Interdisciplinary Applied Mathematics

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3.5 Grooved Channel Flow


Flow in a micromotor or a microbearing is more complicated than the linear Couette flow. We consider a two-dimensional shear-driven grooved channel flow (Figure 3.25) in order to model the geometric complexity of these microdevices. The presence of grooves complicates the geometry, and an analytical solution for this flow does not exist. The flow separates and


Uc


Flow


FIGURE 3.25. Schematic of a grooved channel. This geometry is a prototype for modeling geometric complexity and flow reversal encountered in complex microgeometries.


starts to recirculate in the grooves even for small Reynolds number flows. In the numerical model we assumed that the top wall is moving with a speed , and both surfaces are kept at the same temperature (300 K). We also assumed that the geometry repeats itself along the flow direction. Therefore, the flow is periodic, and only a section of the channel is simulated. In the simulations the Reynolds number is fixed (Re = 5.0), and the Knudsen number is increased by decreasing the channel gap. Therefore, the top wall speed Ux is increased to keep the Reynolds number constant, resulting in an increased flow Mach number according to equation (3.5). The domain is discretized with 12 elements of sixth-order polynomial expansions in each direction, per element.


The streamwise velocity variation across the middle of the channel, normalized by    the    top    wall    velocity    (UTO),    is given    in Figure    3.26.    The    flow


separates due to the presence of the groove and starts to recirculate. This recirculation zone is seen as negative streamwise velocity (i.e., -ff— < 0). Figure 3.26 shows a reduction in the separation zone for slip flows compared to the no-slip case (Kn = 0). The slope of the velocity profile at the top wall is decreased due to the velocity slip effects, resulting in reduction of the shear stress on the top wall. We also see that the net volumetric flowrate, which is proportional to (Udy), is decreased as the Kn is increased.

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