  In this paper, we use the well-known background method to obtain a rigorous
lower bound on the volume flow rate through a helical pipe driven by a pressure
differential in the limit of large Reynolds number. As a consequence, we also
obtain an equivalent upper bound on the friction factor. These bounds are also
valid for toroidal and straight pipes as limiting cases. By considering a
two-dimensional background flow with varying boundary layer thickness along the
circumference of the pipe, we obtain these bounds as a function of the
curvature and torsion of the pipe and therefore capture the geometrical aspects
of the problem. In this paper, we also present a sufficient criterion to find
out which pressure-driven flow and surface-velocity-driven flow problems can be
tackled using the background method.
