Steady motion within a curved pipe

The forcing of fluid through a tube, of uniform cross section (~ a) and coiled to form an arc of a circle of radius R(≫ a), by the application of a steady constant pressure gradient along the tube is discussed for large values of the Dean number D. An asymptotic description is presented for the fully-developed laminar flow within a general symmetric section, in which there is outward centrifuging of the secondary motion in the inviscid core, supplemented by a faster viscous return motion near the pipewalls, while the downpipe velocity increases steadily across the core. A similarity solution of the viscous flow, with the associated core flow thereby being determined, is calculated for a triangular cross section, but for a rectangular tube analytical and numerical arguments are put forward that point fairly conclusively to the non-existence of an attached laminar motion, near the inside bend at least, at high Dean numbers. For if the pressure-gradient is imagined to vary like the mth power of distance across the section, then computed results indicate that local solutions of the boundary layer problem can be found only for (m-1/2). The assertion is backed up by an analytical study for small positive values of (m-1/2). The dynamical properties in the neighbourhood of a flat outside bend, or of a ‘pinched' inside bend, of a tube also need careful consideration, although these may be essentially passive flow regions and some account of the local features there can be given. Near a flat outside bend two possible flow models arise, depending on the local nature of the coreflow.