2.1 Governing equations

The main modelling assumption is that each constituent of the mixture obeys the same balance laws as a single fluid, but mutual interaction of the constituents must be taken into account (Truesdell Reference Truesdell1984). The governing equations for an $n$ -component mixture ( $\unicode[STIX]{x1D6FC}=1,\ldots ,n$ ) then read (see Ruggeri & Simić Reference Ruggeri and Simić2007):

(2.1) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x2202}t}+\text{div}(\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}\boldsymbol{v}_{\unicode[STIX]{x1D6FC}})=\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}},\\ \displaystyle \frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}\boldsymbol{v}_{\unicode[STIX]{x1D6FC}})}{\unicode[STIX]{x2202}t}+\text{div}(\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}\boldsymbol{v}_{\unicode[STIX]{x1D6FC}}\otimes \boldsymbol{v}_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D64F}_{\unicode[STIX]{x1D6FC}})=\boldsymbol{m}_{\unicode[STIX]{x1D6FC}},\\ \displaystyle \frac{\unicode[STIX]{x2202}(\frac{1}{2}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}v_{\unicode[STIX]{x1D6FC}}^{2}+\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FC}})}{\unicode[STIX]{x2202}t}+\text{div}\{({\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}v_{\unicode[STIX]{x1D6FC}}^{2}+\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FC}})\boldsymbol{v}_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D64F}_{\unicode[STIX]{x1D6FC}}\boldsymbol{v}_{\unicode[STIX]{x1D6FC}}+\boldsymbol{q}_{\unicode[STIX]{x1D6FC}}\}=e_{\unicode[STIX]{x1D6FC}},\end{array}\right\}\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}$ , $\boldsymbol{v}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FC}}$ are the mass density, the velocity and the internal energy of the $\unicode[STIX]{x1D6FC}$ -constituent, $\unicode[STIX]{x1D64F}_{\unicode[STIX]{x1D6FC}}$ and $\boldsymbol{q}_{\unicode[STIX]{x1D6FC}}$ are the stress tensor and the heat flux, while the source terms $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}$ , $\boldsymbol{m}_{\unicode[STIX]{x1D6FC}}$ and $e_{\unicode[STIX]{x1D6FC}}$ take into account the mutual interactions of the constituents. Moreover, $v_{\unicode[STIX]{x1D6FC}}=|\boldsymbol{v}_{\unicode[STIX]{x1D6FC}}|$ and similar notations are adopted for other velocities from here on. It is an axiom of the mixture theory that conservation laws of mass, momentum and energy of the whole mixture must hold, imposing the following constraints on the source terms:

(2.2a-c ) $$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D6FC}=1}^{n}\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}=0,\quad \mathop{\sum }_{\unicode[STIX]{x1D6FC}=1}^{n}\boldsymbol{m}_{\unicode[STIX]{x1D6FC}}=\mathbf{0},\quad \mathop{\sum }_{\unicode[STIX]{x1D6FC}=1}^{n}e_{\unicode[STIX]{x1D6FC}}=0.\end{eqnarray}$$

In accordance with our assumption (vii), the stress tensor and the heat flux will be reduced to

(2.3a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D64F}_{\unicode[STIX]{x1D6FC}}=-p_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D644},\quad \boldsymbol{q}_{\unicode[STIX]{x1D6FC}}=\mathbf{0},\end{eqnarray}$$

where $\unicode[STIX]{x1D644}$ is the unit tensor.

For the sake of brevity we shall omit the general analysis of the MT model of mixtures, which can be found elsewhere (Ruggeri & Simić Reference Ruggeri and Simić2007; Madjarević & Simić Reference Madjarević and Simić2013; Madjarević et al. Reference Madjarević, Ruggeri and Simić2014). Instead, the case of a binary reactive mixture will be presented. The governing equations consist of conservation laws of mass, momentum and energy for the mixture, and balance laws of mass, momentum and energy for one constituent (Ruggeri & Simić Reference Ruggeri and Simić2007). Since it is common practice in simple modelling of chemical reactions to monitor the behaviour of the product of the forward reaction, we decided to include the balance laws for the product in the governing equations, so that they read:

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}}{\unicode[STIX]{x2202}t}+\text{div}(\unicode[STIX]{x1D70C}\boldsymbol{v})=0, & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}\boldsymbol{v})}{\unicode[STIX]{x2202}t}+\text{div}\left(\unicode[STIX]{x1D70C}\boldsymbol{v}\otimes \boldsymbol{v}+p\unicode[STIX]{x1D644}+\frac{1}{\unicode[STIX]{x1D70C}z(1-z)}\boldsymbol{J}\otimes \boldsymbol{J}\right)=\mathbf{0}, & \displaystyle\end{eqnarray}$$

(2.6) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\frac{1}{2}\unicode[STIX]{x1D70C}v^{2}+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700}\right)+\text{div}\left\{\!\left(\frac{1}{2}\unicode[STIX]{x1D70C}v^{2}+\unicode[STIX]{x1D70C}\unicode[STIX]{x1D700}+p\right)\boldsymbol{v}+\left(\frac{\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{J}}{\unicode[STIX]{x1D70C}z(1-z)}+\frac{1}{\unicode[STIX]{x1D6FD}}\right)\boldsymbol{J}\right\}=0, & \displaystyle\end{eqnarray}$$

(2.7) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}z}{\unicode[STIX]{x2202}t}+\text{div}(\unicode[STIX]{x1D70C}z\boldsymbol{v}+\boldsymbol{J})=\unicode[STIX]{x1D70F}_{p}, & \displaystyle\end{eqnarray}$$

(2.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}(\unicode[STIX]{x1D70C}z\boldsymbol{v}+\boldsymbol{J})}{\unicode[STIX]{x2202}t}+\text{div}\left(\unicode[STIX]{x1D70C}z\boldsymbol{v}\otimes \boldsymbol{v}+p_{p}\unicode[STIX]{x1D644}+\frac{1}{\unicode[STIX]{x1D70C}z}\boldsymbol{J}\otimes \boldsymbol{J}+\boldsymbol{v}\otimes \boldsymbol{J}+\boldsymbol{J}\otimes \boldsymbol{v}\right)=\boldsymbol{m}_{p}, & \displaystyle\end{eqnarray}$$

and

(2.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}\left(\frac{1}{2}\unicode[STIX]{x1D70C}z\left(\boldsymbol{v}+\frac{\boldsymbol{J}}{\unicode[STIX]{x1D70C}z}\right)^{2}+\unicode[STIX]{x1D70C}z\unicode[STIX]{x1D700}_{p}\right)\nonumber\\ \displaystyle & & \displaystyle \quad +\,\text{div}\left\{\left(\frac{1}{2}\unicode[STIX]{x1D70C}z\left(\boldsymbol{v}+\frac{\boldsymbol{J}}{\unicode[STIX]{x1D70C}z}\right)^{2}+\unicode[STIX]{x1D70C}z\unicode[STIX]{x1D700}_{p}+p_{p}\right)\left(\boldsymbol{v}+\frac{\boldsymbol{J}}{\unicode[STIX]{x1D70C}z}\right)\right\}=e_{p}.\end{eqnarray}$$

Here the following quantities are introduced:

(2.10) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{r}+\unicode[STIX]{x1D70C}_{p}\quad \text{mass density of the mixture,}\\ \displaystyle z=\unicode[STIX]{x1D70C}_{p}/\unicode[STIX]{x1D70C}\quad \text{concentration of the products,}\\ \displaystyle \boldsymbol{v}=\frac{1}{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70C}_{r}\boldsymbol{v}_{r}+\unicode[STIX]{x1D70C}_{p}\boldsymbol{v}_{p})\quad \text{mean velocity of the mixture,}\\ \displaystyle p=p_{r}+p_{p}\quad \text{total pressure of the mixture,}\\ \displaystyle \boldsymbol{J}=\unicode[STIX]{x1D70C}_{p}\boldsymbol{u}_{p}=-\unicode[STIX]{x1D70C}_{r}\boldsymbol{u}_{r}\quad \text{diffusion flux vector,}\\ \displaystyle \boldsymbol{u}_{\unicode[STIX]{x1D6FC}}=\boldsymbol{v}_{\unicode[STIX]{x1D6FC}}-\boldsymbol{v}\;(\unicode[STIX]{x1D6FC}=r,p)\quad \text{diffusion velocities,}\\ \displaystyle \unicode[STIX]{x1D700}_{I}=\frac{1}{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70C}_{r}\unicode[STIX]{x1D700}_{r}+\unicode[STIX]{x1D70C}_{p}\unicode[STIX]{x1D700}_{p})\quad \text{intrinsic (thermal) internal energy density of the mixture,}\\ \displaystyle \unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}_{I}+\frac{1}{2\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70C}_{r}u_{r}^{2}+\unicode[STIX]{x1D70C}_{p}u_{p}^{2})\quad \text{internal energy density of the mixture,}\end{array}\right\}\quad \qquad\end{eqnarray}$$

and

(2.11) $$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D6FD}}=g_{p}-g_{r},\quad \text{with }g_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FC}}+\frac{p_{\unicode[STIX]{x1D6FC}}}{\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}}+\frac{1}{2}u_{\unicode[STIX]{x1D6FC}}^{2},\text{ for }\unicode[STIX]{x1D6FC}=p,r,\end{eqnarray}$$

where $g_{\unicode[STIX]{x1D6FC}}$ are dynamic enthalpies.

