For reactive balance laws derived by kinetic theory for a binary mixture of atoms and diatomic molecules undergoing the exothermic recombination A+A→A2∗→A2A+A\to A_2^*\to A_2, steady one-dimensional combustion waves are classified into parameter regimes by the unburned squared Mach number M02M_0^2 and the dimensionless bond energy E0E_0. Smooth weak deflagration profiles have been computed in the regimes where the burned equilibrium lies below the critical velocity at which the profile system degenerates, but in the two regimes adjacent to that threshold — regime 1(c) (E0>1/4E_0>1/4, M2−<M02≤Meq−M_2^-<M_0^2\le M_{eq}^-) and regime 2(c) (E0≤1/4E_0\le 1/4, M2−<M02≤3/5M_2^-<M_0^2\le 3/5) — the burned equilibrium lies beyond the degeneracy, the steady deflagration is expected to be discontinuous, and the authors report that numerical attempts "have not produced yet physically consistent combustion solutions". The problem is to decide, for these two regime families, whether physically admissible steady deflagration solutions exist: construct them and characterize their structure — smooth reaction zone, terminating discontinuity with its Rankine–Hugoniot and entropy conditions, and selection of the flame eigenvalue — or prove that no physically consistent solution exists, so that no steady deflagration propagates in those parameter ranges.

Background

A deflagration is a combustion wave that advances subsonically into the unburned gas, whereas a detonation is a supersonic combustion wave led by a shock. The model studied here comes from a kinetic theory of a binary mixture of monatomic atoms AA and diatomic molecules A2A_2 undergoing the exothermic two-step recombination A+A→A2∗→A2A+A\to A_2^*\to A_2 through a short-lived excited state (Groppi–Rossani–Spiga, J. Phys. A 33:8819–8833, 2000). In the collision-dominated, slow-reaction regime, a Chapman–Enskog expansion (the standard procedure deriving continuum transport equations from the Boltzmann equation) yields reactive fluid-dynamic balance laws with Fick- and Fourier-type mass-diffusion and heat-conduction fluxes; viscous stress is neglected, so the steady wave problem reduces to three ordinary differential equations (Conforto–Groppi–Monaco–Spiga, Kinet. Relat. Models 4:259–276, 2011). In the frame moving with a steady planar wave, the balance laws for the atomic concentration cc (c=1c=1 unburned, c=0c=0 fully recombined), the scaled gas velocity uu (which equals the specific volume at unit mass flux), and the diffusion flux JJ become the system of Eqs. (13) of the source paper:

dcdx=−L2(2−c)JT,dudx=1(2−c)T  N(c,u,J)1+53M02−103M02u,dJdx=LJ(2−c)T−4μπc2(2−c)2u2 Γ ⁣(32,TactT),\begin{aligned} \frac{dc}{dx} &= -\frac{L}{2}(2-c)\frac{J}{\sqrt{T}},\\ \frac{du}{dx} &= \frac{1}{(2-c)\sqrt{T}}\;\frac{N(c,u,J)}{1+\frac{5}{3}M_0^2-\frac{10}{3}M_0^2u},\\ \frac{dJ}{dx} &= \frac{LJ}{(2-c)\sqrt{T}}-\frac{4\mu}{\sqrt{\pi}}\frac{c^2}{(2-c)^2u^2}\,\Gamma\!\left(\frac{3}{2},\frac{T_{act}}{T}\right), \end{aligned}

with temperature T=(2−c)[(1+53M02)u−53M02u2]T=(2-c)\left[\left(1+\frac{5}{3}M_0^2\right)u-\frac{5}{3}M_0^2u^2\right] and number density n=1(2−c)un=\frac{1}{(2-c)u} (the algebraic relations following from the conservation laws in the nondimensional variables of the unit mass-flux normalization; the pressure is p=nTp=nT, the perfect-gas law in nondimensional form), numerator

