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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