Interdisciplinary Applied Mathematics

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For a comparison we also included similar predictions by the second-order slip boundary condition of (Hsia and Domoto, 1983) (large dashed line). The form of their boundary conditions is similar to Cercignani’s, Deissler’s, and Schamberg’s, and    they    all    become    invalid    at    around    Kn    =    0.1.    This


boundary condition performs worse than even the first-order Maxwell’s boundary condition for large Kn. Only the general slip boundary condition given    by    equation    (2.43)    predicts    the    scaling    of    the    velocity    profiles


accurately.


Flowrate Scaling


In the previous section we analyzed the shape of the velocity profile. Since we have normalized the velocity profile with the local mean velocity, the above analysis cannot predict the volumetric flowrate. In this section we analyze the volumetric flowrate variation in the entire Knudsen regime.


The volumetric flowrate in a channel is a function of channel dimensions, fluid properties (p0, A), and pressure drop, and it can be written as


Q = G    .


V d/X    I


For a channel of height h, using the Navier-Stokes solution and the general slip boundary condition (2.43) we obtain


h3 dP    6Kn


12p0 dx    1 — b Kn


where Kn = A/h.

FIGURE 4.21. Volumetric flowrate (per channel width) per absolute value of the pressure gradient in [m3/(sPa)] as a function of Kn for nitrogen flow. The solid line represents the proposed model.


The flowrate for the continuum and free-molecular flows are both linearly dependent on dP/dx (Kennard, 1938), and thus we choose to normalize the flowrate with the pressure gradient. This quantity is computed based on the DSMC simulations and is shown in Figure 4.21 for nitrogen. For comparison we present the Q/dP/dx predictions obtained using Maxwell’s first-order slip boundary condition (b = 0, dashed lines) and the general slip boundary condition (b = —1, dashed-dotted lines). In both cases the predictions are erroneous. The general slip boundary condition performs the worst for it is asymptotic to a constant value, while the DSMC data show a considerable increase with Kn. The first-order boundary condition follows the DSMC data, however with a significant error.

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