Interdisciplinary Applied Mathematics

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5. Lack of Deterministic Surface Effects: Molecule-wall interactions are specified by the accommodation coefficients (av ,aT); for diffuse reflection, av = 1. Hence, the reflected molecules lose their incoming tangential velocity and are reflected with the tangential velocity of the wall. For av = 0, the tangential velocity of the impinging molecules is not changed. For any other value of av, a combination of these procedures can be applied. This level of wall and boundary interaction treatment is more fundamental than the slip conditions presented in Chapter 2. However, it still lacks the most fundamental way of simulating the molecule-wall interactions, which includes the molecular structure of the walls. Such approaches are obtained by molecular-dynamics simulations (see Section 16.1 and (Tehver et al., 1998)). Also, the accommodation coefficients for every surface and gas pair are    not    available    (see    Section    2.2.2    for details    on    recent    ex

perimental research).

15.1.2 DSMC for Unsteady Flows

Microsystems often experience unsteady or time-periodic flows, which result in    dynamic    variations    of lift    and drag    forces, and    torques    on    the    de

vice components. Computations of unsteady flows require time-dependent numerical simulations, and DSMC provides an effective tool in the transition and free molecular flow regimes (Kn > 0.1). To this end, this section presents the key concepts that need to be addressed in DSMC computations of unsteady flows.

In order to make these ideas specific, in the following we discuss the concepts and conditions relevant to the computations of oscillatory Cou-ette flows, presented in Section 3.3. Lateral oscillations in this prototype geometry require consideration of one-dimensional unsteady flows. In these computations, we utilized the hard sphere (HS) model for molecular collision, and the no-time-counter (NTC) scheme for collision pair selection (Bird, 1994). The choice of the HS model facilitates easy comparisons with the theoretical solutions of the linearized Boltzmann equation, and it also enables easier code implementation. Argon gas is simulated with a reference temperature Te = 273 K. The surfaces are assumed to be fully accommodating. Hence, the particles are reflected from the surfaces according to a Maxwellian distribution with surface velocity and temperature. Simulation parameters are chosen such that the compressibility and viscous heating effects are negligible. Although the gas temperature increases with increased oscillation frequency, the maximum temperature rise in the simulations is less than 2%. More than 100 simulated particles per cell are employed. The entire domain is discretized into 40 to 100 equally spaced cells, to ensure that the cell size (Ay) is smaller than the mean free path for all simulations. A finer grid is used for high Stokes number flows to resolve the flow inside the Stokes layers, while a coarser grid is used for low Stokes number flows. With respect to selecting the time step (At), the following issues need to be considered:

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