Interdisciplinary Applied Mathematics

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b



1UI 2^/s



1


2



V dn )



0



^0


s



(2.39)


The quantities U0 and U0′ for an arbitrary curved surface denote first and second derivatives of the tangential component of the velocity vector along the normal direction to the surface, corresponding to a no-slip solution.


• The parameter b in equation (2.39) is the ratio of the vorticity flux to the wall vorticity, obtained in no-slip flow conditions. The value of b for simple flows can be found analytically.


Similarly, third-order terms in Kn can be matched if c is chosen as


+ (240)


However, the third-order-accurate slip formula is computationally more expensive, since it requires the solutions for the U field. We can obtain a second-order-accurate slip formula by approximating equation (2.35) as


Us — Uw



2 — av Kn dU av l — В Kn dn



2 — av Kn dU av 1 — b Kn dn



+ O(Kn3), (2.41)


where b is the high-order slip coefficient given in equation (2.39). The error for equation (2.41) is O(Kn3), i.e.,


Error = cU0 Kn3 .


Truncated geometric series containing only O(Kn2) terms could have also been used to implement the new second-order slip-boundary condition (see equation (2.37)). The error in this case is also O(Kn3), and is given as


Errors. = [U2 + bUl + (b2 + c]U0) Kn3 .


Since we    do    not    know the magnitude    of    the    U[    and    U22    terms,    it    is    dif


ficult to decide which approach is better. However, we believe that using equation (2.41) is better, since this equation keeps the original form suggested in (2.35). Also for separated flows, equation (2.41) gives no slip at the separation or reattachment points (as predicted from the first-order slip formula), since the shear stress (therefore ^ = 0) is zero at these points. However, the truncated geometric series (equation (2.37)) will give multiplication of infinitesimally small wall shear stress (rwan =    —► 0.0)

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