# Interdisciplinary Applied Mathematics

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V(y) = Ai sinh(^y) + A2 cosh (Фу),

where the constants A1 and A2 are determined by applying the boundary conditions. The complex variable ф can be expressed in terms of the Stokes number, since

ф

3P2

L2

Utilizing the    slip    boundary    condition    (2.42)    with    C2    =    0,    and the gen

eralized slip coefficient C1 from equation (3.8), we obtain the following time-dependent velocity distribution:

u(y,t)    (3.20)

Uo

sinh(V7/3U)

. Kn cosh(V7/ЗУ)

(1 + j[52Cl Kn2) sinh(-y/J/3) + 2a/7/3Ui Kn cos1i(a/J/3)

exp(jwt)

where Y = y/L and Kn = X/L. This is a general solution for the velocity profile that captures the no-slip solution simply by setting Kn = 0, and the first-order slip solution by setting C1 = 1.111 in equation (3.20). Therefore, the above equation is expected to be valid for any Stokes number flow in the continuum and slip flow regimes, since it uniformly captures these well-explored limits.

Figure 3.7 shows variation of the normalized velocity amplitude between the two surfaces. We compare the DSMC results with the predictions of the extended slip model for (a) quasi-steady flows in the entire Knudsen regime, and (b) slip flows for a wide range of Stokes numbers (в < 7.5). The velocity amplitudes are obtained by measuring the magnitude of the maximum velocity at different cross-flow (Y = y/L) locations. Note that the generalized velocity model given by equation (3.20) converges to the first-order slip model    for    Kn    <    0.1.    Hence,    only the predictions    of the    extended    slip

model are shown in the figure. For quasi-steady flows, the velocity amplitude distribution always passes through (y/L,u/u0) = (0.5, 0.5) resulting in a linear velocity distribution with equal amounts of slip on the oscillating and stationary walls. The extended slip model accurately matches the DSMC velocity profile for a wide range of Knudsen numbers (Kn < 12). However, it fails to predict the Knudsen layers that are captured by the DSMC results, as expected. The extended slip model is also valid for high Stokes number flows in the continuum and slip flow regimes due to the use of the Navier-Stokes equations in its derivation. Figure 3.7(b) shows

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