Two-Fluid Hydrodynamic Equations for a Neutral, Disparate-Mass, Binary Mixture

Abstract

In gas mixtures with roughly equal masses, self-collisions and cross-collisions between species are of equal importance in the approach of the mixture to a Maxwellian distribution about a single temperature and velocity and the usual Chapman-Enskog transport theory obtains. In a disparate-mass mixture, however, the effects of self- and cross-collisions are not equal and one has an epochal relaxation, with the temperature relaxation occurring on the longest time scale. The clear distinction between a single molecular relaxation time and a hydrodynamic time scale which makes the usual Chapman-Enskog theory possible is absent for the disparate-mass case, and a more complex ordering of time scales is necessary to obtain hydrodynamic solutions. In the present work, for neutral, disparate-mass mixtures, the Boltzmann collision integrals are replaced by relaxation-type kinetic models and terms in the model equation are ordered according to mass ratio. A set of two-fluid transport equations is then derived by obtaining normal solutions of these model equations. The derived transport equations and coefficients are discussed, and their correspondence to the usual Chapman-Enskog results is shown.

The effect of noncondensables on buoyancy–thermocapillary convection of volatile fluids in confined geometries

“…Most importantly, the Navier-Stokes equation in the dilute approximation is written for the dominant component, rather than for the mixture, and includes an additional term for the cross-collision momentum transport. However, for the problem considered here the cross-collision frequency characterized by the dimensionless parameter Cr is high (Cr ) 1 for c a ) 10 À9 ), and in this limit [26] Hamel's generalized model effectively reduces to the Chapman-Enskog description.…”

“…(1), (6) and (10) in the gas phase essentially represent the leading order of the Chapman-Enskog expansion [25], which is valid when the temperatures of the two components are the same. As argued by Hamel [26], when the masses of the two components are substantially different (for instance, for hexamethyldisiloxane M v % 162 g/mol À1 , while for air M a % 29 g/mol À1 ), a more accurate description would require the introduction of two different temperatures T a -T v and some modifications to all the transport equations. Most importantly, the Navier-Stokes equation in the dilute approximation is written for the dominant component, rather than for the mixture, and includes an additional term for the cross-collision momentum transport.…”

A consistent lattice Boltzmann equation with baroclinic coupling for mixtures

“…As pointed out by Gross and Krook [43], and similarly by Hamel [47,49], one can also linearly combine these two expansions with an adjustable parameter 0 6 b 6 1, that is, a portion of f rð0Þ ; bf rð0Þ , is expressed in terms of f r1ð0Þ , and a portion of f r1ð0Þ ; ð1 À bÞf r1ð0Þ , is expressed in terms f rð0Þ :…”

“…There is a significant amount of literature on gas mixtures within the framework of kinetic theory [37,[41][42][43][44][45][46][47][48][49][50][51][52]. In the Chapman-Enskog analysis for a simple gas, one assumes a clear separation of scales in space and time, that is, to distinguish the spatial and temporal scales which are much larger than the mean free path or mean free time, respectively.…”

Numerical Simulations of Gaseous Mixture Flow in Porous Electrodes for PEM Fuel Cells by the Lattice Boltzmann Method

“…Essentially the key idea is to substitute the previous collisional terms with simplified ones, which are selected with a BGK-like structure. The model obtained is due to Hamel [36][37][38]. In the following, only the equation for a generic species b , a = σ will be considered.…”