# Interdisciplinary Applied Mathematics

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a particle traveling at angle ej = j 60° (0 < j < 6) along the edges of the triangular lattice and the last bit specifying the existence or nonexistence of a particle    at    rest    at    the    lattice    site.    Each particle    (except    a rest    parti

cle) moves one lattice distance in one fundamental time interval. After the particles propagate they then interact according to certain collision rules (Frisch et al., 1986).

Soon it was discovered, however, that there were some difficulties with these CA methods. First, their work requirements increase rapidly with Reynolds number, and in fact, for some cases CA-based computations could be more expensive than corresponding computations with the Navier-Stokes equations (Orszag and Yakhot, 1986). Also, the methods represent the correct incompressible hydrodynamics only in the limit of small Mach number. Finally, CA methods are statistical in nature and suffer from significant noise. The velocity field is computed as the average velocity over a large number N of CA sites, so that there is an error of order 1/VN in the evaluation of this velocity field. The first two arguments are not so critical for microflows, where both the Reynolds number and the Mach number are typically low, except for micronozzles (see Section 6.6). However, the third one, which is similar to the problem in DSMC, is related to the efficiency of these methods, and new theoretical developments have extended C A met h o ds, improving t heir accu racy c o nsiderably.

An effective way to avoid the difficulties with noisiness of CA systems is to use a lattice Boltzmann approach (McNamara and Zanetti, 1988; Higuera and Jimenez, 1989); also see (Succi, 2001; Chen and Doolen, 1998) and references therein. This approach seems to be more efficient than Monte Carlo CA methods for moderate to low Reynolds numbers. The idea is to integrate a kinetic equation for the CA system; here the kinetic equation is for average particle distribution functions along each of the discrete allowed particle velocities at each lattice site. For a two-dimensional CA system there are seven distribution functions (corresponding to the seven bits) at each lattice site. These functions are smooth nonrandom functions governed by nonlinear partial differential equations that are integrated in space-time to obtain the flow description. Velocities are determined as averages over a number    of lattice    sites    of    the    LB    system.    The method    extends easily    to

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