma 4.3 shows that α is not a Riemann invariant for the eigenvalues λ ± . Proposition 4.1 (Genuine nonlinearity) . The eigenvalues λ ± are genuinely nonlinear if α ≥ . If α < they are no more genuinely nonlinear for suitably small values of p and T .Proof. We focus on λ + since R + ∇ λ + = R − ∇ λ − . Since R + ∇ log λ + = R + ∇ λ + λ + , the eigenvalue λ + isgenuinely nonlinear if and only if R + ∇ log λ + >
0. We computelog λ + = log p −
12 log T −
12 log(1 + α ) − log a + 12 log A −
12 log B, (4.13)for A = 52 + φq , B = 32 + φ (cid:18) q − q + 52 (cid:19) = 32 + φ (cid:18) τ + 3 τ + 154 (cid:19) , φ = 12 α (1 − α ) . (4.14)In the following, we exploit many times identity (4.8); we split the proof into three steps.10 About the first three terms in the right-hand side of (4.13) we claim that R + ∇ (cid:20) log p −
12 log T −
12 log(1 + α ) (cid:21) = 1 pA φ T i T + T T ! . (4.15)Indeed, we have (cid:18) ∂∂p − η p η T ∂∂T (cid:19) (cid:18) log p −
12 log T (cid:19) = 1 pA " φq (cid:18) T i T (cid:19) , and, by Lemma 4.3, we have − (cid:18) ∂∂p − η p η T ∂∂T (cid:19) log(1 + α ) = − α ) α (1 − α ) T i T pA = − φ pA T i T .
Then (4.15) follows. (2)
About the last two terms in (4.13), we notice that R + ∇ (cid:18)
12 log A −
12 log B (cid:19) = 12 (cid:18) − α (cid:19) " A q − B (cid:18) q − q + 52 (cid:19) R + ∇ α + φ η p η T (cid:20) q − B − qA (cid:21) ∂ T q = I + II.
About I we notice that 12 A q − B (cid:18) q − q + 52 (cid:19) = − ( q − ) AB = − T T AB , so that, by Lemma 4.3 and (4.14) we have I = − α (1 − α ) (cid:0) − α (cid:1) T T pA B .
About II we have that II = − α (1 − α ) η p η T AB T i T [ qA − A − qB ] . Since qA − A − qB = T i T [1 + φq ] then, by (4.8), we deduce II = − φ η p η T AB T T [1 + φq ] = φ T [1 + φq ] p (cid:2) + φq (cid:3) AB T T [1 + φq ] = φ pA B T T [1 + φq ] . As a consequence R + ∇ (cid:18)
12 log A −
12 log B (cid:19) = φ pA B T T " (1 + φq ) −
12 (1 + α ) (cid:18) − α (cid:19) T i T . (4.16) (3) By (4.15) and (4.16) we deduce that R + ∇ log λ + = 2 pA φ (cid:18) q − q + 54 (cid:19) + φ AB τ " (1 + φq ) −
12 (1 + α ) (cid:18) − α (cid:19) τ . (4.17)The right-hand side of (4.17) is positive for α ≥ . We claim that (4.17) is negative for suitable smallvalues of T and p = p ( T ). To prove the claim, we take T sufficiently small and fix p = p ( T ) in such away that ατ = 1 . (4.18)11herefore, by using notation as in (3.3), p = p ( T ) ∼ τ − κ T e − τ (4.19)for T → p is also small. For T → α ∼ αq ∼ A ∼ B ∼ α < we have R + ∇ log λ + ∼ p ( (cid:20) − τ (cid:21)) = 1144 p (124 − τ ) < τ > Remark 4.1.
The value T i /T > found in (4.20) corresponds, for the value T i = 1 . × K , seeRemark 3.1, to T . . × K . Notice that the term in braces in (4.17) tends exponentially to as T → for every fixed p > . Moreover, it also tends to as p → for every fixed T > . Now, we study in greater detail the inflection locus , which is the set I = (cid:8) ( α, T ); R ± ∇ λ ± = 0 , < α < , T > (cid:9) . Since R + ∇ λ + = R − ∇ λ − , either cases lead to the same result. Lemma 4.4.
The inflection locus I is the zero set of the function f ( α, T ) = 1 + α (1 − α )
54 + T i T + T T ! + α (1 − α ) T T (cid:20) α (1 − α ) (cid:16) + T i T (cid:17)(cid:21) − α (1 − α ) (1 − α ) T T (cid:20) α (1 − α ) (cid:16) + T i T + T T (cid:17)(cid:21) (cid:20) α (1 − α ) (cid:16) + T i T + T T (cid:17)(cid:21) . (4.21) Proof.
Formula (4.21) follows by (4.17) with a direct calculation: the function f is the term in braces inthat formula.Formula (4.17) shows that R + ∇ log λ + > f ( α, T ) > . We refer to Figure 2 for thelocus I . Notice that significant values of the temperature range from 0 to 1 . · K; on the other hand,the ionization degree is low, ranging from 0 to 2 · − . Both ranges are in a physical range. Proposition 4.2.
The inflection locus is an algebraic curve having a singularity at ( α, T ) = (0 , . Nearthis point there exist two branches, whose behavior for T → is, respectively, α ∼ (cid:18) TT i (cid:19) and α ∼ (cid:18) TT i (cid:19) . Proof.
We find by the proof of Proposition 4.1 that ∇ + log λ + changes its sign along the curve ατ = c ( c > . Since, by Lemma 4.4, the inflection locus is an algebraic curve in the ( α, T )-plane, the inflectionlocus behaves like ατ → ατ → ∞ as ( α, T ) → (0 , . First, suppose that ατ → α, T ) → (0 , ατ →
0. About the function f we have that φ (cid:18) q − q + 54 (cid:19) = o (1) , A ∼ , B ∼ , φAB τ ∼ ατ , (1 + φq ) = o (1) ,
12 (1 + α ) (cid:18) − α (cid:19) τ ∼ τ. α × -4 T ( K ) Figure 2: The inflection locus with the level curves α ( T i /T ) = C , C = 1 , ,
5. The level curve with C = 1 was used in the proof of Proposition 4.1. Only the first and the last summand in the expression of f give nonzero contributions and equation f ( α, T ) = 0 can be written as 2 − α τ ∼ . Thus we have α ∼ τ − ; this is the left branch on Figure2. Next, suppose that ατ → ∞ as ( α, T ) → (0 , φ (cid:18) q − q + 54 (cid:19) ∼ ατ , A ∼ ατ , B ∼ ατ , φAB τ ∼ ατ , (1 + φq ) ∼ ατ + α ατ ,
12 (1 + α ) (cid:18) − α (cid:19) τ ∼ τ. In this case we have that equation f ( α, T ) = 0 becomes12 ατ + 2 ατ (cid:18) ατ (cid:19) − ατ ∼ . (4.22)Expression (4.22) cannot hold if ατ ∼ C = 0; then, either ατ ∼ ατ ∼ ∞ . In case ατ ∼ ∞ expression(4.22) can be written as ατ − ατ + α ∼ , which is impossible. In the case ατ ∼ ατ − ατ ∼ α ∼ τ − . This is the right branch on Figure 2, left.From the proof of Proposition 4.2 we deduce that ∇ R ± λ ± is positive outside the bounded region ofFigure 2 and negative inside it.We now provide an explicit sufficient condition on the genuine nonlinearity of the eigenvalues λ ± . Theorem 4.1.
If either α ≤ (cid:16) TT i (cid:17) or T i T ≤ . , then each eigenvalue λ ± is genuinely nonlinear. roof. With the aid of the inequality (1 + a )(1 + b )(1 + c ) ≥ a + b + c , for a, b, c ≥ , we have f ( α, T ) > α (1 − α ) h + T i T + (cid:0) + + (cid:1) T T i − α (1 − α ) T T + α (1 − α ) T T (cid:20) α (1 − α ) (cid:16) + T i T + T T (cid:17)(cid:21) (cid:20) α (1 − α ) (cid:16) + T i T + T T (cid:17)(cid:21) = 1 + α (1 − α ) (cid:20) + T i T + (cid:16) T i T (cid:17) (cid:21) − α (1 − α )60 (cid:16) T i T (cid:17) (cid:20) α (1 − α ) (cid:16) + T i T + T T (cid:17)(cid:21) (cid:20) α (1 − α ) (cid:16) + T i T + T T (cid:17)(cid:21) . Then, both characteristic eigenvalue λ ± are genuinely nonlinear if the following inequality holds: α (1 − α ) (cid:18) T i T (cid:19) ≤
60 + α (1 − α ) "
225 + 180 (cid:18) T i T (cid:19) + 51 (cid:18) T i T (cid:19) . (4.23)Now, assume that the first condition in the statement of the theorem holds; then α (1 − α ) (cid:18) T i T (cid:19) < α (cid:18) T i T (cid:19) ≤ α (1 − α ) ≥ , − α − α ≥ (cid:0) ≤ α < (cid:1) . Then (4.23) is true if (cid:18) T i T (cid:19) ≤
705 + 180 (cid:18) T i T (cid:19) + 51 (cid:18) T i T (cid:19) . Since x − − x − x < ≤ x ≤ . , the claim follows. Remark 4.2.
