Abstract
Phase transitions in a model of van der Waals fluid flows in a nozzle with discontinuous cross-sectional area are investigated. The model admits a physically unstable elliptic region which causes phase transitions, and a jump in cross-sectional area which causes stationary contact waves. Compositions between waves in different characteristic fields are constructed differently in subsonic or supersonic regions. A new kind of waves is introduced which helps establish the existence of solutions of the Riemann problem. There are regions in which the Riemann problem admits a unique solution. Resonant phenomena occur when shock waves associated with different characteristic fields propagate with the same speed. Multiple solutions are also observed even in a strictly hyperbolic region not only by the modeling, but also by the type of a van der Waals fluid.
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1 Introduction
We are interested in this paper the phase transition problem for the following isentropic model of van der Waals fluid flows in a nozzle with discontinuous cross-sectional area
where \(\rho \) is the density, u is the particle velocity, and p is the pressure of the fluid. The cross-sectional area \(a=a(x)\) of the nozzle is assumed to be piecewise constant. The pressure p is a smooth function of the density \(\rho \) and is assumed to be of the van der Waals type with phase transitions; that is, there are four values \(\rho _*,\rho ^*,\alpha ,\beta \) (\(\rho _*<\alpha<\rho ^*<\beta \)) such that
where \((.)'=\hbox {d}/\hbox {d}\rho \), and
Furthermore, we assume that
As shown in Sect. 2, the 1- and 3-characteristic fields of the system (1.1) are not genuinely nonlinear at \(\rho _*\in \Gamma =(0,\alpha )\cup (\beta ,+\infty )\), and the interval \((\alpha ,\beta )\) is the elliptic region of the system (1.1) under the condition (1.2). Note that these assumptions are consistent with the simpler ones in the Lagrange coordinate by considering the pressure
as a function of the specific volume v. Indeed, it is easy to check that
Thus, it holds
which mean that \(v_*=1/\rho ^*,v^*=1/\rho _*\) are two inflection points of the function P(v). Moreover, \(\alpha _v=1/\beta ,\beta _v=1/\alpha \) are local minimum point and local maximum point of the function P(v).
Even when the nozzle is flat, i.e., the cross-sectional area is constant, the Riemann problem may not have any solution. To deal with this nonexistence, we introduce a new kind of waves, called contact phase boundaries. Briefly, a contact phase boundary is a shock wave propagating with the same speed as the fluid. If the nozzle has a discontinuous cross-sectional area, we will show that the Riemann problem admits a unique solution in certain regions. The resonant phenomenon can appear when the solution contains up to three shock waves of the same shock speed. Two admissible stationary waves in strictly hyperbolic region can exist for a fluid of van der Waals type. Thus, multiple solutions may be available due to two reasons: the resonance of the model (1.1) as observed in [15], and the type of a van der Waals fluid by this work.
Note that the Riemann problem for (1.1) with an isentropic and ideal fluid was investigated in [15]. The non-isentropic case with a convex EOS was considered by [2, 23]. Recently, the Riemann problem for (1.1) with an isentropic fluid of van der Waals type without phase transitions was studied in [26]. Note that the model (1.1) may be written in the form of a system of balance laws in non-conservative form. Shock waves and related topics in several systems of this type were investigated in [10, 11, 14, 18]. Furthermore, two-phase flow models involving non-conservative systems of balance laws were considered in [4, 5, 13, 17]. The Riemann problem for various non-conservative models was considered in [2, 8, 21, 23, 25]. The Riemann problem for two-phase flow models was considered in [3, 24]. The related Godunov-type schemes and Riemann solvers were presented in [1, 6, 22]. See also the references therein.
The organization of this paper is as follows. Section 2 is devoted to basic properties of the model (1.1)–(1.4). The Riemann problem for a flat nozzle is considered in Sect. 3. Section 4 is devoted to stationary waves and their admissibility. Finally, the Riemann problem for a discontinuous cross-sectional area is investigated in Sect. 5.
2 Preliminaries
First, it holds for a smooth unknown \(U=(\rho ,u,a)^T\) that
where \((.)'=\hbox {d}/\hbox {d}\rho \), and the function h is defined by
The system (2.1) can therefore be put in the non-conservative form of balance laws
where
Whenever \(p'(\rho )\ge 0\), the matrix A(U) admits three real eigenvalues
where
represents the local sound speed.
The system (1.1) is hyperbolic for \(\rho \in \Gamma =(0,\alpha )\cup (\beta ,+\infty )\). Beside that, the system (1.1) is elliptic for \(\rho \in (\alpha ,\beta )\). The values \(\rho \) in the interval \((0,\alpha )\) will be called the phase-A region, and the values \(\rho \) in the interval \((\beta ,+\infty )\) will be called the phase-B region.
The corresponding eigenvectors can be chosen as
provided \(p'(\rho )\ge 0\). It is convenient to set
Thus, one can see that the system (1.1) is strictly hyperbolic in the domains \(G_{1A}\cup G_{2A}\cup G_{3A}\cup G_{1B}\cup G_{2B}\cup G_{3B}\) and is not strictly hyperbolic on the surfaces \(C_{A}^+\cup C_{A}^-\cup C_{B}^+\cup C_{B}^-\).
Regardless of a, a state \(U=(\rho ,u)\) is called supersonic if \(|u|>c=\sqrt{p'(\rho )}\), subsonic if \(|u|<c\), and sonic if \(|u|=c\). Thus, the regions \(G_{1A}, G_{1B}, G_{3A}\) and \(G_{3B}\) defined in (2.6) are supersonic, the regions \(G_{2A}, G_{2B}\) are subsonic, and the curves \(C_{A}^\pm , C_{B}^\pm \) are sonic.
Evidently, the 2-characteristic field is linearly degenerate. Furthermore, it holds that
where \({\mathcal {A}}(\rho )\) is defined by (1.2). Thus, the 1- and 3- characteristic fields of the system (1.1) are not genuinely nonlinear at \(\rho _*\) with
Next, let us consider shock waves of (1.1). The third equation of the system (1.1) yields the Rankine–Hugoniot relation of the form
where \({\bar{\lambda }}\) is the shock speed, and \([a]:=a_+-a_-\) (\(a_+\) and \(a_-\) are the right-hand state and left-hand state traces). Equation (2.9) implies that \(a_+=a_-\), or \(a_+\ne a_-\), but the shock speed vanishes (\({\bar{\lambda }}=0\)).
If \([a]=0\), the nozzle is flat and the system (1.1) is reduced to the usual gas dynamics equations for an isentropic fluid
The Rankine–Hugoniot relations associated with the system (2.10) read
If \([a]\ne 0\) and \({\bar{\lambda }}=0\), arguing as in [15], we obtain the jump relations
One may deduce from the first equation of (2.11) that
where \(U_0=(\rho _0,u_0,a_0)\) is a given state on one side of the shock. Substituting the last expression for \({\bar{\lambda }}\) in the second equation of (2.11) and after simplifying, we get
Using the fact that the i-Hugoniot curve issued from \(U_0\) is tangent to the i-eigenvector \(r_i(U_0)\), (\(i=1,3\)), we obtain the Hugoniot curves associated with the first- and the third- characteristic fields by
Note that the usual admissibility criterion of shock waves for van der Waals fluids is the Liu entropy condition:
for all U between \(U_0\) and \(U_1\) on the i-Hugoniot curves, \(i=1,3\).
Define the i-shock curves \(\overrightarrow{S_i} (U_0)\) (or \(\overleftarrow{S_i}(U_0)\)) consisting of all right-hand states (or left-hand states) U that can be connected from \(U_0\) (or can connect to \(U_0\)) by a i-shock wave, and satisfy the Liu entropy condition (2.14), \(i=1,3\). It is easy to see that these shock curves can be parameterized by \(\rho \mapsto u(\rho )\) as
where the shock sets \(\Omega _1 (\rho _0)\) and \(\Omega _3^B(\rho _0)\) consist of all values \(\rho \) such that the corresponding point on the Hugoniot curves \( {\mathcal {H}}_1(U_0)\) and \({\mathcal {H}}_3(U_0)\) can be connected to \(U_0\) by a 1-shock from the right and 3-shock from the left, respectively.
Note that \({\overline{\lambda }}(U_0,U)=\dfrac{[\rho u]}{[\rho ]}\). A straightforward calculation yields
Moreover, the Liu entropy condition (2.14) for 1-shocks implies that
for all \(\rho \) between \(\rho _0\) and \(\rho _1\) whenever \(\overline{\lambda _1}(U_0,U)\) and \(\overline{\lambda _1}(U_0,U_1)\) are well defined, that is
The shock sets \(\Omega _1(\rho _0)\) and \(\Omega _3^B(\rho _0)\) will be determined in the next section.
Next, we consider the i-rarefaction waves, i.e., smooth self-similar solutions to the system (1.1). The waves satisfy the following equation
It holds along the 1-integral curve that
The last equations yield
The following lemma provides us with the descriptions of the curve of forward 1-rarefaction waves.
Lemma 2.1
(Forward curve of 1-rarefaction waves) Given a left-hand state \(U_0\), the 1-rarefaction curve \(\overrightarrow{R_1}(U_0)\) consisting of all right-hand states U that can be connected from \(U_0\) by a rarefaction wave is given by
-
Case 1.
If \(\rho _0 \in (0,\rho _*]\), then
$$\begin{aligned} \Pi _1(\rho _0)=(0,\rho _0]. \end{aligned}$$ -
Case 2.
If \(\rho _0 \in (\rho _*,\alpha ]\), then
$$\begin{aligned} \Pi _1(\rho _0)=[\rho _0,\alpha ]. \end{aligned}$$ -
Case 3.
If \(\rho _0 \in [\beta ,+\infty )\), then
$$\begin{aligned} \Pi _1(\rho _0)=[\beta ,\rho _0]. \end{aligned}$$
The curve of backward 3-rarefaction waves is described by the following lemma.
Lemma 2.2
(Backward curve of 3-rarefaction waves) Given a right-hand state \(U_0\), the 3-rarefaction curve \(\overleftarrow{R_3}(U_0)\) consisting of all left-hand states U that can connect to \(U_0\) by a rarefaction wave is:
-
Case 1.
If \(\rho _0 \in (0,\rho _*]\), then
$$\begin{aligned} \Pi _3^B(\rho _0)=(0,\rho _0]. \end{aligned}$$ -
Case 2.
If \(\rho _0 \in (\rho _*,\alpha )\), then
$$\begin{aligned} \Pi _3^B(\rho _0)=[\rho _0,\alpha ]. \end{aligned}$$ -
Case 3.
If \(\rho _0 \in (\beta ,+\infty )\), then
$$\begin{aligned} \Pi _3^B(\rho _0)=[\beta ,\rho _0]. \end{aligned}$$
From assumption (1.4), there exists two values \(\beta ^{-\natural }\) and \(\alpha ^{-\natural }\) (\(\beta ^{-\natural }<\alpha<\beta <\alpha ^{-\natural }\)) such that
We call \(\tau _{\alpha }:[p(\beta ^{-\natural }),p(\alpha )]\rightarrow [\beta ^{-\natural },\alpha ]\) (respectively, \(\tau _{\beta }:[p(\beta ),p(\alpha ^{-\natural })]\rightarrow [\beta ,\alpha ^{-\natural }]\)) is an inverse of the function \(p(\rho )\) in the interval \([\beta ^{-\natural },\alpha ]\) (respectively, \([\beta ,\alpha ^{-\natural }]\)). It holds that the functions \(\tau _{\alpha }(.)\) and \(\tau _{\beta }(.)\) are monotone increasing.
