# Interdisciplinary Applied Mathematics

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1. At has to be significantly smaller than the mean collision time, Atc, to ensure accurate simulations.

2.    The    cell    size    Ay    should be smaller    than    the    mean free path,    A,    and

the molecules should not move across more than one cell between two consecutive time steps. Hence, At <C Ay/vm, where vm = л/2RT is the most probable velocity.

3. The characteristic length scale of the problem, L, and kinematic viscosity, v, in micro- and nanoflows can result in small viscous diffusion time, which scales as L2/v. Hence, At ^ L2/v.

4. The time period of oscillations (T0 = 2п/ш) can become smaller than the mean collision time; hence At ^ 2п/ш, where ш is the frequency of oscillations.

We must indicate that the first three requirements are also valid for steady DSMC computations, while the last requirement is specific for unsteady flows. In all simulations of Section 3.2, the time step was chosen to satisfy all four constraints. The range of the total simulation time was around three to eighteen time periods. This ensured that the transients starting from the quiescent initial conditions decay, and a time-periodic state is achieved. In unsteady DSMC, ensemble-average at each time step replaces the time-average used in steady computations. Ensemble-averaging is performed over 5000 different realizations of the stochastic process for each time step.

In all simulations, the amplitude of the oscillating wall is kept constant at    mo    = 100    m/s,    maintaining    relatively low    Mach    number flows    so    that

compressibility effects are negligible. The gas number density (n0), excitation frequency (ш), and the characteristic system length (L) are adjusted to simulate different combinations of Kn and в.

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