Interdisciplinary Applied Mathematics

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A comparison    between    the    no-slip    and    slip    data    for    the    1-micron    versus


6-micron cases shows enhanced drag reduction in the smaller geometry, where the Knudsen number is higher. Hence, drag reduction increases by rarefaction. This is an important result, which indicates that the microfilters have the potential to allow more mass flowrate than their macroscale counterparts under fixed inlet and exit pressure conditions, and hence they require less power to operate.


Further examination    of    Figure    6.25    shows    that    for    a    fixed    geometry,


both the viscous and form-drag increase with the Reynolds number. The increase is mostly linear for low Reynolds numbers and starts to increase faster than    linearly    for    Re > 10.    We    also    observed    that    the    viscous-drag


is consistently about 50% of the form-drag. This is important in designing microfilter devices, because large drag forces may lead to bursting of thin filter-membranes.

6.5.2 Viscous Heating Characteristics


Work done by the viscous stresses usually becomes important for high-speed flows. For example, in the case of the micronozzles viscous heating effects cannot be neglected. In fact, work done by the viscous stresses causes heat generation, which acts as a volumetric source term in the energy equation. The viscous heating is characterized by


/    (njTji) ■ Ui ds,


J CS


where Tji represents (ij)th component of the viscous stress tensor, nj corresponds to the jth component of the outward surface normal, ui shows the ith component of    the    velocity    on    the    control    surface    (CS)    with    differ


ential area ds. Figure 6.26 shows the viscous heating as a function of the Reynolds    number    for    various    filter    sizes. Smaller    filters    have    shown    sub


stantial viscous heating effects. For example, for the 1-micron filter with Re = 6.95 (M = 0.51), the viscous heating can be as large as 1.2 W/m. In order to keep the Reynolds number within a certain range, the inlet flow speeds are increased substantially, which in return increase the Mach number for small filter dimensions. A comparison of the viscous heating effects shown in Figure 6.26 with the reference exit Mach numbers reveals that the viscous heating effects are important for high-speed flows. Furthermore, increase in the viscous-heating as a function of the Reynolds number (or Mach number, due to the increase in the reference speed) seems to be faster than linear.

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