# Interdisciplinary Applied Mathematics

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S ж n 1/3.    (1.4)

Under standard conditions the mean molecular spacing is S « 3.3 x 10-9 m.

The mean molecular diameter of typical gases, based on the measured coefficient of viscosity and the Chapman-Enskong theory of transport properties for hard sphere molecules (Chapman and Cowling, 1970), is of order 1010 m (see Table 1.1 for various thermophysical properties of common gases). For air under standard conditions, d « 3.7 x 10-10 m (Bird, 1994). Comparison of the mean molecular spacing S and the typical molecular diameter d shows an order of magnitude difference. This leads us to the concept of dilute gas where S/d ^ 1. For dilute gases, binary intermolecular collisions are more likely than simultaneous multiple collisions. On the other hand, dense gases and liquids go through multiple collisions at a given instant, making the treatment of intermolecular collision processes more difficult. The dilute gas approximation, along with the molecular chaos and equipartition of energy principles, leads us to the well-established kinetic theory of gases and formulation of the Boltzmann transport equation starting from the Liouville equation. The assumptions and simplifications of this derivation are given in (Sone, 2002; Cercignani, 1988; Bird, 1994). In Section 15.4 we present an overview of the Boltzmann equation and some benchmark solutions appropriate for microflows, and in Section 15.5 we explain the BBGKY hierarchy that leads from the atomistic to the continuum description.

Momentum and energy transport in a fluid and convergence to a thermodynamic equilibrium state occur due to intermolecular collisions. Hence, the time and length scales associated with the intermolecular collisions are important parameters for many applications. The distance traveled by the molecules between collisions is known as the mean free path A. For a simple gas of hard spherical molecules in thermodynamic equilibrium the mean free path is given in the following form (Bird, 1994):

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