Interdisciplinary Applied Mathematics

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Remark: We presented empirical models for velocity distribution and shear stress for engineering analysis of shear-driven gas flows encountered


FIGURE 3.6.    Variation    of    normalized    shear    stress with    Kn    for    M = 3    flow.

Comparison of DSMC results with Burnett solutions using various boundary conditions. (Courtesy of J.M. Reese.)

in microsystems. The new velocity model is based on a generalized slip coefficient, and it uniformly converges to the first-order slip model for Kn < 0.1 flows, while it accurately matches the DSMC and the linearized Boltzmann results in the transition and the free-molecular flow regimes. Validity of the velocity model    was    established    for    Kn <    12.    The    new    shear    stress    model

is valid in the entire Knudsen regime, and it is asymptotically consistent in the continuum (Kn ^ 0) and free-molecular flow (Kn ^ ж>) regimes. These models are valid for monoatomic dilute hard-sphere gases, and are restricted to low subsonic M < 0.3, nearly isothermal flows, since extreme compressibility effects and viscous heating are neglected in their derivation.

3.3 Oscillatory Couette Flow

Oscillatory Couette flow is the simplest approximation for time-periodic shear-driven gas flows encountered in several microsystem applications, such as microaccelerometers, inertial sensors, and resonant filters. Oscillatory Couette flow can also be interpreted as a variation of Stokes’s second problem (Batchelor, 1998). This classical problem has been investigated extensively using continuum-based flow models for Kn < 0.1 (Veijola and Turowski, 2001); however, rarefaction effects in the transition and free-molecular flow regimes have not been studied in detail.

Let us consider rarefied gas flow between two infinite parallel plates that are at a distance L apart, where the top plate oscillates harmonically with frequency (u>) in the lateral (flow) direction, and the bottom plate is stationary. The two plates are maintained at the same temperature Tw = 273 K. The gas is initially at rest; it has an equilibrium number density n0 and equilibrium temperature Te that is equal to the wall temperature Tw. For a given set of parameters, the oscillatory rarefied Couette flow can be characterized by the Knudsen (Kn), and Stokes (в) numbers. The Stokes number represents balance between the unsteady and viscous effects. It can also be interpreted as the ratio of the diffusion and oscillation characteristic time scales, and it is defined as

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