On collocation schemes for integral equations arising in slender-body approximations of flow past particles with circular cross-section

A collocation method is presented for solving a singular integral equation of the second kind arsing in the slender-body approximation of viscous flow past slender bodies. When the spectral representation of the integral operator is explicitly known, the collocation method is shown to recover the spectrum of the continuous operator. The approximation error is estimated for two discretizations of the integral operator, and convergence is proved. The collocation scheme is validated for several test cases and extended to situations where the spectrum is not explicitly known.