To close the system of equations (2.4)–(2.9), one has to introduce constitutive assumptions about pressure and internal energy density, as well as to determine the structure of the source terms. The latter is dictated by the general principles of extended thermodynamics – Galilean invariance and the entropy principle. Galilean invariance determines the velocity dependence of source terms (Ruggeri & Simić Reference Ruggeri and Simić2007):

(2.12a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{p}=\hat{\unicode[STIX]{x1D70F}}_{p},\quad \boldsymbol{m}_{p}=\hat{\boldsymbol{m}}_{p}+\hat{\unicode[STIX]{x1D70F}}_{p}\boldsymbol{v},\quad e_{p}=\hat{e}_{p}+\hat{\boldsymbol{m}}_{p}\boldsymbol{\cdot }\boldsymbol{v}+\hat{\unicode[STIX]{x1D70F}}_{p}\frac{v^{2}}{2},\end{eqnarray}$$

where $\hat{\unicode[STIX]{x1D70F}}_{p}$ , $\hat{\boldsymbol{m}}_{p}$ and $\hat{e}_{p}$ denote velocity-independent mass, momentum and energy production rate, respectively. Information about the chemical reaction is contained in the source term $\unicode[STIX]{x1D70F}_{p}$ of the mass balance law (2.7). Nevertheless, it is reflected in all other source terms as well, due to Galilean invariance (2.12). The entropy principle imposes a further restriction on the source terms by requiring that every thermodynamic process is compatible with an entropy inequality. Its consequences will be discussed in the sequel.

2.2 Constitutive assumptions and the state variables

Since viscosity and heat conductivity are not taken into account in our model, the constitutive assumptions are related to thermal and caloric equations of state. We shall restrict our study to a binary mixture of ideal gases that consists of reactants and products. As we are analysing the simplest possible model of a reversible chemical reaction, $A_{r}\rightleftharpoons A_{p}$ , according to assumption (i) the masses of the constituents ought to be equal due to mass conservation, $m_{r}=m_{p}=m$ . Consequently, the equations of state have the following form:

(2.13a,b ) $$\begin{eqnarray}p_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}\frac{k}{m}T_{\unicode[STIX]{x1D6FC}},\quad \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FC}}+\int _{T_{0}}^{T_{\unicode[STIX]{x1D6FC}}}c_{v\unicode[STIX]{x1D6FC}}(\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F},\quad \unicode[STIX]{x1D6FC}=r,p,\end{eqnarray}$$

where $k$ is the Boltzmann constant, $c_{v\unicode[STIX]{x1D6FC}}(T_{\unicode[STIX]{x1D6FC}})$ are the specific heats at constant volume and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FC}}$ are the binding energies. Assuming $c_{vp}=c_{vr}=c_{v}=\text{const.}$ , and $\unicode[STIX]{x1D716}_{r}=c_{v}T_{0}$ , the internal energies take the form:

(2.14a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D700}_{p}=c_{v}T_{p}-\unicode[STIX]{x0394}\unicode[STIX]{x1D716},\quad \unicode[STIX]{x1D700}_{r}=c_{v}T_{r},\end{eqnarray}$$

where $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}=\unicode[STIX]{x1D716}_{p}-\unicode[STIX]{x1D716}_{r}$ , and $c_{v}=k/(m(\unicode[STIX]{x1D6FE}-1))$ in accordance with assumption (v).

The temperatures of the constituents are not equal in general, and we have to introduce the average (mean) temperature of the mixture as a global variable. Following Gouin & Ruggeri (Reference Gouin and Ruggeri2008) and Ruggeri & Simić (Reference Ruggeri and Simić2009), we shall define the average temperature using the definition of the intrinsic part of the internal energy density:

(2.15) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{p}c_{v}T_{p}+\unicode[STIX]{x1D70C}_{r}c_{v}T_{r}=(\unicode[STIX]{x1D70C}_{p}c_{v}+\unicode[STIX]{x1D70C}_{r}c_{v})T,\quad \text{that is, }T=zT_{p}+(1-z)T_{r},\end{eqnarray}$$

where we have taken into account the relations $\unicode[STIX]{x1D70C}_{p}=\unicode[STIX]{x1D70C}z$ and $\unicode[STIX]{x1D70C}_{r}=\unicode[STIX]{x1D70C}(1-z)$ . It is also convenient to introduce the difference of temperatures,

(2.16) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}=T_{p}-T_{r},\end{eqnarray}$$

as a state variable, which also serves as a measure of thermal non-equilibrium. In such a way, the temperatures of the constituents can be expressed in terms of $z$ , $T$ and $\unicode[STIX]{x1D6E9}$ , as follows:

(2.17a,b ) $$\begin{eqnarray}T_{p}=T+(1-z)\unicode[STIX]{x1D6E9},\quad T_{r}=T-z\unicode[STIX]{x1D6E9}.\end{eqnarray}$$

The total pressure $p$ and the intrinsic internal energy density $\unicode[STIX]{x1D700}_{I}$ of the mixture can be expressed in terms of $T$ :

(2.18a,b ) $$\begin{eqnarray}p=p_{p}+p_{r}=\unicode[STIX]{x1D70C}\frac{k}{m}T,\quad \unicode[STIX]{x1D700}_{I}=\frac{\unicode[STIX]{x1D70C}_{p}}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D700}_{p}+\frac{\unicode[STIX]{x1D70C}_{r}}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D700}_{r}=\frac{k}{m(\unicode[STIX]{x1D6FE}-1)}T-z\unicode[STIX]{x0394}\unicode[STIX]{x1D716},\end{eqnarray}$$

where we have taken into account assumption (v), that is, $\unicode[STIX]{x1D6FE}_{p}=\unicode[STIX]{x1D6FE}_{r}=\unicode[STIX]{x1D6FE}$ .

Having in mind the structure of the governing equations (2.4)–(2.9), we may fix the set of state variables. Although the first choice is usually $(\unicode[STIX]{x1D70C},\boldsymbol{v},T,\unicode[STIX]{x1D70C}_{p},\boldsymbol{v}_{p},T_{p})$ , using the relations given above, we shall adopt $\boldsymbol{U}=(\unicode[STIX]{x1D70C},\boldsymbol{v},T,z,\boldsymbol{J},\unicode[STIX]{x1D6E9})$ .

2.3 The source terms

In our model, the source terms describe the mutual interactions between the constituents. Their structure, compatible with entropy inequality, was determined by Ruggeri & Simić (Reference Ruggeri and Simić2007) when a non-reactive mixture is considered. However, the extended thermodynamics has its limits and the phenomenological coefficients cannot be determined within its framework. Instead, one has either to take into account experimental results, or to make a link with a more refined approach (e.g. kinetic theory) to determine them (see Madjarević & Simić Reference Madjarević and Simić2013; Madjarević et al. Reference Madjarević, Ruggeri and Simić2014). We shall proceed in the latter way.

2.3.1 The reaction rate

For the reactive mixture considered in this paper, the reaction rate is obtained from the kinetic theory; more precisely, from the Boltzmann equation for chemically reactive mixtures, assuming reactive differential cross-sections with activation energy, and a particular form of the velocity distribution functions in order to obtain an approximate solution to the reactive kinetic equations. Since we are considering an MT mixture, an appropriate form for such an input function should be a Maxwellian distribution in the constituent rest frame,

(2.19) $$\begin{eqnarray}f_{\unicode[STIX]{x1D6FC}}=n_{\unicode[STIX]{x1D6FC}}\left(\frac{m}{2\unicode[STIX]{x03C0}kT_{\unicode[STIX]{x1D6FC}}}\right)^{3/2}\exp \left(-\frac{m}{2kT_{\unicode[STIX]{x1D6FC}}}(c_{\unicode[STIX]{x1D6FC}}-v_{\unicode[STIX]{x1D6FC}})^{2}\right),\quad \unicode[STIX]{x1D6FC}=r,p,\end{eqnarray}$$

where $c_{\unicode[STIX]{x1D6FC}}=|\boldsymbol{c}_{\unicode[STIX]{x1D6FC}}|$ , $\boldsymbol{c}_{\unicode[STIX]{x1D6FC}}$ being the molecular velocity of each constituent, and $n_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6FC}}/m$ the constituent number density. However, the evaluation of the collision integrals applying to the input functions (2.19) becomes a very difficult task and the computations lead to intractable expressions. To overcome such difficulty we approximate function (2.19) through an expansion at the first order around a Maxwellian in the mixture rest frame with temperature $T$ and mean velocity $v$ :

(2.20) $$\begin{eqnarray}\displaystyle G_{\unicode[STIX]{x1D6FC}} & = & \displaystyle n_{\unicode[STIX]{x1D6FC}}\left(\frac{m}{2\unicode[STIX]{x03C0}kT}\right)^{3/2}\exp \left(-\frac{m}{2kT}(c_{\unicode[STIX]{x1D6FC}}-v)^{2}\right)\nonumber\\ \displaystyle & & \displaystyle \times \left[1+\frac{m}{kT}\mathop{\sum }_{i=1}^{3}(c_{\unicode[STIX]{x1D6FC}i}-v_{i})(v_{\unicode[STIX]{x1D6FC}i}-v_{i})+\left(\frac{m}{2kT}(c_{\unicode[STIX]{x1D6FC}}-v)^{2}-\frac{3}{2}\right)\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D6FC}}\right].\end{eqnarray}$$

Here $(v_{\unicode[STIX]{x1D6FC}i}-v_{i})$ and $(c_{\unicode[STIX]{x1D6FC}i}-v_{i})$ represent the spatial components of the diffusion velocity and of the peculiar velocity, respectively, and $\unicode[STIX]{x1D6E5}_{\unicode[STIX]{x1D6FC}}=(T_{\unicode[STIX]{x1D6FC}}-T)/T$ is the normalized temperature difference. The input function (2.20) represents a linearization of the Maxwellian distribution (2.19), with respect to the diffusion velocity and temperature difference.

