The Governing Equations of Gas Dynamics

In this section the conservation equations of gas dynamics are derived from the Boltzmann equation. 

The equation of change, (2.215), becomes particularly simple if refers to a quantity which is conserved in collisions between molecules in accordance with (2.214). Therefore, for conservative quantities (2.215) reduces to  

By letting in (2.216) be m, me, and mc, respectively, the three fundamental conservation equations that are all satisfied by the gas are established. 

By averaging the molecular velocity c yields v = c)m which is the mean velocity of the gas (2.64) as defined in Sect. 2.3.2. Furthermore, the mass density of the gas. 

By inserting (2.220), the mean velocity (2.64), and the mass density (2.58), the momentum equation (2.219) yields  

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There are two levels, discrete particle level and continuum level, for describing and modeling of the macroscopic behaviors of dilute and condensed matters. The physics laws concerning the conservation of mass, momentum, and energy in motion, are common to both levels. For simple dilute gases, the Boltzmann equation, as shown below, provides the governing equation of gas dynamics on the discrete particle level... 

The resulting set of conservation equations derived by kinetic theory corresponds to the governing equations of gas dynamics (sometimes referred to as the Navier-Stokes equations). 

The one-dimensional theory for steady incompressible fluid flow in collapsible tubes (when P — P < 0) was outlined by Shapiro (1977) in a format analogous to that for gas dynamics. The governing equations for the fluid are that for conservation of mass,... 

The equations required to describe the motions of a planetary atmosphere include Newton s second law, the mass continuity equation, the first law of thermodynamics, and an equation of state for the atmospheric gas. These relations are briefly reviewed from a general point of view. More detailed discussions of the governing equations and their applications can be found in texts on dynamical meteorology and geophysical fluid dynamics, such as Pedloskey (1979), Haltiner Williams (1980), Holton (1992), and Salby (1996). 

The governing equations to model the particle formation dynamics are identical for the gas and Uquid phase, whereby of course, the reaction mechanisms and the kinetic parameters (e.g., for the mass transfer) differ. The population balance equation (PBE) can be considered as the master equation for formation of particles by both top-dovm and bottom-up methods (Eq. 5) ... 

The effect of the finite thickness of the turbulent flame was introduced in theories [18, 19]. In [18], the set of governing gas dynamic equations is supplemented by the empirical dependency of flame thickness on the distance to the ignition source, obtained by means of statistical processing of the instantaneous position of a thin wrinkled flame front in model experiments. Semi-empirical the-... 

To properly describe chemical vapor deposition, one must develop a system of equations that encompasses all phenomena involved. This includes a proper representation of reactions in the gas phase, a suitable description of the surface kinetics, and the gas dynamics of a reacting gas mixture. Because the full governing equations are extremely complex and difficult to solve, most authors have examined only limited regimes. For example, we can ignore the gas dynamics... 

Consider the dynamics of a catalytic fluidized bed in which an irreversible gas phase reaction takes place (Aiken and Lapidus, 1974 [17] Cutlip and Shacham, 1999). [9] Partial pressure of reactant in fluid (P), temperature of reactant in fluid (T), partial pressure of reactant at the catalyst surface (Pp) and partial pressure of reacfanf at the catalyst surface (Tp) are governed by the following equations ... 

While electrical breakdown constraints set the lower limit for IMS gas pressure, there also is the upper limit. At some point, the density of gas molecules makes their colhsions with an ion a many-body rather than binary interaction. Eventually, the dynamics becomes governed by laws of viscous fiiction appropriate for hquids. In that regime, the terminal velocity of ions is stiU proportional to E at low E, and mobihty is defined by Equation 1.8. However, formahsms such as Equation 1.10 that relate K to ion structure cease to apply, and becomes independent of gas pressure. ... 

In order to simulate fluid flow, heat transfer, and other related physical phenomena over various length scales, it is necessary to describe the associated physics in mathematical terms. Nearly all the physical phenomena of interest to the fluid dynamics research community are governed by the principles of continuum conservation and are expressed in terms of first- or second-order partial differential equations that mathematically represent these principles (within the restrictions of a continuum-based firamework). However, in case the requirements of continuum hypothesis are violated altogether for certain physical problems (for instance, in case of high Knudsen number rarefied gas flows), alternative formulations in terms of the particle-based statistical tools or the atomistic simulation techniques need to be resorted to. In this entry, we shall only focus our attention to situations in which the governing differential equations physically originate out of continuum conservation requirements and can be expressed in the form of a general differential equation that incorporates the unsteady term, the advection term, the diffusion term, and the source term to be elucidated as follows. 

Another approach for analyzing the stability of the flow is based on wave-theory. In deriving the characteristics of kinematic and dynamic waves in two-component flow, Wallis has shown that the relations between the velocities of these two classes of waves govern the stability of the two stratified layers [74]. It has been shown that the condition of equal kinematic and dynamic waves velocities corresponds to marginal stability. Following this approach, Wu et al. determined the stratified/ nonstratified transition in horizontal gas-liquid flows [38]. The relations between the dispersion equation. Equation 16, and stability criteria Equation 33 on one hand, and the characteristics of kinematic and dynamic waves on the other hand, (for = 0), was shown in Brauner and Moalem Maron [45]and Crowley et al. [47]. 

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