Let us define the mapping
by
It is easy to see that
which yields
for \(i=1,3\). We will show that the shock wave between \(U_0,U_1\) defined by (4.3) satisfies the standard entropy inequality for the general fluid flow in a nozzle (see [9, 12])
where S is the specific entropy and is automatically satisfied for this shock wave. Indeed, the entropy inequality (2.26) (in the sense of distributions) for this shock wave is equivalent to
where \([A]=A_+-A_-\) stands for the jump of a quantity A across the discontinuity, for \(i=1,3\). Since S is constant in the isentropic model, and a, u remain constant across this shock wave, it follows from (2.25) that the left-hand side of (2.27) identically vanishes:
and so (2.26) holds. Thus, this kind of shock waves is admissible and in the following, such a shock wave is referred to as a contact phase boundary. Physically, a contact phase boundary travels with the same velocity as the fluid. In the following, the notation \(W_0(U_0,U)\) will be used to describe a contact phase boundary connecting \(U_0\) to U.
3 Riemann Problem for a Flat Nozzle
In this section, we consider the Riemann problem for a flat nozzle, that is, when \(a=\) constant. This enables us to define Riemann solvers for isentropic van der Waals fluids in the Eulerian coordinates, which will be useful for the descriptions of the constructions of the Riemann problem for non-flat nozzle in Sect. 5. The system (1.1) for constant nozzle becomes
with the initial data of the form
where \((\rho _l,u_l),(\rho _r,u_r)\) are constant states.
3.1 The Shock Sets
We want to employ important calculations of the isentropic ideal van der Waals fluid flows in the Lagrangian coordinates, where we denote by \(p=P(v)\) the pressure as a function of the specific volume v. Given \(U_0(v_0,u_0)\), we call \(\Omega _{v1}(v_0)\) is the 1-shock set, see [16]. It follows from the Liu entropy condition that
for all v between \(v_0\) and \(v_1\) whenever the 1-shock speeds are well defined, that is
We are going to verify that \(\rho _1\in \Omega _1(\rho _0)\) and \(v_1\in \Omega _{v1}(v_0)\) are equivalent. Note that \(P(v)=p(\rho )=p(1/v)\) and v between \(v_0,v_1\) is equivalent with \(\rho =\dfrac{1}{v}\) between \(\rho _0,\rho _1\). It is easy to see that the inequality
\(\rho \dfrac{p(\rho )-p(\rho _0)}{\rho -\rho _0}\le \rho _1\dfrac{p(\rho _1)-p(\rho _0)}{\rho _1-\rho _0}\) for all \(\rho \) between \(\rho _0\) and \(\rho _1\) whenever the condition (2.18) is satisfied is equivalent with the following one
\(\dfrac{P(v)-P(v_0)}{v-v_0}\ge \dfrac{P(v_1)-P(v_0)}{v_1-v_0}\) for all v between \(v_0\) and \(v_1\) whenever the condition (3.4) is satisfied.
The above argument leads us to the following result.
Lemma 3.1
The Liu entropy condition for 1- and 3-shock waves in Lagrangian coordinates and Eulerian coordinates are equivalent.
Let the values \(c,d,b^{-\natural }\) be defined by
Set
It is not difficult to check that for any \(v\in (b^{-\natural },a^{-\natural })\) there exists two lines which are passing through the point with the coordinate v and are also tangent to the graph, which define the two functions denoted by \(\psi ^{\natural }(v)\) and \(\varphi ^{\natural }(v)\) such that
Observe that \(\varphi ^{\natural }(v)<\psi ^{\natural }(v)\).
We define the following value functions
such that
We define \(k_1^{-1},k_2^{-1},k_3^{-1}\) are the inverses of functions \(k_1,k_2,k_3\):
The shock sets are now can be described using the above notations.
Lemma 3.2
(Forward 1-shock set). Given a left-hand state \(U_0\), the 1-shock curve \(\overrightarrow{S_1}(U_0)\) (left-to-right) consisting of all right-hand states U that can be connected from \(U_0\) by a 1-shock satisfies the Liu entropy condition (2.14) is given by
The forward 1-shock set \( \Omega _1(\rho _0)\) is described as follows.
-
(i)
If \(\rho _0\in (b_2,+\infty )\), then
$$\begin{aligned} \Omega _1(\rho _0)=[\rho _0,+\infty ). \end{aligned}$$ -
(ii)
If \(\rho _0\in [\beta ,b_2]\), then
$$\begin{aligned} \Omega _1(\rho _0)=[k_1(\rho _0),k_2(\rho _0)]\cup [\rho _0,+\infty ). \end{aligned}$$ -
(iii)
If \(\rho _0\in (\rho _*,\alpha ]\), then
$$\begin{aligned} \Omega _1(\rho _0)=[k_1(\rho _0),\rho _0]\cup [k_2(\rho _0),+\infty ). \end{aligned}$$ -
(iv)
If \(\rho _0\in (0,\rho _*]\), then
$$\begin{aligned} \Omega _1(\rho _0)=[\rho _0,k_1(\rho _0)]\cup [k_3(\rho _0),+\infty ). \end{aligned}$$
Lemma 3.3
(Backward 3-shock set) Given a right-hand state \(U_0\), the 3-shock curve \(\overleftarrow{S_3}(U_0)\) (right-to-left) consisting of all left-hand states U that can connect to \(U_0\) by a 3-shock satisfies the Liu entropy condition (2.14) is given by
The backward 3-shock set \( \Omega _3^B(\rho _0)\) is described as follows.
-
(i)
If \(\rho _0\in (b_2,+\infty )\), then
$$\begin{aligned} \Omega _3^B(\rho _0)=[\rho _0,+\infty ). \end{aligned}$$ -
(ii)
If \(\rho _0\in [\beta ,b_2]\), then
$$\begin{aligned} \Omega _3^B(\rho _0)=[k_1(\rho _0),k_2(\rho _0)]\cup [\rho _0,+\infty ). \end{aligned}$$ -
(iii)
If \(\rho _0\in (\rho _*,\alpha ]\), then
$$\begin{aligned} \Omega _3^B(\rho _0)=[k_1(\rho _0),\rho _0]\cup [k_2(\rho _0),+\infty ). \end{aligned}$$ -
(iv)
If \(\rho _0\in (0,\rho _*]\), then
$$\begin{aligned} \Omega _3^B(\rho _0)=[\rho _0,k_1(\rho _0)]\cup [k_3(\rho _0),+\infty ). \end{aligned}$$
In the rest of this section, we will define the i-wave curves \(\overrightarrow{W_i}(U_0)\) (\(i=1,3\)) consist of all right-hand states that can be connected from \(U_0\) by i-shock, i-rarefaction or composite of i-shock, i-rarefaction and contact phase boundary waves. Beside that, \(\overleftarrow{W_i}(U_0)\) (\(i=1,3\)) consists of all left-hand states that can connect to \(U_0\) by i-shock, i-rarefaction or composite of i-shock, i-rarefaction and contact phase boundary waves. The following notations will be used:
-
(i)
\(W_i(U_1,U_2)\): a forward i-wave from \(U_1\) to \(U_2\). (It can be i-shock wave (\(W_i=S_i\)) or i-rarefaction wave (\(W_i=R_i\)).)
\(W_i^B(U_1,U_2)\): A backward i-wave from \(U_1\) to \(U_2\). (It can be backward i-shock wave (\(W_i^B=S_i^B\)) or backward i-rarefaction wave (\(W_i^B=R_i^B\)).)
-
(ii)
\(W_i(U_1,U_2)\oplus W_j(U_2,U_3)\) (or \(W_i^B(U_1,U_2)\oplus W_j^B(U_2,U_3)\)) : a forward (or backward) i-wave from \(U_1\) to \(U_2\) is followed by a forward (or backward) j-wave from \(U_2\) to \(U_3\).
3.2 Forward 1-Wave Curves
Given a left-hand state \(U_0(u_0,\rho _0,a_0)\), the 1-wave curve \(\overrightarrow{W_1}(U_0)\) consisting of all right-hand states \(U(u,\rho ,a_0)\) that can be connected from \(U_0\) by a combination of 1-shock waves, 1-rarefaction fans and contact phase boundary with assumption (1.2) has the following description.
3.2.1 Case 1: \(\rho _0\in (b_2,+\infty )\)
-
(i)
If \(\rho \in (b_2,+\infty )\), then the solution is
$$\begin{aligned} {Sol_1}(U_0,U)= S_1(U_0,U), \end{aligned}$$where \(U\in \overrightarrow{S_1}(U_0)\).
-
(ii)
If \(\rho \in [\beta ,\rho _0]\), then the solution is
$$\begin{aligned} {Sol_1}(U_0,U)=R_1(U_0,U), \end{aligned}$$where \(U\in \overrightarrow{R_1}(U_0)\).
-
(iii)
If \(\rho \in [\beta ^{-\natural },\alpha ]\) and \(\varphi _0(\rho )>\rho _0\), then the solution is
$$\begin{aligned} {Sol_1}(U_0,U)=S_1(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ^{-\natural },\alpha ]\) and \(\varphi _0(\rho )<\rho _0\), then the solution is
$$\begin{aligned} {Sol_1}(U_0,U)=R_1(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{R_1}(U_0),\rho _1=\varphi _0(\rho )\).
-
(iv)
If \(\rho \in [\rho _*,\beta ^{-\natural })\), then the solution is
$$\begin{aligned} {Sol_1}(U_0,U)=R_1(U_0,U_1)\oplus S_1(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{R_1}(U_0),U\in \overrightarrow{S_1}(U_1),\rho _1=k_2^{-1}(\rho )\).
-
(v)
If \(\rho \in (0,\rho _*)\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=R_1(U_0,U_1)\oplus S_1(U_1,U_2)\oplus R_1(U_2,U), \end{aligned}$$where \(U_1\in \overrightarrow{R_1}(U_0),U_2\in \overrightarrow{S_1}(U_1),U\in \overrightarrow{R_1}(U_2),\rho _1=b_2,\rho _2=b_1\).
3.2.2 Case 2: \(\rho _0\in [\beta ,b_2]\)
-
(i)
If \(\rho \in (\rho _0,+\infty )\cup [k_1(\rho _0),k_2(\rho _0))\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U), \end{aligned}$$where \(U\in \overrightarrow{S_1}(U_0)\).
-
(ii)
If \(\rho \in [\beta ,\rho _0]\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=R_1(U_0,U), \end{aligned}$$where \(U\in \overrightarrow{R_1}(U_0)\).
-
(iii)
If \(\rho \in [\beta ^{-\natural },\alpha ]\) and \(\rho _0\le \varphi _0(\rho )\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ^{-\natural },\alpha ]\) and \(\rho _0> \varphi _0(\rho )\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=R_1(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{R_1}(U_0),\rho _1=\varphi _0(\rho )\).
-
(iv)
If \(\rho \in [k_2(\rho _0),\beta ^{-\natural })\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=R_1(U_0,U_1)\oplus S_1(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{R_1}(U_0),U\in \overrightarrow{S_1}(U_1),\rho _1=k_2^{-1}(\rho )\).
-
(v)
If \(\rho \in (0,k_1(\rho _0))\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U_1)\oplus R_1(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),U\in \overrightarrow{R_1}(U_1),\rho _1=k_1(\rho _0)\).
3.2.3 Case 3: \(\rho _0\in (\rho _*,\alpha ]\)
-
(i)
If \(\rho \in [k_2(\rho _0),+\infty )\cup [k_1(\rho _0),\rho _0)\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U), \end{aligned}$$where \(U\in \overrightarrow{S_1}(U_0)\).
-
(ii)
If \(\rho \in [\rho _0,\alpha ]\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=R_1(U_0,U), \end{aligned}$$where \(U\in \overrightarrow{R_1}(U_0)\).
-
(iii)
If \(\rho \in [\alpha ^{-\natural },k_2(\rho _0))\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=R_1(U_0,U_1)\oplus S_1(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{R_1}(U_0),U\in \overrightarrow{S_1}(U_1),\rho _1=k_2^{-1}(\rho )\).