Adopting reactive differential cross-sections with activation energy, and taking into account the indistinguishability of the colliding particles, the reaction rate for the mass concentration $z$ of the products results in (Kremer, Bianchi & Soares Reference Kremer, Bianchi and Soares2007)

(2.21) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D70F}}_{p} & = & \displaystyle 2\frac{\unicode[STIX]{x1D70C}^{2}}{m}\sqrt{\frac{\unicode[STIX]{x03C0}kT}{m}}d^{2}\exp \left(-\frac{\unicode[STIX]{x1D716}_{f}}{kT}\right)\left[(1-z)^{2}-z^{2}\exp \left(\frac{\unicode[STIX]{x1D716}_{f}-\unicode[STIX]{x1D716}_{b}}{kT}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\left.(1-z)^{2}\left(\frac{\unicode[STIX]{x1D716}_{f}}{kT}+\frac{1}{2}\right)\unicode[STIX]{x1D6E5}_{r}-(1-z)^{2}\left(\frac{\unicode[STIX]{x1D716}_{b}}{kT}+\frac{1}{2}\right)\unicode[STIX]{x1D6E5}_{p}\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D716}_{f}$ and $\unicode[STIX]{x1D716}_{b}$ denote the forward and the backward activation energies of the chemical reaction. The reaction rate (2.21) shows a second-order form, in the sense that it depends explicitly on the square of the product concentration $z$ and of the reactant concentration $(1-z)$ . This is a direct consequence of the fact that it was derived from the Boltzmann equation with binary reactive encounters. A detailed account of other aspects of the mathematical modelling of reactive flows may be found in Giovangigli (Reference Giovangigli1999).

The thermodynamic equilibrium condition, chemical and thermal, corresponds to $\hat{\unicode[STIX]{x1D70F}}_{p}=0$ and, therefore, we will have thermodynamic equilibrium if

(2.22a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{r}=0,\quad \unicode[STIX]{x1D6E5}_{p}=0,\quad \frac{1-z}{z}=\exp \left(\frac{{\mathcal{Q}}}{2kT}\right),\end{eqnarray}$$

where ${\mathcal{Q}}=\unicode[STIX]{x1D716}_{f}-\unicode[STIX]{x1D716}_{b}=-2m\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ is the reaction heat. The last condition in (2.22) represents the law of mass action of the model and gives a constraint on the chemical equilibrium values of the product concentration $z$ and mixture temperature  $T$ .

2.3.2 The momentum production rate

The momentum production rate in a binary mixture has the form:

(2.23a,b ) $$\begin{eqnarray}\hat{\boldsymbol{m}}_{p}=-\unicode[STIX]{x1D713}_{pp}\left(\frac{\boldsymbol{u}_{p}}{T_{p}}-\frac{\boldsymbol{u}_{r}}{T_{r}}\right),\quad \unicode[STIX]{x1D713}_{pp}=\frac{1}{\unicode[STIX]{x1D70F}_{D}}\unicode[STIX]{x1D70C}z(1-z)T,\end{eqnarray}$$

where $\unicode[STIX]{x1D713}_{pp}$ is a phenomenological coefficient related to the relaxation time $\unicode[STIX]{x1D70F}_{D}$ for diffusion. By linearizing the source term in the neighbourhood of the local equilibrium state, and taking into account the definition of the diffusion flux, one obtains

(2.24) $$\begin{eqnarray}\hat{\boldsymbol{m}}_{p}=-\frac{\boldsymbol{J}}{\unicode[STIX]{x1D70F}_{D}}.\end{eqnarray}$$

2.3.3 The energy production rate

The energy production rate in a binary mixture reads

(2.25a,b ) $$\begin{eqnarray}\hat{e}_{p}=-\unicode[STIX]{x1D703}_{pp}\left(-\frac{1}{T_{p}}+\frac{1}{T_{r}}\right),\quad \unicode[STIX]{x1D703}_{pp}=\frac{1}{\unicode[STIX]{x1D70F}_{T}}\unicode[STIX]{x1D70C}z(1-z)c_{v}T^{2},\end{eqnarray}$$

where $\unicode[STIX]{x1D703}_{pp}$ is a phenomenological coefficient related to the relaxation time $\unicode[STIX]{x1D70F}_{T}$ for the temperature. By linearizing the source term in the neighbourhood of the local equilibrium state, and using the difference of temperatures $\unicode[STIX]{x1D6E9}$ as non-equilibrium variable, one obtains

(2.26) $$\begin{eqnarray}\hat{e}_{p}=-\frac{1}{\unicode[STIX]{x1D70F}_{T}}\unicode[STIX]{x1D70C}z(1-z)c_{v}\unicode[STIX]{x1D6E9}.\end{eqnarray}$$

In expressions (2.24) and (2.26), the relaxation times $\unicode[STIX]{x1D70F}_{D}$ and $\unicode[STIX]{x1D70F}_{T}$ , which are part of the phenomenological coefficients, cannot be determined by the methods of extended thermodynamics. It was shown in Madjarević & Simić (Reference Madjarević and Simić2013) and Madjarević et al. (Reference Madjarević, Ruggeri and Simić2014), using the arguments of kinetic theory, that the relaxation times for diffusion and temperature can be related to diffusivity of the binary mixture. In our notation, the diffusivity $D$ for the hard-sphere model reads:

(2.27) $$\begin{eqnarray}D=\frac{3}{8nd^{2}}\left(\frac{kT}{2\unicode[STIX]{x03C0}}\frac{m_{p}+m_{r}}{m_{p}m_{r}}\right)^{1/2}.\end{eqnarray}$$

Taking into account that $m_{p}=m_{r}=m$ , the relaxation times reduce to the following simple form:

(2.28a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{D}=\frac{m}{kT}D,\quad \unicode[STIX]{x1D70F}_{T}=\frac{2m}{kT}D,\quad D=\frac{3}{8nd^{2}}\left(\frac{kT}{\unicode[STIX]{x03C0}m}\right)^{1/2}.\end{eqnarray}$$

Note that $\unicode[STIX]{x1D70F}_{T}>\unicode[STIX]{x1D70F}_{D}$ , which is a typical result of kinetic theory.

2.4 The basic assumption and reduced form of the governing equations

Equations (2.4)–(2.9), with the source terms given by (2.21), (2.24) and (2.26), constitute a complete set of governing equations for a chemically reacting MT binary mixture. In their derivation we exploited all the simplifying assumptions mentioned in the introduction. However, it was mentioned that assumption (i) implies further simplifications. It is appropriate at this stage to introduce the basic assumption that will imply the reduction of the system of governing equations, and simplification of the remaining ones.

3.1 Governing equations in the moving frame

According to the ZND model, we shall assume that the wave moves in the $x$ -direction, from left to right, and that the flow is one-dimensional, $\boldsymbol{v}=(v,0,0)$ . The wave configuration is supposed to be steady in the shock-attached frame, and we introduce the steady variable $\unicode[STIX]{x1D709}$ and the relative velocity $u$ of the mixture with respect to the shock front:

(3.1a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D709}=x-st,\quad u=v-s.\end{eqnarray}$$

In the steady frame attached to the shock wave, the reaction zone extends from $x_{0}$ to $x_{S}$ , where $x_{0}$ represents the location of the shock front. The state just behind the shock, located at $x=x_{0}$ , is the von Neumann state $N$ , where the chemical reaction is triggered, and the one located at $x=x_{S}$ , at the end of the reaction zone, is the final steady state $S$ , where the mixture reaches chemical and thermal equilibrium. Ahead of the shock front, that is, for $x>x_{0}$ , the quiescent mixture is at rest in its initial state  $I$ , where the rate of the chemical reaction is completely negligible.

Assuming steady, one-dimensional travelling wave structure of the detonation wave, in which the vector of state variables $\boldsymbol{U}=(\unicode[STIX]{x1D70C},u,T,z,\unicode[STIX]{x1D6E9})$ depends on a single independent variable, say $\boldsymbol{U}(\unicode[STIX]{x1D709})=\boldsymbol{U}(x-st)$ , the governing equations are transformed into a set of ordinary differential equations:

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70C}u)=0, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\left[\unicode[STIX]{x1D70C}u^{2}+\unicode[STIX]{x1D70C}\frac{k}{m}T\right]=0, & \displaystyle\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\left[\left(\frac{1}{2}\unicode[STIX]{x1D70C}u^{2}+\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}-1}\unicode[STIX]{x1D70C}\frac{k}{m}T-\unicode[STIX]{x1D70C}z\unicode[STIX]{x0394}\unicode[STIX]{x1D716}\right)u\right]=0, & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70C}zu)=\hat{\unicode[STIX]{x1D70F}}_{p}, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\left[\left(\frac{1}{2}\unicode[STIX]{x1D70C}zu^{2}+\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}-1}\unicode[STIX]{x1D70C}z\frac{k}{m}(T+(1-z)\unicode[STIX]{x1D6E9})-\unicode[STIX]{x1D70C}z\unicode[STIX]{x0394}\unicode[STIX]{x1D716}\right)u\right]=\hat{e}_{p}+\hat{\unicode[STIX]{x1D70F}}_{p}\frac{u^{2}}{2}. & \displaystyle\end{eqnarray}$$

The source terms $\hat{\unicode[STIX]{x1D70F}}_{p}$ and $\hat{e}_{p}$ are used in linearized form (2.21) and (2.26), respectively. Formally, the governing equations of the standard ZND detonation waves are (3.2)–(3.5). Here, they are extended with (3.6), which describes the profile of the temperature difference $\unicode[STIX]{x1D6E9}$ .

