# Interdisciplinary Applied Mathematics

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(Zheng et al., 2002) performed extensive DSMC and Navier-Stokes studies of    pressure-    and    force-driven    Poiseuille    flows    in    the    slip    flow regime,

and compared these with each other and with the kinetic theory predictions. They have shown that the compressible Navier-Stokes equations do not predict the correct flow physics for the force-driven flow case even for Kn = 0.01 flows. Specifically, the Navier-Stokes solutions failed to predict the central minimum in the temperature profile and the nonconstant pressure distribution. In Figure 5.10 we present the temperature distribution predicted across the channel using the DSMC and Navier-Stokes solutions. The temperature distribution predicted by the Navier-Stokes solution gives a temperature maximum in the center of the channel, while the DSMC and the kinetic theory predicts two off-center maxima, with a local temperature minimum at the center of the channel. The inability of the Navier-Stokes equations to predict this behavior indicates a global failure, which cannot be corrected by modifications of the transport coefficients, equation of state, or the slip/jump boundary conditions. (Zheng et al., 2002) have shown that such discrepancies between the DSMC and Navier-Stokes predictions do not exist for the pressure-driven Poiseuille flow.

The temperature minima can also be predicted using the entropic lattice Boltzmann method (LBM), specifically the so-called minimal kinetic model; see Section 15.5. In this nonisorthermal case, the entropic LBM does a good job in capturing the correct temperature variation. Here we present results of (Ansumali et al., 2003), who performed an extensive study of this flow using the 16-velocity minimal kinetic model. The Knudsen number was varied from Kn = 0.001 to Kn = 0.5, while the Mach number was varied from Ma = 0.01 to Ma = 0.3. Some typical simulation results are presented in Figure 5.4.2. Similar to the DSMC results of (Zheng et al., 2002), the minimal kinetic model predicts the temperature minimum. Some important conclusions, which we can draw from the results of (Ansumali et al., 2003) in Figure 5.4.2 are: 7

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