Axisymmetric viscous flow in a conical channel is considered for geometrical systems in which the cross section is multiply-connected. The volumetric flux rates are compared for a 2-flow region and a 1 -flow region of equal cross sectional area, and same mean radial pressure gradient. It is found that for various configurations of the 2-flow system there can be both flux rate enhancement and reduction. As part of an introduction some similar results are presented for Poiseuille flow, and flow between non parallel plane walls the former of which satisfies the exact Navier-Stokes equations.