# Interdisciplinary Applied Mathematics

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In this chapter we present three main numerical methodologies to analyze flows in microdomains:

High-order finite-element (spectral element) methods for Navier-Stokes equations. Formulations for both incompressible and compressible flows in stationary and moving domains are presented.

Meshless methods with random point distribution.

The force coupling method for particulate microflows.

These methods represent three different classes of discretization philosophies. They have been used with success in diverse applications of microsystems, from microfilters, valves, and mixers to self-assembly processes. Clearly, any other discretization method based on finite-differences, finite-elements or finite-volumes can also be used.

##### 14.1 Spectral Element Method: The ^Flow Program

In this section we present the spectral element method (Karniadakis and Sherwin, 1999) implemented in the program p,Flow, which was used in many examples included in this book. A summary of the capabilities of this program or gas flows in different regimes is given in Table 14.1. This is just an example of how a continuum-based approach can be employed to simulate microflows, and thus the spectral element method can be replaced by finite elements, finite volumes, or finite differences. Most of the ideas we present next apply also to these other discretizations. The specific capabilities and issues we cover in this section are to be used as reference in designing a similar program using other continuum-based discretizations.

In gas microflow simulations both the incompressible and the compressible Navier-Stokes equations can be employed to compute the relative effects of compressibility and rarefaction. Strictly speaking, from the theoretical point of view, there is an inconsistency in using the incompressible form of the Navier-Stokes equations with the slip boundary condition (Aoki, 2001). In practice and for very small Reynolds numbers, the limits in    Knudsen    number    for    the    incompressible    models    are dictated    by    the

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