# Interdisciplinary Applied Mathematics

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While more    work    is    required to    fully resolve    such    issues    at    the    funda

mental level, an evaluation of LB models in the context of microflows is instructive. Next, we first present some simulations of the standard LBM method, i.e., the BGK version we described above, and subsequently we present an example obtained using the entropic LBM. We have already presented LBM simulations of a microcavity flow in Chapter 3 as well as simulations using the entropic LBM in Chapter 5. Here we first present results for microchannels obtained by (Nie et al., 1998) using a square lattice and the equations described above. The microchannel has length L = 1000 and height H = 10 in lattice units. A pressure boundary condition was used at the    inlet    and    the    outlet    with    ratio    П =    Pi/Po    =    2.    The    parameter    p0

in equation (15.50) was set by comparing the flowrate from the simulation to experimental results presented in Figure 4.2. The slip velocity and mass flowrate at the outlet of the microchannel are plotted in Figure 15.21. The slip velocity at the outlet was obtained from

U(Y) = Uo(Y — Y2) +Vs,

where Y = y/H.    From    a    least-squares    fit to    the    data    of    Figure    15.21,    Nie

et al. found that

Vs = 8.7 Kn2,

and based on this equation, they obtained an analytical formula for the normalized flowrate (with respect to continuum flowrate):

Mf = 1 + 12VS (Кп)-!±®-.

This formula agrees well with the numerical data shown in Figure 15.21 if we set П = 2. With regard to nonlinear distribution of pressure, for Kn < 0.2 the LBM results agree with the DSMC results presented in Chapter 4. However, for higher Knudsen number values a change in the curvature of the pressure distribution was observed, indicating a slower than linear pressure drop. This latter result, however, has not been verified by DSMC approaches, and it deserves further investigation.

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