Interdisciplinary Applied Mathematics

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5.4 Heat Transfer in Poiseuille Microflows

It is possible to create Poiseuille flow in a microchannel using either a pressure drop or a body force. Although the continuum descriptions for both cases are similar, kinetic theory predictions and statistical simulations for force-driven flows show that the Navier-Stokes solutions fail to predict the important temperature minimum effects, even for Kn « 0.01 flows with M < 0.1    (Zheng    et    al.,    2002).    Therefore,    we    divide    this section    into two

parts, addressing the pressure- and force-driven flows separately.

5-4-1 Pressure-Driven Flows

Heat transfer in pressure-driven gas microflows has important engineering applications. There have been several studies of microflows under constant wall temperature and heat flux conditions. For example, thermally developing slip    flows    under    constant    heat    flux    have    been presented    for    circu

lar and rectangular microchannels in (Ameel et al., 1997; Yu and Ameel, 2002), respectively. A similar problem is studied at the entrance of a conduit by including the heat transfer within the Knudsen layers (Chen and Xu, 2002). Thermally developing flows in circular tubes under constant wall temperature have been studied in the slip flow regime (Larrode et al., 2000). The constant-temperature convective heat transfer under hydrodynamically and thermally fully developed conditions is considered for various geometries and flow regimes, such as slip flow in rectangular channels (Tunc and Bayazitoglu, 2002) and flow in two-dimensional channels for 0 < Kn 10 (Simek and H!adjiconstantinou, 2002).

In this section, we neglect gas compressibility effects and present an analytical treatment of heat transfer for pressure-driven flows. We consider the slip flow regime where the continuum description is valid. Such results were first obtained by (Inman, 1964), who showed that reduced heat transfer rates are obtained in the presence of slip flow. Specifically, Inman obtained the following equation for the Nusselt number Nu for constant heat flux at the wall

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