Interdisciplinary Applied Mathematics

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2/r dx [h2    h av ^ П П



(2.44)


This is identical to the results obtained using equation (2.43) up to second-


order terms in Kn, given below:


U (y)



h? dP у2    у 2 — av Kn


2p dx h2    h av 1 + Kn



(2.45)


This equivalence can be seen by expanding the last term in equation (2.45) as a geometric series expansion in terms of powers of Kn. The leading error in equation (2.45) is therefore proportional to


2 p



dP


dx



Kn3,


where h is the microchannel height.


Remarks: We summarize here observations that will aid in evaluating the proper application and limitations of the slip boundary conditions given by equations (2.42) and (2.43).


1.    The first-order    slip    boundary    condition    should    be    used    for    Kn <


0.1 flows. Since rarefaction effects gradually become important with increased Kn (regular perturbation problem), inclusion of second- and higher-order slip effects into a Navier-Stokes-based numerical model is neither mathematically nor physically inconsistent.


2.    Using the high-order slip boundary conditions with the Navier-Stokes equations can lead to some physical insight. For example, using equation (2.42) for pressure-driven flows with various slip coefficients from Table 2.2 results in different velocity profile and flowrate trends. All the models in Table 2.2, with the exception of equation (2.29), result in increased flowrate due to the second-order slip terms. Although this is a correct trend for flowrate, the velocity distribution predicted by these models become erroneous with increased Kn, as shown in Figures 4.11 and 4.17. This indicates that solely using the high-order slip correction in the transition flow regime is insufficient to predict the velocity profile and the flowrate simultaneously. In Section 4.2, we address this problem by introducing a rarefaction correction parameter that leads to a unified flow model for pressure-driven channel and pipe flows, when combined with the general slip condition (equation (2.43)). The unified model predicts the correct velocity profile, flowrate, and pressure distribution in the entire Knudsen regime (see Section 4.2 for details).

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