# Multicomponent Flow

## Introduction

**Multicomponent flows**
consist of different chemical species
that are mixed at the molecular level and
generally share the same velocity
and temperature.
They differ from
multiphase flows
where the different phases are immiscible
(Drew and Passman 1999)
and only occupy a fraction of the total volume.
The chemical species
may also interact through chemical reactions
and the resulting
multicomponent reactive flows
are observed in various natural phenomena
and engineering applications.

In astronautics, during reentry of a space ship into Earth's atmosphere as illustrated with the Astrium-ESA reentry demonstratror in Figure 1, when the spaceship meets denser parts of the atmosphere, high temperatures arise behind the detached bow shock surrounding the vehicle. Because of these high temperatures, polyatomic gases dissociate, species may ionize, and dissociated molecules may recombine at the body of the spaceship. A detailed knowledge of the resulting multicomponent flow and of the heat fluxes at the vehicle body is of fundamental importance for a proper vehicle design (Anderson 1989).

Combustion of oil, coal or natural gas is still the source of more than 85% of primary energy in the world and a typical domestic premixed laminar flame is presented in Figure 2. It is of the greatest importance to diminish fuel consumption as well as the emission of pollutant in power plants, aircraft engines as well as car engines (Williams 1985; Poinsot and Veynante 2005). This notably requires understanding cycle to cycle variation in piston engines, combustion instabilities in industrial furnaces, ignition and flash back in aero gas turbines and more generally to understand flame structure and dynamics.

In chemical engineering,
chemical reactors may be of
various shapes
and are typically designed
to optimize a given set of chemical reactions.
The corresponding processes may be
highly complex with multiple reactant injections,
heating or cooling devices,
pumps to increase pressure,
homogeneous chemistry as well
as heterogeneous chemistry with catalysts
(Rosner 1986;
Kee et al. 2003;
Schmidt 2009).
Optimizing
reactors' shapes as well as chemical
processes again requires a detailed knowledge of
the corresponding multicomponent
reactive flows.

Last but not least, the study of atmospheric pollution, illustrated in Figure 3 with a picture of pollution above the Eiffel tower in Paris, involves a myriad of trace reactive species. These reactive species are notably responsible for phenomena ranging from urban photochemical smog, acid deposition, stratospheric ozone depletion, to climate change (Seinfeld and Pandis 2006). Investigating multicomponent atmospheric flows including the effect of aerosols and clouds is of the highest importance for the twenty-first century.

\[ \newcommand\scal{{\boldsymbol\cdotp}\mskip1.7mu} \newcommand\dxb{\boldsymbol{\nabla}} \newcommand\vitesse{\boldsymbol{v}} \newcommand\fluxdiff{\boldsymbol{J}} \newcommand\fluxdiffscal{J} \newcommand\vitdiff{\mathbf{v}} \newcommand\vitdiffscal{\mathrm{v}} \newcommand\forcediff{\boldsymbol{d}} \newcommand\heatflux{\boldsymbol{q}} \newcommand\heatfluxscal{q} \newcommand\force{\boldsymbol{\mathrm f}} \newcommand\identite{\boldsymbol{I}} \newcommand\viscous{\boldsymbol{\varPi}} \newcommand\stress{\boldsymbol{\sigma}} \newcommand\compres{\mathrm{z}} \newcommand\tlsc{\mathcal{G}} \newcommand\tlsn{\mathcal{N}} \newcommand\gravite{\boldsymbol{g}} \newcommand\pdemi{\tfrac{1}{2}} \newcommand\pdtiers{\frac{2}{3}} \newcommand\molefraction{\mathsf{x}} \newcommand\massfraction{\mathsf{y}} \newcommand\units{\mathsf{u}} \newcommand\nespe{n} \newcommand\eespe{S} \newcommand\bin{\mathrm{bin}} \newcommand\consv{\mathsf{u}} \newcommand\symev{\mathsf{v}} \newcommand{\doubleindices}[2]{\renewcommand{\arraystretch}{0}\begin{array}{c}\scriptstyle #1\\\scriptstyle #2\end{array}} \]

## Fundamental equations

The fundamental equations governing multicomponent flows may generally be derived from macroscopic theories like the thermodynamics of irreversible processes (de Groot and Mazur 1984), nonequilibrium statistical thermodynamics (Keizer 1987), or from molecular finer theories like statistical mechanics (Bearman and Kirkwood 1958), the kinetic theory of dilute gases (Waldmann 1958; Chapman and Cowling 1970; Ferziger and Kaper 1972; Giovangigli 1999) or the kinetic theory of dense gases (Van Beijeren and Ernst 1973). These equations may be split between conservation equations, thermochemistry, transport fluxes, transport coefficients, and boundary conditions.

### Conservation equations

The equations for conservation of species mass, momentum and energy in a multicomponent flow may be written in the form (de Groot and Mazur 1984; Williams 1985; Rosner 1986; Giovangigli 1999; Kee et al. 2003) \[ \begin{align} \tag{1} & \frac{\partial\rho_k^{}}{\partial t} + \dxb\scal(\rho_k \vitesse) + \dxb\scal\fluxdiff_k = m_k \omega_k, \qquad k\in S, \\[3pt] \tag{2} & \frac{\partial}{\partial t}(\rho \vitesse) + \dxb\scal(\rho \vitesse{\otimes}\vitesse + p \identite) + \dxb\scal\viscous = \sum_{k\in S} \rho_k\force_k, \\[3pt] \tag{3} & \frac{\partial}{\partial t} (\rho e + \pdemi \rho \vitesse\scal\vitesse) + \dxb\scal\bigl( ( \rho e + \pdemi \rho \vitesse\scal\vitesse+p) \vitesse\bigr) + \dxb\scal(\heatflux +\viscous\scal\vitesse) = \sum_{k\in S} (\rho_k\vitesse + \fluxdiff_k)\scal\force_k, \end{align} \] where \(\partial/\partial t\) denotes the time derivative, \(\dxb\) the space derivative operator, \(\rho_k\) the mass density of the \(k\)th species, \(\vitesse\) the mass average flow velocity, \(\fluxdiff_k\) the diffusion flux of the \(k\)th species, \(m_k\) the molar mass of the \(k\)th species, \(\omega_k\) the molar production rate of the \(k\)th species, \(S=\{1,\ldots,n\}\) the set of species indices, \(n\geq1\) the number of species, \(\rho=\sum_{k\in S} \rho_k\) the total mass density, \(p\) the pressure, \(\viscous\) the viscous tensor, \(\force_k\) the force per unit mass acting on the \(k\)th species, \(e\) the internal energy per unit mass and \(\heatflux\) the heat flux. The notation used in Equations (1)-(3) is that of de Groot and Mazur (1984) so that the viscous stress tensor is \(-\viscous\) and the Cauchy stress tensor is given by \(\stress = - p \identite - \viscous\).

The fluxes and the production rates satisfy the mass conservation relations \(\sum_{k\in S} \fluxdiff_k=0\) and \(\sum_{k\in S} m_k \omega_k=0\) and by summing the species equations we recover the total mass conservation equation \(\partial\rho/\partial t + \dxb\scal(\rho \vitesse)=0\). The species governing equations (1) have been written in terms of the species mass densities \(\rho_k\), \(k\in S\), but equivalent formulations are easily written as for instance in terms of the species mass fractions \(\massfraction_k=\rho_k/\rho\). When the force acting on the species reduces to gravity \(\force_k=\gravite\), \(k\in S\), the right members of the momentum conservation equation (2) and of the energy conservation equation (3) are simplified into \(\rho\gravite\) and \(\rho\gravite\scal\vitesse\), respectively. These conservation equations (1)-(3) have to be completed by the relations expressing the thermodynamic properties like \(p\) and \(e\), the chemical production rates \(\omega_k\), \(k\in S\), and the transport fluxes \(\viscous\), \(\fluxdiff_k\), \(k\in S\), and \(\heatflux\) defined in equations (4)-(13).

