Steady motion within a curved pipe

Abstract

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 m th 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.