# Interdisciplinary Applied Mathematics

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A more effective approach would be to use the constraint dynamics to control the density fluctuations at the end of the interface (last bin), while at the same time resorting to a relaxation iterative procedure similar to the DSMC-continuum coupling (see Section 15.1). In the context of liquid flows, we cannot provide a modification of the Navier-Stokes equations to account for    the slip,    and thus    the    constraint,    on    the    MD side,    will    enforce

the no-slip condition as interface to the continuum description. In other words, we interpolate quantities as

Фмъ = ^MD + (1 _ ^№3

Hybrid Iteration 25-27

Hybrid Iteration 20

Hybrid Iteration 10

Hybrid Iteration 1

y

FIGURE 16.5. Convergence history for the obstructed channel problem of Figure 16.4. The velocity profile at different iterations is shown. (Courtesy of N.G. Hadjiconstantinou.)

with в the relaxation parameter. For в = 0 we obtain the alternating Schwarz algorithm used in the work of (Hadjiconstantinou, 1999), which is explained schematically in the plot of Figure 16.4; see also Section 15.2.1. The example is flow in an obstructed channel in which a square region behind the solid block is treated by MD. The continuum iterate receives Dirichlet data on rcont from the molecular solution obtained in the previous iteration cycle (top); the continuum solution on rmoi is subsequently used as Dirichlet    data    on    the    molecular    simulation    (bottom).    A    typical    result

of the convergence history of the iterative process is shown in Figure 16.5. The hybrid solution is compared to the full continuum solution (denoted by a dashed line); the continuum solution is taken as “exact,” since the molecular region is representing bulk fluid far from any boundaries. An initial guess of zero velocity in the molecular region was used. The oscillations around the    “exact”    solution    are    due    to    the    statistical fluctuations    in    the

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