Flow through a network of interconnected channels creates ill-conditioned problems due to the multiply connected topology. Boundary integral equation methods are generally good tools in solving fluid problems by reducing degrees of freedom, but it is less ideal to handle this ill-conditioning due to its non-local representation. A scattering theory has been proposed to address this in 2D by approximating branches in a network as straight channels, in which the solution flow follow a "rapid return to Poiseuille flow" under pressure drop and no-slip wall condition [Wang 2026]. In this talk we will consider the applicability of this approximation in 3D by looking at the asymptotic problems of two specific cases of Stokes flow: when the cylinder is slightly curved (the Dean problem) and when the radius of the cylinder becomes exceedingly thin.

Reference: Haiyang Wang, Fredrik Fryklund, Samuel Potter, and Leslie Greengard. Scattering theory for stokes flow in complex branched structures. Journal of Computational Physics, 564:115148, 2026.