For monatomic hydrogen gas, we have T i = 1 . × K . Thus the λ ± characteristicfields are genuinely nonlinear if T ≥ . × K . For the gas dynamic equations (1.2), the Rankine-Hugoniot conditions for a discontinuity of constantspeed s are s [ ρ ] = [ ρu ] ,s [ ρu ] = [ ρu + p ] ,s [ ρE ] = [ ρuE + pu ] . (5.1)Here, as usual, we denoted [ ρ ] = ρ + − ρ − , where ρ ± are the traces of ρ at x = st from the right andfrom the left, respectively; the same notation is used for the other variables. If [ ρ ] = 0 then [ u ] = 0 by(5.1) and [ p ] = 0 by (5.1) ; in this case, conditions (5.1) describe the jumps of the state variables at acontact discontinuity and s = u ± . From now on we focus on the much more interesting case of shockwaves corresponding to eigenvalues λ ± and assume [ ρ ] = 0. In this case we can eliminate s from the firstequation and substituting it in the other two equations, the conditions (5.1) can be written as ( ( u + − u − ) + ( p + − p − )( v + − v − ) = 0 ,e + − e − + ( p + + p − )( v + − v − ) = 0 . (5.2)In the following we slightly change notation: with reference to (5.2), we drop the + index and write 0instead of the index − . Under this notation, we have the equation of Hugoniot locus of ( α , u , T ) ( ( u − u ) + ( p − p )( v − v ) = 0 , kinetic part, e − e + ( p + p )( v − v ) = 0 , thermodynamic part. (5.3)14or brevity we call (5.3) the kinetic part and (5.3) the thermodynamic part of the Hugoniot locus. Theaim of this section is to give a precise description of this locus and to evaluate, in particular, the variationof the thermodynamical variables across it; this analysis is fundamental for the study of shock waves, see[ ? ]. T In this section we begin the study of the Hugoniot locus by focusing first on the variation of the tempera-ture. Then we study the thermodynamic part of this locus in the ( α, T )-plane. For a contact discontinuity,the thermodynamical part of the Hugoniot locus is given by p = p and α = (cid:16) κp T − e T i T (cid:17) − , see(3.3) , which is excluded in the following discussion. Proposition 5.1 (Variation of the temperature) . The thermodynamic part of the Hugoniot locus (5.3) of ( α , T ) can be expressed as TT = (cid:16) pp (cid:17) (1 + α ) + 2 T i T α (cid:16) p p (cid:17) (1 + α ) + 2 T i T α (5.4) or else, by using only α and T , as T α ) + − α − α ! (cid:18) αα (cid:19) (cid:18) T T (cid:19) e − T i T + T i T + 2 T i T α = T α ) + − α − α ! (cid:18) α α (cid:19) (cid:18) TT (cid:19) e − T i T + T i T + 2 T i T α . (5.5) Proof.
By using the enthalpy, equation (5.3) can be written as H − H −
12 ( p − p )( v + v ) = 0 . (5.6)Now, by (2.17) and (2.19), we have pv = a T (1 + α ) and H = pv + a T i α . As a consequence, we deduce H − H −
12 ( p − p )( v + v ) = 2( pv − p v ) + 12 ( vp − v p ) + a T i ( α − α ) . Then, condition (5.6) is equivalent to2 h a T (1 + α ) − a T (1 + α ) i + 12 (cid:20) a T p p (1 + α ) − a T pp (1 + α ) (cid:21) + a T i ( α − α ) = 0 . If we divide the equation above by a T we obtain (5.4). At last, by (3.9) we have pp = − α − α ! (cid:18) α α (cid:19) (cid:18) TT (cid:19) e − T i T + T i T . (5.7)We deduce (5.5) by inserting (5.7) into (5.4). Remark 5.1 (Polytropic gas) . We explicitly compute
T /T in the case of a polytropic gas, see (5.4) . By (2.10) we obtain T /T = (cid:0) p/p (cid:1) γ − γ e γ − γ ( η − η ) . In the monatomic case γ = , we have TT = (cid:18) pp (cid:19) e ( η − η ) = 4 + pp p p , which coincides with (5.4) for α = α = 0 . Proposition 5.2 ( T as a function of α ) . In the ( α, T ) -plane, the thermodynamic part (5.5) of theHugoniot locus (5.3) of ( α , T ) is the graph of a strictly increasing function T = T ( α ) , for α ∈ (0 , . Moreover, lim α → T ( α ) = 0 , lim α → T ( α ) = ∞ . (5.8) α T ( K ) Figure 3: The thermodynamic part of the Hugoniot locus and the function T = T ( α ), see Proposition5.2. Here T = 300K, α = 3 . × − , p = 1466 . Proof.
We refer to Figure 3 for a graph of the function T . We split the proof into some steps. (i) By (5.5), the thermodynamic part of the Hugoniot locus is the set defined by F ( α, T ) = 0, for F ( α, T ) = T α ) + − α − α ! (cid:18) αα (cid:19) (cid:18) T T (cid:19) e − T i T + T i T + 2 T i T α − T α ) + − α − α ! (cid:18) α α (cid:19) (cid:18) TT (cid:19) e − T i T + T i T + 2 T i T α . (5.9)By differentiating (5.9) with respect to T, and then introducing p and p , we compute F T ( α, T ) = 4(1 + α ) − − α − α ! (cid:18) αα (cid:19) (cid:18) T T (cid:19) (cid:18)
32 + T i T (cid:19) e − T i T + T i T − − α − α ! (cid:18) α α (cid:19) (cid:18) TT (cid:19) (cid:18)
52 + T i T (cid:19) e − T i T + T i T = − " (1 + α ) p p (cid:18)
32 + T i T (cid:19) + (1 + α ) pp T T (cid:18)
52 + T i T (cid:19) − α ) (5.10)= − α )Φ( α, T ) , α, T ) = 14 p p (cid:18)
32 + T i T (cid:19) + 14 pp T (1 + α ) T (1 + α ) (cid:18)
52 + T i T (cid:19) − . (5.11)Obviously, F T ( α, T ) = 0 if and only if Φ( α, T ) = 0 . (ii) We consider 0 < α < , T > F ( α, T ) = 0 . We claim thatΦ( α, T ) > < α < α , T ≤ T or α ≥ α . (5.12)Indeed, by Proposition 5.1 the condition F ( α, T ) = 0 is equivalent to (5.4), namely,14 pp T (1 + α ) T (1 + α ) = 14 p p + T i ( α − α )2 T (1 + α ) − T (1 + α ) T (1 + α ) + 1 . (5.13)Then, if we denote ∆ = 1 − T (1 + α ) T (1 + α ) + T i ( α − α )2 T (1 + α ) , (5.14)we have, for α, T satisfying F ( α, T ) = 0 , Φ( α, T ) = p p (cid:18) T i T (cid:19) + (cid:18)
52 + T i T (cid:19) ∆ − . By (5.13) and (5.14), the quotient Π = p p satisfies the equationΠ + 4∆Π − T (1 + α ) T (1 + α ) = 0 (5.15)and then Π = s + T (1 + α ) T (1 + α ) − . (5.16)As a consequence, we deduceΦ( α, T ) = (cid:18) T i T (cid:19) s + T (1 + α ) T (1 + α ) + ∆2 − . If ∆ > , then Φ( α, T ) > α and T ; hence, assume ∆ ≤
2. If Φ( α, T ) = 0, then ∆ satisfiesthe quadratic equation
154 + 4 T i T + T T ! ∆ + ∆ + (cid:18) T i T (cid:19) T (1 + α ) T (1 + α ) − . (5.17)By (5.14), equation (5.17) can be written as
154 + 4 T i T + T T ! ∆ + T i T "(cid:18) T i T (cid:19) T (1 + α ) T (1 + α ) + α − α α ) = 0 . If α ≥ α , then the left-hand side of the above expression is strictly positive, which is a contradiction.If 0 < α < α , we have (cid:18) T i T (cid:19) T (1 + α ) T (1 + α ) + α − α α ) >
11 + α (cid:20) T T (1 + α ) + α − α (cid:21) . If T ≤ T , the above quantity is positive and hence we reach a contradiction again. This proves(5.12). 17 iii) Consider the function F defined in (5.9); clearly F ( α , T ) = 0 and F T ( α , T ) = 0 by (5.12). Thenit follows from the Implicit Function Theorem that T is a function of α in a neighbourhood of α = α ; moreover, dT /dα = − F α ( α, T ) /F T ( α, T ). By differentiating (5.9) with respect to α andintroducing p and p , we have F α ( α, T ) = 2 T α (1 − α ) (cid:18) α (cid:19) (cid:18) TT (cid:19) e − T i T + T i T − T − α ) α (cid:20) α (2 − α )(1 − α ) (cid:21) (cid:18) T T (cid:19) e − T i T + T i T + 2 T i T = 2 pp T (1 + α ) α (1 − α ) + T (cid:20) α )(2 − α ) α (1 − α ) p p + 2 T i T (cid:21) . (5.18)Thus, by (5.10) and (5.18) we end up with the following expression for dT /dα :1 T dTdα = 4 + 2 T i T + p p (1 + α )(2 − α ) α (1 − α ) + 2 pp T (1 + α ) T α (1 − α )(1 + α ) p p (cid:18)
