INTRODUCTION

WHEN A PLANET or satellite has a weak internal magnetic field or none at all, its interaction with the external plasma can be drastically different from that described in earlier chapters. Such interactions can vary widely, depending on whether or not the body has a significant atmosphere. The simplest case, which is an appropriate starting point for discussion, is that of a body similar to the earth's moon that has, at most, a negligibly thin atmosphere.

PLASMA INTERACTIONS WITH MOONLIKE BODIES

A body like our moon, composed of insulating material and submerged in a flowing plasma, simply absorbs the particles of the plasma that are incident on the body. Thus, the lunar soil contains a record of the composition and energy of the ancient solar wind. At the moon, a bow shock will not form upstream; there is virtually no obstacle as far as the oncoming flow is concerned. The magnetic field diffuses into the very weakly conducting outer layers at a rapid rate, so that it is barely perturbed from its upstream orientation. The most notable features are associated with the plasma-absorption wake left in the plasma behind the body.

Whereas the rest of the interaction is essentially the same as previously described for other bodies, regardless of the properties of the incident plasma flow, and regardless of the orientation of the ambient magnetic field, the wake structure will depend on the upstream variables. If the body's magnetic field is zero, and the flow speed is high compared with the thermal velocity, the wake will persist to large distances; but if the flow is slow relative to the thermal speed, thermal motions perpendicular to the flow direction can refill the empty space within a short distance downstream of the body. The introduction of a magnetic field can either inhibit the refilling of the wake, if the field is nearly parallel to the upstream flow, or have a minimal effect, if it is perpendicular. The cartoons in Figure 8.1 show two hypotheti- cal configurations for the lunar wake in the solar wind for these two magnetic-field orientations.

So far our discussion of the kinetic theory of gases has been limited to kinematic equilibrium properties. We shall see later that molecular collisions are responsible for establishing the equilibrium condition (see Chapter 5). In the absence of equilibrium, intermolecular interactions result in transport of macroscopic gas quantities, such as mass, momentum and energy. Under equilibrium conditions the distribution of molecular velocities is the same Maxwell–Boltzmann distribution at every configuration space location. In other words the effects of molecular collisions cancel each other (the distribution function is constant in time and configuration space) and therefore the details of individual collisions do not play a role in determining the distribution of molecular velocities.

The situation is entirely different if we allow even the slightest deviation from equilibrium. In this case molecular collisions result in the transport of macroscopic quantities (such as mass, momentum and energy) accompanied by a gradual approach to the equilibrium velocity distribution. The details of the macroscopic transport and change of the distribution function are controlled by the specific nature of the molecular collision process. Molecular collisions represent the microscopic process governing all macroscopic transport phenomena.

This chapter lays part of the necessary groundwork for the dynamical description of transport phenomena in gases by examining the details of the most fundamental physical process in the gas: molecular collisions. We begin the discussion of collisional effects by considering in detail the process of two particle (or binary) collisions.

This book provides a comprehensive introduction to the kinetic theory of gases for graduate students and interested researchers. The book is based on graduate level courses I taught in the Department of Aerospace Engineering and in the Department of Atmospheric, Oceanic and Space Sciences of the University of Michigan College of Engineering. These courses were intended to provide a broad introduction to the kinetic theory of gases.

The course on gaskinetic theory and real gas effects has been taught for a long period of time by Professors Thomas C. Adamson, Vi-Cheng Liu, Paul B. Hays, Martin Sichel and others before I was fortunate enough to teach it. I greatly benefited from their experience and lecture notes. The first part of the course (Phasespace distributions, Binary collisions and Elementary transport theory) generally follows the way this course was organized for the last decade or so. The second part is significantly revised and concentrates on the fundamentals of advanced transport theory. Although the course starts at a very elementary level, Chapters 5 and 6 are quite advanced.

This course was intended to provide a comprehensive background for students who eventually intend to carry out independent research in computational fluid dynamics, aerodynamics of high altitude flight, upper atmospheric science, planetary atmospheres and space physics. The reader is expected to have some familiarity with integral calculus, vector and tensor algebra, statistics and classical mechanics.

In its most general form the Boltzmann equation is a seven dimensional non-linear integro-differential equation (see equation (5.29) or (5.30)). The solutions of the Boltzmann equation provide a full description of the phase space distribution function at all times. In most cases, however, it is next to impossible to solve the full Boltzmann equation and one has to resort to various approximate methods to describe the spatial and temporal evolution of macroscopic quantities characterizing the gas. One successful way to find approximate solutions is the Chapman–Enskog method discussed in the previous Chapter. This method uses a power series expansion around the equilibrium distribution function (Maxwellian) to describe slightly non-equilibrium gases. The method assumes that coefficients of increasing powers of random velocity components are proportional to increasing powers of a smallness parameter, thus ensuring rapid convergence.

An alternative, and mathematically equivalent, method was developed by Grad in the late 1940s. In this method transport equations for macroscopic molecular averages are obtained by taking velocity moments of the Boltzmann equation. This seemingly straightforward technique runs into considerable difficulties because the governing equations for the components of the n-th velocity moment also depend on components of the (n+1)-th moment. In order to get a closed transport equation system one has to use closing relations (expressing a higher order velocity moment of the distribution function in terms of the components of lower moments) and thus make implicit assumptions about the distribution function.

It has been observed under certain conditions that a compressible fluid can experience an abrupt change of macroscopic parameters. Examples are detonation waves, explosions, wave systems formed at the nose of projectiles moving with supersonic speeds, etc. In all these cases the wave front is very steep and there is a large pressure rise in traversing the wave, which is called a shock wave.

