# Interdisciplinary Applied Mathematics

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2;

O(Kn3)

OU3 dt

+

(Uo

■V)Us

+

(U3

■V)Uo

+ (U2 -V)Ui

+(Ui ■ V)U2 = -VPs + Re1V2U3.

The boundary conditions for these equations are obtained similarly by substitution of the asymptotic expansion into the slip boundary condition formula:

O(1) :    Uos

O(Kn) :    Uis

Uw:

2a

(2.34)

(U0)

s

O(Kn2) :    U2s

O(Kn3) :    Uss

s

where U/, U», and U/» denote first, second, and third derivatives of the *th-order tangential velocity field along the normal direction to the surface.

A possible solution methodology for slip flow with high-order boundary conditions can be the solution of the Navier-Stokes equations ordee by order. However, this approach is computationally expensive, and there are numerical difficulties associated with accurate calculation of higher-order derivatives of velocity near walls with an arbitrary surface curvature.

We propose a formulation where the governing equations are directly solved without an asymptotic expansion in velocity, as mentioned above. The objective is to establish a methodology to develop slip boundary conditions accurate up to the second-order terms in Kn. First, we introduce a new slip boundary condition

Us — Uw

2 — av Kn    /д U

av 1 — B(Kn) Kn у dn

(2.35)

where B(Kn) is an empirical parameter to be determined. For a general choice of B(Kn), equation (2.35) is first-order accurate in Kn, provided that |B(Kn)| < 1. However, for the continuum flow regime (Kn ^ 0.0)

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