32 + T i T (cid:19) + (1 + α ) pp T T (cid:18)
52 + T i T (cid:19) − α ) . (5.19)The numerator of the right-hand side in (5.19) is strictly positive for every 0 < α < T > α = α by (5.12). We conclude that T ( α )is a strictly increasing function in this neighbourhood. (iv) Let ( α − , α + ) denote the largest (open) interval where the function T ( α ) is defined. We claim thatΦ (cid:0) α, T ( α ) (cid:1) > α ∈ ( α − , α + ) . Indeed, if α ∈ ( α , α + ) then the claim follows by (5.12). If α ∈ ( α − , α ), clearly Φ (cid:0) α, T ( α ) (cid:1) > α . Suppose that there is α such that α − < α < α with Φ (cid:0) α , T ( α ) (cid:1) = 0and Φ (cid:0) α, T ( α ) (cid:1) > α ∈ ( α , α ). Since T ( α ) is increasing in ( α , α ) by (5.19), we have T := T ( α ) < T and F ( α , T ) = 0 . Then (5.12) yields Φ( α , T ) >
0, which is a contradiction. (v)
Next, we claim α − = 0 and α + = 1 . Note that, since T = T ( α ) is strictly increasing in ( α − , α + ), then both limits T ± = lim α → α ± T ( α )exist. Assume by contradiction α + <
1. If T + < ∞ , then F ( α + , T + ) = 0 by continuity and, since α + > α , we have F T ( α + T + ) = 0 by (5.12). But then the function T ( α ) can be extended beyond α + , which is a contradiction. If T + = ∞ , we find by (5.9) that F (cid:0) α, T ( α ) (cid:1) → −∞ as α → α + which is impossible. We conclude that α + = 1; by the same argument, we have α − = 0 . (vi) At last, let us denote T ∞ = lim α → T ( α ) . If T ∞ < ∞ , by (5.9) we find F (cid:0) α, T ( α ) (cid:1) → ∞ as α → α → T ( α ) = ∞ . In the same way we conclude lim α → T ( α ) = 0 . The proposition is completely proved.Since the function T = T ( α ) introduced in Proposition 5.2 is strictly increasing, then it is invertibleand defines a function α = α ( T ). In the following proposition we precise the asymptotic behavior of α ( T )when T is close either to 0 or ∞ . Proposition 5.3 (Asymptotics) . On the Hugoniot locus (5.3) , if T → then α → and more precisely α ∼ α √ − α (cid:20) T i α T (1 + α ) (cid:21) (cid:18) TT (cid:19) e − T i2 T + T i2 T as T → . Furthermore, if T → ∞ then α → and we have − α ∼ − α α (cid:18) TT (cid:19) − e − T i T as T → ∞ . (5.20)18 roof. By (5.5) we see that T → α → . Suppose that α T − e T i T = O (1), i.e. that it tends to a finitenonzero value for T →
0. Then, for T →
0, the left-hand side of (5.5) would be ∼ O (1), a contradiction.Then, we set α ∼ A (cid:16) TT (cid:17) µ e − T i2 T , for some A ≥ . Around the point ( α, T ) = (0 ,
0) we have TT
83 (1 + α ) + 23 − α − α ! (cid:18) αα (cid:19) (cid:18) T T (cid:19) e − T i T + T i T + 43 T i T α ∼ TT + 23 − α α ! A (cid:18) TT (cid:19) µ − e − T i T ∼
83 (1 + α ) + 23 α − α ! A − (cid:18) TT (cid:19) − µ + e T i T + 43 T i T α . If 2 µ − = 1 , then µ = , which is impossible by the first part of the proof. If 2 µ − = − µ + , then µ = 1 , which is also impossible. Then 2 µ − = 0 and hence µ = 34 and A = α ) + T i T α − α α e T i2 T = 2 α √ − α (cid:20) T i α T (1 + α ) (cid:21) e T i2 T . On the other hand, if α → T → ∞ . Suppose that (1 − α ) T = O (1) . Then we have TT
83 (1 + α ) + 23 − α − α ! (cid:18) αα (cid:19) (cid:18) T T (cid:19) e − T i T + T i T + 43 T i T α > O (1) T,
83 (1 + α ) + 23 − α − α ! (cid:18) α α (cid:19) (cid:18) TT (cid:19) e − T i T + T i T + 43 T i T α ∼ O (1) , which is impossible. In this case, by setting 1 − α ∼ B (cid:16) TT (cid:17) − µ for µ > , we have around (1 , ∞ ) TT
83 (1 + α ) + 23 − α − α ! (cid:18) αα (cid:19) (cid:18) T T (cid:19) e − T i T + T i T + 43 T i T α ∼ TT + 23 − α α ! B − (cid:18) TT (cid:19) µ − e − T i T ∼
83 (1 + α ) + 43 α − α ! B (cid:18) TT (cid:19) − µ + e T i T + 43 T i T α . If µ − = − µ + , then µ = 2 which is impossible. Then − µ + = 1 and so we deduce both µ = and B = 4 (cid:16) − α α (cid:17) e − T i T . Remark 5.2.
Let us consider the asymptotics for TT → ∞ , T i T = O (1) and α → , T → . Exchangingthe roles of T and T in the above argument, we have the asymptotic formula α ∼ " α ) + T i T α − α α (cid:18) T T (cid:19) e − T i2 T + T i2 T for TT → ∞ , T i T = O (1) , α → , T → . p and v along the Hugoniot locus In this section we exploit Proposition 5.2 to compute the variation of p and v along the thermodynamicpart (cid:0) α, T ( α ) (cid:1) of the Hugoniot locus. We prove that p increases and v decreases as α increases. Proposition 5.4 ( p and v as functions of α ) . Let T = T ( α ) be the function introduced in Proposition5.2. Then ddα p (cid:0) α, T ( α ) (cid:1) > and ddα v (cid:0) α, T ( α ) (cid:1) < for α ≥ α . roof. By (3.9) and (3.8) we deducelog p = log(1 − α ) − α + 52 log T − T i T + const. (5.21)log v = − log(1 − α ) + 2 log α −
32 log T + T i T + const. (5.22)In the following of the proof, we work on the thermodynamic part (cid:0) α, T ( α ) (cid:1) of the Hugoniot locus; then T = T ( α ) and, with a slight abuse of notation, we denote p ( α ) := p (cid:0) α, T ( α ) (cid:1) and v ( α ) := v (cid:0) α, T ( α ) (cid:1) . (i) First, we consider the variation of p . By (5.21) we compute ddα log p = − α (1 − α ) + (cid:18)
52 + T i T (cid:19) T dTdα , (5.23)and then, by (5.19),12 p α (1 − α ) dpdα = − α (1 − α ) (cid:20) (cid:16) p p (cid:17) (1+ α )(2 − α )4 α (1 − α ) + T i T + (cid:16) pp (cid:17) T (1+ α )2 T α (1 − α ) (cid:21) (cid:16) + T i T (cid:17) (1 + α ) (cid:16) p p (cid:17) (cid:16) + T i T (cid:17) + (1 + α ) (cid:16) pp (cid:17) T T (cid:16) + T i T (cid:17) − (1 + α )=: N ( α ) D ( α ) . (5.24)The denominator D ( α ) is positive by the proof of Proposition 5.2 while N ( α ) = 12 α (1 − α ) (cid:18) T i T (cid:19) (cid:18)
52 + T i T (cid:19) + p p (1 + α ) (2 − α )8 (cid:18)
52 + T i T (cid:19) −
14 (1 + α ) p p (cid:18)
32 + T i T (cid:19) + (1 + α ) . (5.25)Assume p ≥ p . For the last two summands in (5.25) we have −
14 (1 + α ) p p (cid:18)
32 + T i T (cid:19) + (1 + α ) ≥ −
14 (1 + α ) p p
32 + (1 + α ) ≥
58 (1 + α ) . Then, we deduce N ( α ) ≥ T i T α (1 − α ) (cid:20) p p (cid:21) > , for p ≥ p . (5.26)Estimate (5.26) holds in particular for p = p and then shows that p ′ ( α ) > . Then p ′ ( α ) > p ( α ) > p for α ∈ [ α , α ], where α is close to α Now, assume that there exists α ∗ > α suchthat p ′ ( α ∗ ) = 0 and p ( α ) ≥ p for α ∈ [ α , α ∗ ]; then (5.26) also shows that p ′ ( α ∗ ) > , which is acontradiction. Thus we conclude that p ′ ( α ) > α ≥ α . (ii) Now, we consider the variation of v . The result is a counterpart of what proved above but it is notdirectly deduced from it. By (5.22) and (5.19) it follows1 v dvdα = 2 − αα (1 − α ) − (cid:20) T i T + (cid:16) p p (cid:17) (1+ α )(2 − α )4 α (1 − α ) + (cid:16) pp (cid:17) T (1+ α )2 T α (1 − α ) (cid:21) (cid:16) + T i T (cid:17) (1 + α ) (cid:16) p p (cid:17) (cid:16) + T i T (cid:17) + (1 + α ) (cid:16) pp (cid:17) T T (cid:16) + T i T (cid:17) − (1 + α ) =: N ( α ) D ( α ) . As above, D ( α ) is positive, while some computations lead to N ( α ) = (2 − α )(1 + α )4 α (1 − α ) (cid:20) T (1 + α ) T (1 + α ) − (cid:21) + 14 " T (1 + α ) T (1 + α ) pp − (cid:18) T i T (cid:19)