3.2 Algebraic analysis of the detonation wave

As the first step, the algebraic part of the problem will be studied. It will outline the basic structure of the detonation wave, including the important notion of equilibrium Hugoniot curve, peculiar for reversible chemical reactions.

3.2.1 Rankine–Hugoniot conditions

The governing equations of conservative type, equations (3.2)–(3.4), lead to the well-known algebraic Rankine–Hugoniot (RH) conditions that relate a state behind the shock wave to the initial state ahead the shock. Such conditions connect the fluxes of the conserved quantities across the wave front, and are given by

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}u=-\unicode[STIX]{x1D70C}_{0}s, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70C}u^{2}+p=\unicode[STIX]{x1D70C}_{0}s^{2}+p_{0}, & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{2}u^{2}+\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}-1}\frac{p}{\unicode[STIX]{x1D70C}}=\frac{1}{2}s^{2}+\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}-1}\frac{p_{0}}{\unicode[STIX]{x1D70C}_{0}}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}_{0}$ and $p_{0}$ are the mass density and pressure at the unperturbed initial state. Moreover, $z_{0}=0$ and $\unicode[STIX]{x1D6E9}_{0}=0$ at the initial state. Additionally we have assumed that the initial state is at rest, so that $v_{0}=0$ and $u_{0}=-s$ . For each fixed value of the detonation velocity $s$ , the corresponding von Neumann and equilibrium states can be determined from the RH conditions (3.7)–(3.9), with additional proper conditions, as will be explained in the sequel. Introducing the specific volume of the mixture, $\unicode[STIX]{x1D708}=1/\unicode[STIX]{x1D70C}$ , the states $(\unicode[STIX]{x1D708},p)$ in the reaction zone, compatible with a strong detonation wave (Lee Reference Lee2008), are characterized as a function of the product concentration $z$ , with the shock speed $s$ as parameter, by

(3.10a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D708}}{\unicode[STIX]{x1D708}_{0}}=\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}+1}\frac{\displaystyle 1+s^{2}}{\displaystyle s^{2}}-\unicode[STIX]{x1D6F6}(z),\quad \frac{p}{p_{_{0}}}=\frac{\displaystyle 1+s^{2}}{\unicode[STIX]{x1D6FE}+1}+s^{2}\unicode[STIX]{x1D6F6}(z),\end{eqnarray}$$

where $\unicode[STIX]{x1D6F6}(z)$ is defined by

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D6F6}(z)=\frac{\sqrt{\displaystyle (s^{2}-\unicode[STIX]{x1D6FE})^{2}-2(\unicode[STIX]{x1D6FE}+1)(\unicode[STIX]{x1D6FE}-1)z\unicode[STIX]{x0394}\unicode[STIX]{x1D716}s^{2}}}{\displaystyle (\unicode[STIX]{x1D6FE}+1)s^{2}}.\end{eqnarray}$$

In the derivation of (3.10) we used the state equations (2.18); see Marques Junior et al. (Reference Marques Junior, Soares, Bianchi and Kremer2015) for a detailed analysis.

3.2.2 Von Neumann state

The von Neumann state is the state just behind the shock wave. Since the passage of the mixture through the shock front is a non-reactive process, the progress variable remains constant through the shock and the temperature difference does not change. Accordingly, $z=0$ and $\unicode[STIX]{x1D6E9}=0$ at the von Neumann state. Therefore, the state $N$ can be obtained as the non-trivial solution to the RH conditions (3.7)–(3.9) complemented by the conditions $z=0$ and $\unicode[STIX]{x1D6E9}=0$ . Equivalently, the state $N$ can be obtained directly from conditions (3.10), together with the vanishing conditions for $z$ and $\unicode[STIX]{x1D6E9}$ , that is

(3.12a-d ) $$\begin{eqnarray}p_{_{N}}=\frac{2s^{2}}{\unicode[STIX]{x1D6FE}+1}p_{0}-\frac{\unicode[STIX]{x1D6FE}-1}{\unicode[STIX]{x1D6FE}+1}p_{0},\quad \unicode[STIX]{x1D708}_{_{N}}=\frac{\unicode[STIX]{x1D6FE}-1}{\unicode[STIX]{x1D6FE}+1}\unicode[STIX]{x1D708}_{0}+\frac{2}{s^{2}}\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}+1}\unicode[STIX]{x1D708}_{0},\quad z_{_{N}}=0,~\unicode[STIX]{x1D6E9}_{_{N}}=0.\end{eqnarray}$$

3.2.4 Equilibrium final state

The final state $S$ at the end of the reaction zone represents the state for which the reactive mixture reaches chemical and thermal equilibrium. It can be determined from expressions (3.10) with $z$ and $\unicode[STIX]{x1D6E9}$ satisfying the equilibrium conditions (2.22). Therefore, $S$ is characterized by expressions (3.10) together with

(3.13a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}=0,\quad \left(\frac{1+s^{2}}{\unicode[STIX]{x1D6FE}+1}+s^{2}\unicode[STIX]{x1D6F6}(z)\right)\left(\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}+1}\frac{1+s^{2}}{\displaystyle s^{2}}-\unicode[STIX]{x1D6F6}(z)\right)=-\frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{\displaystyle \ln \left(\frac{1-z}{z}\right)},\end{eqnarray}$$

where we have used $\unicode[STIX]{x1D6F6}(z)$ defined by (3.11). Observe that the second condition in (3.13) represents the law of mass action given in (2.22) parametrized by the shock velocity  $s$ .

Additionally, since the state $S$ represents the end point of the continuous reacting flow, in which the mixture reaches equilibrium and all state variables become constant, it can be obtained by integrating equations (3.2)–(3.6) from $\unicode[STIX]{x1D709}=0$ to $\unicode[STIX]{x1D709}=\unicode[STIX]{x1D709}_{S}$ for which all the production rates vanish, as predicted by the equilibrium conditions (2.22) or (3.13).

3.2.6 Hugoniot diagram

The detonation solution can be represented in the $(\unicode[STIX]{x1D708},p)$ -plane through the Hugoniot diagram (Fickett Reference Fickett1985; Lee Reference Lee2008; Conforto et al. Reference Conforto, Monaco and Ricciardello2014; Marques Junior et al. Reference Marques Junior, Soares, Bianchi and Kremer2015) straightly connected to the RH conditions (3.7)–(3.9). As shown in figure 1, this is a qualitative representation of the solution in terms of Rayleigh lines and Hugoniot curves for fixed product concentration, as well as in terms of the equilibrium Hugoniot curve, which constitutes an important ingredient when the chemical reaction is reversible and the reactive mixture reaches a final equilibrium state (Lee Reference Lee2008; Marques Junior et al. Reference Marques Junior, Soares, Bianchi and Kremer2015).

Figure 1. Hugoniot diagram showing unreacted ( $z=0$ ) and complete reaction ( $z=1$ ) Hugoniot curves (solid black curves), Hugoniot curve corresponding to a fixed $z_{eq}$ value (solid grey curve), equilibrium Hugoniot curve (dashed curve) and Rayleigh lines (solid black straight lines).

For fixed values of the material properties, say the specific heat ratio $\unicode[STIX]{x1D6FE}$ , the binding energy difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ and the activation energy $\unicode[STIX]{x1D716}_{f}$ , each Rayleigh line gathers all states obtained for a fixed value of the wave velocity $s$ corresponding to mass and momentum conservation. Each Hugoniot curve for fixed product concentration collects all states along which the total energy of the mixture is preserved. On the other hand, the equilibrium Hugoniot curve accommodates all final equilibrium states for fixed values of the material properties. The necessity for the equilibrium Hugoniot curve comes from the fact that we analyse a reversible chemical reaction. Such a curve is the locus of all thermal and chemical equilibrium states, parametrized by the wave velocity  $s$ .

In the Hugoniot diagram, we highlight two items, namely the $CJ$ state, corresponding to the tangency point of the Rayleigh line to the equilibrium Hugoniot curve, and the equilibrium final state $(\unicode[STIX]{x1D708}_{eq},p_{eq})$ for a fixed value of $s$ ( $s\geqslant s_{j}$ ), obtained as the intersection point of the corresponding Rayleigh line with the equilibrium Hugoniot curve.

For the considered value $s>s_{j}$ , we also plot in figure 1 the fixed-concentration Hugoniot curve obtained for the corresponding value $z=z_{eq}$ (solid grey curve). The equilibrium state $(\unicode[STIX]{x1D708}_{eq},p_{eq})$ for such $s>s_{j}$ , point $S$ in figure 1, is then the intersection of three curves: the corresponding Rayleigh line, the equilibrium Hugoniot curve, and the $z=z_{eq}$ Hugoniot curve. Although in figure 1 it seems that the latter two curves almost coincide, they do not and they must be distinguished.