In the perfect gas approximation shock waves are discontinuity surfaces separating two distinct gas states. In higher order approximations (such as the Navier–Stokes equation) the shock wave is a region where physical quantities change smoothly but rapidly. In this case the shock has a finite thickness, generally of the order of the mean free path.

Since the shock wave is a more or less instantaneous compression of the gas, it cannot be a reversible process. The energy for compressing the gas flowing through the shock wave is derived from the kinetic energy of the bulk flow upstream of the shock wave. It can be shown that a shock wave is not an isentropic phenomenon: the gas experiences an increase in its entropy.

The simplest case for studying shock waves is a normal plane shock wave, when the gas flows parallel to the x axis and all physical quantities depend only on the x coordinate. In this case the normal vector of the shock surface is parallel to the flow direction.

So far we have considered gases which were in kinematic equilibrium. It was always assumed that the molecules did not have any organized motion (bulk motion), and that the gas temperature and the number density of the molecules were constant everywhere. Under such conditions there is no net particle transport from one place to another.

In this chapter we relax our earlier assumptions and let the macroscopic quantities (gas bulk velocity, temperature, pressure, concentration, etc.) have slow spatial variations. These configuration space gradients result in macroscopic transport phenomena, such as diffusion or heat flow. Fluid dynamics describes these transport phenomena using empirical transport coefficients, such as viscosity, diffusion coefficient or heat conductivity.

In this chapter we shall use the elementary mean free path method to describe macroscopic transport phenomena and to calculate approximate values for the transport coefficients. This simple and remarkably successful method is quite general because it does not depend on the form of the distribution function and, therefore, does not assume equilibrium.

The mean free path method is based on the assumption that the gas is not in equilibrium, but the deviation from equilibrium is so small that locally the distribution of molecular velocities can be approximated by Maxwellians. However, the macroscopic parameters of the local Maxwellians slowly vary with configuration space location. Mathematically this assumption can be translated into two basic conditions.

The first condition is that the variation of all macroscopic quantities is small within a mean free path.

In free molecular flows, occurring at very low gas densities, the interaction between gas molecules is neglected and only the interaction between the gas molecules and the solid surface is taken into account. In Chapter 4 we discussed the flow of highly rarefied gases in the case of ‘internal’ flows which are typically associated with vacuum installations. There is, however, a much more interesting situation which arises with respect to high altitude flights at hypersonic velocities. This situation involves the aerodynamics of high speed rarefied gas flow past submerged bodies.

In this chapter we consider the interaction of solid bodies with rarefied gas flows with Knudsen numbers (the ratio of the molecular mean free path and the characteristic size of the body) larger than unity. This regime, which is called free molecular interaction, is quite different from normal gasdynamic flows around solid bodies and therefore very different mathematical methods must be applied.

Typical mean free path values in the terrestrial upper atmosphere are shown in Figure 7.1 (when calculating the mean free path an average cross section value of σ = 10–19 m2 was used). Inspection of Figure 7.1 reveals that above approximately 150 km the condition Kn ≫ 1 is satisfied (at this altitude λ∽100 m, which is large compared to typical space vehicles). In the h ≥ 150 km region the interaction is free molecular but the number density is still high enough to make macroscopic averages meaningful (the atmospheric density profile is shown in Figure 1.2).

In the preceding chapter a highly simplified discussion of molecular transport in non-equilibrium dilute gases was presented. Even though this discussion was highly simplified, it helped us to illuminate the basic physical processes resulting in the transport of macroscopic physical quantities.

In this chapter the theory of macroscopic transport will be treated in a more rigorous way. Our discussion will be based on the Boltzmann equation, a non-linear seven dimensional partial differential equation which describes the evolution of the phase space distribution function in non-equilibrium gases. Conservation equations for macroscopic physical quantities will be obtained in the next chapter as the velocity moments of the Boltzmann equation.

In this chapter our discussions will be based on the following fundamental assumptions:

1. (i) We assume that the gas density is low enough to ensure that the mean free path is large compared to the effective range of the intermolecular forces. This assumption makes it possible to apply the principle of molecular chaos in the treatment of binary encounters. Molecular chaos means that the velocities of the colliding particles are uncorrelated, i.e. particles which have already collided with each other will have many encounters with other molecules before they meet again. A direct consequence of this assumption is the time irreversibility of the Boltzmann equation.

2. (ii) The mean free path is short compared to the dimensions of the problem. This assumption eliminates consideration of wall effects and other edge-related problems.

3. […]

The objective of gaskinetic theory is to explain and predict the macroscopic properties of gases from the properties of their microscopic constituents. Macroscopic properties include the equation of state, specific heats, transport coefficients (such as viscosity, diffusion coefficient, thermal conductivity), etc. The fundamental hypothesis of kinetic theory is that solids, liquids and gases are composed of finite particles (molecules) which are in various states of motion. The term molecule is taken generally to mean a regular arrangement of atoms, or it may refer to a free, atom, an electron or an ion.

Statistical mechanics, like kinetic theory, also aims to explain and predict the macroscopic properties of matter in terms of the microscopic particles of which it is composed. The main difference between gaskinetic theory and statistical mechanics is the manner in which particle interactions are treated. In gaskinetic theory the details of the molecular interactions are taken into account. Statistical mechanics avoids these complicated and sometimes poorly known details and replaces them with certain powerful and general statistical ideas.

Brief history of gaskinetic theory

The history of the kinetic theory of gases can be traced back to ancient Greece, where around 400 B.C. Democritus hypothesized that matter is composed of small indivisible particles which he called atoms. The atoms of different materials were assumed to have different sizes and shapes, so that atoms represent the smallest quantity of a substance that retains its properties.