Interdisciplinary Applied Mathematics

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For example, it may, however, be misleading to identify the flow regimes as “slip” and “continuum,” since the “no-slip” boundary condition is just an empirical finding, and the Navier-Stokes equations are valid for both the slip and the continuum flow regimes. Nevertheless, this identification was first made for rarefied gas flow research almost a century ago, and we will follow this terminology throughout this book.


In the transition reg ime (Kn > 0.1) the constitutive laws that define the stress tensor and the heat flux vector break down (Chapman and Cowling, 1970), requiring higher-order corrections to the constitutive laws, resulting in the Burnett or Woods equations (Woods, 1993). It is also possible to use the Boltzmann equation directly, which is valid at the microscopic level (see Section 15.4). The Burnett and Woo ds equations are derived from the Boltzmann equation based on the Chapman-Enskog expansion of the velocity distribution function f, including terms up to Kn2 in the following form:


f = fo(1 + a Kn +bKn2),    (2.14)


where a and b are functions of gas density, temperature, and macroscopic velocity vector, and fo is the equilibrium (Maxwellian) distribution function (Chapman and Cowling, 1970):


3/2    /    2


(2.15)



fo


m N    / mv


2-кквТо)    ^ V 2/гдТо

FIGURE 2.2. A plot of the Maxwellian distribution showing the most probable velocity and the mean thermal velocity, equation (2.15).


which is plotted in Figure 2.2. Here m is the molecular mass, kB is the Boltzmann constant, T0 is the temperature, and v is the mean thermal velocity of the molecules. The zeroth-order solution of equation (2.14) is the equilibrium solution, where flow gradients vanish; i.e., the viscous stress tensor and the heat flux vector vanish, giving the Euler equations (Chapman and Cowling, 1970). Therefore, Kn = 0 corresponds to the Euler equations. This is a singular limit in transition from the Navier-Stokes equations to the Euler equations, where the infinitesimally small viscosity (or heat conduction coefficient) vanishes.

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