4.1 Steady detonation equations and hierarchy of the models

The governing equations (3.2)–(3.6) will be written in dimensionless form using the following scaled variables (subscript $0$ refers to the unperturbed initial state in front of the shock wave):

(4.1a-g ) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D70C}}=\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D70C}_{0}},\quad \tilde{u} =\frac{u}{\displaystyle \sqrt{\frac{p_{0}}{\unicode[STIX]{x1D70C}_{0}}}},\quad \tilde{T}=\frac{T}{T_{0}},\quad \tilde{\unicode[STIX]{x1D6E9}}=\frac{\unicode[STIX]{x1D6E9}}{T_{0}},\quad \unicode[STIX]{x0394}\tilde{\unicode[STIX]{x1D716}}=\frac{\,\unicode[STIX]{x0394}\unicode[STIX]{x1D716}\,}{\displaystyle \frac{p_{0}}{\unicode[STIX]{x1D70C}_{0}}},\quad \tilde{\unicode[STIX]{x1D716}}_{f}=\frac{\,\unicode[STIX]{x1D716}_{f}\,}{\displaystyle \frac{p_{0}}{\unicode[STIX]{x1D70C}_{0}}},\quad \tilde{\unicode[STIX]{x1D709}}=\frac{\unicode[STIX]{x1D709}}{\displaystyle \unicode[STIX]{x1D70F}_{0}\sqrt{\frac{p_{0}}{\unicode[STIX]{x1D70C}_{0}}}},\phantom{\mathop{\sum }_{\sum }}\end{eqnarray}$$

where the reference time $\unicode[STIX]{x1D70F}_{0}$ is given by

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{0}=\frac{1}{2n_{0}d^{2}}\sqrt{\frac{m}{\unicode[STIX]{x03C0}kT_{0}}}\quad (n_{0}=\unicode[STIX]{x1D70C}_{0}/m).\end{eqnarray}$$

Although the choice (4.2) is taken for convenience, $\unicode[STIX]{x1D70F}_{0}$ and relaxation times $\unicode[STIX]{x1D70F}_{D}$ and $\unicode[STIX]{x1D70F}_{T}$ (2.28) can be related to the so-called collision time scale, i.e. the mean free time of particles between collisions in equilibrium (Bardos, Golse & Levermore Reference Bardos, Golse and Levermore1993). Omitting, for brevity, the tilde in the quantities (4.1), the dimensionless governing equations of the steady detonation are as follows:

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70C}u)=0, & \displaystyle\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70C}u^{2}+\unicode[STIX]{x1D70C}T)=0, & \displaystyle\end{eqnarray}$$

(4.5) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\left[\left(\frac{1}{2}\unicode[STIX]{x1D70C}u^{2}+\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}-1}\unicode[STIX]{x1D70C}T-\unicode[STIX]{x1D70C}z\unicode[STIX]{x0394}\unicode[STIX]{x1D716}\right)u\right]=0, & \displaystyle\end{eqnarray}$$

(4.6) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D70C}zu)=\tilde{\unicode[STIX]{x1D70F}}_{p} & \displaystyle\end{eqnarray}$$

and

(4.7) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\text{d}}{\text{d}\unicode[STIX]{x1D709}}\left[\left(\frac{1}{2}\unicode[STIX]{x1D70C}zu^{2}+\frac{\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x1D6FE}-1}\unicode[STIX]{x1D70C}z\left(T+(1-z)\unicode[STIX]{x1D6E9}\right)-\unicode[STIX]{x1D70C}z\unicode[STIX]{x0394}\unicode[STIX]{x1D716}\right)u\right]\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{\unicode[STIX]{x1D70F}_{0}}{\unicode[STIX]{x1D70F}_{T}}\frac{1}{\unicode[STIX]{x1D6FE}-1}\unicode[STIX]{x1D70C}z(1-z)\unicode[STIX]{x1D6E9}+\tilde{\unicode[STIX]{x1D70F}}_{p}\frac{u^{2}}{2},\end{eqnarray}$$

where

(4.8) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D70F}}_{p} & = & \displaystyle \unicode[STIX]{x1D70C}^{2}\sqrt{T}\exp \left(-\frac{\unicode[STIX]{x1D716}_{f}}{T}\right)\left\{(1-z)^{2}-z^{2}\exp \left(-\frac{2\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{T}\right)\right.\nonumber\\ \displaystyle & & \displaystyle -\left.(1-z)^{2}\left[\frac{\unicode[STIX]{x1D716}_{f}}{T}+\frac{1}{2}+(1-z)\frac{2\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{T}\right]\frac{\unicode[STIX]{x1D6E9}}{T}\right\}\end{eqnarray}$$

and

(4.9) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D70F}_{0}}{\unicode[STIX]{x1D70F}_{T}}=\frac{2}{3}\unicode[STIX]{x1D70C}\sqrt{T}.\end{eqnarray}$$

The governing equations consist of the conservation laws (4.3)–(4.5) of mass, momentum and energy for the mixture, and the balance laws (4.6) and (4.7) of mass and energy for the product. The balance law of momentum for the product is dropped from the set of governing equations in accordance with the basic assumption.

4.1.1 Governing equations solved with respect to first derivatives

The governing equations (4.3)–(4.7) can be solved with respect to first derivatives:

(4.10) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\unicode[STIX]{x1D70C}}{\text{d}\unicode[STIX]{x1D709}}=\frac{(\unicode[STIX]{x1D6FE}-1)\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{u(u^{2}-\unicode[STIX]{x1D6FE}T)}\tilde{\unicode[STIX]{x1D70F}}_{p}, & \displaystyle\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}u}{\text{d}\unicode[STIX]{x1D709}}=-\frac{(\unicode[STIX]{x1D6FE}-1)\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D70C}(u^{2}-\unicode[STIX]{x1D6FE}T)}\tilde{\unicode[STIX]{x1D70F}}_{p}, & \displaystyle\end{eqnarray}$$

(4.12) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}T}{\text{d}\unicode[STIX]{x1D709}}=\frac{(\unicode[STIX]{x1D6FE}-1)\unicode[STIX]{x0394}\unicode[STIX]{x1D716}(u^{2}-T)}{\unicode[STIX]{x1D70C}u(u^{2}-\unicode[STIX]{x1D6FE}T)}\tilde{\unicode[STIX]{x1D70F}}_{p}, & \displaystyle\end{eqnarray}$$

(4.13) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}z}{\text{d}\unicode[STIX]{x1D709}}=\frac{1}{\unicode[STIX]{x1D70C}u}\tilde{\unicode[STIX]{x1D70F}}_{p}, & \displaystyle\end{eqnarray}$$

(4.14) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\unicode[STIX]{x1D6E9}}{\text{d}\unicode[STIX]{x1D709}}=\frac{(\unicode[STIX]{x1D6FE}-1)\unicode[STIX]{x0394}\unicode[STIX]{x1D716}(1-z)-\unicode[STIX]{x1D6FE}[T+(1-2z)\unicode[STIX]{x1D6E9}]}{\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70C}z(1-z)u}\tilde{\unicode[STIX]{x1D70F}}_{p}-\frac{\unicode[STIX]{x1D70F}_{0}}{\unicode[STIX]{x1D70F}_{T}}\frac{\unicode[STIX]{x1D6E9}}{\unicode[STIX]{x1D6FE}u}. & \displaystyle\end{eqnarray}$$

This form will be useful for the analysis of certain qualitative aspects of the solutions.

4.1.2 Hierarchy of the models

In § 5 we shall provide a comparative analysis of the detonation profiles obtained by the solutions to (4.3)–(4.7), with those obtained through the solutions to the simpler models. This analysis will be achieved through the hierarchy of models obtained through the simplification of the reaction rate (4.8). Namely, $\tilde{\unicode[STIX]{x1D70F}}_{p}$ can be split into a single-temperature (ST) part $\tilde{\unicode[STIX]{x1D70F}}_{p}^{ST}$ , which does not depend on $\unicode[STIX]{x1D6E9}$ , and an MT part $\tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}$ , which depends on $\unicode[STIX]{x1D6E9}$ , as follows:

(4.15) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D70F}}_{p}=\tilde{\unicode[STIX]{x1D70F}}_{p}^{ST}(z,T,\unicode[STIX]{x0394}\unicode[STIX]{x1D716})+\tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}(z,T,\unicode[STIX]{x1D6E9}),\end{eqnarray}$$

where

(4.16) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D70F}}_{p}^{ST}(z,T,\unicode[STIX]{x0394}\unicode[STIX]{x1D716})=\unicode[STIX]{x1D70C}^{2}\sqrt{T}\exp \left(-\frac{\unicode[STIX]{x1D716}_{f}}{T}\right)\left\{(1-z)^{2}-z^{2}\exp \left(-\frac{2\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{T}\right)\right\}, & \displaystyle\end{eqnarray}$$

(4.17) $$\begin{eqnarray}\displaystyle & \displaystyle \tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}(z,T,\unicode[STIX]{x1D6E9})=-\unicode[STIX]{x1D70C}^{2}\sqrt{T}\exp \left(-\frac{\unicode[STIX]{x1D716}_{f}}{T}\right)(1-z)^{2}\left[\frac{\unicode[STIX]{x1D716}_{f}}{T}+\frac{1}{2}+(1-z)\frac{2\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{T}\right]\frac{\unicode[STIX]{x1D6E9}}{T}. & \displaystyle\end{eqnarray}$$

In such a way the following hierarchy of the models can be obtained and analysed in the sequel:

(MT)

the MT model is described by the system (4.3)–(4.7), with the complete reaction rate term (4.8);

(qMT)

the quasi-MT model consists of the same equations (4.3)–(4.7) as for the MT model, but with simplified reaction rate $\tilde{\unicode[STIX]{x1D70F}}_{p}=\tilde{\unicode[STIX]{x1D70F}}_{p}^{ST}$ that does not take into account the contribution of the different temperatures; and

(ST)

the ST model consists of (4.3)–(4.6), assumes that the reaction rate $\tilde{\unicode[STIX]{x1D70F}}_{p}=\tilde{\unicode[STIX]{x1D70F}}_{p}^{ST}$ and assumes a common temperature for both constituents throughout the whole process – this model, along with reversible chemical reaction, corresponds to the model studied in (Marques Junior et al. Reference Marques Junior, Soares, Bianchi and Kremer2015).