### Thermodynamics

In the framework of ideal gas mixture thermodynamics, the pressure \(p\), the internal energy per unit mass \(e\) and the entropy per unit mass \(s\) may be written (Guggenheim 1962) \[ \begin{equation} \tag{4} p = \sum_{k\in S} R T \frac{\rho_k}{m_k}, \qquad \rho e = \sum_{k\in S} \rho_k e_k(T), \qquad \rho s = \sum_{k\in S} \rho_k s_k(T,\rho_k), \end{equation} \] where \(R\) is the gas constant, \(T\) the absolute temperature, \(e_k\) the internal energy per unit mass of the \(k\)th species, and \(s_k\) the entropy per unit mass of the \(k\)th species. The internal energy \(e_k\) and entropy \(s_k\) of the \(k\)th species are given by \[ \begin{equation} \tag{5} e_k = e_k^{\rm st} + \int_{T^{\rm st}}^T \!\! c_{vk}(T')\, dT', \qquad s_k = s_k^{\rm st} + \int_{T^{\rm st}}^T \!\! \frac{c_{vk}(T^\prime)}{ T^\prime} \, dT^\prime - \frac{R }{ m_k} \log \frac{\rho_k \, R T^{\rm st}}{ m_k \, p^{\rm st} }, \qquad k\in S, \end{equation} \] where \(e_k^{\rm st}\) is the formation energy of the \(k\)th species at the standard temperature \(T^{\rm st}\), \(c_{vk}\) the constant volume specific heat of the \(k\)th species, and \(s_k^{\rm st}\) the formation entropy of the \(k\)th species at the standard temperature \(T^{\rm st}\) and standard pressure \(p^{\rm st}\). Introducing the mean molar weight \(m\) of the mixture, defined by \(\rho/m = \sum_{k\in S} \rho_k/m_k\), the ideal gas state law may also be written \(p = \rho R T/m\). Other thermodynamic functions are directly expressed in terms of energy and entropy as for instance the enthalpy \(\rho h = \sum_{k\in S} \rho_k h_k(T)\) and the Gibbs function \(\rho g = \sum_{k\in S} \rho_k g_k(T,\rho_k)\) with \(h_k(T) = e_k(T) + R T/m_k\) and \(g_k(T,\rho_k) = h_k(T) - T s_k(T,\rho_k)\), \(k\in S\). Thermodynamic data required for each species of the mixture reduce to the temperature dependent specific heat \(c_{vk}(T)\)---often evaluated in polynomial form---and the two integration constants \(e_k^{\rm st}\) and \(s_k^{\rm st}\) that represent the formation energy and entropy of the \(k^{\rm th}\) species at the standard state. The elemental composition of the chemical species is also required for evaluation the species mass as well as for chemical equilibrium calculations (Guggenheim 1962; Williams 1985).

Thermodynamics
of fluid systems are
classically introduced with the concept of
*local state*,
that is, the classical
laws of
*thermostatics*
are applied
locally and instantaneously
at any point in the fluid system
(de Groot and Mazur 1984).
More satisfactory
*nonequilibrium*
thermodynamics
are obtained
from molecular frameworks
like statistical mechanics or the kinetic theory of gases
and have a wider range
of validity
(de Groot and Mazur 1984;
Keizer 1987;
Giovangigli 1999).
The physical justification
of the existence of a local state
indeed arises from the Boltzmann equation
which shows that
the species distribution functions are essentially Maxwellian
distributions
when collisions are dominant.

Thermodynamics may further be generalized to encompass the situation of nonideal fluids which are such that the compressibility factor \(\compres = pm/(\rho R T)\) deviates from unity. Nonideal thermodynamics are especially important for supercritical fluids and generally at high pressure (Guggenheim 1962; Giovangigli and Matuszewski 2012). A typical exemple of nonideal thermodynamics is that of a fluid governed by Van der Waals equation of state (Guggenheim 1962).

### Chemistry

A chemical mechanism involving \(n^{\hskip-0.04em {\rm r}}\geq 1\) elementary reactions for \(n\geq 1\) species may be written \[ \begin{equation} \tag{6} \sum_{k\in S} \nu_{ki}^{\rm f} \; {\mathfrak M}_k \ \rightleftarrows \ \sum_{k\in S} \nu_{ki}^{\rm b} \; {\mathfrak M}_k, \qquad i\in {\mathfrak R}, \end{equation} \] where \({\mathfrak M}_k\) is the chemical symbol of the \(k\)th species, \(\nu_{ki}^{\rm f}\) and \(\nu_{ki}^{\rm b}\) the forward and backward stoichiometric coefficients of the \(k\)th species in the \(i\)th reaction, and \({\mathfrak R}=\{ 1,\ldots,n^{\hskip-0.04em {\rm r}}\}\) the set of reaction indices. The species molar production rates are in the form \[ \begin{equation} \tag{7} \omega_k= \sum_{i=1}^m (\nu_{ki}^{\rm b} - \nu_{ki}^{\rm f}) \biggl[ {\cal K}_i^{\rm f} \prod_{l\in S} \Bigl(\frac{\rho_l^{}}{m_l}\Bigr)^{\nu_{li}^{\rm f}} - {\cal K}_i^{\rm b} \prod_{l\in S} \Bigl(\frac{\rho_l^{}}{m_l}\Bigr)^{\nu_{li}^{\rm b}} \biggr], \qquad k\in S, \end{equation} \] where \(\mathcal{K}_i^{\rm f}\) and \(\mathcal{K}_i^{\rm b}\) are the forward and backward rate constants of the \(i\)th reaction, respectively.

These rates may be obtained from the mass action law or from the kinetic theory of dilute gases when the chemical characteristic times are larger than the mean free times of the molecules and the characteristic times of internal energy relaxation (Giovangigli 1999; Nagnibeda and Kustova 2009). The reaction constants \(\mathcal{K}_i^{\rm f}\) and \(\mathcal{K}_i^{\rm b}\) are functions of temperature and are Maxwellian averaged values of molecular chemical transition probabilities and this implies the reciprocity relations \[ \begin{equation} \tag{8} {\mathcal K}_i^{\rm e}(T) = \frac{ {\mathcal K}_i^{\rm f} (T) }{ {\mathcal K}_i^{\rm b} (T)}, \qquad \log {\mathcal K}_i^{\rm e}(T) = - \sum_{k\in S} (\nu_{ki}^{\rm b} - \nu_{ki}^{\rm f}) \frac{ m_k g_k(T, m_k) }{ R T}, \qquad i\in{\mathfrak R}, \end{equation} \] where \(\mathcal{K}_i^{\rm e}(T)\) is the equilibrium constant of the \(i\)th reaction (Giovangigli 1999). The forward reaction constants \({\mathcal K}_i^{\rm f}\), \(i\in{\mathfrak R}\), are usually evaluated with Arrhenius law \begin{equation} \tag{9} {\mathcal K}_i^{\rm f} = \mathfrak{A}_i T^{\mathfrak{b}_i} \exp \bigl( - \mathfrak{E}_i/R T \bigr), \qquad i\in{\mathfrak R}, \end{equation} where \(\mathfrak{A}_i\) is the preexponential factor, \(\mathfrak{b}_i\) the temperature exponent and \(\mathfrak{E}_i\) the activation energy of the \(i\)th reaction. The data required for each chemical reaction then reduce to the stoichiometric coefficients \(\nu_{ki}^{\rm f}\) and \(\nu_{ki}^{\rm b}\), and the Arrhenius constants \(\mathfrak{A}_i\), \(\mathfrak{b}_i\), and \(\mathfrak{E}_i\), assuming that the species thermodynamic is known. The chemical reaction stoichiometric coefficients \(\nu_{ki}^{\rm f}\) and \(\nu_{ki}^{\rm b}\) are such that atomic elements are conserved.

The size of detailed chemical reaction mechanisms has been steadily increasing over the past years ranging from a few species and reactions to several thousand of species interacting through tens of thousands of chemical reactions as for instance for bio-fuel or atmospheric pollution.

The law of mass action does not hold for nonideal fluids and the proper form for nonideal rates of progress has been obtained by Marcelin (1910). The nonideal rates may directly be expressed in terms of activities or chemical potentials. There are also perturbations of the chemical source terms due to the perturbed species distribution functions in the Navier-Stokes regime in the framework of the kinetic theory of reacting gases (Giovangigli 1999; Nagnibeda and Kustova 2009).