-
(iv)
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(\varphi _0(\rho )>\rho _0\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=R_1(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{R_1}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(k_1(\rho _0)\le \varphi _0(\rho )\le \rho _0\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(k_1(\rho _0)> \varphi _0(\rho )\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U_1)\oplus R_1(U_1,U_2)\oplus W_0(U_2,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),U_2\in \overrightarrow{R_1}(U_1), \rho _1=k_1(\rho _0),\rho _2=\varphi _0(\rho )\).
-
(v)
If \(\rho \in (0,k_1(\rho _0))\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U_1)\oplus R_1(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),U\in \overrightarrow{R_1}(U_1),\rho _1=k_1(\rho _0)\).
3.2.4 Case 4: \(\rho _0\in (0,\rho _*]\)
-
(i)
If \(\rho \in [k_3(\rho _0),+\infty )\cup (\rho _0,k_1(\rho _0)]\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U), \end{aligned}$$where \(U\in \overrightarrow{S_1}(U_0)\).
-
(ii)
If \(\rho \in (0,\rho _0]\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=R_1(U_0,U), \end{aligned}$$where \(U\in \overrightarrow{R_1}(U_0)\).
-
(iii)
If \(\rho \in [\alpha ^{-\natural },k_3(\rho _0))\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U_1)\oplus R_1(U_1,U_2)\oplus S_1(U_2,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),U_2\in \overrightarrow{R_1}(U_1),U\in \overrightarrow{S_1}(U_2),\rho _1=k_1(\rho _0),\rho _2=k_2^{-1}(\rho )\).
-
(iv)
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(\rho _0<\varphi _0(\rho )<k_1(\rho _0)\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(\varphi _0(\rho )\ge k_1(\rho _0)\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U_1)\oplus R_1(U_1,U_2)\oplus W_0(U_2,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),U_2\in \overrightarrow{R_1}(U_1), \rho _1=k_1(\rho _0),\rho _2=\varphi _0(\rho )\).
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(\rho _0\ge \varphi _0(\rho )\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=R_1(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{R_1}(U_0),\rho _1=\varphi _0(\rho )\).
-
(v)
If \(\rho \in (k_1(\rho _0),\alpha ]\), then the solution is
$$\begin{aligned} Sol_1(U_0,U)=S_1(U_0,U_1)\oplus R_1(U_1,U), \end{aligned}$$where \(U_1\in \overrightarrow{S_1}(U_0),U\in \overrightarrow{R_1}(U_1),\rho _1=k_1(\rho _0)\).
3.3 Backward 3-Wave Curves
Given a right-hand state \(U_0(u_0,\rho _0,a_0)\), the 3-wave curve (backward) \(\overleftarrow{W_3}(U_0)\) consisting of all left-hand states \(U(u,\rho ,a_0)\) that can be connected from \(U_0\) by a combination of 3-shock waves, 3-rarefaction fans and contact phase boundary with assumption (1.2) has the following description.
3.3.1 Case 1: \(\rho _0\in (b_2,+\infty )\).
-
(i)
If \(\rho \in (b_2,+\infty )\), then the solution is
$$\begin{aligned} \mathop {Sol_3^B}(U_0,U)= S_3^B(U_0,U), \end{aligned}$$where \(U\in \overleftarrow{S_3}(U_0)\).
-
(ii)
If \(\rho \in [\beta ,\rho _0]\), then the solution is
$$\begin{aligned} \mathop {Sol_3^B}(U_0,U)=R_3^B(U_0,U), \end{aligned}$$where \(U\in \overleftarrow{R_3}(U_0)\).
-
(iii)
If \(\rho \in [\beta ^{-\natural },\alpha ]\) and \(\varphi _0(\rho )>\rho _0\), then the solution is
$$\begin{aligned} \mathop {Sol_3^B}(U_0,U)=S_3^B(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ^{-\natural },\alpha ]\) and \(\varphi _0(\rho )<\rho _0\), then the solution is
$$\begin{aligned} \mathop {Sol_3^B}(U_0,U)=R_3^B(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{R_3}(U_0),\rho _1=\varphi _0(\rho )\).
-
(iv)
If \(\rho \in [\rho _*,\beta ^{-\natural })\), then the solution is
$$\begin{aligned} \mathop {Sol_3^B}(U_0,U)=R_3^B(U_0,U_1)\oplus S_3^B(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{R_3}(U_0),U\in \overleftarrow{S_3}(U_1),\rho _1=k_2^{-1}(\rho )\).
-
(v)
If \(\rho \in (0,\rho _*)\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=R_3^B(U_0,U_1)\oplus S_3^B(U_1,U_2)\oplus R_3^B(U_2,U), \end{aligned}$$where \(U_1\in \overleftarrow{R_3}(U_0),U_2\in \overleftarrow{S_3}(U_1),U\in \overleftarrow{R_3}(U_2),\rho _1=b_2,\rho _2=b_1\).
3.3.2 Case 2: \(\rho _0\in [\beta ,b_2]\)
-
(i)
If \(\rho \in (\rho _0,+\infty )\cup [k_1(\rho _0),k_2(\rho _0))\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U), \end{aligned}$$where \(U\in \overleftarrow{S_3}(U_0)\).
-
(ii)
If \(\rho \in [\beta ,\rho _0]\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=R_3^B(U_0,U), \end{aligned}$$where \(U\in \overleftarrow{R_3}(U_0)\).
-
(iii)
If \(\rho \in [\beta ^{-\natural },\alpha ]\) and \(\rho _0\le \varphi _0(\rho )\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ^{-\natural },\alpha ]\) and \(\rho _0> \varphi _0(\rho )\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=R_3^B(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{R_3}(U_0),\rho _1=\varphi _0(\rho )\).
-
(iv)
If \(\rho \in [k_2(\rho _0),\beta ^{-\natural })\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=R_3^B(U_0,U_1)\oplus S_3^B(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{R_3}(U_0),U\in \overleftarrow{S_3}(U_1),\rho _1=k_2^{-1}(\rho )\).
-
(v)
If \(\rho \in (0,k_1(\rho _0))\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U_1)\oplus R_3^B(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),U\in \overleftarrow{R_3}(U_1),\rho _1=k_1(\rho _0)\).
3.3.3 Case 3: \(\rho _0\in (\rho _*,\alpha ]\)
-
(i)
If \(\rho \in [k_2(\rho _0),+\infty )\cup [k_1(\rho _0),\rho _0)\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U), \end{aligned}$$where \(U\in \overleftarrow{S_3}(U_0)\).
-
(ii)
If \(\rho \in [\rho _0,\alpha ]\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=R_3^B(U_0,U), \end{aligned}$$where \(U\in \overleftarrow{R_3}(U_0)\).
-
(iii)
If \(\rho \in [\alpha ^{-\natural },k_2(\rho _0))\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=R_3^B(U_0,U_1)\oplus S_3^B(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{R_3}(U_0),U\in \overleftarrow{S_3}(U_1),\rho _1=k_2^{-1}(\rho )\).
-
(iv)
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(\varphi _0(\rho )>\rho _0\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=R_3^B(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{R_3}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(k_1(\rho _0)\le \varphi _0(\rho )\le \rho _0\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(k_1(\rho _0)> \varphi _0(\rho )\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U_1)\oplus R_3^B(U_1,U_2)\oplus W_0(U_2,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),U_2\in \overleftarrow{R_3}(U_1), \rho _1=k_1(\rho _0),\rho _2=\varphi _0(\rho )\).
-
(v)
If \(\rho \in (0,k_1(\rho _0))\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U_1)\oplus R_3^B(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),U\in \overleftarrow{R_3}(U_1),\rho _1=k_1(\rho _0)\).
3.3.4 Case 4: \(\rho _0\in (0,\rho _*]\)
-
(i)
If \(\rho \in [k_3(\rho _0),+\infty )\cup (\rho _0,k_1(\rho _0)]\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U), \end{aligned}$$where \(U\in \overleftarrow{S_3}(U_0)\).
-
(ii)
If \(\rho \in (0,\rho _0]\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=R_3^B(U_0,U), \end{aligned}$$where \(U\in \overleftarrow{R_3}(U_0)\).
-
(iii)
If \(\rho \in [\alpha ^{-\natural },k_3(\rho _0))\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U_1)\oplus R_3^B(U_1,U_2)\oplus S_3^B(U_2,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),U_2\in \overleftarrow{R_3}(U_1),U\in \overleftarrow{S_3}(U_2),\rho _1=k_1(\rho _0),\rho _2=k_2^{-1}(\rho )\).
-
(iv)
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(\rho _0<\varphi _0(\rho )<k_1(\rho _0)\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),\rho _1=\varphi _0(\rho )\).
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(\varphi _0(\rho )\ge k_1(\rho _0)\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U_1)\oplus R_3^B(U_1,U_2)\oplus W_0(U_2,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),U_2\in \overleftarrow{R_3}(U_1), \rho _1=k_1(\rho _0),\rho _2=\varphi _0(\rho )\).
If \(\rho \in [\beta ,\alpha ^{-\natural })\) and \(\rho _0\ge \varphi _0(\rho )\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=R_3^B(U_0,U_1)\oplus W_0(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{R_3}(U_0),\rho _1=\varphi _0(\rho )\).
-
(v)
If \(\rho \in (k_1(\rho _0),\alpha ]\), then the solution is
$$\begin{aligned} Sol_3^B(U_0,U)=S_3^B(U_0,U_1)\oplus R_3^B(U_1,U), \end{aligned}$$where \(U_1\in \overleftarrow{S_3}(U_0),U\in \overleftarrow{R_3}(U_1),\rho _1=k_1(\rho _0)\).
Given the left-hand state \(U_0\), the 1-wave curve \(\overrightarrow{W_1}(U_0)\)
is continuous, onto, monotone decreasing in each of interval \((0,\alpha ],[\beta ,+\infty )\) and satisfying
Similar, given the right-hand state \(U_0\), the 3-wave curve \(\overleftarrow{W_3}(U_0)\)
is continuous, onto, monotone decreasing in each of interval \((0,\alpha ],[\beta ,+\infty )\) and satisfying
Theorem 3.4
(Existence of solution) Under assumption (1.2), the Riemann problem (3.1)–(3.2) admits a unique solution in the class of shock waves satisfying the Liu entropy condition, rarefaction fans and contact phase boundary.
Proof
The left-hand state \(U_L\) and the right-hand state \(U_R\) are given. We consider the intersections of the 1-wave curve \(\overrightarrow{W_1}(U_L)\) and the backward 3-wave curve \(\overleftarrow{W_3}(U_R)\). It implies to consider the equation: \(u_1(\rho )=u_3(\rho )\).
With \(u_1,u_3\in C((0,\alpha ]\cup [\beta ,+\infty );{\mathbb {R}})\), we have the following cases:
-
(i)
\(u_1(\alpha )>u_3(\alpha )\):
The function \(u_1(.)\) is decreasing, and the function \(u_3(.)\) is increasing in the interval \((0,\alpha ]\); then, \(\lbrace \rho :u_1(\rho )=u_3(\rho )\rbrace \cap (0,\alpha ]=\varnothing \).
The function \(u_1(.)\) is decreasing, and the function \(u_3(.)\) is increasing in the interval \([\beta ,+\infty )\). Moreover, we have
$$\begin{aligned} \begin{aligned}&u_1(\beta )>u_3(\beta ),\\&\mathop {\lim }\limits _{\rho \rightarrow +\infty }u_1(\rho )=-\infty ,\\&\mathop {\lim }\limits _{\rho \rightarrow +\infty }u_3(\rho )=+\infty . \end{aligned} \end{aligned}$$(3.15)It shows that two curves intersect at a unique point in the interval \((\alpha ^{-\natural },+\infty )\).
