# Interdisciplinary Applied Mathematics

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E = UModel “ Ulb,

and they are not normalized with UModel since it changes as a function of y. The    model    of equation    (2.29)    has    the    smallest    overall    deviations    of    at

most 0.035 units from the linearized Boltzmann solution ULB.

The second-order slip coefficients cited in Table 2.2 are all positive, with the exception of our model, which corresponds to a Taylor series expansion of the velocity profile including the second-order terms. Therefore, our model uses both the slope of the local velocity profile and the change in the slope in order to predict (extrapolate) the slip velocity near the wall. The volumetric flowrate (equation (4.14)) calculated by other models gives enhancement of the volumetric flowrate compared to the first-order predictions. This is    consistent    with    the    experimental    data    and it is    due    to    the

positive contribution of the second-order slip coefficient.

• In the model of equation (2.29) the second-order slip contribution leads to ca reduction in the volumetric flowrate compared to the first-order model.

• Other models with second-order slip conditions can predict flowrate accurately, but only cat the expense of accuracy in the velocity profile.

This will be demonstrated further in Section 4.2, where we examine in detail flowrate modeling issues for a wide range of Kn.

Remark: The second-order slip boundary condition given by equation (2.29) is based on equation (2.26), where terms higher than (d2U/dn2) are neglected. Solutions of Navier-Stokes equations for long channels result in parabolic velocity profiles, with vanishing contribution for derivatives higher than two (see Section 4.2.1). Therefore, the boundary conditions given by equations (2.29) and (2.26) are identical for this problem. Implementation of our second-order model in the slip flow regime was performed by obtaining the necessary slip information at a distance A away from the surface. For flows with higher Knudsen number in the transition regime we switch our model to the general boundary condition (2.43), the validity of which we investigate in detail in Section 4.2.

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