### Transport fluxes

The transport fluxes \(\viscous\), \(\fluxdiff_k\), \(k\in S\), and \(\heatflux\) due to macroscopic variable gradients may be obtained from various macroscopic and molecular theories (Waldman 1958; Mori 1958; Chapman and Cowling 1970; Ferziger and Kaper 1972; Keizer 1987; Ern and Giovangigli 1994; Giovangigli 1999; Nagnibeda and Kustova 2009) and are in the form \[ \begin{align} \tag{10} \viscous = {}& - \kappa (\dxb\scal\vitesse) \identite - \eta \bigl( \dxb\vitesse + \dxb\vitesse^t - \pdtiers (\dxb\scal\vitesse) \identite \bigr) , \\[3pt] \tag{11} \fluxdiff_k = {}& - \sum_{l \in S} \rho_k D_{kl} \forcediff_l - \rho Y_k \theta_k \dxb\log T, \qquad k \in S, \\[3pt] \tag{12} \heatflux = {}& - \widehat\lambda \,\dxb T - p \sum_{k \in S} \theta_k \forcediff_k + \sum_{k \in S} h_k \fluxdiff_k, \end{align} \] where \(\kappa\) denotes the bulk viscosity (sometimes termed the volume viscosity), \(\eta\) the shear viscosity, \(\identite\) the three dimensional identity tensor, \(D_{kl}\), \(k,l\in S\), the multicomponent diffusion coefficients, \(\forcediff_k\), \(k\in S\), the species diffusion driving forces, \(\theta_k\), \(k\in S\), the species thermal diffusion coefficients, \(\widehat\lambda\) the partial thermal conductivity, and \({}^t\) the transposition operator. The mass fluxes may also be expressed in terms of the species diffusion velocities \(\vitdiff_k\), \(k\in\eespe\), defined by \(\fluxdiff_k = \rho_k \vitdiff_k\), \(k\in\eespe\). The first term in the expression (10) of the viscous tensor $\viscous$ represents a resistance to compression and the second term a resistance to shear. Incidentally, the bulk viscosity \(\kappa\) is of the same order of magnitude than the shear viscosity \(\eta\) for polyatomic gases and its impact on fast flows has been established (Billet et al. 2008; Bruno and Giovangigli 2011). The first term in the expression (11) of the diffusion flux \(\fluxdiff_k\) yields diffusion effects due to mole fraction gradients, pressure gradients, and differences between specific forces acting on the species. The second term represents diffusion arising from temperature gradients and is termed the Soret---or Ludwig Soret---effect. The first term in the expression (12) of the heat flux $\heatflux$ represents represents Fourier's law, the second term corresponds to the Dufour effect, that is, heat diffusion due to concentration gradients, which is the analog of the Soret effect, and the third represents the transfer of energy due to species molecular diffusion. The matrix of diffusion coefficients \(D=(D_{kl})_{k,l\in\eespe}\) is symmetric positive semi-definite and the entropy production due to diffusive processes reads \((p/T) \langle D\forcediff,\forcediff\rangle\) with \(\forcediff= (\forcediff_1,\ldots,\forcediff_\nespe)^t\). Letting \(\massfraction = (\massfraction_1,\ldots,\massfraction_\nespe)^t\) where \(\massfraction_k\) is the mass fraction of the \(k\)th species, \(\theta= (\theta_1,\ldots,\theta_\nespe)^t\), and \(\langle,\rangle\) the scalar product, the diffusion matrix \(D\) and the thermal diffusion coefficients \(\theta\) satisfy the mass conservation constraints \(D\massfraction=0\) and \(\langle\theta,\massfraction\rangle=0\) guaranteeing that \(\sum_{k\in S} \fluxdiff_k=0\). The species diffusion driving force \(\forcediff_k\), \(k\in\eespe\), may be written \[ \begin{equation} \tag{13} \forcediff_k = \dxb \molefraction_k + ( \molefraction_k-\massfraction_k ) \dxb \log p + \frac{\rho_k}{p} (\force - \force_k), \qquad k \in S, \end{equation} \] where \(\molefraction_k\), \(k\in\eespe\), denote the species mole fractions, and \(\force = \sum_{k\in\eespe} \massfraction_k \force_k\) the averaged force. When gravity is the only force acting on the mixture the diffusion driving forces reduce to \(\forcediff_k = \dxb \molefraction_k + ( \molefraction_k-\massfraction_k ) \dxb \log p\). One may equivalently use the unconstrained diffusion driving forces \(\widehat\forcediff_k = (\dxb p_k - \rho_k \force_k)/p\), \(k \in S\), where \(p_k\) denotes the partial pressure of the $k$th species, since \(\forcediff_k = \widehat\forcediff_k - \massfraction_k (\dxb p - \rho \force)/p\). Many equivalent alternative formulation may be derived for multicomponent fluxes as for instance in terms of thermal diffusion ratios but are beyond the scope of the present short article (Waldman 1958; Chapman and Cowling 1970; Ferziger and Kaper 1972; Ern and Giovangigli 1994; Giovangigli 1999).

Historically, the multicomponent fluxes have first been written from empirical laws prior to being derived from the kinetic theory of gases or statistical mechanics. Moreover, even if the structure of multicomponent transport fluxes may be derived empirically or in the framework of macroscopic theories, only the kinetic theory of gases yield the multicomponent transport coefficients.

### Transport coefficients

The evaluation of the
transport coefficients
\(\kappa\),
\(\eta\),
\(\widehat\lambda\),
\(D=(D_{kl})_{k,l\in S}\),
and
\(\theta=(\theta_k)_{k\in S}\)
requires solving
linear systems derived from the
variational solution of systems of
Boltzmann linearized integral equations
(Waldman 1958;
Chapman and Cowling 1970;
Ferziger and Kaper 1972;
Ern and Giovangigli 1994).
The mathematical structure of the transport linear systems as well as
fast iterative algorithms for
evaluating the transport coefficients
have been obtained
(Ern and Giovangigli 1994;
Ern and Giovangigli 1996).
In practice,
for any
coefficient
\(\mu\),
the linear system takes on either
a regular form or a singular form
(Ern and Giovangigli 1994; Giovangigli 1999).
The singular form may be written
\[
\begin{equation}
\tag{14}
\left\{
\begin{array}{l}
G \alpha =
\beta,
\\[2pt]
\langle\tlsc, \alpha \rangle = 0,
\end{array}
\right.
\end{equation}
\]
where
the system matrix \(G\) is symmetric
positive semi-definite
with nullspace
spanned by a
vector
\(\tlsn\),
where \(\tlsc\) denotes
the constraint vector,
$\alpha$ and $\beta$
the unknown and right hand side vectors,
and the well posedness conditions
\(\langle\tlsn,\beta\rangle=0\)
and
\(\langle\tlsn,\tlsc\rangle\neq0\)
hold
(Ern and Giovangigli 1994).
The symmetry properties of the linear systems
and of the transport coefficients
are inherited from
the symmetry properties of the Boltzmann collision operator
(Waldmann 1958;
Ferziger and Kapper 1972;
Ern and Giovangigli 1994;
Giovangigli 1999).
The regular case is
simpler with
\(G\) symmetric positive definite
and without constraint
(Ern and Giovangigli 1994).
The
coefficient \(\mu\)
is then
obtained with
a scalar product
\(\mu = \langle\alpha,\beta'\rangle\).
Direct or iterative
numerical algorithms may be used to
solve the transport linear systems
but are out of the scope
of the present
article.
It is also possible to use
interpolation
*empirical*
expressions that
are typically in the form
\(\eta
=
\sum_{k\in S}
\molefraction_k \eta_k\)
where
\(\eta\) denotes the mixture viscosity,
\(\molefraction_k\) the mole fraction of the
\(k\)th species and
\(\eta_k\) the viscosity of the
\(k\)th species
(Ern and Giovangigli 1994).
Finally, there exists library of computer programs
which may be used for evaluating the
multicomponent transport coefficients
(Ern and Giovangigli (EGLIB)).

In order to illustrate multicomponent diffusion, the Stefan-Maxwell equations associated with the species diffusion velocities \(\vitdiff_k\), \(k\in\eespe\) are presented. These equations, obtained at the leading order from the kinetic theory of gases, are in the form \[ \begin{equation} \tag{15} \forcediff_k = \sum_{\doubleindices{l \in \eespe}{l \ne k}} \frac{\molefraction_k \molefraction_l }{{\cal D}^\bin_{kl} } \,\vitdiff_l \, - \, \sum_{\doubleindices{l \in \eespe}{l \ne k}} \frac{\molefraction_k \molefraction_l }{ {\cal D}^\bin_{kl} } \vitdiff_k, \qquad k \in \eespe, \end{equation} \] where \({\cal D}^\bin_{kl}(T,p)\) denotes the binary diffusion coefficient of the species pair \((k,l)\). These equations must also be completed by the constraint \(\sum_{k \in \eespe} \massfraction_k \vitdiff_k = 0\) associated with mass conservation. An elementary derivation of these equations has been given by Williams (Williams 1958a). The resulting expression for the species diffusion velocities in terms of the mole fraction gradients appears to be complex and couples all species. This complex dependence on concentration gradients is illustrated by the Duncan and Toor experiment on ternary diffusion processes (Duncan and Toor 1962) where reverse diffusion has been observed in full agreement with the Stefan-Maxwell equations.

### Boundary conditions

The description of general reactive flow boundary conditions may be found in the literature (Oran and Boris 1987; Kee et al. 2003). Dirichlet boundary conditions are typically associated with inflow phenomena in infinite length domains, isothermal walls, or classical velocity adherence conditions. Neumann boundary conditions are often associated with symmetry boundaries, adiabatic walls, or nonreactive walls.

When a gaseous mixture is in contact with a solid body or a liquid layer, the interfacial equations are also the boundary conditions of the gas phase equations. Typical interfacial equations may involve conservation jump relations for species mass, momentum and energy, continuity of some variables like temperature or tangential velocity, heterogeneous surface chemistry involving catalysts or solid species, adherence conditions, elastic as well as thermal interactions with solid structures. The species boundary conditions at a reactive interface are for instance in the form \[ \begin{equation} \tag{16} \rho_k^{} ( \vitesse + \vitdiff_k^{} ) {\cdot} {\boldsymbol n} = m_k^{} \widehat\omega_k^{}, \qquad k\in S, \end{equation} \] where \(\widehat\omega_k^{}\) are the surface production rates. These rates may take into account catalysis, film deposition or surface ablation (Kee et al. 2003; Ern et al. 1996). Interaction with boundaries may also involve fluid-structure interaction, evaporation, triple points, free boundaries, and radiative heat losses.

