Laminar dispersed two-phase flows at low concentration - I. Generalized system of equations

Abstract

To represent multidimensional flows of particle-fluid mixtures, Eulerian two-fluid models are currently used nowadays. Even if there are no pure turbulence effects in the carrier phase flow, many closure laws are required in order to supplement the conservation equations. To furnish a systematic method for deriving such closure laws that are valid at least for spherical solid inclusions, a generalised system of equations is proposed in this first paper (hereafter referred to as part I) of a sequence. It is based on the coupling of two sets of equations, one for each phase: the continuous phase is represented by an extension of a hierarchy of equations proposed by Lundgren for treating flows in porous media, and the dispersed phase by an adaptation of the well-known B.B.G.K.Y. hierarchy. The first-order equations of both hierarchies correspond to the conservation equations of standard two-fluid models; they contain the usual unknown terms. In our approach these terms appear to be provided by the second-order equations. Unfortunately, as is usual in other similar statistical theories, the second-order equations contain extra unknown terms which figure in the third-order equations and so on. Formulating closure equations is replaced by the broader problem of truncating the generalised system of equations via a perturbation method based on diluteness.