Rarefied gas dynamics is a branch of fluid mechanics where the continuum assumption is no longer accurate, as a characteristic length scale in the gas (e.g. radius of a body moving in a gas, radius of a tube conducting a gas, etc.) becomes comparable to the mean free path of gaseous particles. Consequently, the gas cannot be described as a continuum. Instead, the gas should be described by its microstate determined by the velocity and position of each particle. Since it would be an impossible computational task in storing the very large volumes of information involved in tracking the behavior of every single particle, the statistical or kinetic theory of gases must be used. This theory consists in obtaining the macroscopic state of the gas described by quantities such as density, bulk velocity, temperature, pressure tensor, and heat flux from the microstate of the gas. To describe non-equilibrium phenomena in rarefied gases, the Boltzmann transport equation must be used, which is the appropriate mathematical tool for this purpose.
Brief history The history of rarefied gas dynamics starts with the kinetic theory of gases. In 1860, Maxwell published results on the distribution of molecular velocities for a gas in equilibrium, which is called the Maxwell-Boltzmann distribution. As a result, the assumption that all molecules move with the same speed was abandoned and the random nature of molecular motion was recognized. In 1872, Boltzmann derived an integro-differential equation (Boltzmann equation) to describe the evolution of the velocity distribution function in space and time. In the same paper, Boltzmann proved the H-theorem demonstrating that intermolecular collisions always produce entropy. In other words, the Boltzmann equation is irreversible. In 1912, Hilbert proved the existence and uniqueness of a solution of the Boltzmann Equation. A connection between the kinetic theory and fluid dynamics was done by Chapman and Enskog who derived the Euler and Navier-Stokes equations based upon a series expansion of the Boltzmann equation with respect to the Knudsen number. This approach allowed to derive expressions for the viscosity and thermal conductivity directly from intermolecular interaction potentials. Furthermore, considering higher order terms of the expansion, Burnett derived more general constitutive equations for the pressure tensor and heat flux. Then, Grad proved the equivalence of the equations of fluid dynamics to an asymptotic form of the Boltzmann equation. The scientific branch of rarefied gas dynamics consolidated in the 19th century and became forefront with space exploration. As a result, the first international symposium on rarefied gas dynamics was held in Nice, France in July 1958.
Knudsen number To determine if a gas can be classified as rarefied, the non-dimensional Knudsen number is usually calculated. The Knudsen number Kn {\displaystyle {\ce {\mathrm {Kn} }}} is defined as the ratio of the mean free path λ {\displaystyle \lambda } , to a characteristic length scale L {\displaystyle L} in the flow, i.e. Kn {\displaystyle {\ce {\mathrm {Kn} }}} = λ / L {\displaystyle \lambda /L} . The value of the Knudsen number tells you whether a molecular modeling approach must be used or a macroscopic description is sufficient. High and intermediate Knudsen numbers mean that the gas is rarefied, while very small Knudsen numbers imply that the gas is a continuum. Various flow regimes can be distinguished based on the value of the Knudsen number. Namely, a gas flow regime can be characterized as continuum (or viscous), slip, transitional, and free-molecular. The following rough ranges of the Knudsen number for these regimes are usually established:
Kn {\displaystyle {\ce {\mathrm {Kn} }}}
< 0.01 {\displaystyle <0.01} for continuum or viscous regime;
0.01 < {\displaystyle 0.01<}
Kn {\displaystyle {\ce {\mathrm {Kn} }}}
< 0.10 {\displaystyle <0.10} for slip flow regime;
0.1 < {\displaystyle 0.1<} Kn {\displaystyle {\ce {\mathrm {Kn} }}} < 10 {\displaystyle <10} for transitional regime;
Kn {\displaystyle {\ce {\mathrm {Kn} }}} > 10 {\displaystyle >10} for free-molecular regime. This division between the regimes can vary depending on flow type and required accuracy. The mathematical approach and tool to model a gas flow depends on the flow regime. In the continuum regime, the gas is not rarefied and hence can be described by the Euler or Navier-Stokes equations. In the slip flow regime, the Navier-Stokes equations are applied too, but the velocity no-slip and temperature continuity boundary conditions are replaced by the velocity slip and temperature jump conditions. All approaches to model gas flows in the transitional regime are based on the Boltzmann transport equation. Since the intermolecular collisions are neglected in the free-molecular regime, the Boltzmann equation is applied in its simplified form.
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