### Simplified models

The complete system of fundamental equations governing multicomponent reactive flows presented in the previous sections may be used to model various flows, but, in a number of situations, simplifications may be introduced following different ideas (Giovangigli 1999).

A first idea is to simplify the reactive aspects of the flow under consideration. In this situation, the number of species and chemical reactions are decreased and the resulting set of partial differential equations is simplified. The transport fluxes and transport property evaluation may accordingly be simplified. A typical example is that of a single irreversible chemical reaction (Williams 1985). Another type of chemistry simplification is associated with the idea of a slow manifold. In this framework, it is assumed that the state of the mixture, after some fast relaxation process that may be discarded, belongs to a manifold associated with a much slower dynamics. The manifold is then parametrized by a small set of parameters, typically some concentrations or thermal parameters, that are governed by a reduced system of partial differential equations. In combustion science for instance, slow manifolds have first been defined by solely looking at the source terms (Peters 1985; Mass and Pope 1992) and then defined through the calculation of libraries of flamelets thereby involving diffusive processes (Gicquel et al. 2000; Van Oijen et al. 2001; Bykov and Maas 2007; Auzillon et al. 2012). The chemical equilibrium model may also be seen as an ultimate simplified slow manifold model where the slow variables are the atomic mass densities, momentum and energy.

A second idea is to simplify the fluid dynamics aspects of the problem. This may be a geometrical simplification in the problem, a similarity assumption in the flow, or a simplification resulting from an asymptotic limit. As typical examples, we mention continuously stirred reactors, quasi one-dimensional flows, creeping flows, boundary layer flows, viscous shock layer flows, mixing layer flows, inviscid flows, or small Mach number flows.

Finally, a third idea is to simplify the coupling between chemistry and fluid dynamics. However, such a simplification is only feasible in very particular situations, since the coupling arises through various terms in the complete equations. Two typical situations are that of an incompressible limit, like the thermo-diffusive approximation in flame theory, or the dilution limit where a dilutant is in large concentration and the reactive species are trace species. Of course, all ideas may also be used simultaneously, so that the whole family of resulting models is very large.

## Mathematical structure and numerical methods

A convenient vector notation is introduced in order to recast the multicomponent flow governing equations into a compact form. The mathematical structure of multicomponent flow equations is then addressed by using symmetrized equations. Such a structure is important for theoretical as well as numerical purposes. Finally, Computational reactive Fluid Dynamics---which is nowadays a major tool in understanding of complex flows---is discussed.

### Vector notation

The equations governing multicomponent flows can be recast in a compact vector form often used to describe numerical methods and required to discuss its mathematical structure. The conservative variable \(\consv\) associated with Equations (1)-(3) is given by \[ \consv = \bigl(\rho_1,\ldots,\, \rho_n,\, \rho v_1,\, \rho v_2,\, \rho v_3, \, \rho e + \pdemi \rho \vitesse\scal\vitesse \bigr)^t, \] the convective flux in the \(i\)th direction by \(F_i = \bigl(\rho_1 v_i,\ldots ,\, \rho_n v_i, \, \rho v_1v_i+\delta_{i1}p, \, \rho v_2v_i+\delta_{i2}p, \, \rho v_3v_i+\delta_{i3}p, \, ({\cal E} + \pdemi \rho \vitesse\scal\vitesse+p)v_i \bigr)^t\), the dissipative flux in the \(i\)th direction by \(F_i^{\rm dis} = \bigl(\fluxdiffscal_{1 i} ,\ldots,\, \fluxdiffscal_{n i}, \, \varPi_{i1}, \, \varPi_{i2}, \, \varPi_{i3}, \, {\mathcal Q}_i + \sum_{j\in C} \varPi_{ij}v_j\bigr)^t\), and the source term by \(\Omega = \bigl(m_1 \omega_1,\ldots, m_n \omega_n, 0,0,0,0\bigr)^t\), where \(\delta_{ij}\) denotes the Kronecker symbol, \(\vitesse = (v_1,v_2,v_3)^t\), \(\fluxdiff_k = (\fluxdiffscal_{k1},\fluxdiffscal_{k2},\fluxdiffscal_{k3})^t\), \(\heatflux = (\heatfluxscal_1,\heatfluxscal_2,\heatfluxscal_3)^t\), \(\viscous=(\varPi_{ij})_{i,j\in C}\), and \(C=\{1,2,3\}\) the set of direction indices. Letting \(A_i = \partial_\consv F_i\), \(i\in C\), the Jacobian matrices of convective fluxes, \(B_{ij}\), \({i,j\in C}\), the dissipation matrices such that \(F_i^{\rm dis} = - \sum_{j\in C} B_{ij}(\consv) \partial_j^{} \consv\), the governing equations may then be recast in the form of a quasilinear system of partial differential equations \begin{equation} \tag{17} \frac{\partial \consv}{\partial t} + \sum_{i\in C} A_i(\consv) \partial_i^{} \consv = \sum_{i,j\in C} \partial_i^{} \bigl(B_{ij}(\consv)\partial_j^{} \consv\bigr) + \Omega(\consv), \end{equation} where \(\partial_i^{}\) denotes the derivative operator in the \(i\)th spatial direction. The convective terms are then the first order terms \(\sum_{i\in C} A_i(\consv) \partial_i^{} \consv\) whereas the dissipative terms are the second order terms \(\sum_{i,j\in C} \partial_i^{} \bigl(B_{ij}(\consv)\partial_j^{} \consv\bigr)\).

### Hyperbolic-parabolic structure

The system of partial differential equations (17) may be symmetrized by using the entropic variable (Hughes et al. 1986; Giovangigli and Massot 1998) \[ \symev = - \bigl(\partial_\consv (\rho s) \bigr)^t = (1/ T) \bigl( g_1-\pdemi {\boldsymbol v}\scal{\boldsymbol v},\ldots, \, g_{n} - \pdemi{\boldsymbol v}\scal{\boldsymbol v}, \, v_1, v_2, \, v_3, \, -1\bigr)^t. \] Performing the change of variable \(\consv=\consv(\symev)\) and letting \(\widetilde A_0=\partial_\symev \consv\), \(\widetilde A_i = A_i \partial_\symev \consv\), \(\widetilde B_{ij} = B_{ij} \partial_\symev \consv\), \(\widetilde{\Omega}=\Omega\), the system (17) is transformed into \begin{equation} \tag{18} \widetilde A_0(\symev) \frac{\partial \symev}{\partial t} + \sum_{i\in C}\widetilde A_i(\symev) \partial_i^{} \symev = \sum_{i,j\in C} \partial_i^{} \bigl( \widetilde B_{ij}(\symev) \partial_j^{} \symev \bigr) + \widetilde{\Omega}(\symev), \end{equation} where \(\widetilde A_0(\symev)\) is symmetric positive definite, \(\widetilde A_i(\symev),\, i\in C\), are symmetric, where we have the reciprocity relations \(\widetilde B_{ij}(\symev)^t= \widetilde B_{ji}(\symev),\, i,j\in C\), and where \(\widetilde B(\symev,w)= \sum_{i,j\in C} \widetilde B_{ij}(\symev)w_iw_j\) is symmetric positive semi-definite for any vector \(w= (w_1,w_2,w_3)^t\) (Giovangigli and Massot 1998).

From the symmetrized form (18) it is classically established that the first order differential operator \(\widetilde A_0(\symev) \partial/\partial_t + \sum_{i\in C}\widetilde A_i(\symev) \partial_i^{}\) associated with convection is hyperbolic whereas the second order operator \(\widetilde A_0(\symev) \partial/\partial_t^{} - \sum_{i,j\in C}\widetilde B_{ij}(\symev) \partial_i^{}\partial_j^{}\) associated with dissipative phenomena is degenerate parabolic. Such a symmetric structure is the consequence of the underlying kinetic framework, that is, of symmetry properties deduced from the Boltzmann collision operator (Giovangigli 1999). Moreover, there is an important coupling stability condition between the hyperbolic and the parabolic operators, the Kawashima-Shizuta condition which physically states that all waves associated with multicomponent Euler equations are damped by dissipative processes (Shizuta and Kawashima 1985; Giovangigli and Massot 1998; Giovangigli and Matuszewski 2013). It is also possible to split the variables between hyperbolic and parabolic variables (Kawashima and Shizuta 1988; Giovangigli and Massot 1998).

The symmetrized forms may notably be used for mathematical purposes like existence theorems or asymptotic stability results (Vol'Pert and Hudjaev 1972; Kawashima and Shizuta 1988; Giovangigli 1999). They may also be used for finite element formulations based on Streamline Upwind Petrov-Galerkin techniques (Hughes et al. 1986).

### Computational reactive fluid dynamics

Computational Fluid Dynamics is now a major tool in understanding of complex flows (Oran and Boris 1987; Ferziger and Peric 1996; Godlewski and Raviart 1996; Laney 1998; Chung 2002; Anderson 2009; Pletcher et al. 2013). Numerical simulation of compressible flows is a difficult task that requires a solid background in fluid mechanics and numerical analysis. The nature of compressible flows may be very complex, with features such as shock fronts, boundary layers, turbulence, acoustic waves, or instabilities.

