On the significance of finite propagation speeds in multicomponent reacting systems

A continuum model of diffusion in a reacting mixture is built around a constitutive relation for momentum exchange by frictional interaction between diffusing species. The resulting continuity and momentum equations incorporate a finite relaxation time for diffusion. Under appropriate conditions, linearization of these equations produces a coupled system of hyperbolic equations for small disturbances of a stationary state. Fick's law can be recovered from the linear equations by assuming instantaneous relaxation of the flux, provided that the stationary state is uniform. Fick's law is generally inconsistent with the momentum equation when the stationary state is nonuniform. The stability of uniform stationary solutions predicted by the parabolic system obtained when Fick's law is used is compared with the stability predicted by the hyperbolic system. When the former predicts stability and has at least one pair of complex-conjugate roots, the latter may predict that the stationary state is unstable. Thus, inclusion of relaxation in the model can lead to qualitatively different predictions of stability.