32 + T i T (cid:19) . (5.27)20e claim that the term in the second bracket in (5.27) is negative for α ≥ α , i.e., T (1 + α ) T (1 + α ) pp − (cid:18) T i T (cid:19) = v v − (cid:18) T i T (cid:19) ≤ , for α ≥ α . (5.28)This would conclude the proof of the lemma: indeed, the first summand in (5.27) is strictly negativebecause T ( α ) is a strictly decreasing function in (0 ,
1) by Proposition 5.2.To prove (5.28), let us denote ˆΠ = pp . By (5.15), we have that ˆΠ satisfies the quadratic equation T (1 + α ) T (1 + α ) ˆΠ −
4∆ ˆΠ − , where ∆ is defined in (5.14). Since ˆΠ ≥ α ≥ α by the previous step (i) , we deduce T (1 + α ) T (1 + α ) ˆΠ = 2∆ + s + T (1 + α ) T (1 + α )and then T (1 + α ) T (1 + α ) (cid:18) pp (cid:19) − (cid:18) T i T (cid:19) = 2∆ + s + T (1 + α ) T (1 + α ) − (cid:18) T i T (cid:19) . (5.29)We have that ∆ ≥ α ∈ (0 ,
1) because the function T ( α ) is strictly decreasing in (0 ,
1) byProposition 5.2. If ∆ = 0 , then (5.29) is strictly negative. Suppose that (5.29) vanishes for some∆ > . Then s + T (1 + α ) T (1 + α ) = 4 (cid:18) T i T (cid:19) − T (1 + α ) T (1 + α ) = 16 (cid:18) T i T (cid:19) (cid:20) T i T + T (1 + α ) T (1 + α ) − T i ( α − α )2 T (1 + α ) (cid:21) . However, the above equation can never be satisfied, since T i T − T i ( α − α )2 T (1+ α ) > α ∈ (0 , In this subsection we focus on the variation of the velocity u along the Hugoniot locus (5.3) of ( α , u , T );these results are used in the forthcoming paper [ ? ] to study the shock tube problem. By (3.8), equation(5.3) can be written as( u − u ) = − p v (cid:18) pp − (cid:19) (cid:18) vv − (cid:19) = − p v (cid:20) T (1 + α ) T (1 + α ) − pp − vv + 1 (cid:21) . Therefore, by (3.8), a point of the Hugoniot locus of ( α , u , T ) lies in the intersection of the set G ( α, u, T ) := (cid:18) pp − (cid:19) (cid:18) vv − (cid:19) + ( u − u ) a T (1 + α ) = 0 (5.30)with (5.3) . For simplicity, we drop the dependence of G on ( α , u , T ); of course, we have G ( α , u , T ) =0. Since u is fixed in the following, we understand the set G ( α, u, T ) as a set in the ( α, T )-plane, seeFigure 4. 21 roposition 5.5 (The kinetic part of the Hugoniot locus) . Fix ( α , u , T ) and u = u ; then, the set G ( α, u, T ) = 0 is constituted by the graphs of two continuous functions T − ( α ) < T + ( α ) , for α ∈ (0 , ,satisfying T − ( α ) < T < T + ( α ) . The function T + ( α ) is differentiable and its graph is located in theregion p ( α, T ) > p . Moreover, we have lim α → T ± ( α ) = ∞ ; for α → , on the set G ( α, u, T ) = 0 we have − α ∼ − α α e − T i T (cid:18) TT (cid:19) − along T + and − α ∼ − α α e − T i T (cid:18) TT (cid:19) − along T − . (5.31) Near α = 1 , both curves are located under the thermodynamic part T = T ( α ) of the Hugoniot locus of ( α , T ) . α T ( K ) Figure 4: Solid line: the set G ( α, u, T ) = 0 in the plane ( α, T ) for fixed u ; dash-point line: thethermodynamic part of the Hugoniot locus. Here T = 300K, α = 3 . × − and u = 0; weused u = 38000 m s − instead of the value u = 8100 m s − of [5], to make the intersection clearer. Proof.
By (5.7) and (3.9) we can eliminate p from the expression of G and write G ( α, u, T ) = pp + vv − T (1 + α ) T (1 + α ) − − ( u − u ) a T (1 + α )= α (1 − α ) α (1 − α ) (cid:18) TT (cid:19) e − T i T + T i T + α (1 − α ) α (1 − α ) (cid:18) T T (cid:19) e − T i T + T i T − T (1 + α ) T (1 + α ) − − ( u − u ) a T (1 + α ) . (5.32)The function T ( T /T ) e − T i T + T i T is strictly increasing and valued in [0 , ∞ ). Then, for every α > T ∗ = T ∗ ( α ) such that, by (3.9), p ∗ p = − α − α ! α α ! (cid:18) T ∗ T (cid:19) e − T i T ∗ + T i T = 1 , for p ∗ = p (cid:0) α, T ∗ ( α ) (cid:1) ; moreover, T ∗ ( α ) = T . Therefore G (cid:0) α, u, T ∗ ( α ) (cid:1) = − ( u − u ) a T (1+ α ) < . For every α fixed we have that G ( α, u, T ) → ∞ for both T → T → ∞ .We conclude that thereare at least two values T ± = T ± ( α ), with T − < T ∗ < T + , such that G (cid:0) α, u, T ± ( α ) (cid:1) = 0. Note that p > p T = T + ( α ) and p < p along T = T − ( α ) . Moreover, by a direct computation we find T ∂G∂T ( α, u, T ) = α (1 − α ) α (1 − α ) (cid:18)
52 + T i T (cid:19) (cid:18) TT (cid:19) e − T i T + T i T − α (1 − α ) α (1 − α ) (cid:18)
32 + T i T (cid:19) (cid:18) T T (cid:19) e − T i T + T i T − α α T ∂ G∂T ( α, u, T ) = α (1 − α ) α (1 − α ) "(cid:18)
52 + T i T (cid:19) (cid:18)
32 + T i T (cid:19) − T i T TT (cid:19) e − T i T + T i T + α (1 − α ) α (1 − α ) "(cid:18)
32 + T i T (cid:19) (cid:18)
52 + T i T (cid:19) + T i T T T (cid:19) e − T i T + T i T > . Then, for every α ∈ (0 ,
1) the equation G ( α, u, T ) = 0 has exactly the two zeroes T ± ( α ).We have, by (2.18), (3.9) and (5.30), ∂G∂T ( α, u, T ) = 1 T "(cid:18)
52 + T i T (cid:19) pp − (cid:18)
32 + T i T (cid:19) vv − T (1 + α ) T (1 + α ) = 1 T "(cid:18)
32 + T i T (cid:19) (cid:18) pp − vv (cid:19) + (cid:18) − vv (cid:19) + ( u − u ) a T (1 + α ) . (5.33)On the graph of T = T + ( α ) we proved that p > p and then (5.3) shows that v < v ; as a consequence,(5.33) yields ∂G∂T ( α, u, T ) > T + ( α ) is differentiable.At last, we prove the asymptotic behavior of the set G ( α, u, T ) = 0 for α close to 1. It is easy to seefrom (5.32) that on this set we have T → ∞ if α →
1. We set 1 − α ∼ B (cid:16) TT (cid:17) − µ for µ >
0. By (5.32) wehave 2 Bα − α (cid:18) TT (cid:19) − µ e T i T + 1 − α α B (cid:18) T T (cid:19) − µ e − T i T ∼ TT (1 + α ) + 1 + ( u − u ) a T (1 + α ) . If T − µ and T µ − are equally large as T → ∞ , we have µ = 2 . Then both these terms are O (1) T , which is a contradiction. In the case − µ = 1 , we have µ = and B = − α α e − T i T . Otherwise µ = and B = − α α e − T i T . By Proposition 5.3, we conclude that the graphs of both T ± are located under thethermodynamic part T = T ( α ) of the Hugoniot locus of ( α , T ) near α = 1 . Lemma 5.1 (Intersection of Hugoniot loci) . Fix ( α , u , T ) and u = u . Then, in the region α <α < there is at least one point of intersection between the set G ( α, u, T ) = 0 and the thermodynamicpart (5.3) of the Hugoniot locus of ( α , T ) . The same result holds also in the region < α < α if ( u − u ) ≤ a T i α . Proof.
We refer to Figure 4. By the proof of Proposition 5.5 we know that G ( α , u, T ) <
0, while byProposition 5.2 we have T ( α ) = 0. Then, in a neighborhood of ( α , T ) , the curve T = T ( α ) is locatedbetween the two curves T = T − ( α ) and T = T + ( α ).In the region α < α <
1, it follows by (5.31) and (5.20) that there is at least one point of intersectionbetween the curves T = T + ( α ) and T = T ( α ).On the other hand, Proposition 5.3 and (5.32) imply G ( α, u, T ) ∼ T (1+ α ) h T i α − ( u − u ) a i as α, T → G ( α, u, T ) = 0 and thegraph of the function T ( α ) intersects in the region 0 < α < α provided ( u − u ) ≤ a T i α . At last, from the results obtained in this section we conclude with the following main result.
