Interdisciplinary Applied Mathematics

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(a) Dynamic response of medium

(b) Velocity histories at various points

(c) Phase variation throughout medium

(d) Log-plot of velocity amplitude

FIGURE 3.8. Details of the flow dynamics for Kn = 0.1 and в = 5.0.

Shear Stress Model

In this section, we extend the shear stress model developed for plane Couette flows in Section 3.2.2 to predict the shear stress on the oscillating wall. Our main assumption is that quasi-steady oscillatory flows should behave more or less like the steady Couette flow. Hence, an effective viscosity derived from the steady Couette flow results can be used as the viscosity coefficient for the quasi-steady oscillatory flows, an assumption that we validate later using the DSMC data. Therefore, using the new velocity model for steady flows    and    the    shear    stress    model    given    by    equation (3.12),    we

define an “effective viscosity” as

where the    subscript    c indicates    plane Couette    flow,    and    uc    is    the    plane

Couette flow    velocity    profile    given    by    equation    (3.7)    with    the    slip    coefficient C1.    The    coefficients    a, b, and c are    due    to    the    shear    stress    model

given in (3.12). For quasi-steady oscillatory flows, we employ this “effective

FIGURE 3.9. Time history of the normalized shear stress at the laterally oscillating wall, predicted by the model of equation (3.27) and the unsteady DSMC simulations at (a) Kn = 2.5, в = 0.1 and (b) Kn = 2.5, в = 0.25.

viscosity” coefficient, and define the shear stress at the oscillating plate as


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