Conservation equations continuum derivation

This chapter is organized into two main parts. To give the reader an appreciation of real fluids, and the kinds of behaviors that it is hoped can be captured by CA models, the first part provides a mostly physical discussion of continuum fluid dynamics. The basic equations of fluid dynamics, the so-called Navier-Stokes equations, are derived, the Reynolds Number is defined and the different routes to turbulence are described. Part I also includes an important discussion of the role that conservation laws play in the kinetic theory approach to fluid dynamics, a role that will be exploited by the CA models introduced in Part II. 

Euler-Euler models assume interpenetrating continua to derive averaged continuum equations for both phases. The probability that a phase exists at a certain position at a certain time is given by a phase indicator function, which, for steady-state processes, is equivalent to the volume of fraction of the correspondent phase (volume-of-fluid technique). The phase-averaging process introduces further unknowns into the basic conservation equations their description requires empirical and problem-dependent input (94). In principal, Euler-Euler models are applicable to all multiphase flows. Advantages and disadvantages of both methods are compared, e.g., in Refs. 95 and 96. 

All conservation equations in continuum mechanics can be derived from the general transport theorem. Define a variable F(t) as a volume integral over an arbitrary volume v(t) in an r-space... 

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 [l]-[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 (see Appendix D and [4] and [5]), One of several exceptions to the last statement is the analysis of von Karman [6], whose results are quoted in [7] and are extended in a more recent publication [8] to a point where the equivalence of the continuum-theory and kinetic-theory results becomes apparent [9]. This appendix is based on material in [8]. 

The conservation equations for continuous flow of species K will be derived by using the idea of a control volume r t) enclosed by its control surface o t) and lying wholly within a region occupied by the continuum here t denotes the time. In this appendix only, the notation of Cartesian tensors will be used. Let i = 1, 2, 3) denote the Cartesian coordinates of a point in space. In Cartesian tensor notation, the divergence theorem for any scalar function belonging to the Kth continuum a (x, t), becomes... 

Derivations of conservation equations from the viewpoint of kinetic theory usually do not exhibit explicitly the diffusion terms, such as diffusion stresses, that appear on the right-hand sides of equations (49), (50), and (51), since it is unnecessary to introduce quantities such as afj specifically in these derivations. Kinetic-theory developments work directly with the left-hand sides of equations (49), (50), and (51). Transport coefficients (Appendix E) are defined only in terms of these kinetic-theory quantities because prescriptions for calculating the individual continua transports, afj and qf, are unduly complex. Moreover, measurement of diffusion stresses is feasible only by direct measurement of diffusion velocities, followed by use of equation (24). Therefore, it has not been fruitful to study the diffusion terms which, in a sense, may be viewed as artifacts of the continuum approach. 

The transport theorem is employed deriving the conservation equations in continuum mechanics. 

The simulations of the gas-liquid flow are based on the Eulerian two fluid model originally derived by Ishii [21]. In this approach, each phase is treated as a continuum. After averaging the general transport equations, we get the following set of multi-phase conservation equations [19,22] ... 

The Reynolds transport theorem is a convenient device to derive conservation equations in continuum mechanics. Toward derivation of the general population balance equation, we envisage the application of this theorem to the deforming particle space continuum defined in the previous section. We assume that particles are embedded on this continuum at every point such that the distribution of particles is described by the continuous density function / (x, r, t). Let i//(x, r) be an extensive property associated with a single particle located at (x, r). 

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