-
(ii)
\(u_1(\beta )<u_3(\beta )\)
Similar to (i), two curves intersect at a unique point in the interval \((0,\beta ^{-\natural })\).
-
(iii)
\(u_1(\alpha )<u_3(\alpha )\) and \(u_1(\beta )>u_3(\beta )\):
The function \(u_1(.)\) is decreasing, and the function \(u_3(.)\) is increasing in the interval \((0,\alpha ]\cup [\beta ,+\infty )\) and
$$\begin{aligned} \begin{aligned}&u_3(\beta )=u_3(\beta ^{-\natural })<u_1(\beta )=u_1(\beta ^{-\natural }),\\&u_3(\alpha )=u_3(\alpha ^{-\natural })>u_1(\alpha )-u_1(\alpha ^{-\natural }). \end{aligned} \end{aligned}$$(3.16)This indicates that there exist two values: \(\rho _m\in [\beta ,\alpha ^{-\natural }]\) and \(\rho '_m\in [\beta ^{-\natural },\alpha ]\), where \(\rho '_m=\varphi _0(\rho _m)\) such that
$$\begin{aligned} \begin{aligned}&u_1(\rho _m)=u_3(\rho _m),\\&u_1(\rho '_m)=u_3(\rho '_m). \end{aligned} \end{aligned}$$(3.17)In this case, the two intersection points correspond to the solution having a contact phase boundary.
The proof is completed. \(\square \)
4 Stationary Waves and the Monotonicity Criterion
4.1 Stationary Waves
Let us now consider the stationary waves associated with the 2-characteristic field. It follows from the relations in (2.12) that
From the first relation in (4.1), we can see that u and \(u_0\) are the same sign. Then, the second relation shows that
which is well defined whenever
Since
is the increasing function of \(\rho \), there exists a value \({\overline{\rho }}(U_0)\) such that the condition (4.3) is equivalent to \(\rho \in (0,{\overline{\rho }}(U_0)]\setminus (\alpha ,\beta )\).
The illustration of Riemann solutions for a flat nozzle
We define:
where \(u=u_2(\rho )\) is given by (4.2).
For any given point \(U_0\), we define the stationary curve \(\overrightarrow{W_2}(U_0)\) consisting of all points U that can be connected from \(U_0\) by a 2-stationary wave.
The curve \(\overrightarrow{W_2}(U_0)\) can be parameterized by \(\rho \mapsto u_2(\rho )\), where \(\rho \in (0,{\overline{\rho }}(U_0)]\setminus (\alpha ,\beta )\).
If \(u_0=0\), it is easy to see that \(\overrightarrow{W_2}(U_0)=\lbrace U_0\rbrace \). For \(u_0\ne 0\), we consider for \(u_0>0\), it is similar for the case \(u_0<0\). It holds that
By a straightforward calculation, we have
Set
It holds that
Note that
From now, to simplify the notations, we define:
4.1.1 Case (I)
Assume that a given left-hand state \(U_0\) satisfies
Assumption (4.12) is equivalent to
Thus, there exists a unique value \(\rho _{F_1}\) such that
Consider \(u_0>0\), we have
Lemma 4.1
Given a left-hand state \(U_0\) \((u_0\ne 0)\) in the hyperbolic regions such that
where h(.) is defined by (4.4) and \(\rho _{F_1}\) is defined by (4.14). The stationary curve \(\overrightarrow{W_2}(U_0)\) is defined by (4.6) consisting of all points U that can be connected from \(U_0\) by a 2-stationary wave. The following conclusions hold:
-
(i)
If \(a<\dfrac{a_0\rho _0 |u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}\), then \(\overrightarrow{W_2}(U_0)=\varnothing \).
-
(ii)
If \(a=\dfrac{a_0\rho _0 |u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1}\rbrace \subset C_A^+\cup C_A^-,\quad \rho _{m_1}=\rho _{F_1}\).
-
(iii)
If \(a>\dfrac{a_0\rho _0 |u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2}\rbrace ,\quad U_{m_1}\in G_{1A}\cup G_{3A},\quad U_{m_2}\in G_{2A},\quad \rho _{m_1}<\rho _{F_1}<\rho _{m_2}\).
Proof
We note that under assumption (4.12), the left-hand and right-hand states of the stationary waves are only in phase-A.
We consider for \(u_0>0\), it is similar for \(u_0<0\). It is not difficult to see that
where the function \(\kappa (.)\) is defined in (4.11). It holds that
If \(a<\kappa (\rho _{F_1})\), then \(Q(\rho _{F_1};U_0)<0\). From (4.15), we can see that \(Q(\rho ;U_0)<0\), \(\rho \in (0,{\overline{\rho }}(U_0)]\). It holds that \(\overrightarrow{W_0}(U_0)=\varnothing \). This proves (i).
To prove (ii), we note that if \(a=\kappa (\rho _{F_1})\), then \(Q(\rho _{F_1};U_0)=0\). From (4.15), we can see that \(Q(\rho ;U_0)=0\) if and only if \(\rho =\rho _{F_1}\). It holds that there exists a unique right-hand state \(U_{m_1}\in \overrightarrow{W_2}(U_0)\). Thus,
then, \(U_{m_1}\in C_A^+\cup C_A^-\). This establishes (ii).
Finally, if \(a>\kappa (\rho _{F_1})\), then \(Q(\rho _{F_1};U_0)>0\). From (4.15), we can see that there exist exactly two values \(\rho _{m_1},\rho _{m_2}\) (with \(\rho _{m_1}<\rho _{F_1}<\rho _{m_2}\)) such that \(Q(\rho _{m_1};U_0)=Q(\rho _{m_2};U_0)=0\). It holds that \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2}\rbrace \). Note that
then \(U_{m_1}\in G_{1A}\cup G_{3A}\) and \(U_{m_2}\in G_{2A}\). This proves (iii). \(\square \)
4.1.2 Case (II)
Assume that a given left-hand state \(U_0\) satisfies
We can see that assumption (4.19) is equivalent to
where the function F is defined in (4.8).
There exists a unique value \(\rho _{F_1}\) such that
Consider \(u_0>0\), we have
Lemma 4.2
Given a left-hand state \(U_0\) \((u_0\ne 0)\) in the hyperbolic regions such that
where \(\rho _*\) is defined by (1.2), h(.) is defined by (4.4), and \(\rho _{F_1}\) is defined by (4.21). The stationary curve \(\overrightarrow{W_2}(U_0)\) is defined by (4.6) consisting of all points U that can be connected from \(U_0\) by a 2-stationary wave. The following conclusions hold:
-
(i)
If \(a<\dfrac{a_0\rho _0 |u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}\), then \(\overrightarrow{W_2}(U_0)=\varnothing \).
-
(ii)
If \(a=\dfrac{a_0\rho _0 |u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1}\rbrace \subset C_B^+\cup C_B^-,\quad \rho _{m_1}=\rho _{F_1}\).
-
(iii)
If \(\dfrac{a_0\rho _0 |u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}<a<\dfrac{a_0\rho _0|u_0|}{\beta |u_2(\beta )|}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2}\rbrace ,\quad U_{m_1}\in G_{1B}\cup G_{3B},\quad U_{m_2}\in G_{2B},\quad \beta<\rho _{m_1}<\rho _{F_1}<\rho _{m_2}<{\overline{\rho }}(U_0)\).
-
(iv)
If \(\dfrac{a_0\rho _0|u_0|}{\beta |u_2(\beta )|}<a<\dfrac{a_0\rho _0|u_0|}{\alpha |u_2(\alpha )|}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1}\rbrace \subset G_{2B},\quad \rho _{F_1}<\rho _{m_1}<{\overline{\rho }}(U_0)\).
-
(v)
If \(a>\dfrac{a_0\rho _0|u_0|}{\alpha |u_2(\alpha )|}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2}\rbrace ,\quad U_{m_1}\in G_{1A}\cup G_{3A},\quad U_{m_2}\in G_{2B},\quad 0<\rho _{m_1}<\alpha ,\quad \rho _{F_1}<\rho _{m_2}<{\overline{\rho }}(U_0)\).
The proof of Lemma 4.2 is similar to the proof of Lemma 4.1.
4.1.3 Case (III)
Assume that a given left-hand state \(U_0\) satisfies
Assumption (4.23) is equivalent to
where the function F is defined in (4.8).
Thus, there exist exactly three values \(\rho _{F_1},\rho _{F_2},\rho _{F_3}\) such that
Consider \(u_0>0\), it holds that
Lemma 4.3
Given a left-hand state \(U_0\) \((u_0\ne 0)\) in the hyperbolic regions such that
where \(\rho _*\) is defined by (1.2), h(.) is defined by (4.4), and \(\rho _{F_1},\rho _{F_2},\rho _{F_3}\) are defined by (4.25). The stationary curve \(\overrightarrow{W_2}(U_0)\) is defined by (4.6) consisting of all points U that can be connected from \(U_0\) by a 2-stationary wave. The following conclusions hold:
-
(i)
If \(a<\mathop {\min }\left\{ \dfrac{a_0\rho _0 |u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}},\dfrac{a_0\rho _0 |u_0|}{\rho _{F_3}\sqrt{p'(\rho _{F_3})}}\right\} \), then \(\overrightarrow{W_2}(U_0)=\varnothing \).
-
(ii)
If \(\dfrac{a_0\rho _0|u_0|}{\beta |u_2(\beta )|}<a<\mathop {\min }\left\{ \dfrac{a_0\rho _0|u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}},\dfrac{a_0\rho _0|u_0|}{\alpha |u_2(\alpha )|}\right\} \), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1}\rbrace \subset G_{2B},\quad \rho _{F_3}<\rho _{m_1}<{\overline{\rho }}(U_0)\).
-
(iii)
If \(a=\dfrac{a_0\rho _0|u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}<\dfrac{a_0\rho _0 |u_0|}{\rho _{F_3}\sqrt{p'(\rho _{F_3})}}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1}\rbrace \subset C_A^+\cup C_A^-,\quad \rho _{m_1}=\rho _{F_1}\).
-
(iv)
If \(a=\dfrac{a_0\rho _0|u_0|}{\rho _{F_3}\sqrt{p'(\rho _{F_3})}}<\dfrac{a_0\rho _0 |u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1}\rbrace \subset C_B^+\cup C_B^-,\quad \rho _{m_1}=\rho _{F_3}\).
-
(v)
If \(a=\dfrac{a_0\rho _0|u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}=\dfrac{a_0\rho _0 |u_0|}{\rho _{F_3}\sqrt{p'(\rho _{F_3})}}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2}\rbrace ,\quad U_{m_1}\in C_A^+\cup C_A^-,\quad U_{m_2}\in C_B^+\cup C_B^-,\quad \rho _{m_1}=\rho _{F_1},\quad \rho _{m_2}=\rho _{F_3}\).
-
(vi)
If \(\dfrac{a_0\rho _0|u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}<a<\dfrac{a_0\rho _0 |u_0|}{\rho _{F_3}\sqrt{p'(\rho _{F_3})}}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2}\rbrace ,\quad U_{m_1}\in G_{1A}\cup G_{3A},\quad U_{m_2}\in G_{2A},\quad 0<\rho _{m_1}<\rho _{F_1}<\rho _{m_2}<\rho _{F_2}\).
-
(vii)
If \(\dfrac{a_0\rho _0 |u_0|}{\rho _{F_3}\sqrt{p'(\rho _{F_3})}}<a<\mathop {\min }\left\{ \dfrac{a_0\rho _0|u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}},\dfrac{a_0\rho _0|u_0|}{\beta |u_2(\beta )|}\right\} \), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2}\rbrace ,\quad U_{m_1}\in G_{1B}\cup G_{3B},\quad U_{m_2}\in G_{2B},\quad \beta<\rho _{m_1}<\rho _{F_3}<\rho _{m_2}<{\overline{\rho }}(U_0)\).
