Interdisciplinary Applied Mathematics

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order of magnitude is reported. Even though the inlet Mach numbers of

о


d


II


О


d


II


X


<D



x


CD



H


CD


«D


Y (No Slip Flow)



E-7«


<D


Y (Slip Flow)


FIGURE 5.9. Temperature profiles in a pressure-driven channel flow for continuum and rarefied flows    as    a function    of    Mach    number.    The top    row    shows    the


insulated channel,    and    the    bottom    row    shows    specified    heat    flux    that    counter


balances the viscous heating effects, so that overall, dTs/dx = 0, and therefore there are no thermal creep effects; Re = 1.0 and Pr = 0.7.


the flows    are    very    small,    exit    Mach numbers    up    to M =    0.70    have    been


observed. For such situations, the flow in the microchannels cannot be assumed incompressible, and thus the above analysis will not be strictly valid. In general, it is theoretically inconsistent (Aoki, 2001) to use the incompressible flow model with slip boundary conditions caused by rarefaction but the analysis here is meant to highlight approximately the heat transfer effects.

5.4-2 Force-Driven Flows


It is    possible    to drive    channel flows    using a    body    force.    The    force-    and


pressure-driven flows are hydrodynamically similar. In either case, the pressure gradient or the force field will be balanced by the viscous shear on the channel walls,    and    for    compressible flows    part    of    the    force    will    be    used to accelerate the fluid in the streamwise direction. However, the two driving forces are very different at the microscopic level. The external force accelerates individual particles, while the pressure gradient induces a collective flow (Zheng et al., 2002). The energy equation for these two cases also shows some differences. For example, the pressure creates a cooling effect by flow expansion (the first term on the right-hand side in equation (2.7)), while the body force affects the kinetic energy of the sytstem (Panton, 1984). For pressure-driven compressible flows the expansion cooling negates the viscous heating (last term in equation (2.7)), while viscous heating may play a crucial role in force-driven flows.

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