Interdisciplinary Applied Mathematics

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The Reynolds equation derived from the linearized Boltzmann equation has a striking resemblance to the Navier-Stokes-based Reynolds model. Both models utilize the mass flowrate of linear Poiseuille and Couette flows (divided by RT). This enables a generalized Reynolds equation, with smooth transition between the continuum (Kn ^ 0) and the free-molecular flow (Kn ^ ж) limits. Fukui and Kaneko obtained a uniformly valid Reynolds equation by employing the Boltzmann solutions for Qp and Qp for different Knudsen number regimes (Fukui and Kaneko, 1988; Fukui and Kaneko, 1990). A uniform approximation of Qp is given in (Alexander et ah, 1994):


Q^(Kn) « 1 + hllA + bg +    ,    (6.11)


where A = 1.318889 and B = 0.387361. The above equation is asymptotically correct in the no-slip and free-molecular flow limits. However, the model looses its accuracy in the transition flow regime with nominal error of ±5%. Also, equation (6.11) does not include the accommodation coefficient dependence on the flowrate. Fukui and Kaneko developed a database using analytical and numerical approximations for Q p and Qp valid for various Knudsen number regimes and accommodation coefficients (Fukui and Kaneko, 1990). They employed this database in the generalized Reynolds equation. Also, Veijola et al. (1998) developed a uniform approximation for Fukui and Kaneko’s database, including the effects of the accommodation coefficients in the following form:


Q p(D> av)



D 6    a.



1



1.34



In ( 1+4.1 ) +-^



D



6.4


| 1.3(1—<t.„)    | 0.64avD°’lr


+1 + 0.08Л183 +1 + 1.2£>а72


where QP (D, av) is the Poiseuille flowrate, not normalized with the continuum flowrate (D ^ ж) limit, and av is the accommodation coefficient. The above formula is valid for D > 0.01 and 0.7 < av < 1 with maximum error of ±1%. Another important contribution of this work is its inclusion of possible differences in the accommodation coefficients on the top and bottom walls of the slider geometry. This feature allows the study of sliders with metal and silicon top and bottom surfaces, respectively, corresponding to different accommodation coefficients (Veijola et al., 1998).

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