-
(viii)
If \(\dfrac{a_0\rho _0|u_0|}{\alpha |u_2(\alpha )|}<a<\dfrac{a_0\rho _0|u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2}\rbrace ,\quad U_{m_1}\in G_{1A}\cup G_{3A},\quad U_{m_2}\in G_{2B},\quad \rho _{F_2}<\rho _{m_1}<\alpha ,\quad \rho _{F_3}<\rho _{m_2}<{\overline{\rho }}(U_0)\).
-
(ix)
If \(a>\dfrac{a_0\rho _0|u_0|}{\rho _{F_2}\sqrt{p'(\rho _{F_2})}}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2}\rbrace ,\quad U_{m_1}\in G_{1A}\cup G_{3A},\quad U_{m_2}\in G_{2B},\quad 0<\rho _{m_1}<\rho _{F_1},\quad \rho _{F_3}<\rho _{m_2}<{\overline{\rho }}(U_0)\).
-
(x)
If \(\mathop {\max }\left\{ \dfrac{a_0\rho _0|u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}},\dfrac{a_0\rho _0|u_0|}{\beta |u_2(\beta )|}\right\}<a<\dfrac{a_0\rho _0|u_0|}{\alpha |u_2(\alpha )|}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2},U_{m_3}\rbrace ,\quad U_{m_1}\in G_{1A}\cup G_{3A},\quad U_{m_2}\in G_{2A}\quad U_{m_3}\in G_{2B},\quad 0<\rho _{m_1}<\rho _{F_1}<\rho _{m_2}<\rho _{F_2},\quad \rho _{F_3}<\rho _{m_3}<{\overline{\rho }}(U_0)\).
-
(xi)
If \(\mathop {\max }\left\{ \dfrac{a_0\rho _0|u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}},\dfrac{a_0\rho _0 |u_0|}{\rho _{F_3}\sqrt{p'(\rho _{F_3})}}\right\}<a<\dfrac{a_0\rho _0|u_0|}{\beta |u_2(\beta )|}\), then \(\overrightarrow{W_2}(U_0)=\lbrace U_{m_1},U_{m_2},U_{m_3},U_{m_4}\rbrace \),\( U_{m_1}\in G_{1A}\cup G_{3A},\quad U_{m_2}\in G_{2A},\quad U_{m_3}\in G_{1B}\cup G_{3B},\quad U_{m_4}\in G_{2B}\),\(0<\rho _{m_1}<\rho _{F_1}<\rho _{m_2}<\rho _{F_2},\quad \beta<\rho _{m_3}<\rho _{F_3}<\rho _{m_4}<{\overline{\rho }}(U_0)\).
The proof of Lemma 4.3 is similar to the proof of Lemma 4.1, so we omit the proof.
Corollary 4.4
-
(i)
Given a left-hand state \(U_0\) \((u_0\ne 0)\) in D defined by
$$\begin{aligned} D:=\lbrace U_0|2h(\beta )<u_0^2+2h(\rho _0)<2h(\rho _*)+p'(\rho _*),\quad \rho _0\in (0,+\infty )\setminus (\alpha ,\beta )\rbrace ,\nonumber \\ \end{aligned}$$(4.27)where \(\rho _*\) is given in (1.2), h(.) is defined by (4.4), and \(\rho _{F_1},\rho _{F_2},\rho _{F_3}\) are defined by (4.25). If
$$\begin{aligned} \mathop {\max }\left\{ \dfrac{a_0\rho _0 |u_0|}{\rho _{F_1}\sqrt{p'(\rho _{F_1})}},\dfrac{a_0\rho _0 |u_0|}{\rho _{F_3}\sqrt{p'(\rho _{F_3})}}\right\}<a<\dfrac{a_0\rho _0 |u_0|}{\beta |u_2(\beta )|}, \end{aligned}$$(4.28), then there are four right-hand states: \(U_{m_1},U_{m_2},U_{m_3},U_{m_4}\in \overrightarrow{W_2}(U_0)\) with \(0<\rho _{m_1}<\rho _{F_1}<\rho _{m_2}<\rho _{F_2},\quad \beta<\rho _{m_3}<\rho _{F_3}<\rho _{m_4}<{\overline{\rho }}(U_0)\) and
$$\begin{aligned} U_{m_1}\in G_{1A}\cup G_{3A},\quad U_{m_2}\in G_{2A},\quad U_{m_3}\in G_{1B}\cup G_{3B},\quad U_{m_4}\in G_{2B} \end{aligned}$$.
-
(ii)
Given a right-hand state U \((u\ne 0)\) in D, where D is defined by (4.27), for each fixed value a, it is always possible to choose \(a_0\) such that
$$\begin{aligned} \mathop {\max }\left\{ \dfrac{a\rho |u|}{\rho _{E_1}\sqrt{p'(\rho _{E_1})}},\dfrac{a\rho |u|}{\rho _{E_3}\sqrt{p'(\rho _{E_3})}}\right\}<a_0<\dfrac{a\rho |u|}{\beta |u_2(\beta )|}, \end{aligned}$$(4.29)where \(\rho _{E_1},\rho _{E_2},\rho _{E_3}\) are the intersections of the curve \({\mathcal {L}}(\rho _0):u_0^2+2h(\rho _0)=u^2+2h(\rho )\) and the curves \(C_A^+\cup C_A^-\cup C_B^+\cup C_B^-\), and then, there are four left-hand states:
$$\begin{aligned} \begin{aligned}&U_{0_1}\in {\mathcal {L}}(\rho _0)\cap (G_{1A}\cup G_{3A}),\quad U_{0_2}\in {\mathcal {L}}(\rho _0)\cap G_{2A},\\&\quad U_{0_3}\in {\mathcal {L}}\cap (G_{1B}\cup G_{3B}),\quad U_{0_4}\in {\mathcal {L}}\cap G_{2B},\\&0<\rho _{0_1}<\rho _{E_1}<\rho _{0_2}<\rho _{E_2}, \quad \beta<\rho _{0_3}<\rho _{F_3}<\rho _{0_4}, \end{aligned} \end{aligned}$$(4.30)that can connect to U by a stationary wave.
Domain D is defined by (4.27)
4.2 The Monotonicity Criterion
In this section, we require that any stationary waves in a Riemann solution must satisfy the following admissible criterion.
(MC): Along any stationary curve \(\overrightarrow{W_2}(U_0)\), the cross-sectional area a is a monotone function of \(\rho \). The total variation of the cross-sectional area component of any Riemann solution must not exceed \(|a_R-a_L|\), where \(a_R\) and \(a_L\) are right-hand and left-hand states of the cross-sectional area.
Without loss of generality, we assume for simplicity in the rest of this section that
We define the admissible stationary waves curve \(\overrightarrow{W'_2}(U_0)\) consisting of all point U that can be connected from \(U_0\) by a admissible stationary wave under the condition (MC). Obvious, we see that
Lemma 4.5
Under the condition (MC), given a left-hand state \(U_0\) \((u_0\ne 0)\) in the hyperbolic regions, assume that \(U_0\in G_{iA}\cup G_{iB}\) then \(\overrightarrow{W'_2}(U_0)\subset G_{iA}\cup G_{iB}\) \((i=1,2,3)\).
Proof
Differentiating both sides of the equation \(a\rho u_2(\rho )=a_0 \rho _0 u_0\) with respect to \(\rho \) gives us
where \((.)'=\hbox {d}/\hbox {d}\rho \), and \(u'_2(\rho )=\mathop {\text {sgn}}(u_0)\dfrac{-h'(\rho )}{\sqrt{u_0^2-2(h(\rho )-h(\rho _0))}}\).
Besides, a straightforward calculation leads us to
Observe that \(u_0^2-p'(\rho _0)=-\dfrac{a'(\rho _0)\rho _0 u_0^2}{a_0}\). Thus, it follows from the condition (MC) that
On the other hand, it follows from (4.6) that
which establishes the results of Lemma 4.5. \(\square \)
For each \(\rho _0\in (0,\alpha ]\cup [\beta ,+\infty )\) is given, we define the functions \(M(,;\rho _0), N(.;\rho _0)\) of \(\rho \) in the intervals \((0,\alpha ]\cup [\beta ,+\infty )\) as follows
It is easy to see that
where the function \({\mathcal {A}}(.)\) is defined by (1.2), (1.3), so we have
Moreover, for straightforward calculation, it holds that
or
Thus,
or in other words, if \(\rho >\rho _0\), then \(N(\rho ;\rho _0)<M(\rho ;\rho _0)\) (or \(N(\rho ;\rho _0)>M(\rho ;\rho _0)\)) in the interval that the function \(N(.;\rho _0)\) is increasing (or decreasing). Moreover, in phase-B, the function \(M(.;\rho _0)\) is increasing function of \(\rho \) and \(M(\rho _0;\rho _0)=\mathop {\lim }\limits _{\rho \rightarrow \rho _0}N(\rho ;\rho _0)\); then, for all \(\rho >\rho _0\) it holds that \(M(\rho ;\rho _0)>N(\rho ;\rho _0)\) and the function \(N(.;\rho _0)\) is increasing function of \(\rho \).
Lemma 4.6
-
(I)
Given a left-hand state \(U_0\) in the hyperbolic regions, the following conclusions hold:
-
(i)
If \(U_0\in G_{1A}\cup G_{1B}\cup G_{2A}\cup G_{2B}\), then
$$\begin{aligned} \overline{\lambda _1}(U_0,U)<0,\quad U\in \overrightarrow{S_1}(U_0). \end{aligned}$$(4.43) -
(ii)
If \(U_0\in G_{3A}\cup G_{3B}\), then
-
there exists \(\rho _{h_1}:=\rho _{h_1}(U_0)>\rho _0\) such that
$$\begin{aligned} \overline{\lambda _1}(U_0,U)<0,\quad U\in \overrightarrow{S_1}(U_0), \,\rho >\rho _{h_1}, \end{aligned}$$(4.44) -
there exists \(\rho _{l_1}:=\rho _{l_1}(U_0)<\rho _{h_1}\) such that
$$\begin{aligned} \overline{\lambda _1}(U_0,U)>0,\quad U\in \overrightarrow{S_1}(U_0), \,|\rho -\rho _0|<|\rho _{l_1}-\rho _0|. \end{aligned}$$(4.45)
Specifically, if \(U_0\in G_{3B}\), then there exists a unique point \(\mathop {U_1}\limits ^\sim \in \overrightarrow{S_1}(U_0)\cap G_{2B}^+\), where \( \rho _{l_1}(U_0)<\mathop {\rho _1}\limits ^\sim := \mathop {\rho _1}\limits ^\sim (U_0)<\rho _{h_1}(U_0)\) such that
$$\begin{aligned} \begin{aligned}&\overline{\lambda _1}(U_0,U)>0,\quad U\in \overrightarrow{S_1}(U_0), \rho _0<\rho<\mathop {\rho _1}\limits ^\sim ,\\&\overline{\lambda _1}(U_0,\mathop {U_1}\limits ^\sim )=0,\\&\overline{\lambda _1}(U_0,U)<0,\quad U\in \overrightarrow{S_1}(U_0),\rho >\mathop {\rho _1}\limits ^\sim . \end{aligned} \end{aligned}$$(4.46) -
-
(i)
-
(II)
Given a right-hand state \(U_0\) in the hyperbolic regions, the following conclusions hold:
-
(i)
If \(U_0\in G_{2A}\cup G_{2B}\cup G_{3A}\cup G_{3B}\), then
$$\begin{aligned} \overline{\lambda _3}(U_0,U)>0,\quad U\in \overleftarrow{S_3}(U_0). \end{aligned}$$(4.47) -
(ii)
If \(U_0\in G_{1A}\cup G_{1B}\), then
-
there exists \(\rho _{h_3}:=\rho _{h_3}(U_0)>\rho _0\) such that
$$\begin{aligned} \overline{\lambda _3}(U_0,U)>0,\quad U\in \overleftarrow{S_3}(U_0), \,\rho >\rho _{h_3}, \end{aligned}$$(4.48) -
there exists \(\rho _{l_3}:=\rho _{l_3}(U_0)<\rho _{h_3}\) such that
$$\begin{aligned} \overline{\lambda _3}(U_0,U)<0,\quad U\in \overleftarrow{S_3}(U_0), \,|\rho -\rho _0|<|\rho _{l_3}-\rho _0|. \end{aligned}$$(4.49)
Specifically, if \(U_0\in G_{1B}\), then there exists a unique point \(\mathop {U_3}\limits ^\sim \in \overleftarrow{S_3}(U_0)\cap G_{2B}^-\), where \( \rho _{l_3}(U_0)<\mathop {\rho _3}\limits ^\sim := \mathop {\rho _3}\limits ^\sim (U_0)<\rho _{h_3}(U_0)\) such that
$$\begin{aligned} \begin{aligned}&\overline{\lambda _3}(U_0,U)<0,\quad U\in \overleftarrow{S_3}(U_0), \rho _0<\rho <\mathop {\rho _3}\limits ^\sim ,\\&\overline{\lambda _3}(U_0,\mathop {U_3}\limits ^\sim )=0,\\&\overline{\lambda _3}(U_0,U)>0,\quad U\in \overleftarrow{S_3}(U_0),\rho >\mathop {\rho _3}\limits ^\sim . \end{aligned} \end{aligned}$$(4.50) -
-
(i)
Proof
We only prove case (I) where the left-hand state is given, the proof for case (II) is similar. We recall that
-
(i)
If \(U_0\in G_{1A}\cup G_{1B}\), then \(u_0<0\) so that \(\overline{\lambda _1}(U_0,U)<0\).
