Interdisciplinary Applied Mathematics

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0



Slip velocity


0.5


0.25


1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (a)


1 1 1 1 1 1


_ — О


— 0- — —


// •


— — — — — ♦ — -в


В —


1 // /


-*11/ |



в = 0.1 & 0.25



— id


II


/ +


— -7- —


в = 0.25 &1.0 — -0—


……………


low в (<=0.25) в =1.0 в = 2.5 в = 5.0 в = 7.5


……

Kn



0.5



0.25



0



10



low p (<=0.25) в =1.0 в = 2.5 в = 5.0


в = 7.5 Л



A


‘Ж’



в = 0.25 &1.0






FIGURE 3.20. Rarefaction and Stokes number effects on the slip velocity at the moving wall.


system to the dissipated energy per cycle. In the laterally oscillating plate problem, the energy dissipation per cycle is due to the viscous dissipation (work done by the viscous forces), which is given as


1 r2n


D = — Tw(ujt)uWig(ujt)d(ujt).    (3.38)


O о


Therefore, accurate characterization of the shear stress (tw ) and gas velocity (uw,g) on the wall as functions of the Knudsen and Stokes numbers is required to estimate the energy dissipation. For quasi-steady oscillatory Couette flow in the continuum flow regime, the energy dissipation per cycle is

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