Theorem 5.1 (The Hugoniot locus) . Fix ( α , u , T ) and u = u . Then there exists a unique point ( α, T ) with α ∈ ( α , , such that ( α, u, T ) belongs to the Hugoniot locus (5.3) of ( α , u , T ) . roof. We must prove that there is a unique point of intersection between the thermodynamic part T = T ( α ) of the Hugoniot locus of ( α , T ) and the set G ( α, u, T ) = 0, where u = u is fixed. To thisaim, it is sufficient to prove that ddα G (cid:0) α, u, T ( α ) (cid:1) > , (5.34)since then the point of intersection obtained in Lemma 5.1 is unique.By (5.32) and (5.33), on the set G ( α, u, T ) = 0 we have ddα G (cid:0) α, u, T ( α ) (cid:1) = − pα (1 − α ) p + (2 − α ) vα (1 − α ) v − TT (1 + α )+ "(cid:18)
52 + T i T (cid:19) pp − (cid:18)
32 + T i T (cid:19) vv − T (1 + α ) T (1 + α ) T dTdα and then by (5.19) we deduce α (1 − α ) dGdα (cid:0) α, u, T ( α ) (cid:1) = − pp + (2 − α )(1 + α ) vv − α (1 − α ) T (1 + α ) T (1 + α ) + "(cid:18)
52 + T i T (cid:19) pp − (cid:18)
32 + T i T (cid:19) vv − T (1 + α ) T (1 + α ) × pp + (cid:16) vv (cid:17) (1 + α )(2 − α ) + 4 α (1 − α ) (cid:16) T i T (cid:17) T (1+ α ) T (1+ α ) (cid:16) + T i T (cid:17) pp + (cid:16) + T i T (cid:17) vv − T (1+ α ) T (1+ α ) =: N ( α ) D ( α ) . As above, the denominator D ( α ) is positive by the proof of Proposition 5.2. After some algebraiccomputations we obtain N ( α ) = T (1 + α ) T (1 + α ) " (cid:18) pp − (cid:19) + 5 (cid:18) − vv (cid:19) (2 − α )(1 + α ) + 2 α (1 − α ) T (1 + α ) T (1 + α ) "(cid:18)
32 + T i T (cid:19) (cid:18)
52 + T i T (cid:19) (cid:18) pp − vv (cid:19) − T i T (cid:26) T (1 + α ) T (1 + α ) − (cid:27) . (5.35)By (5.4), the thermodynamic part of the Hugoniot locus can be written as4 T (1 + α ) + p p T (1 + α ) + 2 T i α = 4 T (1 + α ) + pp T (1 + α ) + 2 T i α , or, equivalently, as pp − vv = 4 (cid:20) T (1 + α ) T (1 + α ) − (cid:21) + 2 T i ( α − α ) T (1 + α ) . Moreover, on the thermodynamic part of the Hugoniot locus we have pp ≥ vv ≤ N ( α ) > α ∈ ( α , We define the amount of entropy in the interval ( a, b ) by H (cid:0) t ; [ a, b ] (cid:1) = Z ba ρ ( x, t ) S ( x, t ) dx. We want to compute the variation of H when a (single) shock with constant velocity s is present in theregion under consideration, namely, a < st < b . We use the notation introduced at the beginning ofSection 5, see (5.2). 24 emma 6.1. Assume that a shock front with constant velocity s is passing in the interval [ a, b ] . Then ρ + ( s − u + ) = ρ − ( s − u − ) and with m = ρ ± ( s − u ± ) we have ddt H (cid:0) t ; [ a, b ] (cid:1) = − m [ S ] + − − [ ρuS ] ba . Proof.
By (4.3) we have ddt H (cid:0) t ; [ a, b ] (cid:1) + [ ρuS ] ba = − s [ ρS ] + − + [ ρuS ] + − . By the Rankine-Hugoniot condition (5.1) we have ρ + ( s − u + ) = ρ − ( s − u − ). Then s [ ρS ] + − − [ ρuS ] + − = s ( ρ + S + − ρ − S − ) − ( ρ + u + S + − ρ − u − S − )= ρ + ( s − u + ) S + − ρ − ( s − u − ) S − = m [ S ] + − and the lemma is proved.We notice that by (5.1) we have m = − p + − p − v + − v − , which shows that m is the Lagrangian shock speed .With the help of the previous lemma we now provide a result about the increase of entropy across ashock front.
Proposition 6.1.
Let ( p − , u − , S − ) and ( p + , u + , S + ) be connected by a shock wave and v pp ( p ± , S ± ) > .If | p + − p − | is sufficiently small, then the entropy increases across the shock front.Proof. We begin by considering an asymptotic expression of [ S ] + − = S + − S − for small | p + − p − | . By(2.4), we have (cid:18) ∂H∂S (cid:19) p = T, (cid:18) ∂H∂p (cid:19) S = v. Thus, on the one hand we have H + − H − = T − ( S + − S − ) + v − ( p + − p − ) + 12 (cid:18) ∂v∂p − (cid:19) S ( p + − p − ) + 16 ∂ v∂p − ! S ( p + − p − ) + O (1)( S + − S − ) + O (1)( S + − S − )( p + − p − ) + O (1)( p + − p − ) . Here we used the shortcut ( ∂v/∂p − ) S = ( ∂v/∂p )( p − , S − ). On the other hand, by (5.3) we have H + − H − = 12 ( v + + v − )( p + − p − ) = v − ( p + − p − ) + 12 ( v + − v − )( p + − p − )= v − ( p + − p − ) + 12 (cid:18) ∂v∂p − (cid:19) S ( p + − p − ) + 14 ∂ v∂p − ! S ( p + − p − ) + O (1)( S + − S − )( p + − p − ) + O (1)( p + − p − ) . Combining the above two expressions and omitting some technical details, we conclude that T − ( S + − S − ) = 112 ∂ v∂p − ! S ( p + − p − ) + O (1)( p + − p − ) . (6.1)By (4.4), the assumption v pp ( p ± , S ± ) > λ + , which is characterized by m >
0; for shock waves of the other family (i.e., corresponding to λ − , then m <
0) the proof is analogous. Since the shock is a Lax shock, then it is compressive, i.e., p + < p − .By Lemma 6.1, the entropy increases across the shock front if and only if S + − S − < . By (6.1), thisinequality is implied once more by the condition v pp ( p ± , S ± ) > emark 6.1. The expression (6.1) was first obtained by H. Bethe in 1942, see [8, (3.44)] and [6, § (2.17) with (2.13) . Theorem 4.1 implies the following result.
Theorem 6.1.
If either α ≤ (cid:16) TT i (cid:17) or T i T ≤ . , then the entropy increases across a shock frontwhose amplitude | p + − p − | is sufficiently small. We write system (1.2) as U t + F ( U ) x = 0 or, for smooth solutions, U t + A ( U ) U x = 0 , where A ( U ) = F ′ ( U ). Let V be a set of new unknowns related to U by U = U ( V ) . We have V t + B ( V ) V x = 0 , where B ( V ) = ( U ′ ) − A ( U ) U ′ and U ′ = U ′ ( V ) denotes the Jacobian matrix of U . Clearly, if R ( U ) is acharacteristic vector field of A ( U ) , then (cid:0) U ′ ( V ) (cid:1) − R (cid:0) U ( V ) (cid:1) is a characteristic field of B ( V ).Some properties of system (1.2) were already established in Lemma 4.2 as far as the coordinates( p, u, T ) were concerned. In the following, the thermodynamic variables α, T are very useful. Recalling(3.9), we consider the transformation of variables puT = − α κα T e − T i T uT . With a slight abuse of notation, we still denote by R ± and R , as in Lemma 4.2, the eigenvectors of theJacobian matrix of the flux in coordinates ( α, u, T ). Proposition 7.1 (Characteristic fields) . The characteristic vector fields with respect to ( α, u, T ) coordi-nates are expressed as R ± = ± α (1 − α + α (1 − α ) (cid:16) + T i T (cid:17) T i Tpλ ± α (1 − α ) (cid:16) + T i T (cid:17) + α (1 − α ) (cid:16) + T i T (cid:17) T , R = α (1 − α )2 ( + T i T )0 T . (7.1) Proof.
