# Interdisciplinary Applied Mathematics

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R and particle radius a and cFS = 1. The fact that the continuum slightly overpredicts the mean particle velocity was attributed by Drazer et al. to the transverse random motion of the particles in the MD simulation. At later times it is possible for the particle to execute an intermittent stick-slip motion, especially for poorly wetting fluid-wall systems. For cFS < 0.7 the particle is eventually adsorbed to the tube wall, and in the stick regime almost all    the    fluid    atoms    between    the    particle    and the    wall    have    been

squeezed out. This total depletion of fluid atoms would require an infinite force in the continuum limit. This phenomenon is also encountered in capillary drying, in which liquid is suddenly ejected from the gap formed between two hydrophobic surfaces when the width falls below a critical value (Lum et al., 1999). The robustness of continuum calculations in this context has been demonstrated also in (Israelachvili, 1992a; Vergeles et al., 1996) for spheres approaching a plane wall; see also Section 10.5.

###### 1.2.1 Molecular Magnitudes

In this section we present relationships for the number density of molecules n, mean molecular spacing S, molecular diameter d, mean free path A, mean collision time tc, and mean-square molecular speed c for gases.

The    number    of    molecules    in    one    mole    of gas is a    constant    known    as

Avogadro’s number 6.02252 x 1023/mole, and the volume occupied by a mole of gas at a given temperature and pressure is constant irrespective of the composition of the gas (Vincenti and Kruger, 1977). This leads to the perfect gas relationship given by

where p is the pressure, T is the temperature, n is the number density of the gas, and kB is the Boltzmann constant (kB = 1.3805 x 10-23 J/K). This ideal gas law is valid for dilute gases at any pressure (above the saturation pressure and below the critical point). Therefore, for most of the microscale gas flow applications we can predict the number density of the molecules at a given temperature and pressure using equation (1.3). At atmospheric pressure and 0°C the number density is n « 2.69 x 1025 m-3. If we assume that all these molecules are packed uniformly, we obtain the mean molecular spacing as

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