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In this paper, we use the well-known background method to obtain a rigorouslower bound on the volume flow rate through a helical pipe driven by a pressuredifferential in the limit of large Reynolds number. As a consequence, we alsoobtain an equivalent upper bound on the friction factor. These bounds are alsovalid for toroidal and straight pipes as limiting cases. By considering atwo-dimensional background flow with varying boundary layer thickness along thecircumference of the pipe, we obtain these bounds as a function of thecurvature and torsion of the pipe and therefore capture the geometrical aspectsof the problem. In this paper, we also present a sufficient criterion to findout which pressure-driven flow and surface-velocity-driven flow problems can betackled using the background method.

Pressure-driven flows in helical pipes: bounds on flow rate and friction factor