N=−E01−c2−c+J2[E0−L(2−c)(1+53M02−53M02u)u]+56M02(u2−1)+52(u−1)(1−53M02u),N=-E_0\frac{1-c}{2-c}+\frac{J}{2}\left[E_0-L(2-c)\left(1+\frac{5}{3}M_0^2-\frac{5}{3}M_0^2u\right)u\right]+\frac{5}{6}M_0^2(u^2-1)+\frac{5}{2}(u-1)\left(1-\frac{5}{3}M_0^2u\right),

where M0M_0 is the Mach number (ratio of flow speed to sound speed) of the unburned gas, E0E_0 the dimensionless chemical bond energy released by recombination, TactT_{act} the activation temperature of the reaction, LL the Lewis number (ratio of thermal diffusivity to mass diffusivity), Γ(3/2,⋅)\Gamma(3/2,\cdot) the complementary incomplete gamma function entering the transition-state reaction rate, and μ>0\mu>0 a dimensionless flame eigenvalue (combining heat-conductivity and reaction-rate constants) that is not known in advance: it must be selected so that a trajectory of the system connects the metastable, slowly reacting unburned equilibrium (c,u,J)=(1,1,0)(c,u,J)=(1,1,0) as x→−∞x\to-\infty to the burned equilibrium (0,ueq−,0)(0,u_{eq}^-,0) as x→+∞x\to+\infty — the classical eigenvalue-flame problem, solved numerically by shooting along the unstable manifold of the burned state.

The burned equilibrium velocities follow algebraically from the conservation laws (the Rankine–Hugoniot jump conditions — the algebraic relations expressing conservation of mass, momentum and energy across a stationary discontinuity — applied to the steady balance laws):

ueq±=38M02[1+53M02∓(M02−1)2−1615E0M02],u_{eq}^{\pm}=\frac{3}{8M_0^2}\left[1+\frac{5}{3}M_0^2\mp\sqrt{(M_0^2-1)^2-\frac{16}{15}E_0M_0^2}\right],

giving the weak (−-) and strong (++) deflagration branches; the strong branch is excluded as physically inadmissible by the standard argument that a subsonic-to-supersonic transition cannot occur through an exothermic wave in this setting. Weak deflagrations exist only up to the Chapman–Jouguet-type bound

Meq−=1+815E0−64225E02+1615E0M_{eq}^{-}=1+\frac{8}{15}E_0-\sqrt{\frac{64}{225}E_0^2+\frac{16}{15}E_0}

(the value of M02M_0^2 at which the burned state becomes sonic). The (E0,M02)(E_0,M_0^2) parameter half-plane is then partitioned into regimes by the additional thresholds M1=35(3+E0)M_1=\frac{3}{5(3+E_0)} and M2−=35(2+E0−1+4E0+E02)M_2^{-}=\frac{3}{5}\left(2+E_0-\sqrt{1+4E_0+E_0^2}\right): deflagration families 1(a–c) for E0>1/4E_0>1/4 with M02M_0^2 in ]0,M1]]0,M_1], ]M1,M2−]]M_1,M_2^-], ]M2−,Meq−]]M_2^-,M_{eq}^-], families 2(a–d) for E0≤1/4E_0\le 1/4 with M02M_0^2 in ]0,M1]]0,M_1], ]M1,M2−]]M_1,M_2^-], ]M2−,3/5]]M_2^-,3/5], ]3/5,Meq−]]3/5,M_{eq}^-], and detonation family 3(a–b) for M02≥Meq+M_0^2\ge M_{eq}^+ (Conforto–Groppi–Monaco–Spiga, Acta Appl. Math. 122:127–140, 2012).

