# Interdisciplinary Applied Mathematics

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plexor directs the fluid to the lower channel of the selected row. (iii) The column multiplexor releases the vertical valves of the chamber, allowing the pressurized fluid to    flow    through    the    chamber    and purge    its contents.    (C)    Demonstration    of

microfluidic memory display: Individual chambers are selectively purged to spell out “C I T”. (Courtesy of S. Quake.)

FIGURE 1.32. Simulation levels in microsystem design.

Designer

FIGURE 1.33. The coupled circuit-fluidic device simulator. Microfluidic systems, including the control electronics, can be simulated using accurate numerical models for all microcomponents.

Part I:

Gas Flows

## 2

### Governing Equations and Slip Models

In this chapter we first present the basic equations of fluid dynamics both for incompressible and compressible flows, and discuss appropriate nondi-mensionalizations for low-speed and high-speed flows. Although most of the flows encountered in microsystems applications are typically of low speed, micropropulsion applications may involve high-speed supersonic flows (see Section 6.6). Subsequently, we consider the compressible Navier-Stokes equations and develop a general boundary condition for velocity slip. This applies to a regime for which Kn < 1, and it corresponds to a second-order correction in Knudsen number. It improves Maxwell’s original first-order formula, which is limited to Kn < 0.1. The validity of this model is assessed in Chapter 4 with DSMC data, linearized Boltzmann equation solutions, as well as with experimental results. A more rigorous derivation of the governing equations from the Boltzmann equation is given in Section 15.4.2.

##### 2.1 The Basic Equations of Fluid Dynamics

Consider fluid flow in the nondeformable control volume 0 bounded by the control    surface    d0    with    n    the    unit    outward    normal.    The equations

of motion can then be derived in an absolute reference frame by applying the principles of mechanics and thermodynamics (Batchelor, 1998). They can be formulated in integral form for mass, momentum, and total energy,

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