Interdisciplinary Applied Mathematics

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(b) Velocity histories at various points



(c) Phase variation throughout medium



(d) Log-plot of velocity amplitude


FIGURE 3.13. Details of the flow dynamics for Kn = 1.0 and в = 5.0.


where uw = uo sin (wf) and к = у 2к™т • Here Te is the initial equilibrium temperature, kB is the Boltzmann constant, and £ and Z are the stream-wise and spanwise components of the molecular velocity, respectively. Diffuse reflections of gas molecules from the surfaces require that the reflected molecules have a Maxwellian distribution f0, characterized by the velocity and    temperature    of    the    plates.    Here,    we    assume that    the amplitude    of


velocity oscillations is less than the speed of sound. This enables linearization of the collisionless Boltzmann equation and the boundary conditions. Following the work of Sone, the velocity distribution function can be decomposed into its equilibrium and fluctuating components as follows (Sone, 1964; Sone, 1965):


f = fo (1 + ф),    (3.31)


where ф is the normalized fluctuation. We can obtain the linearized forms of the Boltzmann equation and the boundary conditions by substituting equation (3.31) into equations (3.28), (3.29a), and (3.30a) and neglecting


(a) Dynamic response of medium



(b) Velocity histories at various points



u/u0


(c) Phase variation throughout medium



(d) Log-plot of velocity amplitude


FIGURE 3.14. Details of the flow dynamics for Kn = 5.0 and в = 2.5.


all the higher order terms in ф, i.e.,


дф дф


~я7 = °> dt dy


(3.32a)


Фо = Ф (У = 0) = 0; п> 0,

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