# Interdisciplinary Applied Mathematics

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Beyond x/h = 3.65, favorable pressure gradients are established. The flow goes through a developing region, followed by a typical compressible channel flow behavior far downstream in the channel, as shown in Figure 6.15 for x/h > 4.2. In contrast to the low Mach number flows, a decrease in the temperature of the fluid near the center of the channel is observed. This shows that the thermal energy of the fluid is converted into kinetic energy, and there is considerable heat transfer from the walls to the fluid.

Detailed observations on separated internal flows, and comparisons of the continuum-based slip flow simulations with the DSMC, can be found in (Beskok, 2001). DSMC predictions for a backward-facing step in the slip and transition flow regimes can be found in (Xue and Chen, 2003). Both studies indicate substantial separation zones for Kn < 0.1 flows. In their studies, Xue and Chen (2003) did not observe flow separation for Kn > 0.1. They attributed this to the rarefaction effects. However, this behavior may be due to the reduction in the Reynolds number of the flow, described by equation (6.18). Increasing the Knudsen number to the transition flow regime (Kn > 0.1) requires Reyf < 10, since M < 1. However, the separa-

FIGURE 6.16. Streamwise velocity distribution along the backward-facing channel at five selected locations (normalized with the speed of sound). Predictions of both    DSMC    (symbol) and pFlow    (lines)    are    presented    (see    Table 6.1    for    a

description of the symbols).

tion zone becomes quite small for Re^ < 10 flows, even in the continuum flow regime.

##### 6.3 Separated External Flows

In this section, we present simulations of slip flow past a circular cylinder, as a prototype of an external flow around a microprobe. Uniform flow past a cylinder with a slip surface has also been studied in (Gampert, 1978), for attached flows using an approximate boundary layer analysis. Following (Beskok and Karniadakis, 1994), we simulate both attached and separated flows; the simulations are performed at two values of Knudsen number: Kn ^ 0 (corresponding to the no-slip) and Kn = 0.015, at Re = 10. Separation of flow is observed with a small circulation bubble; the slip flow direction is reversed inside the separation zone. In Figure 6.17 we plot the magnitude of velocity slip distribution along the cylinder periphery for Reynolds number Re =1 (attached flow; triangles) and Re = 10 (separated flow; circles). The velocity slip increases with the Reynolds number, but it decreases substantially in the separated (almost stagnant) region. This velocity slip is proportional to the shear stress ts, which is plotted in Figure

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