We computeD( p, u, T )D( α, u, T ) = − T κα e − T i T − α κα T (cid:16) + T i T (cid:17) e − T i T T e − T i T κα = pα (1 − α ) and − T κα e − T i T − α κα T (cid:16) + T i T (cid:17) e − T i T − = − κα T e − T i T α (1 − α )2 T (cid:16) + T i T (cid:17) . The result follows by what we pointed out just before the proposition.26ow, we can consider integral curves. We denote with (cid:0) α ± ( s ) , u ± ( s ) , T ± ( s ) (cid:1) (7.2)an integral curve of the characteristic field R ± defined in (7.1), i.e., ( ˙ α ± ( s ) , ˙ u ± ( s ) , ˙ T ± ( s )) = R ± ( α, u, T ).Indeed, we first focus on the projection of the integral curves on the ( α, T )-plane. We have, with a slightabuse of notation, ˙ T ( s )˙ α ( s ) = dTdα = 1 + α (1 − α ) (cid:16) + T i T (cid:17) α (1 − α )2 T i T T, (7.3)and R ± ∇ λ ± = ± α (1 − α ) T i T (cid:20) + α (1 − α ) (cid:16) + T i T (cid:17) (cid:21) ∂λ ± ∂α + 1 + α (1 − α ) (cid:16) + T i T (cid:17) α (1 − α )2 T i T T ∂λ ± ∂T = ± α (1 − α ) T i T (cid:20) + α (1 − α ) (cid:16) + T i T (cid:17) (cid:21) dλ ± dα . (7.4)We observe that the ordinary differential equation (7.3) does not depend on u and can be uniquely solvedby a function T = T ( α ) once the initial data T ( α ) = T is fixed. The projection on the ( α, T )-planeof any such characteristic curve is the same once T is fixed. By inserting T ( α ) in the second equation˙ u ± = R ( α, u ± , T ) one finds u ± = u ± ( α ).Recalling (3.14), with a slight abuse of notation we still denote η ( α, T ) = − − αα + (cid:18)
52 + T i T (cid:19) (1 + α ) (7.5)as a representative for the dimensionless entropy. Since η is a Riemann invariant for the characteristicfields R ± by Lemma 4.2, then it is constant along the integral curves of those fields in ( α, u, T )-space [3,Theorem 7.6.2]; of course, this can be directly verified from (7.5). In other words, the projections on the( α, T )-plane of those integral curves coincide with the isentropes η ( α, T ) = const. Proposition 7.2 (Integral curves in the ( α, T )-plane) . For any η ∈ R the equation (cid:18) − α ∞ α ∞ (cid:19) −
52 (1 + α ∞ ) + η = 0 (7.6) has a unique solution α ∞ ∈ (0 , . The two integral curves of the characteristic fields R ± , correspondingto η , have the same projection on the ( α, T ) -plane, which is the graph of a function T η = T η ( α ) for α ∈ (0 , \ { α ∞ } . The function T η is strictly increasing in both intervals (0 , α ∞ ) and ( α ∞ , ; moreover, lim α → T η ( α ) = 0 , lim α → α ∞ ∓ T η ( α ) = ±∞ , lim α → − T η ( α ) = 0 and for T = T η ( α ) we have the asymptotic expression α ∼ e − Ti T + ( η − ) as α → . (7.7) Proof.
Notice that the function η can assume any real value when α varies in (0 , η ( α, T ) = η for TT i , with η given by (7.5), we have T η ( α ) := (1 + α ) T i (cid:0) − αα (cid:1) − (1 + α ) + η = (1 + α ) T i d ( α ) . (7.8)27he denominator d ( α ) of T η is a strictly decreasing function of α ∈ (0 , α → d ( α ) = + ∞ and lim α → − d ( α ) = −∞ . Then, there is a unique value α ∞ ∈ (0 ,
1) satisfying (7.6). The properties ofthe function T η easily follow. Since T i T η ( α ) ∼ α −
52 + η as α → , the above asymptotic form is obtained. α T ( K ) Figure 5: Integral curves of the characteristic fields R ± in the ( α, T ) plane. We plotted the graphs ofthe functions T η (right) for η = 0 , ,
10; the vertical lines represent the asymptotes at α = α ∞ . Remark 7.1.
Notice that T η ( α ) > for α ∈ (0 , α ∞ ) and the value α ∞ only depends on η and noton T i . Moreover, since the function T η increases in the interval ( α ∞ , and lim α → − T η ( α ) = 0 , T η ( α ) < for α ∈ ( α ∞ , , which shows that the temperature is negative and the states with α > α ∞ arenot physically admissible. Moreover, we remark that, by (3.9) , for α → α ∞ − , on the integral curve wehave p ( α, T ) → ∞ and, by (2.17) and (3.9) , we also have ρ = a (1 + α ) Tp = a (1 + α ) κ α − α T − e T i T → .As a consequence, from now on we understand the function T η to be defined only for α ∈ (0 , α ∞ ) ;moreover, we drop the superscript η when it is not strictly needed. Now, we study the convexity of the integral curves η = const. Lemma 7.1 (Convexity of the integral curves) . The projection of any integral curve on the ( α, T ) -planeis strictly convex for T i T ≤ and strictly concave for small α. Proof.
Instead of directly computing the derivatives of T η from (7.8) we exploit the implicit functiontheorem. We note that η α = α (1 − α ) + (cid:16) + T i T (cid:17) , η T = − T i T (1 + α ) and then η αα = 2 2 α − α (1 − α ) , η αT = − T i T , η T T = 2 T i T (1 + α ) . (7.9)Consequently, since T η solves the implicit equation η (cid:0) α, T η ( α ∞ ) (cid:1) = η , we find that it satisfies d T η ( α ) dα = − η αα η T − η αT η α η T + η T T η α η T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( α, T η ( α ) ) . η αα η T − η αT η α η T + η T T η α = 2(1 + α ) T i T ( (2 α − α ) α (1 − α ) T i T − (cid:20) α (1 − α ) + 52 + T i T (cid:21) T i T + (cid:20) α (1 − α ) + 52 + T i T (cid:21) ) = 2(1 + α ) α (1 − α ) T i T ( − (cid:20) − α − α (1 − α ) (cid:21) T i T + 4 (cid:20) α (1 − α ) (cid:21) ) . If T i T ≤
4, then the numerator is positive and this proves the former part of the statement. To prove thelatter, for α → T η ( α ) ∼ T i α by (7.8) and then the principal term of the numerator is − T T α <
0. The lemma is completely proved.The next result deals with rarefaction waves. For brevity we call the rarefactioncurves corresponding to the eigenvalue λ − while are those corresponding to theeigenvalue λ + . Theorem 7.1 (Variation of the pressure along the integral curves) . We have ddα p (cid:0) α, T ( α ) (cid:1) > . Consider the projections on the ( α, T ) -plane of the rarefaction curves corresponding to the eigenvalues λ ± through the point ( α , T ) ; suppose that they are contained in the region where R ± ∇ λ ± > . Then theprojection of the 1-rarefaction curve is the integral curve through the same point defined for α ∈ (0 , α ] and that of the 2-rarefaction curve is the same integral curve defined for α ∈ [ α , α ∞ ) .Proof. By (3.9) we have log p = log(1 − α ) − α + log T − T i T + const. By differentiation togetherwith (7.3) we deduce1 p dpdα (cid:12)(cid:12)(cid:12)(cid:12) ( α, T ( α ) ) = − α − α − α + (cid:18) T ( α ) + T i T ( α ) (cid:19) d T ( α ) dα = 2 α (1 − α ) − (cid:18) T ( α ) T i (cid:19) " α (1 − α ) (cid:18)
52 + T i T ( α ) (cid:19) > . Since we are assuming R ± ∇ λ ± > , then formula (7.4) shows dλ − dα < , which yields that the 1-characteristic speed is strictly decreasing in α. Thus the 1-rarefaction curve corresponds to the part ofthe integral curve where α ∈ (0 , α ] . In the same way, since dλ + dα > , we conclude that the 2-rarefactioncurve corresponds to the part [ α , α ∞ ) . Remark 7.2.
The above theorem shows that the pressure is strictly increasing along the 2-rarefactioncurve, which looks contradictory. However, here the state ( α , T ) is the back state of the 2-rarefactionwave; then the pressure is strictly decreasing along the 2-rarefaction curve from the front to the backstate.Theorem 4.1 shows that R ± ∇ λ ± > in the region where α ≤ (cid:0) T /T i (cid:1) . If the point ( α , T ) islocated in this region and α is sufficiently small, we conclude by (7.7) that the 1-rarefaction curve andthe 2-rarefaction curve from the front to the back state stay in this region as α → . We now continue the analysis of the integral curves for the characteristic fields R ± by considering thebehavior of the component u . By (7.2) and (7.1) we have dudα = ∓ + α (1 − α ) (cid:16) + T i T (cid:17) α (1 − α )2 T i T pλ . (7.10)29 roposition 7.3 (Integral curves in the ( α, u )-plane) . The integral curves of the characteristic fields R ± through a given point in the ( α, u ) -plane are graphs of functions u ± = u ± ( α ) for α ∈ (0 , α ∞ ) . Thefunction u + is decreasing while u − is increasing and they satisfy lim α → u ± ( α ) = const. , lim α → α ∞ − u ± ( α ) = ∓∞ . Moreover, we have u ± ( α ) = O (1) (cid:18) α ∞ − α (cid:19) for α → α ∞ − . Proof.