4.2 Properties of the solution

The analysis of MT detonation waves possesses certain particular features which distinguish it from the standard analysis based upon reactive Euler equations. Such features will be qualitatively analysed in the sequel.

4.2.1 Equilibrium state

As mentioned above, the MT model of the steady detonation wave describes a continuous solution that connects the von Neumann state $\boldsymbol{U}_{N}$ with the equilibrium one $\boldsymbol{U}_{eq}$ . Thus, we analyse the boundary value problem for (4.3)–(4.7) in a (formally) unbounded domain $\unicode[STIX]{x1D709}\in [0,\infty )$ , with the following boundary data:

(4.18) $$\begin{eqnarray}\boldsymbol{U}(0)=\boldsymbol{U}_{N},\end{eqnarray}$$

(4.19a,b ) $$\begin{eqnarray}\lim _{\unicode[STIX]{x1D709}\rightarrow \unicode[STIX]{x1D709}_{S}}\boldsymbol{U}(\unicode[STIX]{x1D709})=\boldsymbol{U}_{eq}\quad \text{and}\quad \lim _{\unicode[STIX]{x1D709}\rightarrow \unicode[STIX]{x1D709}_{S}}\frac{\text{d}\boldsymbol{U}}{\text{d}\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D709})=\mathbf{0},\quad 0<\unicode[STIX]{x1D709}_{S}<\infty ,\end{eqnarray}$$

where $\unicode[STIX]{x1D709}_{S}$ denotes the value of the steady variable at which the process terminates. Note that condition (4.19) leads to a necessary condition for equilibrium.

Statement 1. If the system (4.3)–(4.7) is in an equilibrium state characterized by relations (4.19), then the state variables obey the following relations:

(4.20a,b ) $$\begin{eqnarray}\frac{1-z_{eq}}{z_{eq}}=\exp \left(-\frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{T_{eq}}\right)\quad \text{and}\quad \unicode[STIX]{x1D6E9}_{eq}=0.\end{eqnarray}$$

Proof. The terminal condition (4.19) implies that production terms in governing equations (4.6) and (4.7) vanish in equilibrium. In particular, the equilibrium condition for (4.6) implies that the reaction rate vanishes, $\tilde{\unicode[STIX]{x1D70F}}_{p}=0$ , which provides a relation between equilibrium values $z_{eq}$ , $T_{eq}$ and $\unicode[STIX]{x1D6E9}_{eq}$ . However, the equilibrium condition for (4.7) requires simultaneous vanishing of the reaction rate $\tilde{\unicode[STIX]{x1D70F}}_{p}$ and the temperature difference $\unicode[STIX]{x1D6E9}$ , which leads to $\unicode[STIX]{x1D6E9}_{eq}=0$ . As a consequence, we have

(4.21) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{eq}=0\quad \Rightarrow \quad \tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}(z_{eq},T_{eq},\unicode[STIX]{x1D6E9}_{eq})=0,\end{eqnarray}$$

so that the ST part $\tilde{\unicode[STIX]{x1D70F}}_{p}^{ST}$ of the reaction rate vanishes independently, implying the law of mass action

(4.22) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D70F}}_{p}^{ST}(z_{eq},T_{eq},\unicode[STIX]{x0394}\unicode[STIX]{x1D716})=0\quad \Rightarrow \quad \frac{1-z_{eq}}{z_{eq}}=\exp \left(-\frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{T_{eq}}\right),\end{eqnarray}$$

which completes the proof.

Having in mind the analysis developed in § 3, it becomes obvious that the terminal state of the system has to be the equilibrium one, i.e.  $\boldsymbol{U}(\unicode[STIX]{x1D709}_{S})=\boldsymbol{U}_{S}=\boldsymbol{U}_{eq}$ , where the equilibrium state is uniquely determined by the intersection of the Rayleigh line and the equilibrium Hugoniot curve, together with the independent condition $\unicode[STIX]{x1D6E9}_{S}=\unicode[STIX]{x1D6E9}_{eq}=0$ . The additional condition (4.20b ) is peculiar for the MT model and has to be considered with particular attention, as will be shown in the sequel.

4.2.2 Non-monotonic detonation profiles

Further qualitative analysis will be based upon twofold characterization of the equilibrium state, given by the terminal condition (4.19). A distinguished role in this analysis will be played by the temperature difference $\unicode[STIX]{x1D6E9}$ . To facilitate the formulation of the forthcoming statements, we shall introduce the following notation for the state variables:

(4.23a,b ) $$\begin{eqnarray}\boldsymbol{U}=(\boldsymbol{U}^{ST},\unicode[STIX]{x1D6E9}),\quad \boldsymbol{U}^{ST}=(\unicode[STIX]{x1D70C},u,T,z).\end{eqnarray}$$

In the case of reversible chemical reaction, typical ZND detonation profiles are monotonic (Marques Junior et al. Reference Marques Junior, Soares, Bianchi and Kremer2015), except in the neighbourhood of the Chapman–Jouguet solution, the solution that connects the von Neumann state $N$ with the Chapman–Jouguet state $CJ$ . However, equilibrium conditions (4.20) of the MT model imply the possibility of occurrence of non-monotonic profiles. The existence of such profiles will be demonstrated first through a qualitative analysis. It will then be confirmed by numerical computation in the next section.

Statement 2 (Crossing condition).

Assume that there exists $\unicode[STIX]{x1D709}_{C}\in (0,\infty )$ such that the state vector $\boldsymbol{U}^{ST}(\unicode[STIX]{x1D709}_{C})=(\unicode[STIX]{x1D70C}_{C},u_{C},T_{C},z_{C})$ satisfies the equilibrium condition (4.20a ), i.e.  $z_{C}=z_{eq}$ , $T_{C}=T_{eq}$ . Then, the following two cases are possible: either

1. (i) $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{C})=\unicode[STIX]{x1D6E9}_{C}=\unicode[STIX]{x1D6E9}_{eq}=0$ and the equilibrium state is reached at $\unicode[STIX]{x1D709}_{C}=\unicode[STIX]{x1D709}_{S}$ ; or

2. (ii) $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{C})\neq 0$ , which implies $\text{d}\boldsymbol{U}(\unicode[STIX]{x1D709}_{C})/\text{d}\unicode[STIX]{x1D709}\neq \mathbf{0}$ , and the process will continue for $\unicode[STIX]{x1D709}>\unicode[STIX]{x1D709}_{C}$ .

Proof. If $z_{C}$ and $T_{C}$ satisfy the equilibrium condition (4.20a ), i.e. the law of mass action, then $\unicode[STIX]{x1D70C}_{C}$ and $u_{C}$ are uniquely determined by (3.10) and (3.7) for the fixed value of $s$ . Moreover, $\tilde{\unicode[STIX]{x1D70F}}_{p}^{ST}(z_{C},T_{C},\unicode[STIX]{x1D6E9}_{C})=0$ . In case (i) it is obvious that the equilibrium state is reached, since both equilibrium conditions (4.20) are satisfied. In case (ii), $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{C})\neq 0$ implies $\tilde{\unicode[STIX]{x1D70F}}_{p}(\unicode[STIX]{x1D709}_{C})=\tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}(z_{C},T_{C},\unicode[STIX]{x1D6E9}_{C})\neq 0$ . Consequently, by a simple inspection of the system (4.10)–(4.14), one may conclude that $\text{d}\boldsymbol{U}(\unicode[STIX]{x1D709}_{C})/\text{d}\unicode[STIX]{x1D709}\neq \mathbf{0}$ , which completes the proof.

The crossing condition has a simple interpretation. If the state variables $\boldsymbol{U}^{ST}$ satisfy the law of mass action (4.20a ), and lie on the Rayleigh line and the equilibrium Hugoniot curve simultaneously, the ST part of the reaction rate will vanish, $\tilde{\unicode[STIX]{x1D70F}}_{p}^{ST}(\unicode[STIX]{x1D709}_{C})=0$ . However, if $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{C})\neq 0$ , the process is not terminated, since the derivatives of the state variables do not yet vanish, and the curve $\boldsymbol{U}^{ST}(\unicode[STIX]{x1D709})$ only crosses the equilibrium values at $\unicode[STIX]{x1D709}_{C}$ . This motivates the denomination crossing condition. Its importance lies in the fact that it may occur for certain values of the parameters, in contrast to the standard ST case.

Statement 3 (Local extremum).

Assume that there exists $\unicode[STIX]{x1D709}_{E}\in (0,\infty )$ such that the derivative of the state vector $\boldsymbol{U}^{ST}$ vanishes, $\text{d}\boldsymbol{U}^{ST}(\unicode[STIX]{x1D709}_{E})/\text{d}\unicode[STIX]{x1D709}=(0,0,0,0)$ . Then, the following two cases are possible: either

1. (i) $\text{d}\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{E})/\text{d}\unicode[STIX]{x1D709}=0$ , which implies $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{E})=0$ , and the equilibrium state is reached at $\unicode[STIX]{x1D709}_{E}=\unicode[STIX]{x1D709}_{S}$ , or

2. (ii) $\text{d}\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{E})/\text{d}\unicode[STIX]{x1D709}\neq 0$ , which implies $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{E})\neq 0$ , and the process will continue for $\unicode[STIX]{x1D709}>\unicode[STIX]{x1D709}_{E}$ .

