The idea that materials may be treated as continua allows one to formulate equations of conservation for mass, momentum and energy in which all variables are continuous functions of space. (See Conservation Equations.) Thus, with the aid of the continuum hypothesis, it is possible to speak of the density of a material at a point in space whereas, from the molecular viewpoint, there actually may be no molecule at the point so that the concept of density has no meaning. The idea that materials may be treated as continua is founded upon the fact that in any element of volume that is small on a practical scale, there are a very large number of molecules (approximately 1016 in a cubic millimetre). Thus, for many purposes, it is possible to find a sufficiently small volume still containing a sufficiently large number of molecules that the discrete molecular nature of matter does not reveal itself.
However, under certain circumstances, the continuum hypothesis is inappropriate. Such circumstances occur when the distance between the molecules or, more correctly, the mean free path that they travel between collisions with other molecules, λ, is comparable with some physical dimension of the container of the flow channel, d (λ/d ~ 1). Naturally, this most often arises when the density of the gas is very low (so that the mean free path is large) and when the gas interacts with solid surfaces with a small-scale structure such as a porous solid or a capillary tube. In such circumstances, the gas molecules may interact as frequently with the solid surface as they do with other molecules and one has what is called a transition regime. As the density of the gas is reduced further, the collisions of molecules with walls completely dominate the processes and one reaches the free-molecule or Knudsen regime when λ/d >> 1. The ratio λ/d is termed the Knudsen Number, Kn.
For flow of a gas in the free-molecule or Knudsen regime through a tube of circular cross-section of radius R and length L, it is possible to show that the amount-of-substance flow through the tube is (Kauzmann, 1966)
under the influence of a pressure difference (p1 − p2) where u is the mean velocity of the gas molecule and M is the molecular mass.
It should be noted that this flow is independent of the viscosity of the gas and of its density and proportional to the third power of the radius of the tube, unlike for viscous flow where it is proportional to the fourth power.
In the transition regime, the behaviour is naturally intermediate between the continuum and free-molecule behaviour and is generally described in terms of slip at the walls so that the normal boundary condition of continuum mechanics, that there is no relative motion between the wall and the fluid, is abandoned. There are no exact means to treat free-molecule or transition flow since it is necessary to consider the nature of collisions between the molecules and the wall explicitly. Nevertheless, some progress has been made with relatively simple models, including the Dusty Gas Model, that is particularly appropriate for the treatment of porous media where the phenomenon of free-molecule flow often occurs (Cunningham and Williams, 1980).
Cunningham, R. E. and Williams, R. J. J. (1980) Diffusion in Gases and Porous Media, Plenum, New York.
Kauzmann, W. (1966) Kinetic Theory of Gases, Benjamin, New York.