Interdisciplinary Applied Mathematics

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0    0.1    0.2    0.3    0.4    0.5


Y


FIGURE 4.11. The velocity distribution normalized with the local average velocity in half of a microchannel (left). The linearized Boltzmann solution is from (Ohwada et al., 1989a). Error in the solution of the Navier-Stokes equations with various slip flow models at Kn = 0.1 (right). For the legend see Table 2.2.


Using this, we can find a local average velocity, which is used to normalize the local velocity distribution. This is the best choice for normalization of the velocity profile, making the magnitude of the nondimensional velocity distribution O(1) for any Kn value. This normalized velocity distribution is given as


U*(y, Kn)



U(x,y)


U(x)



НЮ2 + !+ gi Kn+2C?2Kn2) l + Ct Kn +2U2 Kn2



(4.15)


where U is the average (across the channel) velocity. The velocity profiles obtained by the various slip boundary conditions and the corresponding deviations from the linearized Boltzmann solution of (Ohwada et al., 1989a), are shown    in    Figure    4.11.    The    error    is positive    if the    model (4.15)    overpre


dicts the velocity at a given point and is negative if the velocity is underpredicted. More specifically, for the slip velocity prediction, the model given by equation (2.29) gives the best agreement with the linearized Boltzmann solution, followed by the first-order model, while Schamberg’s boundary condition performs the worst (see Table 2.2). Hsia and Domoto’s coefficients give the most accurate description of the centerline velocity, while Schamberg’s model once again performs the worst. The maximum errors in Schamberg’s model for centerline and velocity slip predictions are about 0.035 and 0.16 units, respectively. The error units are defined as

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