Proof. If $\text{d}\boldsymbol{U}^{ST}(\unicode[STIX]{x1D709}_{E})/\text{d}\unicode[STIX]{x1D709}=(0,0,0,0)$ at some point $\unicode[STIX]{x1D709}_{E}\in (0,\infty )$ , then from (4.10)–(4.13) one obtains $\tilde{\unicode[STIX]{x1D70F}}_{p}(\unicode[STIX]{x1D709}_{E})=0$ . However, this condition is not sufficient for equilibrium since it only determines one relation between all the state variables $\boldsymbol{U}$ . If case (i) is valid, $\text{d}\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{E})/\text{d}\unicode[STIX]{x1D709}=0$ , from (4.14) it follows that $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{E})=0$ , and the system reaches the equilibrium state at $\unicode[STIX]{x1D709}_{E}=\unicode[STIX]{x1D709}_{S}$ . If case (ii) is valid, $\text{d}\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{E})/\text{d}\unicode[STIX]{x1D709}\neq 0$ implies $\unicode[STIX]{x1D6E9}(\unicode[STIX]{x1D709}_{E})\neq 0$ from (4.14), and the process is not terminated, but continues for $\unicode[STIX]{x1D709}>\unicode[STIX]{x1D709}_{E}$ .

As a conclusion of the analysis of non-monotonic profiles, one fundamental remark has to be given. Either in the case of crossing the equilibrium state (statement 2), or in the case of local extremum (statement 3), the main driving agent for continuation of the process is the temperature difference of the constituents.

5.1 Case 1: Influence of $\unicode[STIX]{x1D716}_{f}$ on the profile

First we consider the influence of the activation energy of the forward reaction $\unicode[STIX]{x1D716}_{f}$ on the structure of the continuous reacting flow profile. This is the case in which the MT assumption strongly reveals its influence and leads to completely new results in a qualitative sense. It is well known (Lee Reference Lee2008) that the activation energy is strictly related to the temperature sensitivity of the chemical reaction. To compare our results with (Marques Junior et al. Reference Marques Junior, Soares, Bianchi and Kremer2015), we adopted the same values of the parameters:

(5.3a-c ) $$\begin{eqnarray}\unicode[STIX]{x0394}\unicode[STIX]{x1D716}=1.0,\quad f=1.5,\quad \unicode[STIX]{x1D716}_{f}\in \{2.0,10.0,30.0\}.\end{eqnarray}$$

Numerically computed profiles are presented in figure 2, for both MT and ST models.

For lower values of $\unicode[STIX]{x1D716}_{f}$ , one can observe two qualitatively different types of profiles. The first one ( $\unicode[STIX]{x1D716}_{f}=2.0$ ) is non-monotonic – it contains a crossing point with the state  $\boldsymbol{U}_{eq}^{ST}$ , as indicated by statement 2, as well as at least one local extremum, as described in statement 3. This does not occur in the ST case at all. Moreover, it can be observed that the MT profiles are driven out of the von Neumann state faster than the ST ones, and this fact can be attributed to the presence of MT part $\tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}$ in the reaction rate.

When the activation energy is increased ( $\unicode[STIX]{x1D716}_{f}=10.0$ ), the profiles become monotonic, like in the ST case. There is still, however, a difference between the MT and ST profiles: the MT profiles are driven out of the von Neumann state faster than the ST ones, but the ST profiles are steeper than the MT ones. Obviously, there exists a threshold value of the activation energy $\unicode[STIX]{x1D716}_{f}$ which separates the non-monotonic profiles from monotonic ones. This issue will be tackled in the sequel.

As in Marques Junior et al. (Reference Marques Junior, Soares, Bianchi and Kremer2015), for very large values of the activation energy ( $\unicode[STIX]{x1D716}_{f}=30.0$ ), the reaction rate is considerably smaller (see (4.8)) and the profile is much wider – the equilibrium state is reached at $\unicode[STIX]{x1D709}_{eq}\approx 10^{6}$ . Both MT and ST profiles are monotonic and have a similar behaviour as in the previous case. However, the difference between the state variables in the MT and ST cases is much less pronounced, e.g. $z^{MT}(\unicode[STIX]{x1D709})-z^{ST}(\unicode[STIX]{x1D709})<10^{-3}$ throughout the profile. For these reasons, the profiles obtained for $\unicode[STIX]{x1D716}_{f}=30.0$ are not displayed in the whole range of $\unicode[STIX]{x1D709}$ , but just in the range that is significant for the first two cases.

Figure 2. Influence of $\unicode[STIX]{x1D716}_{f}$ on the reacting flow: profiles of $z(\unicode[STIX]{x1D709}),~u(\unicode[STIX]{x1D709}),~p(\unicode[STIX]{x1D709}),~T(\unicode[STIX]{x1D709})$ obtained with the MT model (black) and ST model (grey), for different values of $\unicode[STIX]{x1D716}_{f}$ .

The analysis of the temperature profiles of the constituents sheds new light on the MT phenomenon. In the case of non-reactive mixtures, it is usually related to the fact that constituents have disparate masses. In our problem the masses of reactants and products are equal by assumption, $m_{r}=m_{p}=m$ . Nevertheless, there is a temperature difference that cannot be ignored. A careful examination of the profiles, presented in figure 3, reveals that there appears an immediate temperature drop of the product temperature right behind the von Neumann state. This result suggests that the main reason for sustained thermal non-equilibrium between the constituents is not the mass difference, but the chemical reaction and the huge initial difference in the concentration of the constituents.

Figure 3. Influence of $\unicode[STIX]{x1D716}_{f}$ on the reacting flow: profiles of $T(\unicode[STIX]{x1D709})$ , $T_{r}(\unicode[STIX]{x1D709})$ and $T_{p}(\unicode[STIX]{x1D709})$ obtained with the MT model for two different values of $\unicode[STIX]{x1D716}_{f}$ : $\unicode[STIX]{x1D716}_{f}=2.0$ (a) and $\unicode[STIX]{x1D716}_{f}=10.0$ (b).

As a final remark, we would like to compare the non-monotonic profiles that appear in our model with the non-monotonic profiles in the standard model for ZND detonation waves. Namely, in the standard model, the non-monotonic temperature profiles appear for large values of the activation energy, and they are preceded by a longer induction zone and have steeper profiles (see Lee Reference Lee2008, pp. 77–78). Other state variables have monotonic profiles. The main difference between our model and the standard one is the reversibility of the chemical reaction and the MT assumption. They are both properly taken into account in the reaction rate (2.21), and its subsequent dimensionless forms. In contrast to the standard model, the non-monotonic profiles in our case appear in all the state variables and for small values of the activation energy $\unicode[STIX]{x1D716}_{f}$ . Therefore, it seems that reversibility and thermal non-equilibrium ( $\unicode[STIX]{x1D6E9}\neq 0$ ) play the crucial role in generating non-monotonic profiles through their appearance in the reaction rate. It must be emphasized that the scope of this remark is restricted to ZND detonation waves, and that we do not tend to cover processes like Rayleigh flow, in which non-monotonic temperature profiles occur as well.

5.2 Case 2: Influence of $f$ on the profile

The influence of the overdrive degree $f$ on the continuous reacting profiles is analysed for the following values of the parameters:

(5.4a-c ) $$\begin{eqnarray}\unicode[STIX]{x0394}\unicode[STIX]{x1D716}=1.0,\quad f\in \{1.08,1.5,3.0\},\quad \unicode[STIX]{x1D716}_{f}=2.0,\end{eqnarray}$$

and the results are presented in figure 4. As expected, different values of $f$ lead to different terminal equilibrium states. However, there is no qualitative difference between the profiles – only non-monotonic profiles appear. This leads to the conclusion that the major influence on the qualitative structure of the profile still comes from the activation energy $\unicode[STIX]{x1D716}_{f}$ , rather than from the overdrive degree $f$ . This is a consequence of the fact that the activation energy is strictly related to the temperature sensitivity of the chemical reaction.

Figure 4. Influence of $f$ on the reacting flow: profiles of $z(\unicode[STIX]{x1D709}),~u(\unicode[STIX]{x1D709}),~p(\unicode[STIX]{x1D709}),~T(\unicode[STIX]{x1D709})$ obtained with the MT model (black) and ST model (grey), for different values of $f$ .

In the present analysis we took the value $f=1.08$ with the aim to mimic the behaviour of the Chapman–Jouguet solution. It can be observed in the temperature profile illustrated in figure 4 that there occurs a temperature drop in the profile. A similar result has been obtained in (Marques Junior et al. Reference Marques Junior, Soares, Bianchi and Kremer2015), where it was attributed to the prevalence of backward reaction (which is endothermic) for $f=1.0$ . However, this phenomenon appears also for $f>1.0$ and yet has to be analysed carefully.

