# Interdisciplinary Applied Mathematics

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Implementation of second-order slip boundary conditions using equation (2.29) requires obtaining the second derivative of the tangential velocity in the normal direction to the surface (d2U/dn2), which may lead to computational difficulties, especially in complex geometric configurations. To circumvent this difficulty we have proposed in the previous section the following general velocity slip boundary condition.

Us — Uw

2 erv

Ov

Kn dU «

16Kn(‘tb’)s

(2.43)

where b is a general slip coefficient. Notice that the value of b can be determined such that for bKn | < 1 the geometric series obtained from the boundary condition of equation (2.43) matches exactly the second-order equation (2.29), and    thus    for    slip    flow    the    above    boundary    condition    is

second-order accurate in the Knudsen number.

An alternative way of implementing the slip boundary condition is to use equation (2.26) derived directly from the tangential momentum flux analysis. Such a boundary condition has not been tested before, so in Section 4.1.3 we will determine the region of its validity, and in particular at what distance from the wall it should be applied, i.e., A or CX, where C =1 (see Figure 2.5 and (Thompson and Owens, 1975)).

As regards the accuracy of two velocity slip boundary conditions, i.e., equation (2.26) versus equation (2.43), we can analyze the differences for the two-dimensional pressure-driven incompressible flow between parallel plates separated by a distance h in the slip-flow regime. Assuming isothermal conditions and that the slip is given by equation (2.26), the corresponding velocity distribution is

U (y)