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Consider the steady-state flow of an infinite stream of a rarefied gas over a body having a characteristic dimension, R, in the absence of external forces. In this case, the Boltzmann equation may be expressed in the form: 6-1$$ v \cdot \frac{{\partial f}} {{\partial \tilde r}} = \iiint {\left( {f'f'_1 - ff_1 } \right)}gbdbd\varepsilon dv_1 . $$ [ABSTRACT FROM AUTHOR]

The non-uniform state of a binary gas mixture is described by the distribution functions: 12-1$$ f_i = f_i^{(0)} (\tilde r,t)\left( {1 + \Psi _i^{(1)} (C_i ,\tilde r,t)} \right);(i = 1,2), $$ where Ci=(mi/2kT)1/2Vi and $$ f_i^{(0)} (\tilde r,t) $$ is given by: 12-2$$ f_i^{(0)} (\tilde r,t) = n_i (\tilde r,t)\left( {\frac{{m_i }} {{2\pi kT(\tilde r,t)}}} \right)^{3/2} \exp \left( { - \frac{{m_i (v_i - u(\tilde r,t))^2 }} {{2kT(\tilde r,t)}}} \right). $$ [ABSTRACT FROM AUTHOR]

The most advanced analysis possible for boundary value transport problems involving arbitrary Knudsen number and for various molecular interaction laws must be founded on the solution of the Boltzmann equation. In the transition regime, the effects of intermolecular collisions become dominant and their influence on the gas behavior is accurately described only by the full Boltzmann form of the collision operator found in the Boltzmann equation. For this reason, different model forms of the collision integral and model equations will not be considered here. [ABSTRACT FROM AUTHOR]

In this chapter, the description of a particle in a rarefied gas is considered under conditions where the Knudsen number is small. The analysis is restricted to the usual conditions assumed for aerosol particle motion in non-uniform gases. These conditions will be discussed later in detail. The classical sphere drag and thermal force problems are solved as important practical applications of the theory and techniques described here. [ABSTRACT FROM AUTHOR]

In the previous chapter the basic analytical methods for the planar boundary transport problems were considered. All of these schemes are considered to be moment methods. It is very important to note that the simple analysis of some general properties of the Boltzmann equation related to the conservation of moments results in sufficiently accurate expressions for the velocity-slip and temperature-jump coefficients. [ABSTRACT FROM AUTHOR]

Consider the unsteady, non-uniform state of an infinite gas. For this general case, the Boltzmann equation has the form [1]: 4-1$$ Df = - J(ff_1 ), $$, where: $$ Df = \frac{{\partial f}} {{\partial t}} + v \cdot \frac{{\partial f}} {{\partial \tilde r}} + F \cdot \frac{{\partial f}} {{\partial v}}, $$, and: $$ J\left( {ff_1 } \right) = \iiint {\left( {ff_1 - f'f'_1 } \right)}gbdbd\varepsilon dv_1 . $$ [ABSTRACT FROM AUTHOR]

For the last thirty years, the most progress in investigating transport problems has been achieved for one-dimensional, stationary transport problems. In this context, several analytical methods have been developed which have been applied to the classical slip problems. This has resulted in several analytical expressions for the slip coefficients. The recent development of direct numerical solutions for these same problems allows one to estimate the accuracies of the various analytical methods. [ABSTRACT FROM AUTHOR]

The theoretical treatment of gas flow problems for extremely high Knudsen numbers (the free-molecular regime) is very different from that typically encountered in conjunction with the hydrodynamics of media having the higher number densities most people are used to working with. The theoretical frameworks encountered at these extremes are connected via various fundamental relationships between the Kinetic Theory of Gases and the mechanics of media that exhibit continuum type behaviors (the continuum regime). The continuum equations may be obtained from the Boltzmann equation if one uses the summational invariants of encounters as molecular properties with which to construct a system of moment equations. This moment system, however, is indeterminate unless one knows the relationships between the basic hydrodynamic values and the stress tensor or the thermal flux vector. These relationships may be derived by using the Chapman-Enskog method of solution of the Boltzmann equation. It is very important in the continuum regime that such gas characteristics as the viscosity and thermal conductivity coefficients be determined independently of the boundary conditions. [ABSTRACT FROM AUTHOR]

One of the most important concepts relating to the statistics of molecular collisions is that of the scattering cross section which is the basic quantity characteristic of molecular encounters. Consider a uniform flux of molecules that have a constant velocity, g, towards a spherically symmetric force center, O. The parameters of importance to the scattering events are shown in Fig. 3-1. Let Ng be the g -component of the uniform molecular number flux and dΣ = sin(θ)dθdε be the solid angle in which molecules are deflected after encounters. The differential scattering cross section, α(θ,g), is defined by [1]: 3-1$$ \alpha (\theta ,g)d\Omega = \frac{{dN}} {{N_g }}, $$ where dN is the number of molecules deflected per unit time by the central force field in dΣ. Since the incident molecules scattered into dΣ must primarily pass through that portion of the annular differential element represented by dS = bdbdε, dN may be expressed in the form: 3-2$$ dN = N_g bdbd\varepsilon . $$ [ABSTRACT FROM AUTHOR]

Consider a simple gas whose molecules possess only energy of translation, and are subject to no external forces. Let the state of the gas be uniform so that the distribution function, f, is independent of r. For this case, the Boltzmann equation may be expressed in the following simple form: 4-1$$ \frac{{\partial f}} {{\partial t}} = \iint {(f'f'_1 - ff_1 )}g\alpha (\theta ,g)d\Omega dv_1 . $$ [ABSTRACT FROM AUTHOR]

It has been established previously that knowledge of the distribution function gives all the necessary information for a gas. To obtain the basic equation for the distribution function, consider a balance of the number of molecules that are located in the element, dvdr, of the six-dimensional phase space for the time interval, dt. Consider a gas in which each molecule is subject to an external force, mF, that is a function of r and t, but not a function of v. For the time interval, dt, the position of this phase element will change from (v,r) to (v + Fdt,r + vdt). It is important to note that the magnitude of this volume element is unaltered in accordance with the Liouville theorem of classical mechanics [1]. [ABSTRACT FROM AUTHOR]

Systems consisting of what are usually called rarefied gases will be considered in this book. Such systems contain large numbers of molecules; specifically, about 2.7 x1019 cn3 for a gas at standard temperature and pressure (STP) defined to be 0 °C (273.15 K) and 1.0 arm (760 torr). How may the behavior of such huge collections of particles be described? If one were to try to apply the methods of classical mechanics, one would have to construct and solve a system of equations of motion containing as many equations and initial conditions as there are numbers of interacting particles. Obviously, such a problem could not be solved today even with the help of the most advanced computers. On this basis, one might naively suspect that the greater the number of particles, the more difficult the problem. This is not strictly true, however, as it turns out that useful results can be obtained by applying statistical descriptions to such systems. [ABSTRACT FROM AUTHOR]