Taking into account chemical reactions dramatically increases the difficulties, especially when detailed chemical and transport models are considered. Interactions between chemistry and fluid mechanics are especially complex in reentry problems (Anderson 1989), combustion phenomena (Poinsot and Veynante 2005), or chemical vapor deposition reactors (Hitchman and Jensen 1993; Kee et al. 2003). An important aspect of complex chemistry flows is the presence of multiple time scales which may range typically from \(10^{-10}\) second up to several seconds. In the presence of multiple time scales, implicit methods are advantageous, since otherwise explicit schemes are limited by the smallest time scales (Descombes and Massot 2004; Oran and Boris 1987). A second potential difficulty associated with the multicomponent aspect is the presence of multiple space scales. In combustion applications for instance the flame fronts are very thin and typically require space steps of \(10^{-3}\) cm at atmospheric pressure, and even \(10^{-5}\) cm at \(100\) atm, whereas a typical engine scale may be of \(10\) cm or even \(100\) cm. The multiple scales can only be solved by using adaptive grids obtained by successive refinements or by moving grids for unsteady problems (Smooke 1982; Oran and Boris 1987; Bennett and Smooke 1998; Smooke 2013). A goal of simplified models, in addition to decreasing the number of unknowns, is also to suppress the fastest times scales and the steepest gradients in chemical fronts, by eliminating also the most reactive intermediate species.

Nonlinear discrete equations may be solved by using Newton's method or any generalization (Smooke 1982; Smooke 2013). The resulting large sparse linear systems may then be solved by using a Krylov-type method, such as GMRES. Other sophisticated methods involve coupled Newton-Krylov techniques (Knoll et al. 1994), time splitting algorithms (Descombes and Massot 2004; Nonaka et al. 2012), higher order compact discretization schemes (Noskov and Smooke 2005) as well as massively parallel simulations (Chen 2011; Moureau et al. 2011). Characteristic type boundary conditions are often used for the simulation of reactive flows (Poinsot and Veynante 2005). Evaluating aero-thermochemistry quantities is computationally expensive since they involve multiple sums and products. Optimal evaluation requires a low-level parallelization depending on the problem granularity. Moreover, it is preferable, when writing numerical software, to clearly separate the numerical tools from the special type of equations that are under consideration. In the context of multicomponent flows, it is therefore a good idea to write codes for general mixtures and use libraries that automatically evaluate thermochemistry properties (Kee et al. 1980; Cantera) and transport properties (Ern and Giovangigli (EGLIB)).

The resulting reactive flow simulations may then be validated with detailed numerical simulations (Martinez et al. 2014) as well as against detailed experimental measurements including temperature and species concentrations. Laser dianostics are especially useful to analyze flame structure (Kohse-Höinghaus and Jeffries 2002; Smooke 2013), Chemical Vapor Deposition reactors (Fotiadis et al. 1990) and hypersonic wind tunnels (Laufer et al. 1990).

## Extended models

In the previous sections, the fundamental modeling of multicomponent flows, the qualitative properties of the resulting systems of partial differential equations, and numerical methods have been addressed. In many practical situations, however, extended models are required and some of these extensions are briefly addressed in this section, namely turbulence modeling, nonideal thermodynamics, ionized flows, thermodynamic nonequilibrium, chemical equilibrium flows, sprays, and radiation. Non-Newtonian flows, thin films, biological flows, relativistic flows, or quantum fluids which may all be multicomponent, will not be addressed, neither heterogeneous multifluids---associated with multiphase flows---where each phase may also be multicomponent and which are investigated elsewhere in Scholarpedia.

### Turbulent flows

Turbulence is one of the most complex phenomena in fluids and turbulent flows are encountered in practical devices like rockets, aircraft engines, industrial furnaces, chemical power plants as well as in the atmosphere. Turbulence may be characterized by fluctuations of all local flow properties (Frisch 1995; Lesieur et al. 2005; Pope 2000; Peters 2000; Poinsot and Veynante 2005). Turbulent flows may either be investigated by using direct numerical simulation (DNS), when all the physical scales are resolved, or by using filtered equations for Large Eddy simulations (LES) or Reynolds-Averaged Navier-Stokes (RANS) simulations.

The LES or RANS equations for turbulent flows are typically derived by applying a filter or averaging operator, respectively, to the set of fundamental equations presented in the previous sections (Pope 2000; Peters 2000; Poinsot and Veynante 2005). With LES the flow variables are filtered in the spectral space, all frequencies greater than a given cut-off are suppressed and those lower than the cut-off are retained, whereas with RANS all flow quantities are averaged. The unclosed correlations are then expressed using subgrid scale models (Lesieur et al. 2005; Pope 2000; Poinsot and Veynante 2005). Products of fluctuations are typically modeled by gradient like laws whereas the filtered chemical source term models may involve wrinkled and strained fluctuating chemical fronts as well as distributed reaction zones depending on the turbulence intensity (Lesieur et al. 2005; Pope 2000; Poinsot and Veynante 2005).

### Nonideal thermodynamics

Progress in the efficiency of automotive engines, gas turbines and rocket motors have notably been achieved with high pressure combustion (Candel et al. 2006). As pressure is increasing, attractive forces between molecules play a more important role in fluids and lead to nonideal effects so that the compressibility factor \(\compres=pm/(\rho R T)\) deviates from unity. This is the case in particular above the critical pressure where it is possible to continuously change a liquid like fluid into a gas like fluid (Guggenheim 1962).

Nonideal multicomponent fluid thermodynamics are often built from equations of state using the compatibility with ideal gases as a limiting condition (Guggenheim 1962; Giovangigli and Matuszewski 2012). The chemistry sources are influenced by nonidealities as well as multicomponent diffusion which is then driven by the gradient of chemical potentials (Marcelin 1910; Keizer 1987; Giovangigli and Matuszewski 2012). These nonidealities prevent unphysical diffusion in cold dense parts of the fluid. The structure of the resulting set of partial differential equations is further analyzed in (Giovangigli and Matuszewski 2013).

### Plasmas

Partially ionized gas mixtures are related to a wide range of practical applications including laboratory plasmas, high-speed gas flows and atmospheric phenomena (Braginskii 1958; Chapman and Cowling 1970; Ferziger and Kaper 1972; Raizer 1987; Bruno et al. 2003). Another fundamental application is inertial confinement fusion where the thermonuclear fusion of light nuclei is a source of energy (Lindl 1998; Atzeni and Meyer-ter-Vehn 2009). Application of the Chapman-Enskog method to partially ionized gases is feasible for low temperature high density plasmas (Ferziger and Kaper 1972). The interactions between particles at distances greater than the Debye length are considered to be mediated by the electric field while those at shorter distance are considered to be true collisions (Ferziger and Kaper 1972). We refer to Raizer (1987), Zhdanov (2002), Nagnibeda and Kustova (2009), Giovangigli and Graille (2009), Graille et al. (2009), Capitelli et al.(2012), and Capitelli et al.(2013) for a detailed presentation of the multicomponent plasmas governing equations. In particular, in strong magnetic fields, the transport fluxes are found to be anisotropic and different coefficients may be obtained depending on the relative orientation of variables gradients with the magnetic field. The corresponding macroscopic equations have to be completed by the Maxwell equations governing the electric and magnetic fields. Many simplifications are also possible and the physics of plasmas is very rich and complex because of the many characteristic lengths and times involved (Ferziger and Kaper 1972; Raizer 1987).

### Multitemperature flows

Thermodynamic nonequilibrium is of fundamental importance in reentry problems, laboratory and atmospheric plasmas, as well as discharges or strong shock waves (Zeldovitch and Raizer 2002; Zhdanov 2002; Capitelli et al. 2007; Nagnibeda and Kustova 2009). The most general thermodynamic nonequilibrium model is the state to state model where each internal state of a molecule is independent and considered as a separate species (Capitelli et al. 2007; Zhdanov 2002; Nagnibeda and Kustova 2009). When there are partial equilibria between some of these states, species internal temperatures may be defined and the complexity of the model is correspondingly reduced (Zhdanov 2002; Nagnibeda and Kustova 2009). Another example is that of electron temperature in plasmas (Graille et al. 2009). The next reduction step then consists in equating some of the species internal temperatures and ultimately lead to the one temperature flow model presented in the previous sections (Nagnibeda and Kustova 2009).

