Interdisciplinary Applied Mathematics

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Finally, we note that

• It is possible to predict the (pressure-driven) channel flowrate in the early transition flow regime, using ca second-order slip solution of the Navier-Stokes equations. However, this procedure may create erroneous velocity profiles, as shown in Figures 4.11 and 4.17.

We caution the reader about these limitations of the second-order theory

FIGURE 2.4. Profile of a polysilicon mirror surface; scan area 10 x 10 pm. (Courtesy of C. Liu.)

in the context of validation efforts using integral (e.g., mass flowrate versus pressure drop) measurements.

Roughness plays a very important role in microscales, but it is difficult to quantify its effect. In practice, it can be characterized using an atomic force microscope (AFM) for nonconductive surfaces, scanning tunnelling microscope (STM) for conductive surfaces, WYCO interferometer for optical nondestructive evaluation, and scanning electron microscope (SEM). Micron-scale roughness can be produced by wet chemical etching. A typical surface profile for a polysilicon mirror surface is shown in Figure 2.4.

The effects of roughness are difficult to analyze theoretically or numerically, but some progress has been made. (Richardson, 1973) considered a periodically modulated (rough) wall and applied a shear stress-free boundary condition. He showed analytically that the no-slip boundary condition is actually a consequence of surface roughness. In a systematic molecular dynamics study, (Mo and Rosenberger, 1990) investigated the effects of both periodic and random roughness with amplitude A. They found that as the roughness height (amplitude) A increases compared to the mean free path A,    the    velocity slip    at    the    wall    decreases.    Specifically,    they    proposed

a criterion    for    the    no-slip    condition    to be    valid    based on    the    ratio    A/A.

If this    ratio    is    of    order    unity,    that    is, if    the    roughness height    is smaller

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