We denote Λ( α, T ) = + α (1 − α ) (cid:16) + T i T (cid:17) α (1 − α )2 T i T p ( α, T ) λ ( α, T ) . By (7.10), for a fixed α ∈ (0 , α ∞ ), the u -component of the integral curve of R ± satisfying u ( α ) = 0 is u ± ( α ) = ∓ Z αα Λ (cid:0) β, T ( β ) (cid:1) dβ, (7.11)where T ( α ) is defined by (7.8), see Remark 7.1. We refer to Figure 6 for graphs. α u × Figure 6: The integral curves (7.11) of the characteristic field R − in the ( α, u ) plane corresponding tothe parameters η = 0 , ,
10, see (7.8). Here α ∼ u (0) ∼
0. Units in mks.
Since Λ >
0, we deduce that u + is decreasing while u − is increasing in the interval (0 , α ∞ ). We noticethat by (4.9) we have pλ = a p T (1 + α ) vuut + α (1 − α ) (cid:16) ( + T i T ) − + (cid:17) + α (1 − α )( + T i T ) . (7.12)By (7.12), when T → ∞ we have pλ = O (1) √ T uniformly in α and then Λ( α, T ) = O (1) T , where the O (1) term is not singular in α . On the other hand we have, by (7.8) and (7.6), that d ( α ) = 2 log (cid:18) − αα (cid:19) −
52 (1 + α ) − (cid:18) − α ∞ α ∞ (cid:19) + 52 (1 + α ∞ ) = O (1)( α − α ∞ ) for α → α ∞ − . (cid:0) α, T ( α ) (cid:1) = O (1) (cid:16) α ∞ − α (cid:17) for α → α ∞ − . As a consequence, u ± ( α ) = ∓ Z αα Λ (cid:0) β, T ( β ) (cid:1) dβ = O (1) Z αα (cid:18) α ∞ − β (cid:19) dβ = O (1) (cid:18) α ∞ − α (cid:19) → ∞ for α → α ∞ − . When T → pλ = O (1) √ T uniformly in α by (7.12) and then Λ( α, T ) = O (1) T α + O (1) T − for ( α, T ) → (0 , O (1) terms are not singular for α → T ( α ) = O (1) 1log α for α → , whence Λ (cid:0) α, T ( α ) (cid:1) = O (1) q log α for α → . Then − u ± ( α ) = ∓ Z α α Λ (cid:0) β, T ( β ) (cid:1) dβ = O (1) Z α α r log 1 β dβ = O (1) Z α α √ log ββ dβ = O (1) for α → . In the case of very high temperatures one can use an approximate version of Saha’s formula. Moreprecisely, when T i T ∼
0, we approximate Saha’s law (2.13) with n r +1 n e n r = G r +1 g e G r (2 πm e kT ) h . (8.1)We call the corresponding model as the High-Temperature-Limit (HTL) model . By using (8.1) instead of(2.13), we see that formulas (3.2) and (3.3) are replaced by α − α = ¯ κρ T , α = (cid:16) κpT − (cid:17) − , (8.2)respectively. Moreover, we have p = 1 − α κα T , v = Rm κ α − α T − . (8.3)We rewrite now the main results we obtained in the previous sections for the HTL model. The equationof state is again (2.17). On the contrary, due to the approximation procedure, we discard the componentdue to the ionization energy in the internal energy; then, the internal energy, the enthalpy (2.19) and thedimensionless entropy (3.12) become e = 32 Rm (1 + α ) T, H = 52 Rm (1 + α ) T, η = − log p + 2 tanh − α + 52 α + 52 log T + Const.Notice that, by (3.14), the dimensionless entropy η now explicitly depends only on α : η = − − αα + 52 α + const . (8.4)Of course, η depends on both T and p through α . As a consequence, if α is constant, then clearly theentropy is constant. The following result is analogous to Lemma 4.2. Lemma 8.1 (Eigenvalues) . In the HTL model in Lagrangian coordinates, the characteristic speeds are λ ± = ± pa s T (1 + α ) , λ = 0 . (8.5) If we still denote by R ± the eigenvectors corresponding to λ ± as in (4.11) , we have R ± ∇ log λ ± = ± p and then both λ ± are genuinely nonlinear. The eigenvalue λ is linearly degenerate. α = 0 we formally find the eigenvalues of polytropic gas-dynamics with γ = . Alsonotice that the degree of ionization is constant along the ± characteristic directions, i.e., R ± ∇ α = 0.This means that both η and α are Riemann invariants for the fields associated to λ ± ; clearly, they arenot independent by (8.4). A pair of Riemann invariants for λ is still { u, p } .Propositions 7.1–7.3 and formula (7.8) read now as follows. Proposition 8.1 (Integral curves) . In the HTL model, the characteristic vector fields R ± with respectto ( α, u, T ) coordinates are expressed as R ± = ∓ a q T (1+ α )52 T , R = α (1 − α )0 T . (8.6) The integral curves of the fields R ± in the ( α, T ) -plane are the straight lines α = const. More precisely,the integral curves through ( α , u , T ) are u − u = ∓ a p α )( T − T ) , while the integral curvesin the ( p, u, α ) -space through ( p , u , α ) are u − u = ∓ p α ) a κα − α ! (cid:18) p − p (cid:19) . (8.7) Proof.
The first two statements follow directly. About the formulas for integral curves, by (8.6) we have dudT = ∓ a p α )2 T − , whence the formula in the ( α, u, T )-space. At last, it follows from (8.3) that along the integral curve of R ± through ( p , u , α ) we have p = − α κα T ; hence, T = (cid:16) κα p − α (cid:17) and (8.7) follows. Remark 8.1.
The integral curves in the ( p, u, α ) -space have a form similar to those of the polytropic gaswith the adiabatic exponent γ = , where the entropy replaces α ; see [7] or [1, (2.7), (2.8)]. Notice thatin the HTL model the integral curves of the general model reduce to the asymptotes, see Figure 5. Toemphasize the above correspondence we introduce the function H ( α ) = 25 log α + 12 log(1 + α ) −
15 log(1 − α ) . By using this pseudo-entropy, we have, see [10] for an analogous expression for -polytropic gasdynamics, u − u = ∓√ aκ e H ( α ) (cid:18) p − p (cid:19) . Compare η ( α ) = 2 log α − − α ) + 52 α and H ( α ) = 25 log α + 310 log(1 + α ) −
15 log(1 − α ) . Now, we study the Hugoniot loci in the HTL model. The branch corresponding to contact disconti-nuities is given by u = u , p = p and α = (cid:16) κp T − (cid:17) − . Then, we focus on shock waves. Aboutthe thermodynamic part of the Rankine-Hugoniot condition we have the following result, to be comparedwith Proposition 5.1; the proof is left to the reader.
