# Interdisciplinary Applied Mathematics

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The gas molecules are hard spheres of uniform size, and they undergo diffuse reflections with the surfaces (av = 1).

The plate velocity is small compared to the mean thermal velocity. Hence the compressibility effects, temperature fluctuations, and viscous heating effects are negligible.

Based on these, the velocity field in the slip flow regime is given as

Uc(Y)

2 UY

1 + 2^CiKn’

(3.7)

where Kn = X/L, and Y = y/L. Utilizing this solution with Ci = 1.111, it has been shown that equation (3.7) is valid for Kn < 0.25 (Marques et al., 2000).

Steady flow calculations in (Sone et al., 1990) using the linearized Boltzmann equation indicate that the bulk flow velocity profile is essentially linear for all Knudsen numbers. However, a kinetic boundary layer (Knud-sen layer) on the order of a mean free path starts to become dominant between the bulk flow and solid surfaces in the transition flow regime. The velocity distribution and other physical variables are subject to appreciable changes within the Knudsen layer, which can be predicted only by solution of the Boltzmann equation. In order to validate Sone’s linearized Boltzmann solutions, (Bahukudumbi et al., 2003) performed a series of hard-sphere DSMC simulations at various Knudsen numbers and wall speeds. In these simulations, solid surfaces were maintained at 273 K, and they were assumed to be fully accommodating (av = 1). Argon with molecular mass m = 6.63 x 10~26 kg and hard-sphere diameter dhs = 3.66 x 10~10 m was simulated. Since the collision diameter for hard-sphere molecules is independent of the relative velocity of the colliding molecules, the viscosity dependence on temperature is in the form of p0 ж T°’5. This trend is slightly different from a more reliable variable hard-sphere model (VHS), which has viscosity dependence on temperature in the form of p,0 ж To81 for argon molecules. However, it is noted that DSMC simulations for hard-sphere and variable hard-sphere molecules yield similar results when the flow is isothermal. Simulations were performed for a wide range of Knudsen and Mach numbers. A minimum of 20 cells across the channel width was used in the simulations, and the domain discretization always ensured a minimum of 3 cells per mean free path. With this discretization, 10,000 hard-sphere particles were utilized, and the results were sampled for 1.0 x 106 time steps. The simulation time step was one-fifth of the mean collision time (Л/л/2ДТо). These sets of parameters are sufficient to obtain accurate DSMC results (Chen and Boyd, 1996).

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