The Boltzmann Equation and Fluid Dynamics

C. Cercignani    Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

1 Introduction

We say that a gas flow is rarefied when the so-called mean free-path of the gas molecules, i.e., the average distance covered by a molecule between to subsequent collisions, is not completely negligible with respect to a typical geometric length (the radius of curvature of the nose of a flying vehicle, the radius of a pipe, etc.). The most remarkable feature of rarefied flows is that the Navier–Stokes equations do not apply. One must then resort to the concepts of kinetic theory of gases and the Navier–Stokes equations must be replaced by the Boltzmann equation [43].

Thus the Boltzmann equation became a practical tool for the aerospace engineers, when they started to remark that flight in the upper atmosphere must face the problem of a decrease in the ambient density with increasing height. This density reduction would alleviate the aerodynamic forces and heat fluxes that a flying vehicle would have to withstand. However, for virtually all missions, the increase of altitude is accompanied by an increase in speed; thus it is not uncommon for spacecraft to experience its peak heating at considerable altitudes, such as, e.g., 70 km. When the density of a gas decreases, there is, of course, a reduction of the number of molecules in a given volume and, what is more important, an increase in the distance between two subsequent collisions of a given molecule, till one may well question the validity of the Euler and Navier–Stokes equations, which are usually introduced on the basis of a continuum model which does not take into account the molecular nature of a gas. It is to be remarked that, as we shall see, the use of those equations can also be based on the kinetic theory of gases, which justifies them as asymptotically useful models when the mean free path is negligible.

In the area of environmental problems, the Boltzmann equation is also required. Understanding and controlling the formation, motion, reactions and evolution of particles of varying composition and shapes, ranging from a diameter of the order of 0.001 μm to 50 μm, as well as their space-time distribution under gradients of concentration, pressure, temperature and the action of radiation, has grown in importance, because of the increasing awareness of the local and global problems related to the emission of particles from electric power plants, chemical plants, vehicles as well as of the role played by small particles in the formation of fog and clouds, in the release of radioactivity from nuclear reactor accidents, and in the problems arising from the exhaust streams of aerosol reactors, such as those used to produce optical fibers, catalysts, ceramics, silicon chips and carbon whiskers.

One cubic centimeter of atmospheric air at ground level contains approximately 2.5 × 1019 molecules. About a thousand of them may be charged (ions). A typical molecular diameter is 3 × 10−  10 m (3 × 10−  4 μm) and the average distance between the molecules is about ten times as much. The mean free path is of the order of 10−  8 m, or 10−  2 μm. In addition to molecules and ions one cubic centimeter of air also contains a significant number of particles varying in size, as indicated above. In relatively clean air, the number of these particles can be 105 or more, including pollen, bacteria, dust, and industrial emissions. They can be both beneficial and detrimental, and arise from a number of natural sources as well as from the activities of all living organisms, especially humans. The particles can have complex chemical compositions and shapes, and may even be toxic or radioactive.

A suspension of particles in a gas is known as an aerosol. Atmospheric aerosols are of global interest and have important impact on our lives. Aerosols are also of great interest in numerous scientific and engineering applications [175].

A third area of application of rarefied gas dynamics has emerged in the last quarter of the twentieth century. Small size machines, called micromachines, are being designed and built. Their typical sizes range from a few microns to a few millimiters. Rarefied flow phenomena that are more or less laboratory curiosities in machines of more usual size can form the basis of important systems in the micromechanical domain.

A further area of interest occurs in the vacuum industry. Although this area existed for a long time, the expense of the early computations with kinetic theory precluded applications of numerical methods. The latter could develop only in the context of the aerospace industry, because the big budgets required till recently were available only there.

The basic parameter measuring the degree of rarefaction of a gas is the Knudsen number (Kn), the ratio between the mean free path λ and another typical length. Of course, one can consider several Knudsen numbers, based on different characteristic lengths, exactly as one does for the Reynolds number. Thus, in the flow past a body, there are two important macroscopic lengths: the local radius of curvature and the thickness of the viscous boundary layer δ, and one can consider Knudsen numbers based on either length. Usually the second one (Knδ = λ/δ), gives the most severe restriction to the use of Navier–Stokes equations in aerospace applications.

When Kn is larger than (say) 0.01, the presence of a thin layer near the wall, of thickness of the order λ (Knudsen layer), influences the viscous profile in a significant way.

This and other effects are of interest in both high altitude flight and aerosol science; in particular they are all met by a shuttle when returning to Earth. Another phenomenon of importance is the formation of shock waves, which are not discontinuity surfaces, but thin layers (the thickness is zero only if the Euler model is adopted).

When the mean free path increases, one witnesses a thickening of the shock waves, whose thickness is of the order of 6λ. The bow shock in front of a body merges with the viscous boundary layer; that is why this regime is sometimes called the merged layer regime by aerodynamicists. We shall use the other frequently used name of transition regime.

When Kn is large (few collisions), phenomena related to gas-surface interaction play an important role. They enter the theory in the form of boundary conditions for the Boltzmann equation. One distinguishes between free-molecule and nearly free-molecule regimes. In the first case the molecular collisions are completely negligible, while in the second they can be treated as a perturbation.

2 The basic molecular model

According to kinetic theory, a gas in normal conditions (no chemical reactions, no ionization phenomena, etc.) is formed of elastic molecules rushing hither and thither at high speed, colliding and rebounding according to the laws of elementary mechanics. Monatomic molecules of a gas are frequently assumed to be hard, elastic, and perfectly smooth spheres. One can also consider these molecules to be centers of forces that move according to the laws of classical mechanics. More complex models are needed to describe polyatomic molecules.

The rules generating the dynamics of many spheres are easy to describe: thus, e.g., if no body forces, such as gravity, are assumed to act on the molecules, each of them will move in a straight line unless it happens to strike another molecule or a solid wall. The phenomena associated with this dynamics are not so simple, especially when the number of spheres is large. It turns out that this complication is always present when dealing with a gas, because the number of molecules usually considered is extremely large: there are about 2.7 ⋅ 1019 in a cubic centimeter of a gas at atmospheric pressure and a temperature of 0 °C.

Given the vast number of particles to be considered, it would of course be a hopeless task to attempt to describe the state of the gas by specifying the so-called microscopic state, i.e., the position and velocity of every individual sphere; we must have recourse to statistics. A description of this kind is made possible because in practice all that our typical observations can detect are changes in the macroscopic state of the gas, described by quantities such as density, bulk velocity, temperature, stresses, heat-flow, which are related to some suitable averages of quantities depending on the microscopic state.

3 The Boltzmann equation

The exact dynamics of N particles is a useful conceptual tool, but cannot in any way be used in practical calculations because it requires a huge number of real variables (of the order of 1020). The basic tool is the one-particle probability density, or distribution function P(1)(x, ξ, t). The latter is a function of seven variables, i.e., the components of the two vectors x and ξ and time t.

Let us consider the meaning of P(1)(x, ξ, t); it gives the probability density of finding one fixed particle (say, the one labelled by 1) at a certain point (x, ξ) of the six-dimensional reduced phase space associated with the position and velocity of that...