If \(U_0\in G_{2A}\cup G_{2B}\), it follows from the Liu entropy condition (2.14) that
$$\begin{aligned} \overline{\lambda _1}(U_0,U)\le \mathop {\lim }\limits _{\rho \rightarrow \rho _0 }\overline{\lambda _1}(U_0,U) =u_0-\sqrt{p'(\rho _0)}<0. \end{aligned}$$(4.51) -
(ii)
If \(U_0\in G_{3A}\cup G_{3B}\), we consider the function
$$\begin{aligned} f(\rho ,\rho _0)=u_0-\sqrt{\dfrac{\rho }{\rho _0}} \sqrt{\dfrac{p(\rho )-p(\rho _0)}{\rho -\rho _0}}. \end{aligned}$$(4.52)We see that
$$\begin{aligned} \mathop {\lim }\limits _{\rho \rightarrow \rho _0}f(\rho ,\rho _0)=u_0-\sqrt{p'(\rho _0)}>0; \end{aligned}$$(4.53)then, there exists a neighborhood of \(\rho _0\) such that \(f(.,\rho _0)\) is positive. So that we can choose \(\rho _{l_1}\) such that
$$\begin{aligned} \overline{\lambda _1}(U_0,U)>0, \quad U\in \overrightarrow{S_1}(U_0), \, |\rho -\rho _0|<|\rho _{l_1}-\rho _0|. \end{aligned}$$Other hand, it is easy to see that there exists a value \(\rho _m\) which is large enough such that the function \(f(.,\rho _0)\) is decreasing in \((\rho _m,+\infty )\). Moreover,
$$\begin{aligned} \mathop {\lim }\limits _{\rho \rightarrow +\infty }f(\rho ,\rho _0)=-\infty . \end{aligned}$$(4.54)Thus, we can always choose \(\rho _{h_1}\ge \rho _m\) such that (4.44) is satisfied.
Specifically, if \(U_0\in G_{3B}\), then the shock set: \(\Omega _1(\rho _0)=(\rho _0,+\infty )\). We have
$$\begin{aligned} \overline{\lambda _1}(U_0,U)=u_0-\dfrac{\sqrt{N(\rho ;\rho _0)}}{\rho _0},\quad \rho >\rho _0, \end{aligned}$$(4.55)where \(N(.;\rho _0)\) is defined by (4.37).
From (4.42), it implies that \(\overline{\lambda _1}(U_0,U)\) is a decreasing function in the shock set: \(\Omega _1(\rho _0)=(\rho _0,+\infty )\). Moreover,
$$\begin{aligned} \begin{aligned}&\mathop {\lim }\limits _{\rho \rightarrow \rho _0}\overline{\lambda _1}(U_0,U)>0,\\&\mathop {\lim }\limits _{\rho \rightarrow +\infty }\overline{\lambda _1}(U_0,U)=-\infty ; \end{aligned} \end{aligned}$$(4.56)then, there exists a value \(\mathop {\rho _1}\limits ^\sim >\rho _0\) such that (4.46) is satisfied. Beside that, \(\overline{\lambda _1}(U_0,\mathop {U_1}\limits ^\sim )=0\); then,
$$\begin{aligned} u_0=\sqrt{\dfrac{\mathop {\rho _1}\limits ^\sim }{\rho _0}}\sqrt{\dfrac{p(\mathop {\rho _1}\limits ^\sim )-p(\rho _0)}{\mathop {\rho _1}\limits ^\sim -\rho _0}}. \end{aligned}$$Moreover, \(\mathop {\rho _1}\limits ^\sim \in \overrightarrow{S_1}(U_0)\); then,
$$\begin{aligned} \mathop {u_1}\limits ^\sim =u_0-\dfrac{\mathop {\rho _1}\limits ^\sim -\rho _0}{\sqrt{\mathop {\rho _1}\limits ^\sim \rho _0}}\sqrt{\dfrac{p(\mathop {\rho _1}\limits ^\sim )-p(\rho _0)}{\mathop {\rho _1}\limits ^\sim -\rho _0}}. \end{aligned}$$It implies that
$$\begin{aligned} \mathop {u_1}\limits ^\sim =\sqrt{\dfrac{\rho _0}{\mathop {\rho _1}\limits ^\sim }}\sqrt{\dfrac{p(\mathop {\rho _1}\limits ^\sim )-p(\rho _0)}{\mathop {\rho _1}\limits ^\sim -\rho _0}}, \end{aligned}$$or
$$\begin{aligned} \mathop {u_1}\limits ^\sim =\dfrac{\sqrt{N(\mathop {\rho _1}\limits ^\sim ; \rho _0)}}{\mathop {\rho _1}\limits ^\sim }. \end{aligned}$$(4.57)Applying (4.42) for \(\mathop {\rho _1}\limits ^\sim >\rho _0\), we have
$$\begin{aligned} 0<\mathop {u_1}\limits ^\sim =\dfrac{\sqrt{N(\mathop {\rho _1}\limits ^\sim ;\rho _0)}}{\mathop {\rho _1}\limits ^\sim }<\dfrac{\sqrt{M(\mathop {\rho _1}\limits ^\sim ;\rho _0)}}{\mathop {\rho _1}\limits ^\sim } = \sqrt{p'(\mathop {\rho _1}\limits ^\sim )}, \end{aligned}$$(4.58)where \(N(.;\rho _0)\) and \(M(,;\rho _0)\) are defined by (4.37).
It holds that \(\mathop {U_1}\limits ^\sim \in G_{2B}^+\).
\(\square \)
Corollary 4.7
(Admissible stationary waves in domain D)
-
(I)
Given a left-hand state \(U_0\in D\), where D is defined by (4.27), assume that the condition (4.28) is satisfied. The admissible stationary waves curve \(\overrightarrow{W'_2}(U_0)\) consists of all point U that can be connected from \(U_0\) by a admissible stationary wave under the condition (MC). It holds that
-
If \(U_0\in G_{1A}\cup G_{1B}\cup G_{3A}\cup G_{3B}\), then \(\overrightarrow{W'_2}(U_0)=\lbrace U_{m_1},U_{m_3}\rbrace \),
-
If \(U_0\in G_{2A}\cup G_{2B}\), then \(\overrightarrow{W'_2}(U_0)=\lbrace U_{m_2},U_{m_4}\rbrace \),
where \(U_{m_1},U_{m_2},U_{m_3},U_{m_4}\) are defined in Lemma 4.4.
-
-
(II)
Given a right-hand state \(U\in D\), where D is defined by (4.27), assume that the condition (4.29) is satisfied. The admissible stationary waves curve (backward) \(\overleftarrow{W'_2}(U)\) consists of all point \(U_0\) that can connect to U by a admissible stationary wave under the condition (MC). It holds that
-
If \(U\in G_{1A}\cup G_{1B}\cup G_{3A}\cup G_{3B}\), then \(\overleftarrow{W'_2}(U)=\lbrace U_{0_1},U_{0_3}\rbrace \),
-
If \(U\in G_{2A}\cup G_{2B}\), then \(\overleftarrow{W'_2}(U)=\lbrace U_{0_2},U_{0_4}\rbrace \), where \(U_{0_1},U_{0_2},U_{0_3},U_{0_4}\) are defined in (4.30).
-
5 Riemann Problem for a Nozzle with Discontinuous Cross Section
In this section, we will give the solution constructions of the Riemann problem (1.1) with the given left-hand state \(U_L\) and the given right-hand state \(U_R\). We consider only the case \(a_L<a_R\). The solutions consist of the admissible stationary waves under the condition (MC).
To construct solutions of the Riemann problem for (1.1), we will use the notations \(Sol_1,Sol_3^B\) defined in Sect. 3.
For any state \(U_0(\rho _0,u_0;a_0)\), we denote by
the set of states of admissible stationary wave that can be connected from \(U_0\) when the cross-sectional area jumps from \(a_0\) to a.
5.1 Case 1: Supersonic Construction
Due to the order of the characteristic speeds which allow combinations of waves associated with different characteristic fields, the construction starting in supersonic regions will be either based on either a right-hand state with negative velocity or a left-hand state with positive velocity. However, under the transformation \(x\mapsto -x, u\mapsto -u\), a right-hand side state \(U_+=(\rho ,u,a)\) will be transformed to a left-hand state \(U_-=(\rho ,-u,a)\). Thus, it is sufficient to consider only one case when the left-hand state has a positive velocity, since the construction for the case where the right-hand state has a negative velocity can similarly be performed. Furthermore, we can extend this construction to the sonic curves.
