### The conservation equations for independent coexistent continua and for multicomponent reacting gas mixtures

W. Nachbar, F. Williams, S. S. Penner
1959 Quarterly of Applied Mathematics
The equations for conservation of mass, momentum, and energy are derived for a set of independent, coexistent continua obeying the laws of dynamics and thermodynamics. The idea of a control volume and a control surface for each continuum is used in the analysis. The derived results are practically identical with relations obtained previously by Th. von Karman. A direct comparison is conducted between the continuum theory results and those obtained from kinetic theory by assuming that, for each
more » ... ing that, for each of the species, the kinetic theory definitions apply. It is found that the new terms appearing in the conservation equations derived from continuum theory are precisely those which are required to make these equations identical with the results obtained from the kinetic theory of multicomponent, reacting gas mixtures. However, the continuum theory forms of the equations are not useful because they require knowledge of the transport properties for individual species in the mixture. I. Introduction. The equations for conservation of mass, momentum, and energy for a one-component continuum are well known and are derived in standard treatises on fluid mechanics [1-3]. On the other hand, the conservation equations for reacting, multicomponent, gas mixtures are generally obtained as the equations of change for the summational invariants arising in the solution of the Boltzmann equation [4, 5] . One of several exceptions to the last statement is the analysis of von Karman [6] whose results are quoted in a recently published book [7, 8] . Since von Karman's method of analysis has not been described in detail, and since his results seem to differ from the classical relations through the occurrence of higher order terms in the diffusion velocities, it appeared worthwhile to re-examine this problem with some care. The objective of our investigation is the derivation of the conservation laws for multicomponent, reacting, gas mixtures. To this end we invent a physical model consistent with continuum theory. Our model involves the idea of a multicomponent continuum composed of coexistent continua, each obeying the laws of dynamics and thermodynamics, a notion which was first introduced by Stefan in 1871 f-For an n-component gas mixture we presume the existence of n distinct continua within any arbitrary volume, continuum K corresponding to the chemical species K. We shall use the terms continuum K, species K, and component K interchangeably, it being understood that each of these phrases refers to continuum K of the coexistent continua as long as we are following * Contract AF 18(603)-146. f We arrive at the model of simultaneous coexistent continua as the logical transcription to continuum theory of the fact that the entire volume is accessible to all of the different molecules in a gas mixture.