The decisive structural fact is the critical velocity ucrit=12(1+35M02)u_{crit}=\frac{1}{2}\left(1+\frac{3}{5M_0^2}\right) at which the coefficient 1+53M02−103M02u1+\frac{5}{3}M_0^2-\frac{10}{3}M_0^2u of du/dxdu/dx vanishes and the type of the burned equilibrium changes (the equilibrium eigenvalues change sign there). The burned equilibrium ueq−u_{eq}^- crosses ucritu_{crit} exactly at M02=M2−M_0^2=M_2^-. Hence in regime 1(c) (E0>1/4E_0>1/4, M2−<M02≤Meq−M_2^-<M_0^2\le M_{eq}^-) and regime 2(c) (E0≤1/4E_0\le 1/4, M2−<M02≤3/5M_2^-<M_0^2\le 3/5) the burned equilibrium lies beyond this turning point, the smooth connecting profile is obstructed, and the weak deflagration is expected to be a discontinuous wave — a smooth reaction zone possibly terminated by a jump. The 2012 classification paper computed only smooth profiles (a regime-1(b) deflagration with μ≈6208\mu\approx 6208 and a smooth weak detonation) and explicitly deferred the treatment of profiles with jump discontinuities; the source paper (Conforto–Monaco–Ricciardello, Continuum Mech. Thermodyn. 26:503–519, 2014) carried out the systematic parameter study of amplitudes, monotonicity trends, fluxes and Hugoniot geometry (the locus of admissible burned states in the pressure–specific-volume plane), but for regimes 1(c) and 2(c) reports that its numerical attempts "have not produced yet physically consistent combustion solutions", so the structure, admissibility and physical realization of these discontinuous steady deflagration families remain open.

Problem Statement

For the steady one-dimensional deflagration problem of the recombination model above (the pinned ODE system with boundary conditions (c,u,J)→(1,1,0)(c,u,J)\to(1,1,0) as x→−∞x\to-\infty and (c,u,J)→(0,ueq−,0)(c,u,J)\to(0,u_{eq}^-,0) as x→+∞x\to+\infty, flame eigenvalue μ>0\mu>0 to be selected), decide whether physically consistent steady deflagration solutions exist in the two parameter regimes

regime 1(c):E0>14,M2−<M02≤Meq−,\text{regime 1(c):}\quad E_0>\tfrac{1}{4},\quad M_2^-<M_0^2\le M_{eq}^-, regime 2(c):E0≤14,M2−<M02≤35,\text{regime 2(c):}\quad E_0\le\tfrac{1}{4},\quad M_2^-<M_0^2\le\tfrac{3}{5},

where M2−=35(2+E0−1+4E0+E02)M_2^-=\frac{3}{5}\left(2+E_0-\sqrt{1+4E_0+E_0^2}\right) and Meq−=1+815E0−64225E02+1615E0M_{eq}^-=1+\frac{8}{15}E_0-\sqrt{\frac{64}{225}E_0^2+\frac{16}{15}E_0} — the regimes in which the burned equilibrium ueq−u_{eq}^- lies beyond the critical velocity ucrit=12(1+35M02)u_{crit}=\frac{1}{2}\left(1+\frac{3}{5M_0^2}\right) where the profile system degenerates, so that the connecting steady deflagration is expected to be discontinuous, and in which the source paper's numerical attempts have not produced physically consistent combustion solutions.

For these two regime families, determine whether physically consistent steady deflagration solutions exist in the stated parameter ranges. If such solutions exist, characterize their structure: whether each solution consists of a smooth reaction zone terminated by a discontinuity satisfying the Rankine–Hugoniot jump conditions of the steady reactive balance laws together with an entropy condition, the position and type of that discontinuity, and the selection rule for the flame eigenvalue μ\mu. If no admissible profile exists in these regimes — that is, every connecting profile violates positivity of temperature and density, the jump conditions, or the entropy condition — then no steady deflagration propagates there. Physical consistency means: T>0T>0 and number density n=1(2−c)u>0n=\frac{1}{(2-c)u}>0 along the smooth parts; connection to the weak-branch burned equilibrium ueq−u_{eq}^- (strong-deflagration connections to ueq+u_{eq}^+ are excluded as physically inadmissible); and any discontinuity satisfies the reactive Rankine–Hugoniot jump conditions and an entropy inequality. The two regime families are both part of the problem and may be treated separately.

References

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