Relativistic Hydrodynamics 2026, 09: Reaction Fronts

Reaction fronts are discontinuities in the flow described by moving surfaces where a suitable fluid mixture undergoes a chemical transformation (releasing energy & heat). The fluid behind the reaction front can be either compressed & accelerated, or decompressed and decelerated. Reaction fronts are very similar to shock fronts and can be described using the same theory, though the fluids on either side of the front are of course chemically and physically different.

The Adiabat in the (p, h/ρ) plane is the "reaction adiabat" (as opposed to the analogue Taub adiabat in shock-fronts) and represents only the state downstream while the upstream is described through a Taub adiabat. The properties on either side are described through the compression coefficient x, which comes out to a ratio of densities in front and behind the reaction front. The relation between the pressures (and hence velocities) define the type of reaction front. If the pressure is greater before the the front than behind it, it's considered a "detonation", otherwise it's a "deflagration". Detonations can be distinguished according to the magnitude of the fluid velocities on either side of the front relative to the local sound speed. They are either weak, strong, or Chapman-Jouguet. The detonation characteristics can be deduced by counting the number of unknowns typical of the hydrodynamic solution of a detonation wave which are the enthalpy and velocity on both sides of the reaction front, along with the detonation velocity.

Deflagration fronts move subsonically relative to their medium, the front perturbs the unreacted fluid with a precompression wave travelling ahead of the deflagration front. The leading edge of the precompression wave is either a weak discontinuity at local sound speed if the front is subsonic, or a front moving supersonically if it's supersonic. It should be emphasised that the fluid undergoes a chemical or physical transformation when swept by the deflagration front, with no chemical transformations in the precompression front. Deflagrations are either weak, strong or Chapman-Jouguet depending on whether the fluid velocity behind the front is supersonic, subsonic or sonic. The latter case sees the fastest deflagrations, producing a reacted fluid with maximum entropy. Subsonic deflagrations have 10 unknowns, supersonic ones have 12 unknowns.

Reaction fronts with under-determinacy 0 are "evolutionary" and are linearly stable. Those with non-zero under-determinacy can be made evolutionary, through additional boundary conditions, usually prescribing the front velocity as a function of TD-variables on either side.

Perturbations that can leave the reaction fronts are given by the condition

which can only be satisfied for i = a, when the medium ahead of the front is supersonic (qualifying a detonations). The ± cases distinct between deflagrations and Chapman-Jouget detonations (strong)/deflagrations (weak). Through linearized hydrodynamic equations, can be found that no acoustic signals/perturbation can be transmitted upstream of the flow. For Chapman-Jouguet detonation fronts, the downstream fluid variables are always 0 and independent of perturbation, which makes this fronts stable against linear perturbations. Similar for strong detonations, which are indistinguishable from Chapman-Jouguet at first order. Weak detonations need to be analyzed through different, case-by-case methods. Weak and Chapman-Jouguet deflagrations are stable to linear perturbations and corrugations, if a boundary condition is specified for the velocity of the reaction front.