# Interdisciplinary Applied Mathematics

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Some of these    algorithms,    even    for    the    continuum,    have    been    used    ex

tensively in simulating microflows, such as the direct simulation Monte Carlo method (DSMC), while others have only recently been applied to this field, e.g., meshless methods. The force coupling method for particulate microflows    that    we    present    in    Section    14.3.2 is an    example    of    an

efficient method for a very difficult problem involving many moving interfaces. We selected these methods carefully, taking into account the interdisciplinary character of microsystems applications, where full-system simulation requires simultaneous solutions of fluidic, electrical, mechanical, and thermal domains. Therefore, suitable methods for microsystems should be very efficient but also very accurate, since new and unfamiliar physics in the microdomain requires resolution-independent simulation studies. They should also be very flexible, especially in the context of moving domains, and therefore meshless methods seem a good candidate.

In Chapter 15 we present atomistic methods for gas microflows and nanoflows. We start with the most popular approach, direct simulation Monte Carlo (DSMC), for steady and unsteady flows, and summarize practical guidelines that need to be followed in simulations of rarefied gases. We then discuss techniques of coupling DSMC with continuum descriptions, and we analyze microfilters as an example. We also include an overview of the Boltzmann equation as well as benchmark solutions in the slip flow and transitional regimes, that could be useful to the reader for validation of experimental and numerical results. Finally, we present the lattice Boltzmann method, which solves the Boltzmann equation fast but in a constrained subspace. This method too could be a very effective tool for simulating flows in complex microgeometries.

In Chapter 16 we present atomistic methods for liquid microflows and nanoflows. We provide the general formulation of the molecular dynamics (MD) method as well as specific subsections on different types of potentials, thermostats, and practical guidelines for data analysis as well as sources for available software. We then discuss multiscale modeling, i.e., coupling of atomistic and continuum descriptions, and we apply an embedding multiscale method to electroomostic flows at nanoscales. A new method, dissipative particle dynamics (DPD), is presented last; it is a stochastic MD approach for modeling liquid flows at the mesoscopic level, and it is a potentially very effective tool for multiscale modeling of liquids.

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