# Interdisciplinary Applied Mathematics

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There are many possibilities for an effective implementation of boundary conditions. Next we compare two different approaches to implementing the no-slip boundary    condition    for    a    Poiseuille    flow,    following    the    studies    of

(Pivkin, 2005); see Figure 16.12. Similar procedures have been proposed in (Willemsen et al., 2000), for no-slip conditions. We consider a box with dimensions 10 x 10 x 10 (where rc = 1), N = 3,000, a = 3.0, p =3, and Y = 4.5. The walls are composed of three layers of DPD particles and are four times denser than the fluid (squares) in the first case and equal to the wall density (circles) in the second case. The repulsion force coefficient is aij =    25    for    the    particles    in    the    flow and    also    for    the wall particles.    The

results of this comparison are summarized in Figure 16.12 for the velocity profiles and for the corresponding density profiles. In both cases the bounce-back condition is implemented, and the only difference is the density at the wall.

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### Reduced-Order Modeling

In this chapter, we introduce several reduced-order modeling techniques for analyzing microsystems following the discussion of Section 1.7. Specifically, techniques such as generalized Kirchhoffian networks, black box models, and Galerkin methods are described in detail. In generalized Kirchhoffian networks, a complex microsystem is decomposed into lumped elements that    interact    with    each    other as    constituent    parts    of    a    Kirchhoffian    net

work. Techniques such as equivalent circuit representations and description-language-based approaches are described under generalized Kirchhoffian networks. In black box models, detailed results from simulations are used to construct simplified and more abstract models. Methods such as nonlinear static models and linear and nonlinear dynamic models are described under the framework of black box models. Finally, Galerkin methods, where the basic idea is to create a set of coupled ordinary differential equations, are described. Both linear and nonlinear Galerkin methods are discussed. The advantages and limitations of the various techniques are highlighted.

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