Interdisciplinary Applied Mathematics

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In the above equations, Uik,W.ijk ^ ж as r ^ 0, and the integral is singular. Thus, regularization of equation (14.13) is necessary. Using the linear and constant modes given in (Mukherjee, 2000), we obtain


0= —    Wijk (x, y)Ui(y) — Ui(x) — Ui,p(x)(xp(y) — Xp(x))]nj (y)dS


JdB


+ / Uik(x, y)[(Tij(y) — Oij(x)] nj (y)dS,    (14.14)


JdB


where p = 1, 2. If the velocity and the stress are sufficiently smooth, the singularity in equation (14.14) is removed. Therefore, the integration can be evaluated using Gaussian quadrature schemes.


For the numerical implementation, the velocity and the traction are approximated in each cloud by the varying base interpolating polynomial as described above (Li and Aluru, 2003), and can be written in a general form as


NP


uk(y)= Y^ ni (y)uk


I =1


and


NP



Tk (y) = >] ni (y)A



I=1



k,


where к = 1,2, Tk is the traction, uk and tI are the nodal parameters, NI(y) is the approximation function, and NP is the number of points.


The above approximations and their derivatives are substituted into equation    (14.14).    In    order to    evaluate    the    integrals    in    equation    (14.14),


the boundary is decomposed into cells, and equation (14.14) is satisfied on every boundary node. By using Gaussian quadrature and looping over all the boundary nodes, equation (14.14) can be written in matrix form


Au + Bt = 0,


u

p=0


FIGURE 14.13. Sketch of Stokes flow through a step channel. Also shown is the inlet velocity.

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