Interdisciplinary Applied Mathematics

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2.3.2 General Slip Condition


The expansion originally given in (Schaaf and Chambre, 1961) is of first order in Kn. However, for higher Knudsen numbers, second-order corrections to these boundary conditions may become necessary. The velocity slip near the wall is coupled with the first and second variations of the tangential velocity in the normal direction to the wall. Numerical implementation of the slip formula in this form is computationally difficult. Therefore, further simplification of (2.28) without changing the second-order dependence on Kn is desired. For this purpose we assume that the transition from no-slip flow to slip flow occurs smoothly. Thus, a regular perturbation expansion of the    velocity    field    in    terms    of    Kn    is defined in    equation (2.32)    below,


where the no-slip Navier-Stokes velocity field is denoted by U0(x, t), and corrections to    the    velocity    field    due    to different    orders    of    Kn    dependence


are denoted by Ui(x, t) (i = 1, 2, 3 …). We then have


U = U0 + Kn Ui + Kn2 U2 + Kn3 U3 + O(Kn4).    (2.32)


This substitution enables us to rewrite the Navier-Stokes equations for different orders of Kn dependence in the following form:


O(1) :


3U0


dt


+


(Uo


■V)Uo


=


-VPo + Re


-1V2 Uo; (2.33)


O(Kn) :


3Ui


dt


+


(Ui


■V)Uo


+


(Uo


■V)Ui


= -VPi + Re-1V2Ui;


O(Kn2) :


<9U2


dt


+


(Uo


■V)U2


+


(U2


■V)Uo


+ (Ui ■ VUi)


= -VP2 + Re-1V2


U

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