### Chemical equilibrium flows

Chemical equilibrium flows are a limiting model which is of interest for various applications such as chemical vapor deposition reactors (Gokoglu 1988), flows around space vehicles (Anderson 1989; Mottura et al. 1997), or diverging nozzle rocket flows (Williams 1985). These simplified models are valid when the characteristic chemical times are small in comparison with the flow time. The equations governing chemical equilibrium flows may either be derived directly in a kinetic framework (Ern and Giovangigli 1998), or by superimposing chemical equilibrium in the equations presented in the previous sections. Both methods lead to the same conservations equations, transport fluxes, thermodynamics, as well as qualitative properties of transport coefficients but yield different quantitative values for the transport coefficients (Ern and Giovangigli 1998). The chemical equilibrium constraints may then be used to eliminate the chemical unknowns and to reduce the model into a system of partial differential equations governing the slow variables that are the atomic mass densities, momentum and energy (Giovangigli 1999). The chemical equilibrium model may also be seen as an ultimately simplified slow manifold model.

### Sprays and clouds

Many practical devices involve dispersed condensed phases in the form of droplets or solid particles like sprays, aerosols, mists, dusts, clouds, fumes, suspensions, or sooting flames. Each of the condensed phase may itself be multicomponent and may interact with the multicomponent gas. In these situations there are often so many droplets or solid particles that only a statistical description is feasible through the concept of distribution function similar to that used in kinetic theory (Williams 1958b; Williams 1985). The corresponding Lagrangian models typically involve Boltzmann type and kinetic type spray equations as introduced by Williams (1958b, 1985). The coupling between the dispersed condensed phases and the gas phase then arise through vaporization, condensation, sublimation, drag, coalescence, as well as atomization (Williams 1985). The kinetic type equations may then be discretized in a fully Lagrangian way (O'Rourke 1985) as well as in an Eulerian way leading to multifluid models (Laurent and Massot 1990; Fox et al. 2008). When the condensed phases are not dispersed, multiphase flows are obtained (Drew and Passman 1999) and are discussed elsewhere in Scholarpedia.

### Radiation

A radiant heat flux may sometimes be added to the heat flux in the energy conservation equation (Williams 1985; Zeldovitch and Raizer 2002). This radiant heat flux is the integral of the radiant intensity over all frequencies and all solid angles and the radiant intensity is governed by a Boltzmann type equation involving emission, absorption, and scattering coefficients (Williams 1985). Two classical approximated models in radiation transport are the optically thick or optically thin media which lead---neglecting absorption and scattering---to Stefan-Boltzmann type radiation heat loss source terms (Willimas 1985). Radiant effects are also important at boundaries which may absorb and emit radiant heat.

## Examples of multicomponent flows

Three typical numerical simulations of multicomponent reactive flows are presented in this section in order to illustrate the preceding developments, namely a chemical vapor deposition reactor, a direct numerical simulation of a high pressure flame in a mixing layer and a reentry flow.

### A chemical vapor deposition reactor

Chemical Vapor Deposition (CVD) is an industrially important process used to produce solid films with extremely fine compositional control and uniformity. The influence of various operating parameters on product quality and on the chemical process in CVD reactors may be investigated numerically.

We consider a three dimensional reactor where trimethylgallium \(\textrm{Ga}( \textrm{CH}_3)_3\) and Arsine \(\textrm{AsH}_3\) are injected with hydrogen \(\textrm{H}_2\) as a carrier gas (Ern et al. 1996). The geometry is three-dimensional \((x,y,z) \in [-1.5,1.5]{\times}[0,7.2]{\times}[0,10]\) cm, \(x,y,z\) being respectively the vertical, transverse, and streamwise coordinates, and the injected flow rate is 5 l/min at standard conditions with inlet partial pressures of 1 atm for \(\textrm{H}_2\), 1.8 \(10^{-4}\) atm for \({\rm Ga}({\rm C}{\rm H}_3)_3\), and 3.3 \(10^{-3}\) atm for \({\rm As}{\rm H}_3\). The bottom of the reactor \(x=-1.5\) is heated at \(1000\) K on the domain \((y,z)\in [0,7.2]{\times}[4,10]\) cm and is otherwise at room temperature and the crystal may grow on the substrate \((y,z)\in[0,7.2]{\times}[4,5]\) cm. The reaction mechanism in the gas phase involves the 15 species \({\rm Ga}({\rm C}{\rm H}_3)_3\), \({\rm Ga}({\rm C}{\rm H}_3)_2\), \({\rm Ga}{\rm C}{\rm H}_3\), \({\rm Ga}({\rm C}{\rm H}_3)_2{\rm C}{\rm H}_2\), \({\rm Ga}{\rm C}{\rm H}_3{\rm C}{\rm H}_2\), \({\rm Ga}{\rm C}{\rm H}_2\), \({\rm As}{\rm H}_3\), \({\rm As}{\rm H}_2\), \({\rm As}{\rm H}\), \({\rm C}{\rm H}_3\), \({\rm C}{\rm H}_4\), \({\rm C}_2{\rm H}_6\), \({\rm H}\), \({\rm H}_2\) interacting through 17 reactions. The surface reaction mechanism involves the 7 surface species \({\rm H}^{(G)}\), \({\rm C}{\rm H}_3^{(G)}\), \({\rm Ga}{\rm C}{\rm H}_3^{(G)}\), \({\rm H}^{(A)}\), \({\rm C}{\rm H}_3^{(A)}\), \({\rm As}{\rm H}^{(A)}\), \({\rm As}^{(A)}\), and 2 bulk solid species \({\rm Ga}{\rm As}^{(b)}\), \({\rm Ga}{\rm C}^{(b)}\), interacting through 30 surface reactions. The compressible Navier-Stokes equations have been solved for the carrier gas \(\textrm{H}_2\) and the reactive species equations for the trace reactive species. The numerical method combines finite differences, Newton's method, coupled implicit iterations, and generalized conjugate gradient solvers.

Figures 4 illustrates the mole fraction isopleth
of
\({\rm Ga}({\rm C}{\rm H}_3)_3\)
and
\({\rm As}{\rm H}_3\)
in the
symmetry plane of the CVD reactor.
The inlet is on the left of the
symmetry plane and the outlet on the right.
At the bottom of the reactor
\(x=-1.5\)
the substrate
corresponds to the segment
\(z\in[4,5]\)
and the susceptor
to the segment
\(z\in[5,10]\).
Both reactive species
\({\rm Ga}({\rm C}{\rm H}_3)_3\)
and
\({\rm As}{\rm H}_3\)
are gradually
decomposed
with increasing temperature
above the heated substrate and susceptor
and are then carried outside the reactor.
Many intermediate species are formed that
interact chemically with
substrate and lead to crystal growth.
Figure 5 illustrates the mole fraction of
\({\rm As}{\rm H}_2\) which is formed by surface chemistry and
desorption,
of
\({\rm Ga}{\rm C}{\rm H}_3\)
which leads to carbon impurities in the crystal
as well as
\({\rm H}\) mainly present in the hot zone of the reactor.
In CVD systems
thermal diffusion
(Soret effet)
drives heavy reactant sources
away from the hot depletion zone
and plays a significant role in CVD modeling
(Ern et al. 1996).

### A high pressure flame

A two-dimensional Hydrogen/Oxygen flame stabilized behind a splitter plate with a mean pressure of 100 bar is investigated (Ruiz et al. 2012). At such high pressures, above the critical pressure, the fluids are nonideal, and a real gas equation of state is used. The \({\rm O}_2\) fluid is in a liquid-like dense state, whereas the \({\rm H}_2\) stream has a gas-like density. The two-dimensional splitter plate represents the lip of an injector and the operating point is typical of a real engine. The mixture involves the \(n = 8\) species \({\rm H}_2\), \({\rm O}_2\), \({\rm H}_2{\rm O}\), \({\rm H}\), \({\rm O}\), \({\rm O}{\rm H}\), \({\rm H}{\rm O}_2\), \({\rm H}_2{\rm O}_2\) interacting through \(n^{\hskip-0.04em {\rm r}} = 12\) chemical reactions (Ruiz et al. 2012).

Although turbulence is a 3D phenomenon, the flame/flow interaction is mainly 2D in the stabilization region and the simplification to 2D is not a strong limitation. Letting \(h = 0.05\) cm be the splitter height, the computational domain is 11\(h\) long in the x-direction and 10\(h\) in the y-direction. Hydrogen is injected above the splitter at a temperature \(T = 150\) K and a velocity \(u = 125\) m/s. Below the splitter, oxygen is fed at \(T= 100\) K and \(u = 30\) m/s. The shape of the inlet velocity profiles follows a 1/7th power law. Although developed turbulence is generally present in the feeding lines of rocket engines, no velocity perturbation is added to the inflow boundary condition. Yet, strong turbulence levels caused by vortex shedding are observed downstream of the splitter as illustrated in Figure 6 where the temperature field is presented, allowing for a developed turbulent mixing layer and strong flame/turbulence interactions (Ruiz et al. 2011).

### A reentry flow

We consider an hypersonic flow around an Apollo-like spatial vehicle reentering into Earth's atmosphere. The freestream conditions corresponds to a Mach number \(Ma =15\), a pressure of \(35\) Pa, and air at temperature \(T=256\) K. The reaction mechanism involves the \(n = 5\) species \({\rm O}_2\), \({\rm N}_2\), \({\rm O}\), \({\rm N}\), \({\rm N}{\rm O}\), interacting through \(n^{\hskip-0.04em {\rm r}} =4\) reactions (Park et al. 2001; Magin and Degrez 2004). The model is a non equilibrium model with two temperatures associated with rotational-translational energy modes and vibrational-electronic energy modes (Magin and Degrez 2004).

