  In a companion study \cite{patterson2020computing2D}, we present a numerical
method for simulating 2D viscous flow through an open compliant closed channel,
drive by pressure gradient. We consider the highly viscous regime, where fluid
dynamics is described by the Stokes equations, and the less viscous regime
described by the Navier-Stokes equations. In this study, we extend the method
to 3D tubular flow. The problem is formulated in axisymmetric cylindrical
coordinates, an approach that is natural for tubular flow simulations and that
substantially reduces computational cost. When the elastic tubular walls are
stretched or compressed, they exert forces on the fluid. These singular forces
introduce unsmoothness into the fluid solution. As in the companion 2D study
\cite{patterson2020computing2D}, we extend the immersed interface method to an
open tube, and we compute solution to the model equations using the resulting
method. Numerical results indicate that this new method preserves sharp jumps
in the solution and its derivatives, and converges with second-order accuracy
in both space and time.