-
(i)
If \(\Upsilon (U_L,a_L,a_R)\cap G_{3A}\ne \varnothing \), where \(\Upsilon \) is defined by (5.1), we consider a point
$$\begin{aligned} U^{A@}(\rho ^{A@},u^{A@})\in \Upsilon (U_L,a_L,a_R)\cap G_{3A}. \end{aligned}$$(5.2)From Lemma 4.6, there exists \(U^{A@}_{l_1}(\rho ^{A@}_{l_1},u^{A@}_{l_1})\) such that
$$\begin{aligned} \begin{aligned}&U^{A@}_{l_1}\in \overrightarrow{S_1}(U^{A@}),\\&\overline{\lambda _1}(U^{A@},U)>0 \text { if and only if } U\in \overrightarrow{S_1}(U^{A@})\text { and } |\rho -\rho ^{A@}|<|\rho ^{A@}_{l_1}-\rho ^{A@}|. \end{aligned}\nonumber \\ \end{aligned}$$(5.3)If \(\Upsilon (U_L,a_L,a_R)\cap G_{3B}\ne \varnothing \), where \(\Upsilon \) is defined by (5.1), let
$$\begin{aligned} U^{B@}(\rho ^{B@},u^{B@}) \in \Upsilon (U_L,a_L,a_R)\cap G_{3B}. \end{aligned}$$(5.4)From Lemma 4.6, there exists a unique point \({\mathop {U}\nolimits ^\sim }^{B@}({\mathop {\rho }\limits ^\sim }^{B@},{\mathop {u}\limits ^\sim }^{B@}) \in \overrightarrow{S_1}(U^{B@})\cap G_{2B}^+\) such that
$$\begin{aligned} \begin{aligned}&\overline{\lambda _1}(U^{B@},U)>0\quad \text { if } U\in \overrightarrow{S_1}(U^{B@}),\rho ^{B@}<\rho<{\mathop {\rho }\limits ^\sim }^{B@},\\&\overline{\lambda _1}(U^{B@},{\mathop {U}\limits ^\sim }^{B@})=0,\\&\overline{\lambda _1}(U^{B@},U)<0\quad \text { if } U\in \overrightarrow{S_1}(U^{B@}),\rho >{\mathop {\rho }\limits ^\sim }^{B@}.\\ \end{aligned} \end{aligned}$$(5.5)We define the 1- and 2-wave curve \(\Lambda _1\) as follows
$$\begin{aligned} \Lambda _1 = \left\{ \begin{gathered} \Lambda _{1A}:=\lbrace U\in \overrightarrow{W_1}(U^{A@})\mid \rho \le \rho ^{A@}_{l_1}\rbrace \text { if } \Upsilon (U_L,a_L,a_R)\cap G_{3A}\ne \varnothing ,\\ \Lambda _{1B}:=\lbrace U\in \overrightarrow{W_1}(U^{B@})\mid \rho \le {\mathop {\rho }\limits ^\sim }^{B@}\rbrace \text { if } \Upsilon (U_L,a_L,a_R)\cap G_{3B}\ne \varnothing ,\\ \end{gathered} \right. \nonumber \\ \end{aligned}$$(5.6)where \(\Upsilon \) is defined by (5.1) and \(U^{A@},\rho ^{A@}_{l_1}, U^{B@}, {\mathop {\rho }\limits ^\sim }^{B@}\) are defined by (5.2), (5.3), (5.4), (5.5).
If \(\overleftarrow{W_3}(U_R)\cap \Lambda _1\ne \varnothing \), then the Riemann solution beginning with a admissible stationary wave from \(U_L\) followed by a composite 1-wave and finished by a composite 3-wave connects to \(U_R\). We call this intersection point
$$\begin{aligned} U_1\in \overleftarrow{W_3}(U_R)\cap \Lambda _1. \end{aligned}$$(5.7)Specifically,
-
+ if \(U_1\in \Lambda _{1A}\), then the Riemann solution has the following form
$$\begin{aligned} W_2(U_L,U^{A@})\oplus Sol_1(U^{A@},U_1)\oplus Sol^B_3(U_R,U_1), \end{aligned}$$(5.8)where \(U^{A@}\) is defined by (5.2),
-
+ if \(U_1\in \Lambda _{1B}\), then we have the form of the Riemann solution
$$\begin{aligned} W_2(U_L,U^{B@})\oplus Sol_1(U^{B@},U_1)\oplus Sol_3^B(U_R,U_1), \end{aligned}$$(5.9)where \(U^{B@}\) is defined by (5.4).
-
-
(ii)
If \(\lbrace \Upsilon (U_L,a_L,a)\mid a\in (a_L,a_R)\rbrace \cap G_{3B}\ne \varnothing \), then there exist two values \(a_m\) and \(a_n\) such that
$$\begin{aligned} \lbrace \Upsilon (U_L,a_L,a)\mid a\in [a_m,a_n]\rbrace \subset G_{3B},\quad a_L\le a_m<a_n\le a_R. \end{aligned}$$(5.10)Let us define the continuous curve \({\mathcal {M}}_1\) containing all states that can be connected from \(U_L\) by admissible stationary waves when the cross-sectional area jumps from \(a_L\) to a, for all \(a\in [a_m,a_n]\):
$$\begin{aligned} {\mathcal {M}}_1:=\lbrace \Upsilon (U_L,a_L,a)\mid a\in [a_m,a_n]\rbrace , \end{aligned}$$(5.11)where \(\Upsilon \) is defined by (5.1), and \(a_m,a_n\) are defined by (5.10).
From Lemma 4.6, for each \(U_0\in {\mathcal {M}}_1\) then there exists \(\mathop {U_0}\limits ^\sim \in G_{2B}^+\) such that
$$\begin{aligned} \mathop {U_0}\limits ^\sim \in \overrightarrow{S_1}(U_0),\quad \overline{\lambda _1}(U_0,\mathop {U_0}\limits ^\sim )=0. \end{aligned}$$(5.12)We define the continuous curve \({\mathcal {M}}_2\) containing all points that can be connected from each point on \({\mathcal {M}}_1\), respectively, by 1-shock waves with zero-speed:
$$\begin{aligned} \begin{aligned}&\Upsilon ^\sharp (U_L,a):=\lbrace \mathop {U_0}\limits ^\sim \mid U_0\in \Upsilon (U_L,a_L,a)\subset {\mathcal {M}}_1 \rbrace ,\\&{\mathcal {M}}_2:=\lbrace \Upsilon ^\sharp (U_L,a)\mid a\in [a_m,a_n]\rbrace , \end{aligned} \end{aligned}$$(5.13)where \(\Upsilon \) is defined by (5.1), \(\mathop {U_0}\limits ^\sim \) is defined by (5.12), and \(a_m,a_n\) are defined by (5.10).
Let us define the 1- and 2-wave curve \(\Lambda _2\) as following
$$\begin{aligned} \begin{aligned}&\Upsilon ^\circledast (U_L,a):=\lbrace \Upsilon (U_0,a,a_R)\mid U_0\in \Upsilon ^\sharp (U_L,a)\rbrace ,\\&\Lambda _2:=\lbrace \Upsilon ^\circledast (U_L,a)\mid a\in [a_m,a_n]\rbrace , \end{aligned} \end{aligned}$$(5.14)where \(\Upsilon \) is defined by (5.1), and \(a_m,a_n\) are defined by (5.10).
If \(\overleftarrow{W_3}(U_R)\cap \Lambda _2\ne \varnothing \), we call this intersection point
$$\begin{aligned} U_2\in \overleftarrow{W_3}(U_R)\cap \Lambda _2. \end{aligned}$$(5.15)There exist \(U_2^\bullet \in {\mathcal {M}}_2\) defined by (5.13), \(U_2^{\bullet \bullet }\in {\mathcal {M}}_1\) defined by (5.11) and a value \(a\in [a_m,a_n]\) such that
$$\begin{aligned} \begin{aligned}&U_2^{\bullet \bullet }\in \Upsilon (U_L,a_L,a),\\&U_2\in \Upsilon (U_2^\bullet ,a,a_R),\\&U_2^\bullet \in \overrightarrow{S_1}(U_2^{\bullet \bullet }),\quad \overline{\lambda _1}(U_2^{\bullet \bullet },U_2^\bullet )=0, \end{aligned} \end{aligned}$$(5.16)where \(\Upsilon \) is defined by (5.1).
The Riemann solution has the form
$$\begin{aligned} W_2(U_L,U_2^{\bullet \bullet })\oplus S_1(U_2^{\bullet \bullet },U_2^\bullet )\oplus W_2(U_2^\bullet ,U_2)\oplus Sol_3^B(U_R,U_2), \end{aligned}$$(5.17)where \(U_2\) is defined by (5.15), \(U_2^\bullet ,U_2^{\bullet \bullet }\) are defined by (5.16).
-
(iii)
Now, we consider the construction beginning with a 1-wave followed by admissible stationary wave and finished by a 3-wave connects to \(U_R\).
If \(U_L\in G_{3A}\), from Lemma 4.6, there exists \(U_{h_1}(\rho _{h_1},u_{h_1})\) such that
$$\begin{aligned} \begin{aligned}&U_{h_1}\in \overrightarrow{S_1}(U_L),\\&\overline{\lambda _1}(U_L,U)<0 \text { if and only if } U\in \overrightarrow{S_1}(U_L) \text { and } \rho >\rho _{h_1}. \end{aligned} \end{aligned}$$(5.18)If \(U_L\in G_{3B}\), from Lemma 4.6, there exists \(\mathop {U_L}\limits ^\sim (\mathop {\rho _L}\limits ^\sim ,\mathop {u_L}\limits ^\sim ) \in G_{2B}^+\) such that
$$\begin{aligned} \begin{aligned}&\overline{\lambda _1}(U_L,U)>0\quad \text { if } U\in \overrightarrow{S_1}(U_L),\rho _L<\rho<\mathop {\rho _L}\limits ^\sim ,\\&\overline{\lambda _1}(U_L,\mathop {U_L}\limits ^\sim )=0,\\&\overline{\lambda _1}(U_L,U)<0\quad \text { if }U\in \overrightarrow{S_1}(U_L),\rho >\mathop {\rho _L}\limits ^\sim . \end{aligned} \end{aligned}$$(5.19)Let us define the 1- and 2-wave curve \(\Lambda _3\) as follows
$$\begin{aligned} \Lambda _3 := \left\{ \begin{gathered} \lbrace \Upsilon (U_0,a_L,a_R)\mid U_0\in \overrightarrow{S_1}(U_L),\rho _0\ge \rho _{h_1}\rbrace \text { if } U_L\in G_{3A},\\ \lbrace \Upsilon (U_0,a_L,a_R)\mid U_0\in \overrightarrow{S_1}(U_L),\rho _0\ge \mathop {\rho _L}\limits ^\sim \rbrace \text { if } U_L\in G_{3B},\\ \end{gathered} \right. \end{aligned}$$(5.20)where \(\Upsilon \) is defined by (5.1), \(\rho _{h_1}\) is defined by (5.18), and \(\mathop {\rho _L}\limits ^\sim \) is defined by (5.19).
If \(\overleftarrow{W_3}(U_R)\cap \Lambda _3\ne \varnothing \), we let
$$\begin{aligned} U_3\in \overleftarrow{W_3}(U_R)\cap \Lambda _3. \end{aligned}$$(5.21)There exists a point \(U_3^\bullet \in \overrightarrow{S_1}(U_L)\) such that
$$\begin{aligned} \overline{\lambda _1}(U_L,U_3^\bullet )=0,\quad U_3\in \Upsilon (U_3^\bullet ,a_L,a_R). \end{aligned}$$(5.22)If least one point of \(U_R\) or \(U_3\) is not belonging to \(G_{1A}\cup G_{1B}\), then each 3-wave in the composite 3-wave \(Sol_3^B(U_R,U_3)\) has positive speed. The Riemann solution has the following form
$$\begin{aligned} S_1(U_L,U_3^\bullet )\oplus W_2(U_3^\bullet ,U_3)\oplus Sol_3^B(U_R,U_3), \end{aligned}$$(5.23)where \(U_3\) is defined by (5.21), and \(U_3^\bullet \) is defined by (5.22).
The curve \(\Lambda _1\) defined by (5.6) and the illustration of Riemann solution in case (i) supersonic construction
The curve \(\Lambda _2\) defined by (5.14) and the illustration of Riemann solution in case (ii) supersonic construction
The curve \(\Lambda _3\) defined by (5.20) and the illustration of Riemann solution in case (iii) supersonic Construction
The curve \(\Delta _1\) defined by (5.29) and the illustration of Riemann solution in case (i) subsonic construction
The curve \(\Delta _2\) defined by (5.37) and the illustration of Riemann solution in case (ii) subsonic construction
The curve \(\Delta _3\) defined by (5.41) and the illustration of Riemann solution in case (iii) subsonic construction
5.2 Case 2: Subsonic Construction
We consider the construction starting from a given left-hand state in subsonic region \(G_{2A}\cup G_{2B}\).