The wall boundary conditions are that of radiative equilibrium with an emissivity coefficient of \(0.8\), and catalytic surface reactions at the wall are not taken in account. In Figure 7 are presented the Mach numbers around the capsule with an angle of attack of 25 degrees (Lani 2008). The temperature behind the shock is \(5000\) K and the pressure \(10000\) Pa to be compared with \(256\) K and \(35\) Pa in front of the shock.

## References

- J. D. Anderson, Jr.,
*Hypersonics and High Temperature Gas Dynamics*, McGraw-Hill Book Company, New-York, (1989).

- J. D. Anderson, Jr., in
*Computational Fluid Gas Dynamics, An Introduction*, J.F. Wendt Ed., 3rd Ed., Springer, Berlin, (2009).

- S. Atzeni and J. Meyer-ter-Vehn,
*The Physics of Inertial Fusion*, Oxford Science Publications, Oxford, (2009).

- P. Auzillon, O. Gicquel, N. Darabiha, D. Veynante, and B. Fiorina, A Filtered Tabulated Chemistry Model for LES of Stratified Flames, Comb. Flame, 159, (2012) pp. 2704-2717.

- R. J. Bearman and J. G. Kirkwood, The Statistical Mechanics of Transport Processes. XI. Equations of Transport in Multicomponent Systems, J. Chem. Phys., 28 (1958), pp. 136-145.

- B. A. Bennett and M. Smooke, Local Rectangular Refinement with Application to Nonreacting and Reacting Fluid Flow Problems, J. Comp. Phys., 151, (1999) pp. 684-727.

- G. Billet, V. Giovangigli, and G. de Gassowski, Impact of Volume Viscosity on a Shock/Hydrogen Bubble interaction, Combustion Theory and Modelling, 12, (2008), pp. 221-248.

- S.I. Braginskii, Transport processes in a plasma, M.A. Leontovich editor, Rev. Plasma Phys., 1, (1965), pp. 205-311.

- D. Bruno, M. Capitelli, and A. Dangola, Transport Coefficients of Partially Ionized Gases: A Revisitation, AIAA Paper, AIAA-2003-4039, (2003).

- D. Bruno and V. Giovangigli, Relaxation of Internal Temperature and Volume Viscosity, Physics of Fluids, 23, (2011), 093104.

- V. Bykov and U. Maas, Extension of ILDM Method to the Domain of Slow Chemistry, Proc. Comb. Inst., 31, (2007), pp. 465-472.

- S. Candel, M. Juniper, G. Singla, P. Scouflaire, and C. Rolon, Structure and Dynamics of Cryogenic Flames at Supercritical Pressure, Comb. Sci. Tech. 178 (2006), pp. 161-192.

- M. Capitelli, I. Armenise, D. Bruno, M. Cacciatore, R. Celiberto, G. Colonna, O. de Pascale, P. Diomede, F. Esposito, C. Gorse, K. Hassouni, A. Laricchiuta, S. Longo, D. Pagano, D. Pietanza, and M. Rutigliano, Non-Equilibrium Plasma Kinetics: A State-to-State Approach, Plasma Sourc. Sci. Tech., 16, (2007), pp. S30-S44.

- M. Capitelli, G. Colonna, and P. d'Angola,
*Fundamental Aspects of Plasmas Chemical Physics: Thermodynamics*, Springer series on Atomic, Optical and Plasma Physics, 66, (2012).

- M. Capitelli, D. Bruno, and A. Larichiuta,
*Fundamental Aspects of Plasmas Chemical Physics: Transport*, Springer series on Atomic, Optical and Plasma Physics, 74, (2013).

- S. Chapman and T. G. Cowling,
*The Mathematical Theory of Non-Uniform Gases*, Cambridge University Press, Cambridge, (1970).

- J. H. Chen, Petascale Direct Numerical Simulation of Turbulent Combustion---Fundamental Insights Towards Predictive Models, Proc. Comb. Inst., 33, (2011), pp. 99-123.

- T.J. Chung,
*Computational Fluid Dynamics*, Cambridge University Press, Cambridge, (2002).

- S. Descombes and M. Massot, Operator Splitting for Nonlinear Reaction-Diffusion Systems with Entropic Structure: Singular Perturbation and Order Reduction, Numer. Math., 97, (2004), pp. 667--698.

- D. A. Drew and S. L. Passman,
*Theory of Multicomponent Fluids*, Applied Mathematical Sciences 135, Springer, New York, (1999).

- J.B. Duncan and H.L. Toor, An Experimental Study of Three Component Gas Diffusion, AIChE J., 8 (1962), pp. 38-41.

- A. Ern and V. Giovangigli,
*Multicomponent Transport Algorithms*, Lecture Notes in Physics Monographs, M24, (1994).

- A. Ern and V. Giovangigli, The Structure of Transport Linear Systems in Dilute Isotropic Gas Mixtures, Phys. Rev. E, 53, (1996), pp. 485-492.

- A. Ern and V. Giovangigli, The Kinetic equilibrium regime, Physica-A, 260 (1998), pp. 49-72.

- A. Ern and V. Giovangigli, EGlib server and User's manual, http:/\(\!\)/www.cmap.polytechnique.fr/www.eglib.

- A. Ern, V. Giovangigli and M. Smooke, Numerical Study of a Three-Dimensional Chemical Vapor Deposition Reactor with Detailed Chemistry, J. Comp. Phys., 126, (1996), pp. 21-39.

- J. H. Ferziger and H. G. Kaper,
*Mathematical Theory of Transport Processes in Gases*, North Holland Pub. Co., Amsterdam, (1972).

- J. H. Ferziger and M. Peric,
*Computational Methods for Fluid Dynamics*, Springer-Verlag, Berlin, (1996).

- U. Frisch,
*Turbulence*, Cambridge University Press, Cambridge, (1995).

- D. Fotiadis, M. Boekholt, K.F. Jensen, and W. Richter, Flow and Heat Transfer in CVD Reactors: Comparison of Raman Measurements and Finite Element Model Predictions, J. Crystal Growth, 100, (1990), pp. 577-599.

- R. O. Fox, F. Laurent, and M. Massot, Numerical Simulation of Spray Coalescence in an Eulerian Framework : Direct Quadrature Method of Moments and Multi-Fluid Method, J. Comp. Phys., 227, (2008), pp. 3058-3088.

- O. Gicquel, N. Darabiha, and D. Thevenin, Laminar premixed Hydrogen/Air Counterflow Flame Simulations using Flame Prolongation of ILDM with Differential Diffusion, Proc. Comb. Inst., 28, (2000), pp. 1901-1908.

- V. Giovangigli,
*Multicomponent Flow Modeling*, Birkhäuser, Boston, (1999).

- V. Giovangigli and B. Graille, Kinetic Theory of Partially Ionized Reactive Gas Mixtures II, J. Phys. A, 42, (2009), pp. 1-31.

- V. Giovangigli and M. Massot, Asymptotic Stability of Equilibrium States for Multicomponent Reactive Flows, Math. Mod. Meth. Appl. Sci., 8, (1998), pp. 251-297.

- V. Giovangigli and L. Matuszewski, Supercritical Fluid Thermodynamics from Equations of State, Phys. D, 241 (2012) pp. 649-670.

- V. Giovangigli and L. Matuszewski, Mathematical Modeling of Supercritical Multicomponent Reactive Fluids, Math. Mod. Meth. App. Science, 23, (2013), pp. 2193-2251.

- E. Godlewski and P.A. Raviart,
*Numerical Approximation of Hyperbolic Systems of Conservation Laws*, Springer-Verlag, New-York, (1996).

- S. A. Gokoglu, Significance of Vapor Phase Chemical Reactions on CVD Rates Predicted by Chemically Frozen and Local Thermochemical Equilibrium Boundary Layer Theories, J. Electrochem. Soc., 135, (1988), pp. 1562-1570.

- B. Graille, T. Magin and M. Massot, Kinetic Theory of Plasmas: Translational Energy, Math. Mod. Meth. App. Sci., 19, (2009), pp. 527-599.

- S.R. de Groot and P. Mazur,
*Non-equilibrium thermodynamics*, Dover publications, Inc. New York, (1984).

- E. A. Guggenheim,
*Thermodynamics*, North Holland, Amsterdam, (1962).

- T. J. R. Hughes, L. P. Franca, and M. Mallet, A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics, Comp. Meth. Appl. Mech. Eng., 54, (1986), pp. 223-234.

- M.L. Hitchman and K.F. Jensen,
*Chemical Vapor Deposition: Principles and Application*, Academic Press, (1993).

