# Interdisciplinary Applied Mathematics

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(Cl) = (C2) = 0. The transformation essentially shifts the rough boundary to the bulk and makes it flat. It is given by

Y =L[y ~ l/2[g2(a:,g) -gi{x,z) L-[£i{x,z)+&(x,z)    ‘

The work of Meyerovich has addressed mostly high Knudsen numbers, since the theory for small Knudsen numbers has not been developed yet. This is the most difficult regime, since it requires simultaneous account for bulk and boundary scattering, including the interference between bulk and boundary scattering.

Computationally, it is possible to study the additional pressure drop due to inhomogeneities for prototype roughness in microchannels and interwall correlations. For example, it is possible to generate two identical rough walls and shift one with respect to the other. When this shift is zero, the inhomogeneities from different walls are perfectly correlated with each other. With increasing shift, this correlation will gradually decrease and will    disappear    when    the    shift    becomes    much    larger than    the    correlation

length. The dependence of the flow parameters on this shift will mimic the dependence on the interwall correlation of inhomogeneities.

We have used the p,Flow program as well as the DSMC approach to predict the additional pressure drop due to regularized roughness in long microchannels. We have considered geometries with in-phase and out-of-phase types of roughness, as shown in Figure 14.1. Typical results are summarized in Figure 4.13. We compare the total pressure as well as the deviation from a linear drop for different roughness types and with a corresponding smooth channel. For all these cases the mass flowrate was maintained constant and the Reynolds number was Re = 0.44, while the Knudsen number at the outlet    of    the    channel    was    Kn = 0.17.    We    see that in    order to match    the

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