-
(i)
We define the intersection point
$$\begin{aligned} U^\sharp (\rho ^\sharp ,u^\sharp ) \in \overrightarrow{W_1}(U_L)\cap (C_A^+\cup C_B^+). \end{aligned}$$(5.24)If \(\Upsilon (U^\sharp ,a_L,a_R)\cap G_{3A}\ne \varnothing \), where \(\Upsilon \) is defined by (5.1), we let
$$\begin{aligned} U^{A\sharp }(\rho ^{A\sharp },u^{A\sharp }) \in \Upsilon (U^\sharp ,a_L,a_R)\cap G_{3A}. \end{aligned}$$(5.25)From Lemma 4.6, there exists \(U^{A\sharp }_{l_1}(\rho ^{A\sharp }_{l_1},u^{A\sharp }_{l_1})\) such that
$$\begin{aligned} \begin{aligned}&U^{A\sharp }_{l_1}\in \overrightarrow{S_1}(U^{A\sharp }),\\&\overline{\lambda _1}(U^{A\sharp },U)>0 \text { if and only if } U\in \overrightarrow{S_1}(U^{A\sharp })\text { and } |\rho -\rho ^{A\sharp }|<|\rho ^{A\sharp }_{l_1}-\rho ^{A\sharp }|. \end{aligned}\nonumber \\ \end{aligned}$$(5.26)If \(\Upsilon (U^\sharp ,a_L,a_R)\cap G_{3B}\ne \varnothing \), where \(\Upsilon \) is defined by (5.1), we let
$$\begin{aligned} U^{B\sharp }(\rho ^{B\sharp },u^{B\sharp }) \in \Upsilon (U^\sharp ,a_L,a_R)\cap G_{3B}. \end{aligned}$$(5.27)From Lemma 4.6, there exists a state \({\mathop {U}\limits ^\sim }^{B\sharp }({\mathop {\rho }\limits ^\sim }^{B\sharp },{\mathop {u}\limits ^\sim }^{B\sharp }) \in \overrightarrow{S_1}(U^{B\sharp })\cap G_{2B}^+\) such that
$$\begin{aligned} \begin{aligned}&\overline{\lambda _1}(U^{B\sharp },U)>0\quad \text { if } U\in \overrightarrow{S_1}(U^{B\sharp }),\rho ^{B\sharp }<\rho<{\mathop {\rho }\limits ^\sim }^{B\sharp },\\&\overline{\lambda _1}(U^{B\sharp },{\mathop {U}\limits ^\sim }^{B\sharp })=0,\\&\overline{\lambda _1}(U^{B\sharp },U)<0\quad \text { if } U\in \overrightarrow{S_1}(U^{B\sharp }),\rho >{\mathop {\rho }\limits ^\sim }^{B\sharp }. \end{aligned} \end{aligned}$$(5.28)Let us define the 1- and 2-wave curve \(\Delta _1\) as follows
$$\begin{aligned} \Delta _1 = \left\{ \begin{gathered} \Delta _{1A}:=\lbrace U\in \overrightarrow{W_1}(U^{A\sharp })\mid \rho \le \rho ^{A\sharp }_{l_1}\rbrace \text { if } \Upsilon (U^\sharp ,a_L,a_R)\cap G_{3A}\ne \varnothing ,\\ \Delta _{1B}:=\lbrace U\in \overrightarrow{W_1}(U^{B\sharp })\mid \rho \le {\mathop {\rho }\limits ^\sim }^{B\sharp }\rbrace \text { if } \Upsilon (U^\sharp ,a_L,a_R)\cap G_{3B}\ne \varnothing ,\\ \end{gathered} \right. \nonumber \\ \end{aligned}$$(5.29)If \(\overleftarrow{W_3}(U_R)\cap \Delta _1\ne \varnothing \), we let a intersection point
$$\begin{aligned} U_1\in \overleftarrow{W_3}(U_R)\cap \Delta _1. \end{aligned}$$(5.30)The Riemann solution beginning with a composite 1-wave with negative speed for each 1-wave from \(U_L\) followed by admissible stationary wave, followed by a composite 1-wave with positive speed for each 1-wave and finished by a composite 3-wave connects to \(U_R\). Specifically,
-
+ If \(U_1\in \Delta _{1A}\), then it holds the form of Riemann solution
$$\begin{aligned} Sol_1(U_L,U^\sharp )\oplus W_2(U^\sharp ,U^{A\sharp })\oplus Sol_1(U^{A\sharp },U_1)\oplus Sol_3^B(U_R,U_1),\quad \quad \quad \end{aligned}$$(5.31)where \(U^\sharp , U^{A\sharp },U_1\) are defined by (5.24), (5.25), (5.30).
-
+ If \(U_1\in \Delta _{1B}\), then the Riemann solution has the following form
$$\begin{aligned} Sol_1(U_L,U^\sharp )\oplus W_2(U^\sharp ,U^{B\sharp })\oplus Sol_1(U^{B\sharp },U_1)\oplus Sol_3^B(U_R,U_1),\quad \quad \quad \end{aligned}$$(5.32)where \(U^\sharp , U^{B\sharp }, U_1\) are defined by (5.24), (5.27), (5.30).
-
-
(ii)
We consider the intersection point \(U^\sharp \) is defined by (5.24). If \(\lbrace \Upsilon (U^\sharp ,a_L,a)\mid a\in (a_L,a_R)\rbrace \cap G_{3B}\ne \varnothing \), then there exist two values \(a_l, a_h\) such that
$$\begin{aligned} \lbrace \Upsilon (U^\sharp ,a_L,a)\mid a\in [a_l,a_h]\rbrace \subset G_{3B},\quad a_L\le a_l<a_h\le a_R. \end{aligned}$$(5.33)We define the continuous curve \({\mathcal {N}}_1\subset G_{3B}\) containing all states that can be connected from \(U^\sharp \) defined as (5.24) by admissible stationary waves when the cross-sectional area jumps from \(a_L\) to a, for all \(a\in [a_l,a_h]\):
$$\begin{aligned} {\mathcal {N}}_1:=\lbrace \Upsilon (U^\sharp ,a_L,a)\mid a\in [a_l,a_h]\rbrace , \end{aligned}$$(5.34)where \(\Upsilon \) is defined by (5.1), and \(a_l,a_h\) are defined by (5.33).
From Lemma 4.6, for each \(U_0\in {\mathcal {N}}_1\) then there exists \(\mathop {U_0}\limits ^\sim \in G_{2B}^+\) such that
$$\begin{aligned} \mathop {U_0}\limits ^\sim \in \overrightarrow{S_1}(U_0),\quad \overline{\lambda _1}(U_0,\mathop {U_0}\limits ^\sim )=0. \end{aligned}$$(5.35)We define the continuous curve \({\mathcal {N}}_2\) belonging to \(G_{2B}^+\) as follows
$$\begin{aligned} \begin{aligned}&\Upsilon ^\natural (U^\sharp ,a):=\lbrace \mathop {U_0}\limits ^\sim \mid U_0\in \Upsilon (U^\sharp ,a_L,a)\subset {\mathcal {N}}_1\rbrace ,\\&{\mathcal {N}}_2:=\lbrace \Upsilon ^\natural (U^\sharp ,a)\mid a\in [a_l,a_h]\rbrace , \end{aligned} \end{aligned}$$(5.36)where \(\Upsilon \) is defined by (5.1), \(U^\sharp \) is defined by (5.24), \(\mathop {U_0}\limits ^\sim \) is defined by (5.35), and \(a_l,a_h\) are defined by (5.33).
Let us define the 1- and 2-wave curve \(\Delta _2\) as follows
$$\begin{aligned} \begin{aligned}&\Upsilon ^\dagger (U^\sharp ,a):=\lbrace \Upsilon (U_0,a,a_R)\mid U_0\in \Upsilon ^\natural (U^\sharp ,a)\rbrace ,\\&\Delta _2:=\lbrace \Upsilon ^\dagger (U^\sharp ,a)\mid a\in [a_l,a_h]\rbrace , \end{aligned} \end{aligned}$$(5.37)where \(\Upsilon \) is defined by (5.1), and \(a_l,a_h\) are defined by (5.33).
If \(\overleftarrow{W_3}(U_R)\cap \Delta _2\ne \varnothing \), we consider a intersection point
$$\begin{aligned} U_2\in \overleftarrow{W_3}(U_R)\cap \Delta _2. \end{aligned}$$(5.38)It holds that there exist a value \(a\in [a_l,a_h]\) and two states \(U_2^\bullet \in {\mathcal {N}}_2\) and \(U_2^{\bullet \bullet }\in {\mathcal {N}}_1\) such that
$$\begin{aligned} \begin{aligned}&U_2^{\bullet \bullet }\in \Upsilon (U^\sharp ,a_L,a),\\&U_2\in \Upsilon (U_2^\bullet ,a,a_R),\\&U_2^\bullet \in \overrightarrow{S_1}(U_2^{\bullet \bullet }), \quad \overline{\lambda _1}(U_2^{\bullet \bullet },U_2^\bullet )=0, \end{aligned} \end{aligned}$$(5.39)where \(\Upsilon \) is defined by (5.1).
The Riemann solution has the form
$$\begin{aligned} Sol_1(U_L,U^\sharp )\oplus W_2(U^\sharp ,U_2^{\bullet \bullet })\oplus S_1(U_2^{\bullet \bullet },U_2^\bullet )\oplus W_2(U_2^\bullet ,U_2)\oplus Sol_3^B(U_R,U_2),\nonumber \\ \end{aligned}$$(5.40)where \(U^\sharp \) is defined by (5.24), \(U_2\) is defined by (5.38), and \(U_2^\bullet \) and \(U_2^{\bullet \bullet }\) are defined by (5.39).
-
(iii)
With \(U^\sharp \) is defined by (5.24), we define the 1- and 2-wave curve \(\Delta _3\) as follows
$$\begin{aligned} \Delta _3:=\lbrace \Upsilon (U_0,a_L,a_R)\mid U_0\in \overrightarrow{W_1}(U_L),\rho _0\ge \rho ^\sharp \rbrace , \end{aligned}$$(5.41)where \(\Upsilon \) is defined by (5.1).
If \(\overleftarrow{W_3}(U_R)\cap \Delta _3\ne \varnothing \), we call a intersection point
$$\begin{aligned} U_3\in \overleftarrow{W_3}(U_R)\cap \Delta _3. \end{aligned}$$(5.42)Then, there exists a state \(U_3^\bullet (\rho _3^\bullet ,u_3^\bullet ) \in \overrightarrow{W_1}(U_L)\) such that
$$\begin{aligned} U_3\in \Upsilon (U_3^\bullet ,a_L,a_R),\quad \rho _3^\bullet \ge \rho ^\sharp , \end{aligned}$$(5.43)where \(\Upsilon \) is defined by (5.1), and \(\rho ^\sharp \) is defined by (5.24).
Moreover, each 1-wave in composite 1-wave \(Sol_1(U_L,U_3^\bullet )\) has negative speed. Beside that, if least one points of \(U_R\) or \(U_3\) not belonging to \(G_{1A}\cup G_{1B}\), then each 3-wave in the composite 3-wave \(Sol_3^B(U_R,U_3)\) has positive speed.
The Riemann solution has the form
$$\begin{aligned} Sol_1(U_L,U_3^\bullet )\oplus W_2(U_3^\bullet ,U_3)\oplus Sol_3^B(U_R,U_3), \end{aligned}$$(5.44)where \(U_3\) is defined by (5.42), and \(U_3^\bullet \) is defined by (5.43).
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Acknowledgements
The authors would like to thank the reviewers for their very constructive comments and fruitful discussions. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number “101.02-2019.306”.
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Thanh, M.D., Vinh, D.X. The Riemann Problem with Phase Transitions for Fluid Flows in a Nozzle. Bull. Malays. Math. Sci. Soc. 44, 2271–2317 (2021). https://doi.org/10.1007/s40840-020-01059-7
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DOI: https://doi.org/10.1007/s40840-020-01059-7