- S. Kawashima and Y. Shizuta, On the Normal Form of the Symmetric Hyperbolic-Parabolic Systems associated with the Conservation Laws, Tôhoku Math. J., 40, (1988), pp. 449-464.

- R. J. Kee, J. A. Miller, and T. H. Jefferson, CHEMKIN: A General-Purpose, Problem-Independent, Transportable, Fortran Chemical Kinetics Code Package. SANDIA National Laboratories Report, SAND80-8003 (1980).

- R. J. Kee, M. E. Coltrin, and P. Glarborg,
*Chemically Reacting Flow*, Wiley Interscience, (2003).

- J. Keizer,
*Statistical Thermodynamics of Nonequilibrium Processes*, Springer-Verlag, New York, (1987).

- D. A. Knoll, P. R. McHugh, and D. E. Keyes, Newton-Krylov Methods for Low-Mach-Number Compressible Combustion, AIAA J., 34, (1996), pp. 961-967.

- K. Kohse-Höinghaus, and J.B. Jeffries,
*Applied Combustion Diagnostics*, Taylor and Francis, London, (2002).

- C.B. Laney,
*Computational Gasdynamics*, Cambridge University Press, Cambridge, (1998).

- A. Lani, An Object Oriented and High Performance Platform for Aerothermodynamics Simulation, PhD Thesis, Von Karman Institute for Fluid Dynamics, (2008).

- G. Laufer, R.L. McKenzie, and D.G. Fletcher, Method for Measuring Temperature and Densities in Hypersonics Wind Tunnel Air Flows Using Laser-Induced Fluorescence, Applied Optics, 29, (1990), pp. 4873-4883.

- M. Lesieur, O. Métais, and P. Comte,
*Large-Eddy Simulation of Turbulence*, Cambridge University press, Cambridge, (2005).

- J.D. Lindl,
*Inertial Confinement Fusion*, Amer. Inst. Physics, (1998).

- U. Maas and S. B. Pope, Simplifying Chemical Kinetics : Intrinsic Low-Dimensional Manifolds in Composition Space, Comb. Flame, 88, (1992), pp. 239-264.

- T. E. Magin and G. Degrez, Transport Algorithms for Partially Ionized and Unmagnetized Plasmas, J. Comp. Phys., 198, (2004) pp. 424-449.

- M. R. Marcelin, Sur la Mécanique des Phénomènes Irréversibles, Comptes Rendus de l'Académie des Sciences de Paris, Séance du 5 décembre 1910, (1910), pp. 1052-1055.

- L. Mottura, L. Vigevano, and M. Zaccanti, An Evaluation of Roe's Scheme Generalizations for Equilibrium Real Gas Flows, J. Comp. Phys., 138, (1997), pp. 354-399.

- H. Mori, Statistical-Mechanical Theory of Transport in Fluids, Phys. Rev., 112 (1958), pp. 1829-1842.

- V. Moureau, P. Domingo, L. Vervisch, From large-eddy simulation to direct numerical simulation of a lean premixed swirl flame: filtered laminar flame-pdf modeling, Comb. Flame, 158, (2011), pp. 1340-1357.

- P.J. Martinez Ferrer, R. Buttay, G. Lehnasch, and A. Mura, A Detailed Verification Procedure for Compressible Reactive Multicomponent Navier-Stokes Solvers, Comput. Fluids, 89, (2014) pp. 88-110.

- E. Nagnibeda and E. Kustova,
*Non-equilibrium Reacting Gas Flow*, Springer Verlag, Berlin, (2009).

- A. Nonaka, J. B. Bell, M. S. Day, A. S. Almgren and M. L. Minion, A Deferred Correction Coupling Strategy for Low Mach Number Flow with Complex Chemistry, Comb. Theor. Mod., 16, (2012), pp. 1053-1088.

- M. Noskov and M. Smooke, An Implicit Compact Scheme Solver with Application to Chemically Reacting Flows, J. Comp. Phys., 203, (2005) pp. 700-730.

- C. Park, R. L. Jaffe and H. Partridge, Chemical Kinetic Parameters of Hyperbolic Earth Entry, J.Thermophys. Heat Transfer, 15, (2001) pp. 76-90.

- E. Oran and J. P. Boris,
*Numerical Simulation of Reactive Flows*, Elsevier, New York, (1987).

- P. J. O'Rourke, Collective Drop Effects on Vaporizing Liquid Sprays, PhD Thesis, Princeton University, (1981).

- N. Peters, Numerical Simulation and Asymptotic Analysis of a Systematically Reduced Reaction Schemes for Hydrocarbon Flames, in: R. Glowinsky, B. Larrouturou, and R. Temam, Eds., Numertical Simulation of Combustion Phenomena, Springer-Verlag, Berlin, (1985).

- N. Peters,
*Turbulent Combustion*, Cambridge University Press, Cambridge, (2000).

- T. Poinsot and D. Veynante,
*Theoretical and Numerical Combustion*, 2nd Ed., Edwards, Philadelphia, (2005).

- S. Pope,
*Turbulent Flows*, Cambridge University Press, Cambridge, (2000).

- Y. P. Raizer,
*Gas Discharge Physics*, Springer Verlag, Berlin, (1987).

- D. E. Rosner,
*Transport Processes in Chemically Reacting Flow Systems*, Butterworths, Boston, (1986).

- A. Ruiz, B. Cuenot, L. Selle, and T. Poinsot, The Flame Structure of a Turbulent Supercritical Hydrogen/Oxygen Flow Behind a Splitter Plate, AIAA Paper 2011-6121, San Diego, California, USA, (2011).

- J. H. Seinfeld and S. N. Pandis,
*Atmospheric Chemistry and Physics*, John Wiley and Sons, (1998).

- Y. Shizuta and S. Kawashima, Systems of Equations of Hyperbolic-Parabolic Type with Applications to the Discrete Boltzmann Equation, Hokkaido Math. J., 14, (1985), pp. 249-275.

- M. D. Smooke, Solution of Burner Stabilized Premixed Laminar Flames by Boundary Value Methods, J. Comp. Phys., 48, (1982), pp. 72-105.

- M. D. Smooke, The Computation of Laminar Flames, Proc. Comb. Inst. 34, (2013) 65-98.

- H. Van Beijeren and M. H. Ernst, The Modified Enskog Equations for Mixtures, Phys. A, 70 (1973), pp. 225-242.

- J.A. Van Oijen, F A. Lammers, and L.P.H. de Goey, Modeling of Complex Premixed Burner Systems by Using Flamelet-Generated Manifolds, Combust. Flame, 127 (2001), pp. 2124-2134.

- A.I. Vol'Pert, S.I. Hudjaev, On the Cauchy Problem for composite systems of Nonlinear Differential Equations, Math USSR Sbornik., 16, (1972), pp. 517-544.

- L. Waldmann, Transporterscheinungen in Gasen von mittlerem Druck, Handbuch der Physik, 12, (1958) 295-514.

- F.A. Williams, Elementary Derivation of the Multicomponent Diffusion Equation, Amer. J. Phys., 26, (1958a), pp. 467-469.

- F. A. Williams, Spray Combustion and Atomization, Phys. Fluids, 1, (1958) 541-545.

- F. A. Williams,
*Combustion Theory*, 2nd Ed., The Benjamin/Cummings Pub. Co. Inc., Menlo park, (1985).

- Y. B. Zeldovitch and Y. P. Raizer,
*Physics of Shock Waves and High Temperature Hydrodynamic Phenomena*, Dover, New York, (2002).

- V. M. Zhdanov,
*Transport Processes in Multicomponent Plasmas*, Taylor and Francis, London, (2002).

## Recommended Reading

- J. D. Anderson, Jr.,
*Hypersonics and High Temperature Gas Dynamics*, McGraw-Hill Book Company, New-York, (1989).

- V. Giovangigli,
*Multicomponent Flow Modeling*, Birkhäuser, Boston, (1999).

- E. A. Guggenheim,
*Thermodynamics*, North Holland, Amsterdam, (1962).

- R. J. Kee, M. E. Coltrin, and P. Glarborg,
*Chemically Reacting Flow*, Wiley Interscience, (2003).

- E. Nagnibeda and E. Kustova,
*Non-equilibrium Reacting Gas Flow*, Springer Verlag, Berlin, (2009).

- T. J. Poinsot and D. Veynante,
*Theoretical and Numerical Combustion*, 2nd Ed., R.T. Edwards, (2005).

- D. E. Rosner,
*Transport Processes in Chemically Reacting Flow Systems*, Butterworths, Boston, (1986).

- L. D. Schmidt,
*The Engineering of Chemical Reactions*, Oxford University press, Oxford, (2009).

- J. H. Seinfeld and S. N. Pandis,
*Atmospheric Chemistry and Physics*, John Wiley and Sons, (1998).

- F. A. Williams,
*Combustion Theory*, 2nd Ed., The Benjamin/Cummings Pub. Co. Inc., Menlo park, (1985).

- V. M. Zhdanov,
*Transport Processes in Multicomponent Plasmas*, Taylor and Francis, London, (2002).

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