Lemma 8.2 (Variation of the temperature) . In the HTL model, the thermodynamic part of the Hugoniotlocus (5.3) is expressed as T (1 + α ) T (1 + α ) = 4 + pp p p (8.8)32 r else as T α ) + − α − α ! (cid:18) αα (cid:19) (cid:18) T T (cid:19) = T α ) + − α − α ! (cid:18) α α (cid:19) (cid:18) TT (cid:19) . (8.9)The next result is analogous to that given in Proposition 5.2. An important difference that weemphasize in its proof is that the corresponding function T = T ( α ) is not differentiable at α , where thetangent to its graph becomes vertical; see Figure 7. Proposition 8.2 ( T as a function of α ) . In the ( α, T ) -plane, the thermodynamic part of the Hugoniotlocus of ( α , T ) is the graph of a strictly increasing function T = T ( α ) , for α ∈ (0 , . Moreover, thefunction T is differentiable for α ∈ (0 , \ { α } , and tends to , ∞ as α → , , respectively.Proof. Arguing as to obtain (5.19) we have formally dTdα − α − α ! (cid:18) αα (cid:19) (cid:18) T T (cid:19) + 53 − α − α ! (cid:18) α α (cid:19) (cid:18) TT (cid:19) −
83 (1 + α ) = 4 T α − α ) (cid:18) α (cid:19) (cid:18) TT (cid:19) + T
83 + 2(1 − α )3 α (cid:20) α (2 − α )(1 − α ) (cid:21) (cid:18) T T (cid:19) . (8.10)By introducing p and p , we deduce dTdα " (1 + α ) (cid:18) p p (cid:19) + 53 (1 + α ) (cid:18) pp (cid:19) T T −
83 (1 + α ) = 43 (cid:18) pp (cid:19) T (1 + α ) α (1 − α ) + T "
83 + 23 (cid:18) p p (cid:19) (1 + α )(2 − α ) α (1 − α ) . (8.11)Note that in both formulas the coefficient of dT /dα vanishes at ( α , T ), differently from (5.19) wherethe exact Saha’s law is used. By (8.8) we deduce (cid:18) pp (cid:19) T T (1 + α ) = (cid:18) p p (cid:19) (1 + α ) + 4(1 + α ) − α ) T T . (8.12)Then, formulas (8.11) and (8.12) imply1
T dTdα = 1 + (cid:16) p p (cid:17) (1+ α )(2 − α )4 α (1 − α ) + (cid:16) pp (cid:17) T (1+ α )2 T α (1 − α ) (1 + α ) (cid:16) p p − (cid:17) + h α − (1 + α ) T T i . (8.13)We examine the denominator of (8.13), see (5.10), (5.11) and (5.19): to this aim we setΦ( α, T ) = (1 + α ) (cid:18) p p − (cid:19) + 52 (cid:20) α − (1 + α ) T T (cid:21) . Let us recall that Φ( α , T ) = 0 . We have ∂ Φ ∂α ( α , T ) = 2(1 + α ) α (1 − α ) + 52 , ∂ Φ ∂T ( α , T ) = 0 , ∂ Φ ∂T ( α , T ) = 154 T (1 + α ) . Then, in a neighborhood of ( α , T ) the set (cid:8) Φ( α, T ) = 0 (cid:9) is the graph of a function ˜ α = ˜ α ( T ) and˜ α ( T ) − α ∼ − α )8 T h α ) α (1 − α ) + i ( T − T ) (8.14)33s T → T . Let us denote the numerator of (8.13) by Ψ( α, T ) . Since Ψ( α , T ) = 0 , the thermodynamicpart of the Hugoniot locus of ( α , T ) is represented in a small neighborhood of ( α , T ) by the graph ofa function α = α ( T ) and T dαdT = Φ( α, T )Ψ( α, T ) . Clearly, dαdT ( T ) = 0 and Φ T ( α , T ) = 0 yields d αdT ( T ) = 0 . Thus we conclude that α − α = O (1)( T − T ) and, by (8.14), the thermodynamic part of the Hugoniot locus at ( α , T ) stays in the region Φ ( α, T ) > , α = α , in a neighborhood of α ; see Figure 7 on the right. α T ( K ) α T ( K ) Figure 7: The thermodynamic part of the Hugoniot locus for the HTL model. On the left: the function T = T ( α ). On the right: a detail around ( α , T ) of the graph of the function T = T ( α ) and the zeroset Φ( α, T ) = 0 (dashed line), see the proof of Proposition 8.2; the function Φ is positive on the rightof the dashed line. Here α = 0 . T = 300K in order to emphasize the vertical tangent at ( α , T ). Let us denote Φ( α ) = Φ (cid:0) α, T ( α ) (cid:1) with a slight abuse of notation. By (8.13), the ratio Π := p p satisfies the quadratic equation(1 + α )Π + 4 (cid:20) (1 + α ) − (1 + α ) T T (cid:21) Π − (1 + α ) T T = 0 . By setting ∆ = (1 + α ) − (1 + α ) T T , we see that Π is expressed as Π = q +(1+ α )(1+ α ) T T − α ; hence,Φ( α ) = r + (1 + α )(1 + α ) T T − (1 + α ) + 12 ∆ . Notice that Φ( α ) = 0. Suppose that ∆ > α. Then Φ( α ) > < ∆ ≤ α. Then Φ( α ) = 0 implies a quadratic equation for ∆ ∆ + (1 + α )∆ + (1 + α )(1 + α ) T T − (1 + α ) = 0 . (8.15)Note that (1 + α )∆ = (1 + α ) − (1 + α )(1 + α ) T T . Then we find that the equation (8.15) turns out tobe ∆ = 0 , which is a contradiction. Thus we have proved that Φ( α ) > α = α . Finally, by the same argument as proving Proposition 5.2, we can show that T tends to 0 , ∞ as α → , , respectively.Next, we study the variation of both p and v along the Hugoniot locus, see Proposition 5.4. Lemma 8.3 ( p and v as functions of α ) . Let T = T ( α ) be as in Proposition 8.2. Then ddα p (cid:0) α, T ( α ) (cid:1) > and ddα v (cid:0) α, T ( α ) (cid:1) > for α ≥ α . roof. For simplicity we only deal with pressure. By letting T i T ∼ D ( α ) is positiveand N ( α ) is54 α (1 − α ) + 5(1 + α ) (2 − α )16 p p −
38 (1 + α ) p p + (1 + α ) > −
38 (1 + α ) p p + (1 + α ) > , for p > p . Clearly the above expression is positive for all p which are close to p . We conclude the proofby the same argument proving Proposition 5.4.Now, we study the asymptotic behavior of the Hugoniot locus for extreme values. The proof isanalogous to that of Proposition 5.3 and then is omitted.
Proposition 8.3 (Asymptotics) . On the Hugoniot locus (5.3) , if T → then α → and if T → ∞ , then α → . More precisely, we have α ∼ α √ − α (cid:18) TT (cid:19) for T → and − α ∼ − α α (cid:18) TT (cid:19) − for T → ∞ . Now, we examine the kinetic part of the Hugoniot locus.
Proposition 8.4 (The kinetic part of the Hugoniot locus) . In the HTL model, the kinetic part of theHugoniot locus of ( α , u , T ) is ( u − u ) = 3 a T (1 + α )( p − p ) p (4 p + p ) . (8.16) Proof.
The kinetic part (5.3) of the Rankine-Hugoniot condition is ( u − u ) = − ( p − p )( v − v ). By(2.18) and Lemma 8.2 we have vv = (cid:18) p p (cid:19) T (1 + α ) T (1 + α ) = (cid:18) p p (cid:19) pp p p and then (8.16) follows by ( u − u ) = 2 v ( p − p ) p + p . (8.17)Notice that (8.17) is the same formula for -polytropic gasdynamics. In this paper we studied a model, consisting of three equations, for the macroscopic motion of an ionizedgas in one space dimension. The model is closed by a state law and by a further thermodynamicalrelation called Saha’s law. The degree of ionization α and the temperature T are two proper independentthermodynamical variables.We showed that these equations constitute a strictly hyperbolic system of conservation laws; thisimplies that the initial-value problem is well-posed locally in time for sufficiently smooth initial data.The geometric properties of the system are rather complicated and, to the best of our knowledge, havenever been pointed out in the literature. In particular, we found the loss of genuine nonlinearity in abounded region for both forward and backward characteristic fields.About the shock structure of the system, we found that the thermodynamical part of the Hugoniotlocus of a fixed state ( α , T ) constitutes a simple graph in the ( α, T )-plane: more precisely, the temper-ature is an increasing function of the degree of ionization. Along this locus, also the pressure increaseswith the degree of ionization, which shows that the compressive branch of the locus is the part α ≥ α .We notice that, even if the degree of ionization is small, the compressive branch may cross the region35here the forward and backward characteristic fields are not genuinely nonlinear, which also causes thedecrease of the physical entropy.Nevertheless, we established precise conditions which limit the “classical” region where the forwardand backward characteristic fields are genuinely nonlinear and where the physical entropy is strictlyconcave. In such a region, the classical theory of shock waves is valid and the compression branch isadmissible. In a forthcoming paper, we will discuss the electromagnetic shock tube filled with monatomichydrogen gas and we expect that the actual experiments only involve such a classical region.We also proposed the High-Temperature-Limit model , an approximation of the exact model in thecase that the energy (or the temperature) is extremely high when compared with the dissociation energy.In this model, the physical entropy is just a function of the degree of ionization and does not depend onthe temperature. Hence, the degree of ionization is a Riemann invariant. The integral curves (rarefactioncurves) are similar to those of the monatomic ( -polytropic) gas. The shape of the thermodynamic partof the Hugoniot locus is quite different from that of the exact system considered in the previous sections. Acknowledgments
The second author is member of the
Gruppo Nazionale per l’Analisi Matematica, la Probabilit´a e le loroApplicazioni (GNAMPA) of the
Istituto Nazionale di Alta Matematica (INdAM). He also was supportedby the project
Balance Laws in the Modeling of Physical, Biological and Industrial Processes of GNAMPA.
References [1] F. Asakura. Wave-front tracking for the equations of non-isentropic gas dynamics—basic lemmas.
ActaMath. Vietnam. , 38(4):487–516, 2013.[2] H. Bradt.
Astrophysic Processes . Cambridge University Press, Cambridge, 2008.[3] C. M. Dafermos.
Hyperbolic conservation laws in continuum physics . Springer-Verlag, Berlin, third edition,2010.[4] E. Fermi.
Thermodynamics . Dover Publications, 1956.[5] K. Fukuda, R. Okasaka, and T. Fujimoto. Ionization equilibrium of He-H plasma heated by a shock wave(Japanese).
Kaku-Yuga Kenkyu (Studies on Nuclear Fusion) , 19(3):199–213, 1967.[6] L. D. Landau and E. M. Lifshitz.
Course of theoretical physics. Vol. 6 . Pergamon Press, Oxford, secondedition, 1987. Fluid mechanics, Translated from the third Russian edition by J. B. Sykes and W. H. Reid.[7] T. P. Liu. Solutions in the large for the equations of nonisentropic gas dynamics.
Indiana Univ. Math. J. ,26(1):147–177, 1977.[8] R. Menikoff and B. J. Plohr. The Riemann problem for fluid flow of real materials.
Rev. Modern Phys. ,61(1):75–130, 1989.[9] M. N. Saha. Ionization in the solar chromosphere.
Phil. Mag. , 40(238):472–488, 1920.[10] J. Smoller.
Shock Waves and Reaction-Diffusion Equations . Springer-Verlag, New York, second edition, 1994.[11] W. Vincenti and C. J. Kruger.
Introduction to Physical Gas Dynamics . Wiley, New York, 1965.[12] Y. B. Zel’dovich and Y. P. Raizer.
Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena .Dover, New York, 2002..Dover, New York, 2002.