# Interdisciplinary Applied Mathematics

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2 ст v

G v

Kn

1 —6Kn

Assuming this    form    of    velocity    distribution,    the    average    velocity    in    the

channel (U = Q/h) can be obtained as

U (x) = F

dP

dx

MCjhjA

1    ( 2 — av Kn

6 у av ) 1 — bKn

By nondimensionalizing the velocity distribution with the local average velocity the dependence on local flow conditions (F ((dP/dx), pc, h, A)) is eliminated. Therefore, the resulting relation is a function of Kn and y only. Assuming av = 1 (for simplicity), we obtain

U *(y, Kn) = U (x,y)/U(x)

«-Ш2 +

3L +

h ‘

1

Kn -6 Kn

Kn

1 — 6 Kn

(4.18)

A similar analysis has been performed by (Piekos and Breuer, 1995). They used the first-order slip boundary conditions and subsequently separated

FIGURE 4.19. Velocity profile comparisons of the model (equation (4.18)) with DSMC and linearized Boltzmann solutions (Ohwada et al., 1989a). Maxwell’s first-order boundary condition is shown with dashed lines (b = 0), and the general slip boundary condition (b = —1) is shown with solid lines.

this equation into an ж-dependent contribution and a у-dependent contribution to investigate the breakdown of slip flow theory. However, here we will keep the form of equation (4.18) in the following analysis. Equation (4.18)    solely    describes    the    shape    of the    velocity    distribution,    but    it    does

not properly model the flowrate. Flowrate modeling requires additional corrections, as shown in the subsection below.

In Figure 4.19 we plot the nondimensional velocity variation obtained in a series    of    DSMC simulations    for    Kn =    0.1,    Kn    =    1,    Kn    =    5,    and

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