# Interdisciplinary Applied Mathematics

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To examine the accuracy of the results obtained with DSMC, we compare them with solutions of the linearized Boltzmann equation in the slip flow regime obtained in    (Ohwada et    al.,    1989a),    for    Kn    = 0.1.    In    Figure    4.11

the velocity distribution obtained by both methods is plotted normalized with the local average velocity. The differences of DSMC and linearized Boltzmann predictions are almost indistinguishable. We can then use these profiles to examine the accuracy of the analytic solutions obtained using

FIGURE 4.9. Velocity profiles predicted by the Navier-Stokes and DSMC simulations at various x/L locations. The inlet is located at x/L = 0.0.

various slip    boundary    conditions.    We    write    all    proposed    models    in    the    fol

lowing general form: where the first-order (Ci) and second-order (C2) slip coefficients are given in Table 2.2.

The velocity distribution for an isothermal flow in a long channel (h/L ^ 1) of thickness h is predicted by the second-order slip boundary conditions in the following form:

U (x,y)

dP h2 dx 2p

!)2 + (Ю+С.Кп+адК»2

where Kn is the local Knudsen number, and it varies in the streamwise direction.    Hence, the    velocity    profile    is a    function    of    both    x and y.    The

corresponding volumetric flowrate also increases along the channel as the density is decreased from the inlet to the exit. The local volumetric flowrate is

(4.14)

dP h3 /1

Q(x) = — — —[-+ClKnY2C2Kn2

dx 2p 6

channel wall as predicted by the ERROR

FIGURE 4.10. Slip velocity along the Navier-Stokes and DSMC simulations.

Y

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