Interdisciplinary Applied Mathematics

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■■(L-y)


   —i


_(y,t) =



Pouo



<L-y)


t



t



n sin



t


KL~y)


ц


k(L y)



exp(-n 2) dn ,



(3.36a)



n



exp(-n 2) dn .    (3.36b)


The gas velocity Uw,g(t) and shear stress on the oscillating (top) wall are calculated as


uo


uw<g(t) = u(L, t) = —j= sin(wt) exp(—KZr/ 2) dr]


(3.37a)


= sin (<j)t) = uWy9 sin (cot),


Tw



т (L,t) =



Po uo


Pouo


к



sin(wt)



n sin(wt) exp(-K2n 2) dn — tw sin(wt).



(3.37b)


The magnitude of gas velocity on the oscillating plate and the corresponding shear stress are uw = and tw= p№‘§jlkBJ’u, respectively. This shows that


• for oscillatory Couette flows, the magnitude of the gas velocity and shear stress on the oscillating surface reach the same asymptotic limit as their steady counterpart when Kn ^ ж.


Validation of the DSMC results in the free-molecular flow regime is presented in Figure 3.15. We compare the normalized velocity amplitudes obtained from the DSMC with the solution of the linearized collisionless Boltzmann equation at different Stokes numbers. The free-molecular solution plotted in this figure is obtained from equation (3.36a). Overall, a very good agreement between the DSMC results and the free-molecular solution is obtained. The DSMC results in Figure 3.15(a) show statistical scatter associated with a high Knudsen number simulation. However, agreement between the theory and simulations is remarkable in Figures 3.15(b-d).


In Figures 3.16 and 3.17, we compare the dynamic response of the flow obtained from the DSMC results and the collisionless Boltzmann equation solutions for the Kn = 10,в = 1.0, and Kn = 10,в = 2.5 cases. Predictions of the velocity profiles, phase angles, and the slip velocities are presented in the figures. As observed in Figure 3.17(a), both methods capture the “bounded rarefaction layer” equally well. Due to the onset of statistical scatter outside this layer, we plotted the DSMC phase angle only for y/L > 0.65 in Figure 3.17(c). Nevertheless, the DSMC and Boltzmann solutions match remarkably well within the bounded layer, confirming the accuracy of the DSMC results.

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