5.3 Case 3: Influence of $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ on the profile

In Marques Junior et al. (Reference Marques Junior, Soares, Bianchi and Kremer2015), the influence of the binding energy difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ was not analysed at all. Nevertheless, in the context of MT ZND detonation waves, it is of interest to pay attention to the impact of this parameter on the reacting profiles, since it is related to the reaction heat. In fact, higher values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ indicate more violent chemical reactions. Although $s_{j}$ varies with $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ , we kept the overdrive degree  $f$ fixed in all the computations. However, we computed the profiles for two different values of $\unicode[STIX]{x1D716}_{f}$ , so that the final set of parameter values is

(5.5a-c ) $$\begin{eqnarray}\unicode[STIX]{x0394}\unicode[STIX]{x1D716}\in \{1.0,5.0,10.0\},\quad f=1.5\quad \unicode[STIX]{x1D716}_{f}\in \{2.0,10.0\},\end{eqnarray}$$

and the results are presented in figures 5 and 6, for $\unicode[STIX]{x1D716}_{f}=2.0$ and $\unicode[STIX]{x1D716}_{f}=10.0$ , respectively.

It may be observed in figure 5 that for $\unicode[STIX]{x1D716}_{f}=2.0$ all the computed profiles in the MT model are non-monotonic, regardless of the value of $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ . This means that the influence of a low level of activation energy opens the door to non-monotonic profiles, without any regard to the heat of reaction.

Figure 5. Influence of $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ on the reacting flow: profiles of $z(\unicode[STIX]{x1D709}),~u(\unicode[STIX]{x1D709}),~p(\unicode[STIX]{x1D709}),~T(\unicode[STIX]{x1D709})$ obtained with the MT model (black) and ST model (grey), for $\unicode[STIX]{x1D716}_{f}=2.0$ and different values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ .

However, figure 6 reveals that, for $\unicode[STIX]{x1D716}_{f}=10.0$ , the value $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}=1.0$ yields monotonic MT profiles, whereas higher values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ yield non-monotonic profiles. Therefore, when the activation energy is high, the reaction has to be sufficiently violent in order to produce non-monotonic profiles. Moreover, by a continuity argument we may expect that there exists a threshold value of $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ which separates monotonic MT profiles from non-monotonic ones and this aspect will be analysed in the sequel.

Figure 6. Influence of $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ on the reacting flow: profiles of $z(\unicode[STIX]{x1D709}),~u(\unicode[STIX]{x1D709}),~p(\unicode[STIX]{x1D709}),~T(\unicode[STIX]{x1D709})$ obtained with the MT model (black) and ST model (grey), for $\unicode[STIX]{x1D716}_{f}=10.0$ and different values of $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ .

5.4 General remark on overdriven detonations

In view of the analysis related to the Hugoniot diagram (figure 1), and the detailed explanation of pathological and overdriven detonations, it may be observed that non-monotonic profiles of overdriven detonations exactly follow the scenario given by Khokhlov (Reference Khokhlov1989) and Sharpe (Reference Sharpe1999). They pass through the point in which (4.20a ) is satisfied, reach the turning point, and then converge to the equilibrium state. This behaviour is predicted in § 4 in the qualitative analysis of non-monotonic profiles. As already mentioned in the analysis of case 1, in the ST ZND model with reversible reaction, studied by Marques Junior et al. (Reference Marques Junior, Soares, Bianchi and Kremer2015), this behaviour cannot be obtained, but only the monotonic one. Therefore, the MT assumption appears to be a crucial modelling assumption for capturing the non-monotonic behaviour. In particular, for a certain range of values of the parameters, the single reversible reaction (1.1) in conjunction with the MT assumption yield the same qualitative behaviour in the case of overdriven detonation as the two-step reactions $A\rightarrow B\rightarrow C$ with an endothermic phase following the exothermic one.

It is also interesting to relate the turning points in the state space and the corresponding heat release. In the MT ZND model, taking into account (2.18b ), the heat release is $Q=z\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ . The turning point in overdriven detonations corresponds to the maximum heat release (see Sharpe Reference Sharpe1999). Consequently, one has

(5.6) $$\begin{eqnarray}\frac{\text{d}Q}{\text{d}\unicode[STIX]{x1D709}}=\frac{\text{d}z}{\text{d}\unicode[STIX]{x1D709}}\unicode[STIX]{x0394}\unicode[STIX]{x1D716}=0,\end{eqnarray}$$

and this implies $\tilde{\unicode[STIX]{x1D70F}}_{p}=0$ (see (4.10)–(4.14)), which corresponds to the local extremum in our study. However, there remains a qualitative difference between the overdriven detonations in the MT ZND detonation model with a single reversible reaction, and the same detonations in the two-step model – in the former case there appear more than one local extremum in the profile. This will remain an open question for future studies.

5.5 Qualitative analysis of the reaction rate

The analysis in case 3 to a certain extent confirms the observation already given in case 1 – the major influence on the qualitative behaviour of the continuous reacting profile comes from the activation energy $\unicode[STIX]{x1D716}_{f}$ . However, the heat of reaction $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ becomes important, and actually decisive, when the activation energy is large and suppresses the non-monotonic behaviour.

These findings can be supported by the qualitative analysis of the reaction rate $\tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}$ already given in (4.17) by

(5.7) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}(z,T,\unicode[STIX]{x1D6E9})=-\unicode[STIX]{x1D70C}^{2}\sqrt{T}\exp \left(-\frac{\unicode[STIX]{x1D716}_{f}}{T}\right)(1-z)^{2}\left[\frac{\unicode[STIX]{x1D716}_{f}}{T}+\frac{1}{2}+(1-z)\frac{2\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}{T}\right]\frac{\unicode[STIX]{x1D6E9}}{T}.\end{eqnarray}$$

For $\unicode[STIX]{x1D716}_{f}$ small enough, the reaction rate $\tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}$ is large enough to induce the crossing condition and appearance of a local extremum, i.e. the appearance of non-monotonic profiles. When $\unicode[STIX]{x1D716}_{f}$ is increased, the exponential factor in $\tilde{\unicode[STIX]{x1D70F}}_{p}^{MT}$ is dominant and suppresses the non-monotonic behaviour. However, this may be changed if some other parameter, like $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ , becomes sufficiently large to prevail and re-establish the non-monotonic profiles. That is what exactly happened in case 3.

5.6 Threshold in the parameter space

The main difference between MT and ST ZND detonation waves lies in the fact that in the MT case there appear two qualitatively different configurations of continuous reacting profiles – non-monotonic and monotonic ones. The non-monotonic behaviour of state variables is directly related to the MT assumption, since both crossing condition and local extremum are consequences of thermal non-equilibrium, $\unicode[STIX]{x1D6E9}\neq 0$ . However, the parametric analysis revealed that the activation energy $\unicode[STIX]{x1D716}_{f}$ has a major influence on the qualitative behaviour. Therefore, we performed a thorough numerical analysis of the profiles with the aim to determine the threshold in parameter space separating the non-monotonic from the monotonic ones.

Figure 7 presents the threshold in the $(\unicode[STIX]{x1D716}_{f},f)$ -plane for the fixed value of the reaction heat, $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}=1.0$ . It reveals that a threshold exists separating two classes of profiles. It may be observed that the threshold value of the activation energy $\unicode[STIX]{x1D716}_{f}$ increases with the increase of the overdrive degree $f$ , and this relation is approximately linear. The threshold values are the same for all the state variables $\unicode[STIX]{x1D70C}$ , $u$ and $z$ , but they differ for $T$ – there occurs a temperature drop phenomenon for small values of the overdrive degree $f\geqslant 1.0$ (up to $f\approx 1.1$ ). This is indicated in the threshold graph by a narrow dark grey region at the bottom that corresponds to the non-monotonic profiles, but it is related to a temperature drop which occurs in the ST case as well. For other state variables $\unicode[STIX]{x1D70C}$ , $u$ and $z$ , this region should be regarded as light grey.

Figure 7. Threshold behaviour in the $(\unicode[STIX]{x1D716}_{f},f)$ -plane: non-monotonic profiles (white region) and monotonic profiles (light grey region) of all state variables; non-monotonic profiles of the temperature $T$ (dark grey region).

Figure 8 presents the threshold in the $(\unicode[STIX]{x0394}\unicode[STIX]{x1D716},f)$ -plane for the fixed value of the activation energy, $\unicode[STIX]{x1D716}_{f}=10.0$ . Numerical analysis showed that the threshold in this case is not linear, and that there is an upper bound for $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ for which the monotonic profiles appear. Also, the peculiar region of solely temperature non-monotonic profiles is found like in the previous case. It has to be noted that our analysis was performed for the value of $\unicode[STIX]{x1D716}_{f}$ for which two kinds of profiles exist; e.g. for $\unicode[STIX]{x1D716}_{f}=2.0$ there exist only non-monotonic profiles in the region of parameter space in our study.

Figure 8. Threshold behaviour in the $(\unicode[STIX]{x0394}\unicode[STIX]{x1D716},f)$ -plane: non-monotonic profiles (white region) and monotonic profiles (light grey region) of all state variables; non-monotonic profiles of the temperature $T$ (dark grey region).

Finally, figure 9 presents the threshold in the $(\unicode[STIX]{x1D716}f,\unicode[STIX]{x0394}\unicode[STIX]{x1D716})$ -plane for the fixed value of the overdrive degree, $f=1.5$ . Like in the first case, the threshold is approximately linear, and there is a lower bound for the activation energy $\unicode[STIX]{x1D716}_{f}$ for the existence of monotonic profiles. However, since $f>1.1$ there is no region in the parameter space in which only temperature non-monotonic profiles appear.

Figure 9. Threshold behaviour in the $(\unicode[STIX]{x1D716}f,\unicode[STIX]{x0394}\unicode[STIX]{x1D716})$ -plane: non-monotonic profiles (white region) and monotonic profiles (light grey region) of all state variables.