Semi-analytical approach for solving Stokes flow problems with circular boundaries

Abstract

Steady, plane Stokes flow of an incompressible viscous fluid is considered within a circular boundary. To fully capture the circular boundary, the boundary densities in the boundary integral equation (BIE) are expanded in terms of Fourier series. The kernel functions in the BIE are expanded to degenerate kernels by using the separation of field and source points. Therefore, the approach can be considered as a semi-analytical method. Novelly, the improper integrals are transformed to series sum and are easily calculated. The linear algebraic system can be established by matching the boundary conditions at the collocation points. Then, the unknown Fourier coefficients can be easily determined. Four gains are achieved, (1) well-posed mode, (2) free of principal value, (3). Illimination of boundary layer effect, (4). Exponential convergence. Finally, several examples including circular and eccentric domains are presented to demonstrate the validity of the present method. 本文將考慮含圓形邊界的穩態、平面的史托克流問題，其中流體為不可壓縮且具黏滯性。邊界積分方程中的未知密度函數以傅立葉級數做展開，其中的退化核係將基本解中依場、源點分離所導得的級數形式，藉由退化核的內外域表示式可避免主值積分的計算。在邊界上均勻佈點，並配合邊界條件可得一線性代數方程式，其未知的傅立葉係數即可輕易求得，將之代回邊界積分方程中可得場解。本法可視為一半解析法。本法有四大好處:(1)良態模式,(2)無須主值計算,(3)無邊界層效應, (4)指數收斂。最後，將舉幾個例子來驗證此法的可行性。