Interdisciplinary Applied Mathematics

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s


У


y’y* _ (Po)exp


~~ (Po)th ’


where the Poiseuille number is


Po = Cf Re


with Cf the friction factor and Re the Reynolds number. For flow in a circular pipe    the    theoretical    value    is    (Po)th    = 64,    and    for    a two-dimensional

FIGURE 1.15. Ratio of the Poiseuille number obtained experimentally normalized by the Poiseuille number obtained theoretically (equation (1.8)). The dashed line    corresponds    to    normalization    with    no-slip    theory    and the solid    line    to    slip


theory. Here triangles and circles correspond to the nitrogen and helium flow in a 0.51pm microchannel, respectively. (Courtesy of H. Bau.)


channel (Po)th = 96. The data are from the experiments of (Harley et al., 1995) for a microchannel of 0.51pm deep; this is case (JH6) studied in Section 4.1.2. The inlet pressure varied from 1.1 MPa to 3.5 MPa, and the exit pressure was atmospheric. The Mach number was kept below 0.02 for all experiments and the Knudsen number ranged from 0.004 to 0.373. It is interesting to see that the velocity slip effect is more pronounced in the low Reynolds number regime. This is consistent with the fact that the Knudsen number scales inversely proportionally to the Reynolds number (see equation (1.7)).    For    example,    for    Re =    0.012    the    inlet    Knudsen number    was


0.025 and the outlet Knudsen number was 0.373.


For liquids the noncontinuum behavior is more difficult to detect. It is manifested as anomalous diffusion, i.e., different diffusion in the near-wall region than in the bulk, and is associated with the rheology of the liquid. Early studies by (Debye and Cleland, 1959) with paraffin in porous Vycor glass indicated the presence of a slipping adsorbed thin layer. This could either increase or decrease the pressure drop compared to the continuum behavior; both C* < 1 and C* > 1 were measured. A possibility of permanent structural modification of water in a very thin film was reported in (Derjaguin, 1970), but such results have been largely rejected. In a    follow-up    study (Derjaguin    et    al.,    1983),    it    was    demonstrated    that

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